Finite Group Databasebeta

← #29836487

Subgroups — #29836487

Order 15616.

738 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840197C211yesmaximal, characteristic, complemented
2C22×#14519720C211yesmaximal, characteristic, complemented
3D4×D488C211yesmaximal, characteristic, complemented
4#21448086C211yesmaximal, characteristic, complemented
5#21447934C211yesmaximal, characteristic, complemented
6#21447903C211yesmaximal, characteristic, complemented
7#21447871C211yesmaximal, characteristic, complemented
8#21447859C211yesmaximal, characteristic, complemented
9#21447747C211yesmaximal, characteristic, complemented
10#21447742C211yesmaximal, characteristic, complemented
11#21447736C211yesmaximal, characteristic, complemented
12#21447730C211yesmaximal, characteristic, complemented
13#21447681C211yesmaximal, characteristic, complemented
14#21447645C211yesmaximal, characteristic, complemented
15C61×#3285C211yesmaximal, characteristic, complemented
16C22×D488C2211yescharacteristic
17C22×D488C2211yescharacteristic
18D4×D244—21no—
19#16840197—24no—
20C2×#14519720—24no—
21C2×#14519720C2211yescharacteristic
22C2×#14519720C2211yescharacteristic
23C2×#14519720C2211yescharacteristic, complemented
24C2×#14519720C2211yescharacteristic, complemented
25C2×#14519720C2211yescharacteristic, complemented
26C2×#14519720C2211yescharacteristic, complemented
27C2×#14519718C2211yescharacteristic
28C2×#14519718C2211yescharacteristic, complemented
29C2×#14519718C2211yescharacteristic, complemented
30C2×#14519717C2211yescharacteristic, complemented
31C22×#11758185C2211yescharacteristic, complemented
32#16840139C2211yescharacteristic
33#16839998C2211yescharacteristic
34#16839998C2211yescharacteristic
35#16839958—21no—
36#16839898—21no—
37#16839898—21no—
38C4×D488C2211yescharacteristic
39#16839587C2211yescharacteristic, complemented
40C4⋊#11758185C2211yescharacteristic, complemented
41#16839566C2211yescharacteristic, complemented
42#16839566C2211yescharacteristic, complemented
43#16839563C2211yescharacteristic, complemented
44#16839562C2211yescharacteristic
45#16839562C2211yescharacteristic, complemented
46#16839562C2211yescharacteristic, complemented
47C22⋊#11758185C2211yescharacteristic, complemented
48C22⋊#11758185C2211yescharacteristic, complemented
49#16839538C2211yescharacteristic, complemented
50C2×#135×C61C2211yescharacteristic, complemented
51C22×Q16×C61C2211yescharacteristic, complemented
52C61×#548—21no—
53C61×#495C2211yescharacteristic, complemented
54C61×#467C2211yescharacteristic, complemented
55C61×#466C2211yescharacteristic, complemented
56C61×#464—21no—
57C61×#452—21no—
58C61×#448—21no—
59C4×Q16×C61C2211yescharacteristic, complemented
60C8×D4×C61C2211yescharacteristic
61C22×D244—21no—
62C22×D244—21no—
63C2×D4×D61—41no—
64C2×D488—22no—
65C2×D488—22no—
66C2×D488—21no—
67C2×D488—21no—
68C2×D488—21no—
69C2×D488—41no—
70C2×D488—41no—
71C2×D488C2311yescharacteristic
72C2×D488C2311yescharacteristic
73C2×D488C2311yescharacteristic
74#14519774—21no—
75#14519752—41no—
76#14519751—21no—
77#14519751—21no—
78#14519747—41no—
79#14519747—41no—
80C4×D244—21no—
81#14519720—22no—
82#14519720—22no—
83#14519720—22no—
84#14519720—22no—
85#14519720—41no—
86#14519720D414yescomplemented
87#14519718—22no—
88#14519718—22no—
89#14519718—41no—
90#14519717—22no—
91C2×#11758185—22no—
92C2×#11758185C2311yescharacteristic
