Finite Group Databasebeta

← #29833872

Subgroups — #29833872

Order 15616.

1150 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840563C211yesmaximal, characteristic, complemented
2C2×#16840196C211yesmaximal, characteristic, complemented
3C2×#16840191C211yesmaximal, characteristic, complemented
4#21455484C211yesmaximal, characteristic, complemented
5#21447761C211yesmaximal, characteristic, complemented
6#21447760C211yesmaximal, characteristic, complemented
7#21447749C211yesmaximal, characteristic, complemented
8#21447748C211yesmaximal, characteristic, complemented
9#21447740C211yesmaximal, characteristic, complemented
10#21447739C211yesmaximal, characteristic, complemented
11#21447729C211yesmaximal, characteristic, complemented
12#21447528C211yesmaximal, characteristic, complemented
13#357×D61C211yesmaximal, characteristic, complemented
14#21447431C211yesmaximal, characteristic, complemented
15C61×#2184C211yesmaximal, characteristic, complemented
16C2×#14519822—21no—
17#16840563—24no—
18#16840563—24no—
19C2×#14519803—21no—
20C2×#14519801—21no—
21C2×#14519801—21no—
22C2×#14519799C2211yescharacteristic
23C2×#14519796C2211yescharacteristic
24C2×#14519796C2211yescharacteristic
25C2×#14519796C2211yescharacteristic, complemented
26C2×D8×D61C2211yescharacteristic
27C2×D8×D61C2211yescharacteristic
28C2×#14519783C2211yescharacteristic
29#16840341—21no—
30#16840196—24no—
31#16840191—24no—
32C2×Dic488C2211yescharacteristic, complemented
33C2×#14519722C2211yescharacteristic, complemented
34C2×#14519721C2211yescharacteristic, complemented
35C2×#14519721C2211yescharacteristic, complemented
36C2×#14519721C2211yescharacteristic, complemented
37C2×#14519718C2211yescharacteristic, complemented
38C2×#14519716C2211yescharacteristic, complemented
39C2×C16×D61C2211yescharacteristic, complemented
40#16840007—21no—
41#16840006—21no—
42#16840004C2211yescharacteristic, complemented
43#16840003C2211yescharacteristic, complemented
44#16839995C2211yescharacteristic, complemented
45#107×D61C2211yescharacteristic, complemented
46#16839760—21no—
47#16839567C2211yescharacteristic, complemented
48#16839565C2211yescharacteristic, complemented
49#16839564C2211yescharacteristic, complemented
50#16839563C2211yescharacteristic, complemented
51#16839563C2211yescharacteristic, complemented
52#16839562C2211yescharacteristic, complemented
53C22⋊#11758185C2211yescharacteristic, complemented
54#16839541C2211yescharacteristic, complemented
55#16839541C2211yescharacteristic, complemented
56C22×D8×C61C2211yescharacteristic
57C2×D16×C61C2211yescharacteristic, complemented
58C2×D16×C61C2211yescharacteristic, complemented
59C61×#466C2211yescharacteristic, complemented
60C61×#357C2211yescharacteristic, complemented
61C61×#357C2211yescharacteristic, complemented
62C61×C22⋊C16C2211yescharacteristic, complemented
