Finite Group Databasebeta

← #29833788

Subgroups — #29833788

Order 15616.

1150 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#135×D61C211yesmaximal, characteristic, complemented
2C2×#16840193C211yesmaximal, characteristic, complemented
3C2×#16840191C211yesmaximal, characteristic, complemented
4#21455398C211yesmaximal, characteristic, complemented
5#21447761C211yesmaximal, characteristic, complemented
6#21447759C211yesmaximal, characteristic, complemented
7#21447754C211yesmaximal, characteristic, complemented
8#21447749C211yesmaximal, characteristic, complemented
9#21447740C211yesmaximal, characteristic, complemented
10#21447734C211yesmaximal, characteristic, complemented
11#21447729C211yesmaximal, characteristic, complemented
12#21447525C211yesmaximal, characteristic, complemented
13#21447523C211yesmaximal, characteristic, complemented
14C22⋊C16×D61C211yesmaximal, characteristic, complemented
15C61×#2186C211yesmaximal, characteristic, complemented
16C2×#41×D61—21no—
17C2×#14519800C2211yescharacteristic
18C2×#14519799C2211yescharacteristic
19C2×#14519798—21no—
20C2×#14519797—21no—
21C2×#14519796C2211yescharacteristic
22C2×#14519796C2211yescharacteristic, complemented
23#135×D61—24no—
24#135×D61—24no—
25C2×Q16×D61C2211yescharacteristic
26C2×SD16×D61—21no—
27C2×D8×D61C2211yescharacteristic
28C22×C8×D61C2211yescharacteristic
29#16840272—21no—
30#16840193—24no—
31#16840191—24no—
32C2×Dic488C2211yescharacteristic, complemented
33C2×#14519723C2211yescharacteristic, complemented
34C2×#14519722C2211yescharacteristic, complemented
35C2×#14519721C2211yescharacteristic, complemented
36C2×#14519721C2211yescharacteristic, complemented
37C2×#14519718C2211yescharacteristic, complemented
38C2×C16×D61C2211yescharacteristic, complemented
39C2×C16×D61C2211yescharacteristic, complemented
40#16840004C2211yescharacteristic
41#16840003C2211yescharacteristic, complemented
42#16839995C2211yescharacteristic
43#16839990—21no—
44#16839989—21no—
45#16839799C2211yescharacteristic, complemented
46#16839755—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×SD32×C61C2211yescharacteristic, complemented
58C2×D16×C61C2211yescharacteristic, complemented
59C61×#466C2211yescharacteristic, 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—21no—
68C2×#11758224—41no—
69#41×D61—28no—
70#41×D61—44no—
71C2×Q8×D61—21no—
72C2×D4×D61—21no—
73C2×D4×D61—41no—
74C22×C4×D61—41no—
75C22×C4×D61D411yescharacteristic
76#14519800—22no—
77#14519800D414yescomplemented
78#14519799—22no—
79#14519799—22no—
80#14519798—28no—
81#14519797—28no—
82#14519796—22no—
83#14519796—22no—
84#14519796—22no—
85#14519796D414yescomplemented
86C2×Dic244C2311yescharacteristic
87C2×Dic244C2311yescharacteristic
88C2×#11758202—21no—
89Q16×D61—22no—
90Q16×D61—22no—
91C2×#11758190—21no—
92C2×#11758200—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#14519759—41no—
108#14519757—21no—
109#14519756—21no—
110#14519748—21no—
111#14519745—41no—
112D4×Dic61—41no—
113#14519737—41no—
114#14519735—21no—
115Dic488—22no—
116#14519723—22no—
117#14519722—22no—
118#14519721—22no—
119#14519721—22no—
120#14519718—22no—
121C2×#11758185C2311yescharacteristic
122C2×#11758185C2311yescharacteristic
123C16×D61—22no—
124C16×D61—22no—
125#14519711C2311yescharacteristic
126#14519708—21no—
127#14519702—21no—
128#14519700—21no—
129#14519698—21no—
130C2×#41×C61—21no—
131#135×C61—22no—
132#135×C61—22no—