93C2×#11758185C2311yescharacteristic
94C2×#11758185C2311yescharacteristic
95C2×#11758185C2311yescharacteristic
96#14519711C2311yescharacteristic
97#14519699—21no—
98#14519699—21no—
99#14519699—21no—
100C2×#41×C61—21no—
101C22×Q8×C61—21no—
102#135×C61—22no—
103#135×C61—41no—
104C2×Q16×C61—22no—
105C2×Q16×C61—21no—
106C2×Q16×C61—41no—
107C2×Q16×C61C2311yescharacteristic
108C2×Q16×C61C2311yescharacteristic
109C2×SD16×C61—21no—
110C2×SD16×C61—21no—
111C2×D8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114#124×C61—21no—
115#124×C61—41no—
116C22⋊Q8×C61—21no—
117C22⋊Q8×C61—41no—
118#121×C61—21no—
119#121×C61—41no—
120C4×Q8×C61—21no—
121C4×D4×C61—41no—
122C4×D4×C61D411yescharacteristic
123#107×C61C2311yescharacteristic
124C4⋊C8×C61D411yescharacteristic
125#103×C61—21no—
126#103×C61—21no—
127#102×C61—21no—
128C22⋊C8×C61D411yescharacteristic
129C22⋊C8×C61D411yescharacteristic
130C4×C8×C61C2311yescharacteristic
131C23×D61—41no—
132C23×D61—41no—
133#41×C61—42no—
134#41×C61—81no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—42no—
138C2×Q8×C61—41no—
139C2×Q8×C61—81no—
140C2×D4×C61—21no—
141C2×D4×C61—41no—
142C2×D4×C61—41no—
143C2×D4×C61D811yescharacteristic
144C22×C4×C61—41no—
145C22×C4×C61—41no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C2×C22⋊D61—41no—
149C2×C22⋊D61—41no—
150D4×D61—84no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—42no—
155C2×D244—42no—
156C2×D244—41no—
157C2×D244—41no—
158C2×D244—41no—
159C2×D244—41no—
160C2×D244—81no—
161C2×D244—81no—
162C2×C4×D61—41no—
163Q16×C61—21no—
164Q16×C61—21no—
165Q16×C61—21no—
166Q16×C61—42no—
167Q16×C61C2×D412yescomplemented
168SD16×C61—42no—
169SD16×C61—81no—
170D8×C61—21no—
171D8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61C2×D411yescharacteristic
177C2×C8×C61C2×D411yescharacteristic
178C2×C8×C61C2×D411yescharacteristic
179C2×C8×C61C2411yescharacteristic
180C4⋊C4×C61—21no—
181C4⋊C4×C61—41no—
182C4⋊C4×C61—41no—
183C4⋊C4×C61D811yescharacteristic
184C22⋊C4×C61—41no—
185C22⋊C4×C61—41no—
186C22⋊C4×C61—41no—
187C22⋊C4×C61—41no—
188C22⋊C4×C61D811yescharacteristic
189C22⋊C4×C61D811yescharacteristic
190C42×C61—41no—
191C42×C61C2×D411yescharacteristic
192#11758198—41no—
193#11758198—41no—
194#11758198—41no—
195C4⋊Dic61—21no—
196D488—21no—
197D488—21no—
198D488—21no—
199D488—21no—
200D488—21no—
201D488—42no—
202D488—42no—
203D488—81no—
204D488C2×D412yes—
205#11758185—21no—
206#11758185—21no—
207#11758185—21no—
208#11758185C2×D412yescomplemented
209C23×C61—41no—
210C23×C61C2×D811yescharacteristic
211C23×C61C2×D811yescharacteristic
212C22×D61—41no—
213C22×D61—41no—
214C22×D61—41no—
215C22×D61—82no—
216C22×D61—82no—
217C22×D61—82no—
218C22×D61—82no—
219C22×D61—161no—
220C22×D61—161no—
221Q8×C61—22no—
222Q8×C61—41no—
223Q8×C61—41no—
224Q8×C61—41no—
225Q8×C61—82no—
226D4×C61—22no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—82no—
230D4×C61—82no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—82no—
239C2×C4×C61—82no—
240C2×C4×C61C2×D811yescharacteristic
241C2×C4×C61C2×D811yescharacteristic
242C2×C4×C61C2×D811yescharacteristic
243C2×C4×C61C2×D811yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C22⋊D61—82no—
246C22⋊D61—82no—
247C2×Dic61—41no—
248D244—22no—
249D244—41no—
250D244—41no—
251D244—41no—
252D244—41no—
253D244—41no—
254D244—82no—
255D244—82no—
256D244—82no—
257D244—161no—
258C4×D61—82no—
