Finite Group Databasebeta

← #29833830

Subgroups — #29833830

Order 15616.

1140 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#135×D61C211yesmaximal, characteristic
2C2×#16840204C211yesmaximal, characteristic, complemented
3C2×#16840201C211yesmaximal, characteristic, complemented
4#467×D61C211yesmaximal, characteristic, complemented
5#21447761C211yesmaximal, characteristic, complemented
6#21447759C211yesmaximal, characteristic, complemented
7#21447757C211yesmaximal, characteristic, complemented
8#21447754C211yesmaximal, characteristic, complemented
9#21447751C211yesmaximal, characteristic, complemented
10#21447749C211yesmaximal, characteristic, complemented
11#21447746C211yesmaximal, characteristic, complemented
12#21447530C211yesmaximal, characteristic, complemented
13#21447528C211yesmaximal, characteristic, complemented
14#21447428C211yesmaximal, characteristic, complemented
15C61×#2189C211yesmaximal, characteristic, complemented
16C2×#41×D61—21no—
17C2×#14519800C2211yescharacteristic
18C2×#14519799C2211yescharacteristic
19C2×#14519798—21no—
20C2×#14519797—21no—
21C2×#14519796C2211yescharacteristic
22#135×D61—24no—
23#135×D61—24no—
24C2×Q16×D61C2211yescharacteristic
25C2×Q16×D61C2211yescharacteristic, complemented
26C2×SD16×D61—21no—
27C2×D8×D61C2211yescharacteristic
28C22×C8×D61C2211yescharacteristic
29C22⋊Q8×D61—21no—
30#16840204—24no—
31#16840201—24no—
32C2×Dic488C2211yescharacteristic, complemented
33C2×#14519723C2211yescharacteristic, complemented
34C2×#14519723C2211yescharacteristic, complemented
35C2×#14519722C2211yescharacteristic, complemented
36C2×#14519721C2211yescharacteristic, complemented
37C2×#14519720C2211yescharacteristic, complemented
38C2×#14519716C2211yescharacteristic, complemented
39C2×#14519716C2211yescharacteristic, complemented
40#16840004C2211yescharacteristic
41#16840003C2211yescharacteristic, complemented
42#16840002—21no—
43#16840001—21no—
44#16840000C2211yescharacteristic
45#107×D61C2211yescharacteristic, complemented
46#103×D61—21no—
47#16839567C2211yescharacteristic, complemented
48#16839566C2211yescharacteristic, complemented
49#16839565C2211yescharacteristic, complemented
50#16839564C2211yescharacteristic, complemented
51#16839563C2211yescharacteristic, complemented
52#16839562C2211yescharacteristic, complemented
53C22⋊#11758185C2211yescharacteristic, complemented
54#16839541C2211yescharacteristic
55#16839541C2211yescharacteristic
56C2×#135×C61C2211yescharacteristic
57C2×Q32×C61C2211yescharacteristic, complemented
58C2×SD32×C61C2211yescharacteristic, complemented
59C61×#467C2211yescharacteristic, complemented
60C61×#358C2211yescharacteristic, complemented
61C61×#357C2211yescharacteristic, complemented
62C61×C22⋊C16C2211yescharacteristic, complemented