63#14519822—28no—
64#14519822—44no—
65C2×#11758222—21no—
66C2×#11758222D411yescharacteristic
67C2×#11758224—21no—
68C2×#11758224—21no—
69C2×#11758224—21no—
70C2×#11758224—41no—
71C22×C22⋊D61—41no—
72C22×C22⋊D61—41no—
73C2×D4×D61—21no—
74C2×D4×D61—21no—
75C2×D4×D61—41no—
76#14519803—28no—
77#14519801—28no—
78#14519801—28no—
79#14519799—22no—
80#14519799—22no—
81#14519796—22no—
82#14519796—22no—
83#14519796—22no—
84#14519796—22no—
85#14519796—22no—
86#14519796D414yescomplemented
87C2×Dic244C2311yescharacteristic
88C2×Dic244C2311yescharacteristic
89C2×#11758190—21no—
90C2×#11758200—21no—
91C2×#11758200—21no—
92C2×#11758200—21no—
93C2×D488C2311yescharacteristic
94C2×#11758199—21no—
95C2×#11758199—21no—
96D8×D61—22no—
97D8×D61—22no—
98D8×D61—22no—
99D8×D61D414yescomplemented
100#14519783—22no—
101#14519783—22no—
102C2×#11758194D411yescharacteristic
103C2×#11758189D411yescharacteristic
104C2×#11758189D411yescharacteristic
105C2×C8×D61C2311yescharacteristic
106C2×C8×D61C2311yescharacteristic
107#14519757—21no—
108#14519756—21no—
109#14519754—41no—
110#14519750—41no—
111#14519749—41no—
112#14519748—21no—
113#14519748—41no—
114C4⋊C4×D61—21no—
115Dic488—22no—
116#14519722—22no—
117#14519721—22no—
118#14519721—22no—
119#14519721—22no—
120#14519718—22no—
121#14519716—22no—
122C2×#11758185C2311yescharacteristic
123C2×#11758185C2311yescharacteristic
124C16×D61—22no—
125#14519711C2311yescharacteristic
126#14519710—21no—
127#14519702—21no—
128#14519700—21no—
129#14519698—21no—
130C22×D4×C61—21no—
131C2×D8×C61—22no—
132C2×D8×C61—22no—
133C2×D8×C61—21no—
134C2×D8×C61C2311yescharacteristic
135C2×D8×C61C2311yescharacteristic
136C2×D8×C61C2311yescharacteristic
137C22×C8×C61C2311yescharacteristic
138#121×C61—21no—
139D16×C61—22no—
140D16×C61—22no—
141C2×C16×C61C2311yescharacteristic
142C2×C16×C61C2311yescharacteristic
143#107×C61C2311yescharacteristic
144#102×C61—21no—
145C24×C61—41no—
146C23×D61—41no—
147C23×D61—41no—
148C2×D4×C61—24no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—42no—
153C2×D4×C61—41no—
154C2×D4×C61—41no—
155C22×C4×C61C2×D411yescharacteristic
156C2×C22⋊D61—48no—
157C2×C22⋊D61—48no—
158C2×C22⋊D61—41no—
159C2×C22⋊D61—41no—
160C2×C22⋊D61—41no—
161C2×C22⋊D61—41no—
162C2×C22⋊D61—41no—
163C2×C22⋊D61—41no—
164C2×C22⋊D61—41no—
165C2×C22⋊D61D811yescharacteristic
166C2×C22⋊D61D811yescharacteristic
167C22×Dic61—41no—
168C22×Dic61—41no—
169C22×Dic61—41no—
170#11758224—24no—
171#11758224—24no—
172#11758224—48no—
173#11758224—42no—
174#11758224—42no—
175#11758224—42no—
176#11758224—42no—
177D4×D61—24no—
178D4×D61—24no—
179D4×D61—42no—
180D4×D61—48no—
181D4×D61—42no—
182#11758222—22no—
183#11758222—22no—
184#11758222—24no—
185#11758222—42no—
186C2×D244—21no—
187C2×D244D811yescharacteristic
188C2×C4×D61—41no—
189C2×C4×D61—41no—
