Finite Group Databasebeta

← #29835124

Subgroups — #29835124

Order 15616.

568 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456459C211yesmaximal, characteristic, complemented
2#21454670C211yesmaximal, characteristic, complemented
3#21448010C211yesmaximal, characteristic, complemented
4#21447965C211yesmaximal, characteristic, complemented
5#21447955C211yesmaximal, characteristic, complemented
6#21447877C211yesmaximal, characteristic, complemented
7#21447877C211yesmaximal, characteristic, complemented
8#21447747C211yesmaximal, characteristic, complemented
9#21447733C211yesmaximal, characteristic
10#21447694C211yesmaximal, characteristic, complemented
11#21447686C211yesmaximal, characteristic, complemented
12#21447668C211yesmaximal, characteristic, complemented
13#21447665C211yesmaximal, characteristic, complemented
14#21447623C211yesmaximal, characteristic, complemented
15C61×#2237C211yesmaximal, characteristic, complemented
16C22×D488C2211yescharacteristic
17#16840385—21no—
18#16840236—21no—
19C2×#14519720C2211yescharacteristic
20C2×#14519720C2211yescharacteristic
21C2×D976C2211yescharacteristic
22C2×D976C2211yescharacteristic
23#16840166C2211yescharacteristic, complemented
24#16840147C2211yescharacteristic, complemented
25#16840123—21no—
26#16840010—21no—
27#16839998C2211yescharacteristic
28#16839873—21no—
29#16839873—21no—
30#16839865—21no—
31Q16×Dic61C2211yescharacteristic, complemented
32#16839844C2211yescharacteristic
33#16839844C2211yescharacteristic
34#16839844C2211yescharacteristic
35#16839844C2211yescharacteristic
36#16839844C2211yescharacteristic, complemented
37#16839766—21no—
38#16839717C2211yescharacteristic
39#16839589C2211yescharacteristic, complemented
40#16839587C2211yescharacteristic, complemented
41#16839580C2211yescharacteristic, complemented
42#16839580C2211yescharacteristic, complemented
43#16839564C2211yescharacteristic
44#16839564C2211yescharacteristic
45#16839562C2211yescharacteristic
46#16839562C2211yescharacteristic
47#16839562C2211yescharacteristic, complemented
48#16839562C2211yescharacteristic, complemented
49C22⋊#11758185C2211yescharacteristic
50#16839537C2211yescharacteristic, complemented
51C16×Dic61C2211yescharacteristic, complemented
52C61×#467C2211yescharacteristic
53C61×#427C2211yescharacteristic
54C61×#367C2211yescharacteristic, complemented
55C61×#366C2211yescharacteristic, complemented
56C61×#358C2211yescharacteristic, complemented
57C61×#358C2211yescharacteristic, complemented
58C61×C22⋊C16C2211yescharacteristic
59C22×D244—21no—
60C2×#11758201—21no—
61C2×D488—21no—
62C2×D488—22no—
63C2×D488—22no—
64C2×D488C2311yescharacteristic
65C2×D488C2311yescharacteristic
66C2×D488C2311yescharacteristic
67C2×#11758194D411yescharacteristic
68#14519776—21no—
69#14519772—41no—
70#14519766—21no—
71#14519763—41no—
72#14519759—41no—
73#14519757—41no—
74#14519751—21no—
75Q8×Dic61—21no—
76C4×D244—41no—
77C4×D244—41no—
78#14519739—21no—
79#14519739—21no—
80#14519739—21no—
81#14519739—21no—
82#14519739—21no—
83#14519736D411yescharacteristic
84C22⋊C4×D61—41no—
85C22⋊C4×D61—41no—
86#14519720—22no—
87#14519720—22no—
88D976—22no—
89D976—22no—
90C2×#11758185C2311yescharacteristic
91C2×#11758185C2311yescharacteristic
92#14519711C2311yescharacteristic
93#14519710—21no—
94#14519701—21no—
95#14519699—21no—
96#14519697—21no—
97#14519697—21no—
98#14519697—21no—