63C2×#11758226—21no—
64C2×#11758222—21no—
65C2×#11758222—41no—
66C2×#11758224—21no—
67C2×#11758224—41no—
68#41×D61—28no—
69#41×D61—44no—
70C2×Q8×D61—21no—
71C2×Q8×D61—21no—
72C2×D4×D61—21no—
73C2×D4×D61—41no—
74C22×C4×D61—41no—
75C22×C4×D61D411yescharacteristic
76#14519800—22no—
77#14519800D414yes—
78#14519799—22no—
79#14519799—22no—
80#14519798—28no—
81#14519797—28no—
82#14519796—22no—
83#14519796D414yes—
84C2×Dic244C2311yescharacteristic
85C2×Dic244C2311yescharacteristic
86C2×#11758202—21no—
87C2×#11758202—21no—
88Q16×D61—22no—
89Q16×D61—22no—
90Q16×D61—22no—
91Q16×D61—22no—
92C2×#11758190—21no—
93C2×#11758200—21no—
94C2×#11758201—21no—
95SD16×D61—28no—
96C2×D488C2311yescharacteristic
97C2×#11758199—21no—
98D8×D61—22no—
99D8×D61—22no—
100C22×#92418D411yescharacteristic
101C2×C8×D61—22no—
102C2×C8×D61—22no—
103C2×C8×D61D411yescharacteristic
104C2×C8×D61D411yescharacteristic
105C2×C8×D61C2311yescharacteristic
106C2×C8×D61C2311yescharacteristic
107#14519757—21no—
108#14519756—21no—
109#14519755—41no—
110#14519754—41no—
111#14519753—21no—
112C4⋊C4×D61—21no—
113C4⋊C4×D61—41no—
114C22⋊C4×D61—41no—
115Dic488—22no—
116#14519723—22no—
117#14519723—22no—
118#14519722—22no—
119#14519721—22no—
120#14519720—22no—
121#14519716—22no—
122#14519716—22no—
123C2×#11758185C2311yescharacteristic
124C2×#11758185C2311yescharacteristic
125#14519711C2311yescharacteristic
126#14519710—21no—
127#14519702—21no—
128#14519701—21no—
129#14519700—21no—
130C2×#41×C61—21no—
131#135×C61—22no—
132#135×C61—22no—
133C2×Q16×C61C2311yescharacteristic
134C2×Q16×C61C2311yescharacteristic
135C2×SD16×C61—21no—
136C2×D8×C61C2311yescharacteristic
137C22×C8×C61C2311yescharacteristic
138C22⋊Q8×C61—21no—
139Q32×C61—22no—
140SD32×C61—22no—
141C2×C16×C61C2311yescharacteristic
142C2×C16×C61C2311yescharacteristic
143#107×C61C2311yescharacteristic
144#103×C61—21no—
145C23×D61—41no—
146C23×D61D811yescharacteristic
147#41×C61—24no—
148#41×C61—42no—
149C2×Q8×C61—21no—
150C2×Q8×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—41no—
153C22×C4×C61—41no—
154C22×C4×C61C2×D411yescharacteristic
155C2×C22⋊D61—41no—
156C2×C22⋊D61—41no—
157C2×C22⋊D61—41no—
158C22×Dic61—41no—
159C22×Dic61D811yescharacteristic
160#11758226—24no—
161#11758226—42no—
162Q8×D61—24no—
163Q8×D61—42no—
164Q8×D61—42no—
165Q8×D61—42no—
166#11758224—24no—
167#11758224—48no—
168#11758224—42no—
169D4×D61—24no—
170D4×D61—42no—
171D4×D61—48no—
172#11758222—24no—
173#11758222—48no—
174#11758222—42no—
175C2×D244—21no—
176C2×D244—41no—
177C2×C4×D61—22no—
178C2×C4×D61—22no—
179C2×C4×D61—48no—
180C2×C4×D61—41no—
181C2×C4×D61—41no—
182C2×C4×D61—41no—
183C2×C4×D61—41no—
184C2×C4×D61—41no—
185C2×C4×D61D811yescharacteristic
186C2×C4×D61D811yescharacteristic