190C2×C4×D61C2×D411yescharacteristic
191C2×C4×D61C2×D411yescharacteristic
192C2×Dic122—21no—
193C2×Dic122—21no—
194C2×Dic122D811yescharacteristic
195D8×C61—24no—
196D8×C61—21no—
197D8×C61—21no—
198D8×C61—21no—
199D8×C61—21no—
200D8×C61C2×D412yes—
201D8×C61C2×D412yes—
202C2×C8×C61—21no—
203C2×C8×C61—21no—
204C2×C8×C61C2×D411yescharacteristic
205C2×C8×C61C2411yescharacteristic
206C4⋊C4×C61—21no—
207C22⋊C4×C61—41no—
208C22⋊Dic61—41no—
209C22⋊Dic61—41no—
210#11758200—24no—
211#11758200—24no—
212#11758200—42no—
213#11758199—24no—
214#11758199—24no—
215#11758198—41no—
216C4⋊Dic61—21no—
217C4⋊Dic61—41no—
218#11758196—41no—
219#11758196—41no—
220#11758194—22no—
221C2×#92418C2×D411yescharacteristic
222C2×#92418C2×D411yescharacteristic
223Dic244—21no—
224Dic244—21no—
225Dic244—21no—
226Dic244C2×D412yes—
227D488—21no—
228D488C2×D412yes—
229#11758190—24no—
230#11758189—22no—
231#11758189—22no—
232C8×D61—21no—
233C8×D61—21no—
234C8×D61C2×D412yes—
235C8×D61C2×D412yes—
236C16×C61—21no—
237C16×C61—21no—
238#11758185—21no—
239#11758185—21no—
240C23×C61—44no—
241C23×C61—41no—
242C23×C61—41no—
243C23×C61—41no—
244C23×C61—161no—
245C23×C61—161no—
246C23×C61C2×D811yescharacteristic
247C22×D61—44no—
248C22×D61—41no—
249C22×D61—44no—
250C22×D61—161no—
251C22×D61—161no—
252C22×D61—161no—
253C22×D61—161no—
254C22×D61C2×D811yescharacteristic
255C22×D61C2×D811yescharacteristic
256D4×C61—22no—
257D4×C61—22no—
258D4×C61—44no—
259D4×C61—41no—
260D4×C61—41no—
261D4×C61—41no—
262D4×C61—41no—
263D4×C61—161no—
264C2×C4×C61—21no—
265C2×C4×C61—21no—
266C2×C4×C61—41no—
267C2×C4×C61C2×D811yescharacteristic
268C2×C4×C61C22×D411yescharacteristic
269C22⋊D61—22no—
270C22⋊D61—22no—
271C22⋊D61—44no—
272C22⋊D61—44no—
273C22⋊D61—44no—
274C22⋊D61—44no—
275C22⋊D61—44no—
276C22⋊D61—82no—
277C22⋊D61—161no—
278C2×Dic61—44no—
279C2×Dic61—41no—
280C2×Dic61—44no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—82no—
284C2×Dic61C2×D811yescharacteristic
285C2×Dic61C2×D811yescharacteristic
286D244—22no—
287D244—22no—
288D244—41no—
289C4×D61—21no—
290C4×D61—21no—
291C4×D61—44no—
292C4×D61—161no—
293C4×D61#12012yes—
294C4×D61AGL(1,ℤ/8)12yes—
295Dic122—22no—
296Dic122—22no—
297Dic122—41no—
298Dic122—41no—
299Dic122—41no—
300C8×C61—21no—
301C8×C61C22×D411yescharacteristic
302C8×C61C22×D411yescharacteristic
303#92418—21no—
304#92418#12012yes—
305#33526—611nomaximal, Sylow
306C22×C61—21no—
307C22×C61—21no—
308C22×C61—21no—
309C22×C61—42no—
310C22×C61—42no—
311C22×C61—81no—
312C22×C61—81no—
313C22×C61—81no—
314C22×C61—81no—
315C22×C61—81no—
316C22×C61—81no—
317C22×C61—81no—
318C22×C61—164no—
319C22×C61C22×D811yescharacteristic
320C2×D61—21no—
321C2×D61—21no—
322C2×D61—21no—
323C2×D61—21no—
324C2×D61—21no—
325C2×D61—42no—
326C2×D61—81no—
327C2×D61—81no—
328C2×D61—164no—
329C2×D61—164no—
330C2×D61#44712yes—
331C4×C61—21no—