133C2×Q16×C61C2311yescharacteristic
134C2×SD16×C61—21no—
135C2×D8×C61C2311yescharacteristic
136C2×D8×C61C2311yescharacteristic
137C22×C8×C61C2311yescharacteristic
138#121×C61—21no—
139SD32×C61—22no—
140D16×C61—22no—
141C2×C16×C61C2311yescharacteristic
142C2×C16×C61C2311yescharacteristic
143#107×C61C2311yescharacteristic
144#102×C61—21no—
145C23×D61—41no—
146C23×D61D811yescharacteristic
147#41×C61—24no—
148#41×C61—42no—
149C2×Q8×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—41no—
153C2×D4×C61—41no—
154C22×C4×C61—41no—
155C22×C4×C61C2×D411yescharacteristic
156C2×C22⋊D61—41no—
157C2×C22⋊D61—41no—
158C2×C22⋊D61—41no—
159C2×C22⋊D61—41no—
160C2×C22⋊D61—41no—
161C22×Dic61—41no—
162C22×Dic61—41no—
163C22×Dic61D811yescharacteristic
164#11758226—24no—
165#11758226—42no—
166Q8×D61—24no—
167Q8×D61—42no—
168#11758224—24no—
169#11758224—48no—
170#11758224—42no—
171#11758224—42no—
172#11758224—42no—
173D4×D61—24no—
174D4×D61—42no—
175D4×D61—48no—
176#11758222—24no—
177#11758222—48no—
178#11758222—42no—
179C2×D244—21no—
180C2×D244—41no—
181C2×C4×D61—22no—
182C2×C4×D61—22no—
183C2×C4×D61—48no—
184C2×C4×D61—41no—
185C2×C4×D61—41no—
186C2×C4×D61—41no—
187C2×C4×D61D811yescharacteristic
188C2×C4×D61D811yescharacteristic
189C2×C4×D61C2×D411yescharacteristic
190C2×C4×D61C2×D411yescharacteristic
191C2×Dic122—21no—
192C2×Dic122—21no—
193C2×Dic122—41no—
194Q16×C61—21no—
195Q16×C61C2×D412yes—
196SD16×C61—24no—
197D8×C61—21no—
198D8×C61—21no—
199D8×C61—21no—
200D8×C61C2×D412yes—
201C2×C8×C61—21no—
202C2×C8×C61—21no—
203C2×C8×C61C2×D411yescharacteristic
204C2×C8×C61C2411yescharacteristic
205C4⋊C4×C61—21no—
206C22⋊C4×C61—41no—
207C22⋊Dic61—41no—
208C22⋊Dic61—41no—
209#11758202—24no—
210#11758201—24no—
211#11758200—24no—
212#11758200—42no—
213#11758199—24no—
214#11758198—41no—
215#11758198—41no—
216C4⋊Dic61—21no—
217C4⋊Dic61—41no—
218#11758196—41no—
219C4×Dic61—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×C61—41no—
242C23×C61C2×D811yescharacteristic
243C22×D61—22no—
244C22×D61—22no—
245C22×D61—41no—
246C22×D61—44no—
247C22×D61—41no—
248C22×D61—82no—
249C22×D61—82no—
250C22×D61C2×D811yescharacteristic
251C22×D61C2×D811yescharacteristic
252Q8×C61—22no—
253Q8×C61—41no—
254D4×C61—22no—
255D4×C61—44no—
256D4×C61—41no—
257D4×C61—41no—
258D4×C61—41no—
259D4×C61—82no—
260C2×C4×C61—21no—
261C2×C4×C61—21no—
262C2×C4×C61—44no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61C2×D811yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C22⋊D61—44no—
268C22⋊D61—44no—
269C22⋊D61—44no—
270C22⋊D61—82no—
271C22⋊D61—82no—
272C2×Dic61—22no—
273C2×Dic61—41no—
274C2×Dic61—44no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—82no—
278C2×Dic61C2×D811yescharacteristic
279C2×Dic61C2×D811yescharacteristic
280D244—22no—
281D244—44no—
282D244—41no—
283C4×D61—22no—
284C4×D61—21no—
285C4×D61—21no—
286C4×D61—22no—
287C4×D61—44no—
288C4×D61—44no—
289C4×D61—44no—
290C4×D61#12012yes—
291C4×D61AGL(1,ℤ/8)12yes—
292Dic122—22no—
293Dic122—44no—
294Dic122—41no—
295Dic122—41no—
296Dic122—41no—
297C8×C61—21no—
298C8×C61C22×D411yescharacteristic
299C8×C61C22×D411yescharacteristic
300#92418—21no—
301#92418#12012yes—
302#33516—611nomaximal, Sylow
303C22×C61—21no—
304C22×C61—21no—
305C22×C61—21no—
306C22×C61—42no—
307C22×C61—81no—
308C22×C61—81no—
309C22×C61—81no—
310C22×C61—81no—
311C22×C61—81no—
312C22×C61C22×D811yescharacteristic
313C2×D61—21no—
314C2×D61—21no—
315C2×D61—21no—