187C2×C4×D61C2×D411yescharacteristic
188C2×C4×D61C2×D411yescharacteristic
189C2×Dic122—21no—
190C2×Dic122—21no—
191C2×Dic122—41no—
192C2×Dic122—41no—
193Q16×C61—21no—
194Q16×C61—21no—
195Q16×C61—21no—
196Q16×C61C2×D412yes—
197SD16×C61—24no—
198D8×C61—21no—
199D8×C61C2×D412yes—
200C2×C8×C61—21no—
201C2×C8×C61—21no—
202C2×C8×C61C2×D411yescharacteristic
203C2×C8×C61C2411yescharacteristic
204C4⋊C4×C61—21no—
205C4⋊C4×C61—41no—
206C22⋊C4×C61—41no—
207C22⋊Dic61—41no—
208#11758202—24no—
209#11758202—42no—
210#11758201—24no—
211#11758200—24no—
212#11758199—24no—
213#11758198—41no—
214#11758198—41no—
215C4⋊Dic61—21no—
216C4⋊Dic61—41no—
217#11758196—41no—
218#11758196—41no—
219#11758196—41no—
220C2×#92418—22no—
221C2×#92418C2×D411yescharacteristic
222C2×#92418C2×D411yescharacteristic
223Dic244—21no—
224Dic244—21no—
225Dic244—21no—
226Dic244C2×D412yes—
227D488—21no—
228D488C2×D412yes—
229#11758190—24no—
230C8×D61—22no—
231C8×D61—21no—
232C8×D61—21no—
233C8×D61—22no—
234C8×D61C2×D412yes—
235C8×D61C2×D412yes—
236C16×C61—21no—
237C16×C61—21no—
238#11758185—21no—
239#11758185—21no—
240C23×C61—41no—
241C23×C61C2×D811yescharacteristic
242C22×D61—22no—
243C22×D61—22no—
244C22×D61—41no—
245C22×D61—44no—
246C22×D61—41no—
247C22×D61—82no—
248C22×D61—82no—
249C22×D61C2×D811yescharacteristic
250C22×D61C2×D811yescharacteristic
251Q8×C61—22no—
252Q8×C61—41no—
253Q8×C61—41no—
254Q8×C61—41no—
255D4×C61—22no—
256D4×C61—44no—
257D4×C61—41no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—44no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61C2×D811yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C22⋊D61—44no—
267C22⋊D61—44no—
268C22⋊D61—44no—
269C2×Dic61—22no—
270C2×Dic61—41no—
271C2×Dic61—44no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61C2×D811yescharacteristic
275C2×Dic61C2×D811yescharacteristic
276D244—22no—
277D244—44no—
278D244—41no—
279C4×D61—22no—
280C4×D61—21no—
281C4×D61—21no—
282C4×D61—22no—
283C4×D61—44no—
284C4×D61—44no—
285C4×D61—44no—
286C4×D61—82no—
287C4×D61—82no—
288C4×D61#12012yes—
289C4×D61AGL(1,ℤ/8)12yes—
290Dic122—22no—
291Dic122—44no—
292Dic122—41no—
293Dic122—41no—
294Dic122—41no—
295Dic122—82no—
296C8×C61—21no—
297C8×C61C22×D411yescharacteristic
298C8×C61C22×D411yescharacteristic
299#92418—21no—
300#92418#12012yes—
301#33521—611nomaximal, Sylow
302C22×C61—21no—
303C22×C61—21no—
304C22×C61—21no—
305C22×C61—42no—
306C22×C61—81no—
307C22×C61—81no—
308C22×C61C22×D811yescharacteristic
309C2×D61—21no—
310C2×D61—21no—
311C2×D61—21no—
312C2×D61—21no—
313C2×D61—44no—
314C2×D61—42no—
315C2×D61—88no—
316C2×D61—81no—
317C2×D61—81no—
318C2×D61#44712yes—
319C2×D61#51012yescomplemented
320C4×C61—21no—
321C4×C61—42no—
322C4×C61—81no—
323C4×C61—81no—