259C8×C61—21no—
260C8×C61—21no—
261C8×C61—21no—
262C8×C61C22×D411yescharacteristic
263C8×C61H(5,𝔽2)11yescharacteristic
264#33831—611nomaximal, Sylow
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—21no—
268C22×C61—21no—
269C22×C61—21no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61#50912yescomplemented
274C22×C61C22×D811yescharacteristic
275C2×D61—42no—
276C2×D61—81no—
277C2×D61—81no—
278C2×D61—81no—
279C2×D61—81no—
280C2×D61—81no—
281C2×D61—81no—
282C2×D61—81no—
283C2×D61—81no—
284C2×D61—164no—
285C2×D61—164no—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—21no—
289C4×C61—21no—
290C4×C61—42no—
291C4×C61—81no—
292C4×C61—81no—
293C4×C61D4211yescharacteristic
294C4×C61#57611yescharacteristic
295Dic61—81no—
296C2×#509—611no—
297C22×SD32—611no—
298#3285D6111yescharacteristic, complemented
299D4×D8—611no—
300#2250—611no—
301#2220—611no—
302#2215—611no—
303#2210—611no—
304#2207—611no—
305#2187—611no—
306#2186—611no—
307#2185—611no—
308#2184—611no—
309#2175—611no—
310#2168—611no—
311C2×C61—22no—
312C2×C61—41no—
313C2×C61—161no—
314C2×C61D4×D811yescharacteristic
315C2×C61C2×#50911yescharacteristic
316C2×C61#341611yescharacteristic
317D61—82no—
318D61—161no—
319D61—161no—
320C2×#135C2×D6111yescharacteristic, complemented
321C22×Q16C2×D6111yescharacteristic, complemented
322C22×D8—611no—
323C22×D8—611no—
324#548—21no—
325D42—1221no—
326#509—1224no—
327C2×SD32—611no—
328C2×SD32—611no—
329C2×SD32—611no—
330C2×SD32—611no—
331C2×SD32—611no—
332C2×SD32—611no—
333C2×SD32—1224no—
334C2×D16—611no—
335C2×D16—611no—
336C2×D16—611no—
337C2×M32—611no—
338C22×C16—611no—
339#495C2×D6111yescharacteristic, complemented
340#493—611no—
341#467C2×D6111yescharacteristic, complemented
342#466—611no—
343#466—611no—
344#466C2×D6111yescharacteristic, complemented
345#464—21no—
346#459—1221no—
347#452—21no—
348#448—21no—
349#447—1221no—
350#447—1221no—
351C4×Q16C2×D6111yescharacteristic, complemented
352C4×D8—611no—
353C8×D4C2×D6111yescharacteristic
354#366—611no—
355C4⋊C16—611no—
356#358—611no—
357#358—611no—
358#357—611no—
359#357—611no—
360#357—611no—
361#357—611no—
362C22⋊C16—611no—
363C22⋊C16—611no—
364#346—611no—
365C61#3383111yescharacteristic, complemented, Sylow
366C2×#41—21no—
367C22×Q8—21no—
368C22×D4—1221no—
369C22×D4—1221no—
370C22×D4—2441no—
371#135—22no—
372#135—41no—
373C2×Q16—22no—
374C2×Q16—21no—
375C2×Q16—41no—
376C2×Q16C22×D6111yescharacteristic
377C2×Q16C22×D6111yescharacteristic
378C2×SD16—21no—
379C2×SD16—21no—
380C2×D8—611no—
381C2×D8—611no—
382C2×D8—611no—
383C2×D8—1222no—
384C2×D8—1222no—
385C2×D8—1221no—
386C2×D8—1221no—
387C2×D8—1221no—
388C2×D8—2441no—
389C2×D8—2441no—
390C2×D8C22×D6111yescharacteristic
391C22×C8C22×D6111yescharacteristic
392C22×C8C22×D6111yescharacteristic
393#127—1221no—
394#124—21no—
395#124—41no—
396C22⋊Q8—21no—
397C22⋊Q8—41no—
398#121—21no—
399#121—41no—
400#121—1221no—
401#121—1221no—
402#121—2441no—
403#120—2441no—
404#120—2441no—
405C4×Q8—21no—
406C4×D4—41no—
407C4×D4—1221no—
408C4×D4C22⋊D6111yescharacteristic
409SD32—614no—
410SD32—1222no—
411SD32—1222no—
412SD32—1222no—
413SD32—1222no—
414SD32—2441no—
415D16—1222no—
416D16—1222no—
417D16—2441no—
418M32—1222no—
419C2×C16—611no—
420C2×C16—611no—
421C2×C16—611no—
422C2×C16—611no—
423C2×C16—1222no—
424#107—611no—
425#107C22×D6111yescharacteristic
426C4⋊C8C22⋊D6111yescharacteristic
427#103—21no—