316C2×D61—21no—
317C2×D61—44no—
318C2×D61—42no—
319C2×D61—88no—
320C2×D61—81no—
321C2×D61—81no—
322C2×D61#44712yes—
323C2×D61#50812yescomplemented
324C4×C61—21no—
325C4×C61—42no—
326C4×C61—81no—
327C4×C61C2×#12011yescharacteristic
328C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
329Dic61—21no—
330Dic61—42no—
331Dic61—81no—
332Dic61—81no—
333Dic61#44712yes—
334C22×#135—611no—
335C2×#508—611no—
336C2×#508—611no—
337#3050—611no—
338#2189—611no—
339#2189—611no—
340#2188—611no—
341#2187—611no—
342#2186—611no—
343#2186D6111yescharacteristic, complemented
344#2185—611no—
345#2184—611no—
346#2138—611no—
347#2138—611no—
348C2×C22⋊C16—611no—
349C2×C61—41no—
350C2×C61—82no—
351C2×C61—161no—
352C2×C61C2×#44711yescharacteristic
353C2×C61C2×#50811yescharacteristic
354C2×C61#341411yescharacteristic
355D61—22no—
356D61—22no—
357D61—82no—
358C22×#41—1221no—
359C2×#135—611no—
360C2×#135—611no—
361C2×#135—611no—
362C2×#135—611no—
363C2×#135—1221no—
364C2×#135—1221no—
365C2×#135—1224no—
366C2×#135—1224no—
367C2×#135C2×D6111yescharacteristic
368C22×Q16—611no—
369C22×SD16—1221no—
370C22×D8—611no—
371C23×C8—611no—
372#525—1221no—
373#508—1224no—
374#508—1224no—
375C2×Q32—611no—
376C2×Q32—611no—
377C2×SD32—611no—
378C2×SD32—611no—
379C2×SD32—611no—
380C2×SD32C2×D6111yescharacteristic, complemented
381C2×D16—611no—
382C2×D16C2×D6111yescharacteristic, complemented
383C22×C16—611no—
384C22×C16—611no—
385#467—611no—
386#467—611no—
387#466—611no—
388#466C2×D6111yescharacteristic, complemented
389#465—1221no—
390#465—1221no—
391#427—611no—
392#416—1221no—
393#358—611no—
394#358—611no—
395#358—611no—
396#358C2×D6111yescharacteristic, complemented
397#357—611no—
398#357—611no—
399#357—611no—
400#357C2×D6111yescharacteristic, complemented
401C22⋊C16—611no—
402C22⋊C16—611no—
403C22⋊C16—611no—
404C22⋊C16C2×D6111yescharacteristic, complemented
405C61#3351611yescharacteristic, complemented, Sylow
406C2×#41—21no—
407C2×#41—1221no—
408C2×#41—1221no—
409C2×#41—1221no—
410C2×#41—1228no—
411C2×#41—1221no—
412C2×#41—2441no—
413C2×#41—2441no—
414C2×#41—2444no—
415C22×Q8—1221no—
416C22×D4—1221no—
417C22×D4—2441no—
418C23×C4—611no—
419C23×C4—2441no—
420#135—22no—
421#135—22no—
422#135—614no—
423#135—614no—
424#135—1222no—
425#135—1222no—
426#135—1222no—
427#135—1222no—
428#135—1222no—
429#135—1222no—
430#135—1228no—
431#135—1228no—
432C2×Q16—611no—
433C2×Q16—611no—
434C2×Q16—1221no—
435C2×Q16—1222no—
436C2×Q16—1222no—
437C2×Q16C22×D6111yescharacteristic
438C2×SD16—21no—
439C2×SD16—1221no—
440C2×SD16—1221no—
441C2×SD16—1221no—
442C2×SD16—1228no—
443C2×SD16—1221no—
444C2×D8—611no—
445C2×D8—1221no—
446C2×D8—1222no—
447C2×D8—1222no—
448C2×D8C22×D6111yescharacteristic
449C2×D8C22×D6111yescharacteristic
450C22×C8—611no—
451C22×C8—611no—
452C22×C8—611no—
453C22×C8—611no—
454C22×C8—611no—
455C22×C8—1222no—
456C22×C8—1222no—
457C22×C8C22×D6111yescharacteristic
458#123—2441no—
459C22⋊Q8—1221no—
460C22⋊Q8—1221no—
461#121—21no—
462#121—1221no—
463#120—2441no—
464C4×D4—2441no—
465C4×D4—2441no—
466#117—1221no—
467Q32—1222no—
468Q32—1222no—
469SD32—22no—
470SD32—1222no—
471SD32—1222no—
472SD32—1222no—
473D16—22no—
474D16—1222no—
475C2×C16—611no—
476C2×C16—611no—
477C2×C16—1222no—
478C2×C16—1222no—
479C2×C16C22×D6111yescharacteristic
480C2×C16C22×D6111yescharacteristic
481#107—611no—
482#107C22×D6111yescharacteristic
483#106—1221no—
484#103—1221no—
485#103—1221no—
486#102—21no—