324C4×C61C2×#12011yescharacteristic
325C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
326Dic61—21no—
327Dic61—42no—
328Dic61—81no—
329Dic61—81no—
330Dic61#44712yes—
331C22×#135—611no—
332C2×#510—611no—
333C2×#510—611no—
334C2×#467—611no—
335#2189—611no—
336#2189—611no—
337#2189—611no—
338#2189D6111yescharacteristic, complemented
339#2188—611no—
340#2188—611no—
341#2187—611no—
342#2187—611no—
343#2139—611no—
344#2139—611no—
345#2112—611no—
346C2×C61—41no—
347C2×C61—82no—
348C2×C61C2×#44711yescharacteristic
349C2×C61C2×#51011yescharacteristic
350C2×C61#341411yescharacteristic
351D61—22no—
352D61—22no—
353D61—82no—
354C22×#41—1221no—
355C2×#135—611no—
356C2×#135—611no—
357C2×#135—611no—
358C2×#135—1221no—
359C2×#135—1221no—
360C2×#135—1224no—
361C2×#135—1224no—
362C2×#135C2×D6111yescharacteristic
363C22×Q16—611no—
364C22×Q16—611no—
365C22×SD16—1221no—
366C22×D8—611no—
367C23×C8—611no—
368C2×C22⋊Q8—1221no—
369#510—1224no—
370#510—1224no—
371C2×Q32—611no—
372C2×Q32—611no—
373C2×Q32—611no—
374C2×Q32C2×D6111yescharacteristic, complemented
375C2×SD32—611no—
376C2×SD32—611no—
377C2×SD32—611no—
378C2×SD32C2×D6111yescharacteristic, complemented
379C2×M32—611no—
380C2×M32—611no—
381#467—611no—
382#467—611no—
383#467—611no—
384#467—1221no—
385#467—1221no—
386#467C2×D6111yescharacteristic, complemented
387C2×#107—611no—
388C2×#103—1221no—
389#358—611no—
390#358—611no—
391#358—611no—
392#358C2×D6111yescharacteristic, complemented
393#357—611no—
394#357—611no—
395#357—611no—
396#357C2×D6111yescharacteristic, complemented
397C22⋊C16—611no—
398C22⋊C16—611no—
399C22⋊C16—611no—
400C22⋊C16C2×D6111yescharacteristic, complemented
401C61#3352111yescharacteristic, complemented, Sylow
402C2×#41—21no—
403C2×#41—1221no—
404C2×#41—1221no—
405C2×#41—1228no—
406C2×#41—1221no—
407C2×#41—2441no—
408C2×#41—2441no—
409C2×#41—2444no—
410C22×Q8—1221no—
411C22×Q8—1221no—
412C22×D4—1221no—
413C22×D4—2441no—
414C23×C4—611no—
415C23×C4—2441no—
416#135—22no—
417#135—22no—
418#135—614no—
419#135—614no—
420#135—1222no—
421#135—1222no—
422#135—1222no—
423#135—1222no—
424#135—1228no—
425#135—1228no—
426C2×Q16—611no—
427C2×Q16—611no—
428C2×Q16—1222no—
429C2×Q16—1222no—
430C2×Q16—1221no—
431C2×Q16—1221no—
432C2×Q16—1222no—
433C2×Q16—1222no—
434C2×Q16C22×D6111yescharacteristic
435C2×Q16C22×D6111yescharacteristic
436C2×SD16—21no—
437C2×SD16—1221no—
438C2×SD16—1221no—
439C2×SD16—1228no—
440C2×SD16—1221no—
441C2×D8—611no—
442C2×D8—1221no—
443C2×D8—1222no—
444C2×D8—1222no—
445C2×D8C22×D6111yescharacteristic
446C22×C8—611no—
447C22×C8—611no—
448C22×C8—611no—
449C22×C8—611no—
450C22×C8—611no—
451C22×C8—1222no—
452C22×C8—1222no—
453C22×C8C22×D6111yescharacteristic
454C22⋊Q8—21no—
455C22⋊Q8—1221no—
456C22⋊Q8—1221no—
457C22⋊Q8—1221no—