332C4×C61—81no—
333C4×C61C2×#12011yescharacteristic
334C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
335Dic61—21no—
336Dic61—42no—
337Dic61—81no—
338Dic61—81no—
339Dic61#44712yes—
340C2×#576—611no—
341C2×#509—611no—
342C2×#508—611no—
343#3061—611no—
344#2189—611no—
345#2189—611no—
346#2187—611no—
347#2187—611no—
348#2186—611no—
349#2186—611no—
350#2184—611no—
351#2184D6111yescharacteristic, complemented
352#2139—611no—
353C2×#357—611no—
354#2113—611no—
355C2×C61—41no—
356C2×C61—82no—
357C2×C61—82no—
358C2×C61—161no—
359C2×C61C2×#44711yescharacteristic
360C2×C61#341511yescharacteristic
361C2×C61#341611yescharacteristic
362D61—22no—
363D61—41no—
364D61—82no—
365C2×H(5,𝔽2)—1221no—
366#576—1224no—
367#576—1224no—
368C2×AGL(1,ℤ/8)—1221no—
369C2×AGL(1,ℤ/8)—1221no—
370C2×AGL(1,ℤ/8)—1221no—
371C2×#135—611no—
372C2×#135—611no—
373C2×#135—611no—
374C2×#135—611no—
375C22×D8—611no—
376C22×D8—611no—
377C22×D8C2×D6111yescharacteristic
378C2×#131—611no—
379#537—1221no—
380#509—1224no—
381#508—1224no—
382C2×Q32—611no—
383C2×SD32—611no—
384C2×SD32—611no—
385C2×SD32—611no—
386C2×SD32—611no—
387C2×D16—611no—
388C2×D16C2×D6111yescharacteristic, complemented
389C2×D16C2×D6111yescharacteristic, complemented
390C2×M32—611no—
391C22×C16—611no—
392#468—1221no—
393#468—1221no—
394#467—611no—
395#467—611no—
396#466—611no—
397#466C2×D6111yescharacteristic, complemented
398C2×#107—611no—
399#417—1221no—
400#358—611no—
401#358—611no—
402#357—611no—
403#357—611no—
404#357—611no—
405#357—611no—
406#357C2×D6111yescharacteristic, complemented
407#357C2×D6111yescharacteristic, complemented
408C22⋊C16—611no—
409C22⋊C16—611no—
410C22⋊C16—611no—
411C22⋊C16C2×D6111yescharacteristic, complemented
412C61#3352611yescharacteristic, complemented, Sylow
413H(5,𝔽2)—1228no—
414H(5,𝔽2)—2444no—
415C2×#41—611no—
416C2×#41—1221no—
417C2×#41—1221no—
418C2×#41—1221no—
419C2×#41—1221no—
420C2×#41—2441no—
421C22×D4—21no—
422C22×D4—1221no—
423C22×D4—1221no—
424C22×D4—2441no—
425C22×D4—2441no—
426C22×D4—2441no—
427AGL(1,ℤ/8)—1228no—
428AGL(1,ℤ/8)—1228no—
429AGL(1,ℤ/8)—1228no—
430#135—614no—
431#135—1222no—
432#135—1222no—
433#135—1222no—
434#135—1222no—
435#135—1222no—
436#135—1222no—
437#135—1222no—
438C2×Q16—611no—
439C2×Q16—611no—
440C2×SD16—1221no—
441C2×SD16—1221no—
442C2×SD16—1221no—
443C2×SD16—1221no—
444C2×D8—22no—
445C2×D8—22no—
446C2×D8—21no—
447C2×D8—614no—
448C2×D8—611no—
449C2×D8—1222no—
450C2×D8—1221no—
451C2×D8—1221no—
452C2×D8—1222no—
453C2×D8—1222no—
454C2×D8C22×D6111yescharacteristic
455C2×D8C22×D6111yescharacteristic
456C2×D8C22×D6111yescharacteristic
457#131—1222no—
458#131—1222no—
459C2×M16—611no—
460C2×M16—611no—
461C2×M16—611no—
462C22×C8—611no—
463C22×C8—611no—
464C22×C8C22×D6111yescharacteristic
465C22⋊Q8—1221no—
466C22⋊Q8—1221no—
467C22⋊Q8—2441no—
468#121—21no—
469#121—1221no—