428#103—21no—
429#102—21no—
430#102—1221no—
431#102—1221no—
432#102—1221no—
433C22⋊C8C22⋊D6111yescharacteristic
434C22⋊C8C22⋊D6111yescharacteristic
435C4×C8C22×D6111yescharacteristic
436C24—2441no—
437C24—2441no—
438#41—42no—
439#41—81no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—42no—
443C2×Q8—41no—
444C2×Q8—81no—
445C2×D4—21no—
446C2×D4—41no—
447C2×D4—41no—
448C2×D4—1221no—
449C2×D4—1221no—
450C2×D4—1221no—
451C2×D4—4881no—
452C2×D4—4881no—
453C2×D4—4884no—
454C2×D4—2442no—
455C2×D4—2442no—
456C2×D4—2441no—
457C2×D4—2441no—
458C2×D4—2441no—
459C2×D4—2441no—
460C2×D4—2441no—
461C2×D4—2441no—
462C2×D4#1175819911yescharacteristic
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—2441no—
466C22×C4C2×C22⋊D6111yescharacteristic
467C22×C4C2×C22⋊D6111yescharacteristic
468Q16—21no—
469Q16—21no—
470Q16—21no—
471Q16—42no—
472Q16D4×D6112yescomplemented
473SD16—42no—
474SD16—81no—
475D8—21no—
476D8—21no—
477D8—612no—
478D8—1221no—
479D8—1221no—
480D8—1221no—
481D8—1221no—
482D8—1221no—
483D8—4881no—
484D8—2442no—
485D8—2442no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8C2×C22⋊D6111yescharacteristic
491C2×C8C2×C22⋊D6111yescharacteristic
492C2×C8C2×C22⋊D6111yescharacteristic
493C2×C8C23×D6111yescharacteristic
494C4⋊C4—21no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—1221no—
498C4⋊C4#1175819911yescharacteristic
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—2441no—
504C22⋊C4—2441no—
505C22⋊C4—2441no—
506C22⋊C4#1175819911yescharacteristic
507C22⋊C4#1175819911yescharacteristic
508C42—41no—
509C42C2×C22⋊D6111yescharacteristic
510C16—612no—
511C16—1221no—
512C16—1221no—
513C16—1221no—
514C23—41no—
515C23—9761no—
516C23—9761no—
517C23—4882no—
518C23—4882no—
519C23—4882no—
520C23—4882no—
521C23—2441no—
522C23—2441no—
523C23—2441no—
524C23C2×#1175819911yescharacteristic
525C23C2×#1175819911yescharacteristic
526Q8—22no—
527Q8—41no—
528Q8—41no—
529Q8—41no—
530Q8—82no—
531D4—22no—
532D4—41no—
533D4—41no—
534D4—82no—
535D4—82no—
536D4—1222no—
537D4—9761no—
538D4—4882no—
539D4—4882no—
540D4—4882no—
541D4—4882no—
542D4—4882no—
543D4—2441no—
544D4—2441no—
545D4—2441no—
546D4—2441no—
547D4—2441no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—4882no—
558C2×C4—2441no—
559C2×C4C2×#1175819911yescharacteristic
560C2×C4C2×#1175819911yescharacteristic
561C2×C4C2×#1175819911yescharacteristic
562C2×C4C2×#1175819911yescharacteristic
563C2×C4C22×C22⋊D6111yescharacteristic
564C8—21no—
565C8—21no—
566C8—21no—
567C8C2×D4×D6111yescharacteristic
568C8#1451982211yescharacteristic
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—9764no—
578C22—9764no—
579C22—4881no—
580C22—4881no—
581C22—4881no—
582C22—4881no—
583C22—4881no—
584C22—4881no—
585C22—4881no—
586C22—4881no—
587C22—2442no—
588C22#1684019712yescomplemented
589C22C22×#1175819911yescharacteristic
590C4—21no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—42no—
595C4—81no—
596C4—81no—
597C4—4881no—
598C4D4×C22⋊D6111yescharacteristic
599C4#1684056511yescharacteristic
600C2—22no—
601C2—41no—
602C2—161no—
603C2—9761no—
604C2—9761no—
605C2—4882no—
606C2D4×#1175819911yescharacteristic
607C2C2×#1684019711yescharacteristic
608C2#2145896311yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3285C2
socleC22×C61C22×D8
hypercenter#3285D61
hypocenterC61#33831
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3285