458C22⋊Q8—2441no—
459C22⋊Q8—2441no—
460C2×C4⋊C4—1221no—
461C2×C4⋊C4—2441no—
462C2×C22⋊C4—2441no—
463Q32—22no—
464Q32—1222no—
465Q32—1222no—
466Q32—1222no—
467SD32—22no—
468SD32—1222no—
469SD32—1222no—
470SD32—1222no—
471M32—1222no—
472M32—1222no—
473C2×C16—611no—
474C2×C16—611no—
475C2×C16C22×D6111yescharacteristic
476C2×C16C22×D6111yescharacteristic
477#107—611no—
478#107—1221no—
479#107C22×D6111yescharacteristic
480#103—21no—
481#103—1221no—
482#103—1221no—
483#103—1221no—
484C24—611no—
485C24—2441no—
486#41—24no—
487#41—42no—
488#41—1224no—
489#41—1224no—
490#41—1224no—
491#41—2442no—
492#41—2448no—
493#41—2442no—
494#41—2442no—
495#41—2448no—
496C2×Q8—21no—
497C2×Q8—21no—
498C2×Q8—1224no—
499C2×Q8—1221no—
500C2×Q8—1221no—
501C2×Q8—2441no—
502C2×Q8—2441no—
503C2×Q8—2442no—
504C2×Q8—2442no—
505C2×Q8—2442no—
506C2×D4—21no—
507C2×D4—41no—
508C2×D4—1224no—
509C2×D4—1221no—
510C2×D4—2441no—
511C2×D4—2441no—
512C2×D4—2442no—
513C2×D4—2448no—
514C2×D4—2441no—
515C2×D4—2441no—
516C22×C4—41no—
517C22×C4—611no—
518C22×C4—611no—
519C22×C4—611no—
520C22×C4—611no—
521C22×C4—611no—
522C22×C4—1222no—
523C22×C4—1222no—
524C22×C4—2441no—
525C22×C4—2448no—
526C22×C4—2441no—
527C22×C4—2441no—
528C22×C4—2441no—
529C22×C4—2441no—
530C22×C4—2441no—
531C22×C4D4×D6111yescharacteristic
532Q16—21no—
533Q16—21no—
534Q16—21no—
535Q16—612no—
536Q16—1224no—
537Q16—1221no—
538Q16—1221no—
539Q16—1221no—
540Q16—2442no—
541Q16D4×D6112yes—
542SD16—24no—
543SD16—1224no—
544SD16—1224no—
545SD16—1224no—
546D8—21no—
547D8—612no—
548D8—1224no—
549D8—1221no—
550D8D4×D6112yes—
551C2×C8—21no—
552C2×C8—21no—
553C2×C8—611no—
554C2×C8—612no—
555C2×C8—612no—
556C2×C8—611no—
557C2×C8—1222no—
558C2×C8—1222no—
559C2×C8—1222no—
560C2×C8—1221no—
561C2×C8—1221no—
562C2×C8D4×D6111yescharacteristic
563C2×C8C23×D6111yescharacteristic
564C4⋊C4—21no—
565C4⋊C4—41no—
566C4⋊C4—1221no—
567C4⋊C4—2441no—
568C4⋊C4—2441no—
569C4⋊C4—2441no—
570C4⋊C4—2441no—
571C22⋊C4—41no—
572C22⋊C4—2441no—
573C22⋊C4—2441no—
574C22⋊C4—2441no—
575C16—21no—
576C16—21no—
577C16—1221no—
578C16—1221no—
579C23—41no—
580C23—611no—
581C23—611no—
582C23—1222no—
583C23—1222no—
584C23—4882no—
585C23—4882no—
586C23—2441no—
587C23—2444no—
588C23—2441no—
589C23D8×D6111yescharacteristic
590Q8—22no—
591Q8—41no—
592Q8—41no—
593Q8—41no—
594Q8—1222no—
595Q8—4882no—
596Q8—2444no—
597Q8—2441no—
598Q8—2441no—
599Q8—2441no—
600D4—22no—
601D4—44no—
602D4—41no—
603D4—1222no—
604D4—2444no—
605D4—2444no—
606D4—2444no—
607D4—2444no—
608D4—2441no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—611no—
616C2×C4—612no—
617C2×C4—612no—
618C2×C4—611no—
619C2×C4—1222no—
620C2×C4—1222no—
621C2×C4—1222no—
622C2×C4—1221no—
623C2×C4—1221no—
624C2×C4—4882no—
625C2×C4—4882no—
626C2×C4—2444no—