470#121—2441no—
471#121—2441no—
472#121—2441no—
473C2×C4⋊C4—1221no—
474Q32—1222no—
475SD32—1222no—
476SD32—1222no—
477SD32—1222no—
478SD32—1222no—
479D16—22no—
480D16—22no—
481D16—1222no—
482M32—1222no—
483C2×C16—611no—
484C2×C16—611no—
485C2×C16—1222no—
486C2×C16C22×D6111yescharacteristic
487C2×C16C22×D6111yescharacteristic
488#107—611no—
489#107—1221no—
490#107C22×D6111yescharacteristic
491#103—1221no—
492#103—1221no—
493#102—21no—
494#102—1221no—
495C24—41no—
496C24—2441no—
497C24—2441no—
498#41—1222no—
499#41—1222no—
500#41—1224no—
501#41—1224no—
502#41—1224no—
503#41—2442no—
504#41—2442no—
505#41—2442no—
506#41—2442no—
507#41—2442no—
508#41—2448no—
509C2×Q8—611no—
510C2×Q8—1221no—
511C2×Q8—1221no—
512C2×D4—24no—
513C2×D4—21no—
514C2×D4—21no—
515C2×D4—21no—
516C2×D4—42no—
517C2×D4—41no—
518C2×D4—41no—
519C2×D4—611no—
520C2×D4—611no—
521C2×D4—611no—
522C2×D4—1224no—
523C2×D4—1224no—
524C2×D4—1221no—
525C2×D4—2442no—
526C2×D4—2441no—
527C2×D4—2441no—
528C2×D4—2442no—
529C2×D4—2448no—
530C2×D4—2448no—
531C2×D4—2448no—
532C2×D4—2441no—
533C2×D4—2441no—
534C2×D4—2441no—
535C2×D4—2441no—
536C2×D4—2441no—
537C22×C4—611no—
538C22×C4—611no—
539C22×C4—2441no—
540C22×C4—2441no—
541C22×C4—2441no—
542C22×C4—2441no—
543C22×C4—2441no—
544C22×C4D4×D6111yescharacteristic
545Q16—612no—
546Q16—1221no—
547Q16—1221no—
548Q16—1221no—
549SD16—1224no—
550SD16—1224no—
551SD16—1224no—
552SD16—2442no—
553D8—24no—
554D8—21no—
555D8—21no—
556D8—21no—
557D8—21no—
558D8—612no—
559D8—1224no—
560D8—1224no—
561D8—1221no—
562D8D4×D6112yes—
563D8D4×D6112yes—
564M16—1222no—
565M16—1222no—
566M16—1222no—
567C2×C8—21no—
568C2×C8—21no—
569C2×C8—611no—
570C2×C8—612no—
571C2×C8—612no—
572C2×C8—611no—
573C2×C8—1221no—
574C2×C8—1221no—
575C2×C8D4×D6111yescharacteristic
576C2×C8C23×D6111yescharacteristic
577C4⋊C4—21no—
578C4⋊C4—1221no—
579C4⋊C4—2441no—
580C4⋊C4—2441no—
581C4⋊C4—2441no—
582C22⋊C4—41no—
583C22⋊C4—2441no—
584C22⋊C4—2441no—
585C22⋊C4—2441no—
586C16—21no—
587C16—21no—
588C16—1221no—
589C16—1221no—
590C23—44no—
591C23—41no—
592C23—41no—
593C23—41no—
594C23—161no—
595C23—161no—
596C23—611no—
597C23—611no—
598C23—9761no—
599C23—9761no—
600C23—9761no—
601C23—9761no—
602C23—2444no—
603C23—2444no—
604C23—2441no—
605C23D8×D6111yescharacteristic
606Q8—1222no—
607Q8—1222no—
608Q8—2441no—
609Q8—2441no—
610Q8—2441no—
611D4—22no—
612D4—22no—
613D4—44no—
614D4—41no—
615D4—41no—
616D4—41no—
617D4—41no—
618D4—161no—
619D4—1222no—
620D4—1222no—
621D4—1222no—
622D4—1222no—
623D4—9761no—
624D4—4882no—
625D4—2444no—
626D4—2444no—
627D4—2444no—
628D4—2444no—
629D4—2444no—
630D4—2441no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—41no—
634C2×C4—611no—
635C2×C4—612no—
636C2×C4—612no—
637C2×C4—611no—
638C2×C4—1221no—
639C2×C4—1221no—
640C2×C4—9761no—
641C2×C4—4882no—
642C2×C4—2444no—