Complements

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

NH ≅ G/Ntype
1. C2×#16840197C2semidirect
2. C22×#14519720C2semidirect
3. D4×D488C2semidirect
4. #21448086C2semidirect
5. #21447934C2semidirect
6. #21447903C2semidirect
7. #21447871C2semidirect
8. #21447859C2semidirect
9. #21447747C2semidirect
10. #21447742C2semidirect
11. #21447736C2semidirect
12. #21447730C2semidirect
13. #21447681C2semidirect
14. #21447645C2semidirect
15. C61×#3285C2semidirect
23. C2×#14519720C22semidirect
24. C2×#14519720C22semidirect
25. C2×#14519720C22semidirect
26. C2×#14519720C22semidirect
28. C2×#14519718C22semidirect
29. C2×#14519718C22semidirect
30. C2×#14519717C22semidirect
31. C22×#11758185C22semidirect
39. #16839587C22semidirect
40. C4⋊#11758185C22semidirect
41. #16839566C22semidirect
42. #16839566C22semidirect
43. #16839563C22semidirect
45. #16839562C22semidirect
46. #16839562C22semidirect
47. C22⋊#11758185C22semidirect
48. C22⋊#11758185C22semidirect
49. #16839538C22semidirect
50. C2×#135×C61C22semidirect
51. C22×Q16×C61C22semidirect
53. C61×#495C22semidirect
54. C61×#467C22semidirect
55. C61×#466C22semidirect
59. C4×Q16×C61C22semidirect
86. #14519720D4semidirect
167. Q16×C61C2×D4semidirect
208. #11758185C2×D4semidirect
273. C22×C61#509semidirect
298. #3285D61semidirect
320. C2×#135C2×D61semidirect
321. C22×Q16C2×D61semidirect
339. #495C2×D61semidirect
341. #467C2×D61semidirect
344. #466C2×D61semidirect
351. C4×Q16C2×D61semidirect
365. C61#33831semidirect
472. Q16D4×D61semidirect
588. C22#16840197semidirect

Maximal subgroups

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

subgroupindex
C2×#168401972
C22×#145197202
D4×D4882
#214480862
#214479342
#214479032
#214478712
#214478592
#214477472
#214477422
#214477362
#214477302
#214476812
#214476452
C61×#32852
#3383161