627C2×C4—2444no—
628C2×C4—2444no—
629C2×C4—2444no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4D8×D6111yescharacteristic
634C2×C4C2×D4×D6111yescharacteristic
635C8—21no—
636C8—612no—
637C8—1221no—
638C8C2×D4×D6111yescharacteristic
639C8C2×D4×D6111yescharacteristic
640C22—21no—
641C22—21no—
642C22—21no—
643C22—42no—
644C22—81no—
645C22—81no—
646C22—612no—
647C22—612no—
648C22—1221no—
649C22—1221no—
650C22—1221no—
651C22—1221no—
652C22—4888no—
653C22—4881no—
654C22—4881no—
655C22—2444no—
656C22—2442no—
657C22C2×D8×D6111yescharacteristic
658C4—21no—
659C4—42no—
660C4—81no—
661C4—81no—
662C4—612no—
663C4—1221no—
664C4—4881no—
665C4—4881no—
666C4—2442no—
667C4#120×D6111yescharacteristic
668C4AGL(1,ℤ/8)×D6111yescharacteristic
669C2—41no—
670C2—82no—
671C2—1222no—
672C2—1222no—
673C2—4882no—
674C2#447×D6111yescharacteristic
675C2#510×D6111yescharacteristic
676C2#2145895211yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2189C2
socleC22×C61C22×D8
prime radicalC2×#135×D61C2
hypercenter#2189D61
hypocenterC61#33521
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2189

Complements

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

NH ≅ G/Ntype
2. C2×#16840204C2semidirect
3. C2×#16840201C2semidirect
4. #467×D61C2semidirect
5. #21447761C2semidirect
6. #21447759C2semidirect
7. #21447757C2semidirect
8. #21447754C2semidirect
9. #21447751C2semidirect
10. #21447749C2semidirect
11. #21447746C2semidirect
12. #21447530C2semidirect
13. #21447528C2semidirect
14. #21447428C2semidirect
15. C61×#2189C2semidirect
25. C2×Q16×D61C22semidirect
32. C2×Dic488C22semidirect
33. C2×#14519723C22semidirect
34. C2×#14519723C22semidirect
35. C2×#14519722C22semidirect
36. C2×#14519721C22semidirect
37. C2×#14519720C22semidirect
38. C2×#14519716C22semidirect
39. C2×#14519716C22semidirect
41. #16840003C22semidirect
45. #107×D61C22semidirect
47. #16839567C22semidirect
48. #16839566C22semidirect
49. #16839565C22semidirect
50. #16839564C22semidirect
51. #16839563C22semidirect
52. #16839562C22semidirect
53. C22⋊#11758185C22semidirect
57. C2×Q32×C61C22semidirect
58. C2×SD32×C61C22semidirect
59. C61×#467C22semidirect
60. C61×#358C22semidirect
61. C61×#357C22semidirect
62. C61×C22⋊C16C22semidirect
319. C2×D61#510semidirect
338. #2189D61semidirect
374. C2×Q32C2×D61semidirect
378. C2×SD32C2×D61semidirect
386. #467C2×D61semidirect
392. #358C2×D61semidirect
396. #357C2×D61semidirect
400. C22⋊C16C2×D61semidirect
401. C61#33521semidirect

Maximal subgroups

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

subgroupindex
C2×#135×D612
C2×#168402042
C2×#168402012
#467×D612
#214477612
#214477592
#214477572
#214477542
#214477512
#214477492
#214477462
#214475302
#214475282
#214474282
C61×#21892
#3352161