487#102—1221no—
488C24—611no—
489C24—2441no—
490#41—24no—
491#41—42no—
492#41—1224no—
493#41—1224no—
494#41—1224no—
495#41—2442no—
496#41—2448no—
497#41—2442no—
498#41—2442no—
499#41—2442no—
500#41—2442no—
501#41—2448no—
502C2×Q8—21no—
503C2×Q8—1224no—
504C2×Q8—1221no—
505C2×Q8—1221no—
506C2×Q8—2441no—
507C2×Q8—2442no—
508C2×D4—21no—
509C2×D4—21no—
510C2×D4—41no—
511C2×D4—41no—
512C2×D4—1224no—
513C2×D4—1221no—
514C2×D4—2441no—
515C2×D4—2441no—
516C2×D4—2442no—
517C2×D4—2448no—
518C2×D4—2441no—
519C2×D4—2441no—
520C2×D4—2441no—
521C2×D4—2441no—
522C22×C4—41no—
523C22×C4—611no—
524C22×C4—611no—
525C22×C4—611no—
526C22×C4—611no—
527C22×C4—611no—
528C22×C4—1222no—
529C22×C4—1222no—
530C22×C4—2441no—
531C22×C4—2448no—
532C22×C4—2441no—
533C22×C4—2441no—
534C22×C4—2441no—
535C22×C4—2441no—
536C22×C4D4×D6111yescharacteristic
537Q16—21no—
538Q16—612no—
539Q16—1224no—
540Q16—1221no—
541Q16—1221no—
542Q16—1221no—
543Q16D4×D6112yes—
544SD16—24no—
545SD16—1224no—
546SD16—1224no—
547SD16—1224no—
548SD16—2442no—
549D8—21no—
550D8—21no—
551D8—21no—
552D8—612no—
553D8—1224no—
554D8—1221no—
555D8D4×D6112yes—
556C2×C8—21no—
557C2×C8—21no—
558C2×C8—611no—
559C2×C8—612no—
560C2×C8—612no—
561C2×C8—611no—
562C2×C8—1222no—
563C2×C8—1222no—
564C2×C8—1222no—
565C2×C8—1221no—
566C2×C8—1221no—
567C2×C8D4×D6111yescharacteristic
568C2×C8C23×D6111yescharacteristic
569C4⋊C4—21no—
570C4⋊C4—1221no—
571C4⋊C4—2441no—
572C4⋊C4—2441no—
573C22⋊C4—41no—
574C22⋊C4—2441no—
575C22⋊C4—2441no—
576C22⋊C4—2441no—
577C22⋊C4—2441no—
578C42—2441no—
579C16—21no—
580C16—21no—
581C16—1221no—
582C16—1221no—
583C23—41no—
584C23—41no—
585C23—611no—
586C23—611no—
587C23—1222no—
588C23—1222no—
589C23—4882no—
590C23—4882no—
591C23—2441no—
592C23—2444no—
593C23—2441no—
594C23D8×D6111yescharacteristic
595Q8—22no—
596Q8—41no—
597Q8—1222no—
598Q8—2444no—
599Q8—2441no—
600Q8—2441no—
601Q8—2441no—
602D4—22no—
603D4—44no—
604D4—41no—
605D4—41no—
606D4—41no—
607D4—82no—
608D4—1222no—
609D4—4882no—
610D4—4882no—
611D4—2444no—
612D4—2444no—
613D4—2444no—
614D4—2444no—
615D4—2441no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—611no—
622C2×C4—612no—
623C2×C4—612no—
624C2×C4—611no—
625C2×C4—1222no—
626C2×C4—1222no—
627C2×C4—1222no—
628C2×C4—1221no—
629C2×C4—1221no—
630C2×C4—4882no—
631C2×C4—2444no—
632C2×C4—2444no—
633C2×C4—2444no—
634C2×C4—2444no—
635C2×C4—2441no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4D8×D6111yescharacteristic
639C2×C4C2×D4×D6111yescharacteristic
640C8—21no—
641C8—612no—
642C8—1221no—
643C8C2×D4×D6111yescharacteristic
644C8C2×D4×D6111yescharacteristic
645C22—21no—
646C22—21no—
647C22—21no—
648C22—42no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—612no—
655C22—612no—
656C22—1221no—
657C22—1221no—
658C22—1221no—
659C22—1221no—
660C22—4888no—
661C22—4881no—
662C22—4881no—
663C22—2444no—
664C22—2442no—
665C22C2×D8×D6111yescharacteristic
666C4—21no—
667C4—42no—
668C4—81no—
669C4—612no—
670C4—1221no—
671C4—4881no—
672C4—4881no—
673C4—2442no—
674C4#120×D6111yescharacteristic
675C4AGL(1,ℤ/8)×D6111yescharacteristic
676C2—41no—
677C2—82no—
678C2—161no—
679C2—1222no—
680C2—1222no—
681C2—4882no—
682C2#447×D6111yescharacteristic
683C2#508×D6111yescharacteristic
684C2#2145895111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2186C2
socleC22×C61C22×D8
hypercenter#2186D61
hypocenterC61#33516
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2186