643C2×C4—2444no—
644C2×C4—2444no—
645C2×C4—2441no—
646C2×C4—2441no—
647C2×C4—2441no—
648C2×C4D8×D6111yescharacteristic
649C2×C4C2×D4×D6111yescharacteristic
650C8—21no—
651C8—612no—
652C8—1221no—
653C8C2×D4×D6111yescharacteristic
654C8C2×D4×D6111yescharacteristic
655C22—21no—
656C22—21no—
657C22—21no—
658C22—42no—
659C22—42no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22—164no—
668C22—612no—
669C22—1221no—
670C22—1221no—
671C22—1221no—
672C22—1221no—
673C22—1221no—
674C22—9764no—
675C22—9764no—
676C22—4881no—
677C22—4881no—
678C22—2442no—
679C22C2×D8×D6111yescharacteristic
680C4—21no—
681C4—81no—
682C4—612no—
683C4—1221no—
684C4—4881no—
685C4—4881no—
686C4—2442no—
687C4#120×D6111yescharacteristic
688C4AGL(1,ℤ/8)×D6111yescharacteristic
689C2—41no—
690C2—82no—
691C2—82no—
692C2—161no—
693C2—1222no—
694C2—4882no—
695C2—2441no—
696C2#447×D6111yescharacteristic
697C2#2145895611yescharacteristic
698C2#2145896511yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2184C2
socleC22×C61C22×D8
hypercenter#2184D61
hypocenterC61#33526
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2184

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16840563C2semidirect
2. C2×#16840196C2semidirect
3. C2×#16840191C2semidirect
4. #21455484C2semidirect
5. #21447761C2semidirect
6. #21447760C2semidirect
7. #21447749C2semidirect
8. #21447748C2semidirect
9. #21447740C2semidirect
10. #21447739C2semidirect
11. #21447729C2semidirect
12. #21447528C2semidirect
13. #357×D61C2semidirect
14. #21447431C2semidirect
15. C61×#2184C2semidirect
25. C2×#14519796C22semidirect
32. C2×Dic488C22semidirect
33. C2×#14519722C22semidirect
34. C2×#14519721C22semidirect
35. C2×#14519721C22semidirect
36. C2×#14519721C22semidirect
37. C2×#14519718C22semidirect
38. C2×#14519716C22semidirect
39. C2×C16×D61C22semidirect
42. #16840004C22semidirect
43. #16840003C22semidirect
44. #16839995C22semidirect
45. #107×D61C22semidirect
47. #16839567C22semidirect
48. #16839565C22semidirect
49. #16839564C22semidirect
50. #16839563C22semidirect
51. #16839563C22semidirect
52. #16839562C22semidirect
53. C22⋊#11758185C22semidirect
54. #16839541C22semidirect
55. #16839541C22semidirect
57. C2×D16×C61C22semidirect
58. C2×D16×C61C22semidirect
59. C61×#466C22semidirect
60. C61×#357C22semidirect
61. C61×#357C22semidirect
62. C61×C22⋊C16C22semidirect
86. #14519796D4semidirect
99. D8×D61D4semidirect
351. #2184D61semidirect
388. C2×D16C2×D61semidirect
389. C2×D16C2×D61semidirect
397. #466C2×D61semidirect
406. #357C2×D61semidirect
407. #357C2×D61semidirect
411. C22⋊C16C2×D61semidirect
412. C61#33526semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168405632
C2×#168401962
C2×#168401912
#214554842
#214477612
#214477602
#214477492
#214477482
#214477402
#214477392
#214477292
#214475282
#357×D612
#214474312
C61×#21842
#3352661