Complements

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

NH ≅ G/Ntype
1. C2×#135×D61C2semidirect
2. C2×#16840193C2semidirect
3. C2×#16840191C2semidirect
4. #21455398C2semidirect
5. #21447761C2semidirect
6. #21447759C2semidirect
7. #21447754C2semidirect
8. #21447749C2semidirect
9. #21447740C2semidirect
10. #21447734C2semidirect
11. #21447729C2semidirect
12. #21447525C2semidirect
13. #21447523C2semidirect
14. C22⋊C16×D61C2semidirect
15. C61×#2186C2semidirect
22. C2×#14519796C22semidirect
32. C2×Dic488C22semidirect
33. C2×#14519723C22semidirect
34. C2×#14519722C22semidirect
35. C2×#14519721C22semidirect
36. C2×#14519721C22semidirect
37. C2×#14519718C22semidirect
38. C2×C16×D61C22semidirect
39. C2×C16×D61C22semidirect
41. #16840003C22semidirect
45. #16839799C22semidirect
47. #16839567C22semidirect
48. #16839566C22semidirect
49. #16839565C22semidirect
50. #16839564C22semidirect
51. #16839563C22semidirect
52. #16839562C22semidirect
53. C22⋊#11758185C22semidirect
57. C2×SD32×C61C22semidirect
58. C2×D16×C61C22semidirect
59. C61×#466C22semidirect
60. C61×#358C22semidirect
61. C61×#357C22semidirect
62. C61×C22⋊C16C22semidirect
77. #14519800D4semidirect
85. #14519796D4semidirect
323. C2×D61#508semidirect
343. #2186D61semidirect
380. C2×SD32C2×D61semidirect
382. C2×D16C2×D61semidirect
388. #466C2×D61semidirect
396. #358C2×D61semidirect
400. #357C2×D61semidirect
404. C22⋊C16C2×D61semidirect
405. C61#33516semidirect

Maximal subgroups

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

subgroupindex
C2×#135×D612
C2×#168401932
C2×#168401912
#214553982
#214477612
#214477592
#214477542
#214477492
#214477402
#214477342
#214477292
#214475252
#214475232
C22⋊C16×D612
C61×#21862
#3351661