99#14519697—21no—
100#14519697—21no—
101#14519687D411yescharacteristic
102#14519687D411yescharacteristic
103C8×Dic61C2311yescharacteristic
104C8×Dic61C2311yescharacteristic
105C2×Q16×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C22⋊Q8×C61—21no—
108#117×C61—21no—
109C2×C16×C61C2311yescharacteristic
110C2×C16×C61C2311yescharacteristic
111#107×C61C2311yescharacteristic
112#107×C61C2311yescharacteristic
113#107×C61C2311yescharacteristic
114#106×C61—21no—
115#103×C61—21no—
116C23×D61—41no—
117C2×Q8×C61—21no—
118C22×C4×C61C2×D411yescharacteristic
119C2×C22⋊D61—41no—
120C2×D244—24no—
121C2×D244—21no—
122C2×D244—21no—
123C2×D244—21no—
124C2×D244—42no—
125C2×D244—41no—
126C2×C4×D61—41no—
127C2×C4×D61—41no—
128C2×C4×D61—41no—
129C2×C4×D61—41no—
130C2×C4×D61—41no—
131C2×Dic122—41no—
132Q16×C61—21no—
133Q16×C61—21no—
134C2×C8×C61—21no—
135C2×C8×C61—21no—
136C2×C8×C61C2×D411yescharacteristic
137C2×C8×C61C2411yescharacteristic
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—41no—
142C22⋊C4×C61—41no—
143C22⋊C4×C61—41no—
144C42×C61—41no—
145C22⋊Dic61D811yescharacteristic
146C22⋊Dic61D811yescharacteristic
147#11758201—42no—
148#11758198—41no—
149#11758198—41no—
150#11758198—41no—
151#11758198—41no—
152#11758198—41no—
153#11758198—41no—
154#11758198—41no—
155#11758198—41no—
156C4⋊Dic61—21no—
157C4⋊Dic61—41no—
158C4⋊Dic61D811yescharacteristic
159C4⋊Dic61D811yescharacteristic
160#11758196—41no—
161C4×Dic61—41no—
162C4×Dic61C2×D411yescharacteristic
163C4×Dic61C2×D411yescharacteristic
164#11758194—41no—
165C2×#92418C2×D411yescharacteristic
166C2×#92418C2×D411yescharacteristic
167D488—24no—
168D488—21no—
169D488—21no—
170D488—21no—
171D488—21no—
172D488#4112yes—
173D488#4112yes—
174C16×C61—21no—
175C16×C61—21no—
176#11758185—21no—
177#11758185—21no—
178C23×C61C2×D811yescharacteristic
179C22×D61—41no—
180C22×D61—41no—
181C22×D61—44no—
182C22×D61—41no—
183C22×D61—161no—
184C22×D61—161no—
185Q8×C61—41no—
186Q8×C61—41no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61C2×D811yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C22⋊D61—161no—
196C2×Dic61—41no—
197C2×Dic61C2×D811yescharacteristic
198C2×Dic61C2×D811yescharacteristic
199C2×Dic61C2×D811yescharacteristic
200C2×Dic61C2×D811yescharacteristic
201D244—22no—
202D244—22no—
203D244—44no—
204D244—41no—
205D244—41no—
206D244—41no—
207D244—41no—
208C4×D61—161no—
209C4×D61—161no—
210C4×D61—161no—
211C4×D61—161no—
212C4×D61—161no—
213Dic122—161no—
214C8×C61—21no—
215C8×C61C2×#4111yescharacteristic
216C8×C61C2×#4111yescharacteristic
217#92418—21no—
218#92418—21no—
219#33678—611nomaximal, Sylow
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—21no—
223C22×C61C22×D811yescharacteristic
224C2×D61—42no—
225C2×D61—42no—
226C2×D61—81no—
227C2×D61—81no—
228C2×D61—81no—
229C2×D61—81no—
230C2×D61—81no—
231C2×D61—81no—
232C2×D61—81no—
233C2×D61—164no—
234C4×C61—21no—
235C4×C61—81no—
236C4×C61—81no—
237C4×C61—81no—
238C4×C61—81no—
239C4×C61#52511yescharacteristic
240C4×C61#57511yescharacteristic
241Dic61—21no—
242Dic61—21no—
243Dic61—21no—
244Dic61—21no—
245Dic61—81no—
246#3175—611no—
247#2948—611no—
248#2237D6111yescharacteristic, complemented
249#2234—611no—
250#2226—611no—
251#2224—611no—
252#2210—611no—
253#2210—611no—
254#2187—611no—
255#2184—611no—
256#2177—611no—
257#2175—611no—
258C4×SD32—611no—
259C4×D16—611no—
260#2164—611no—
261C2×C61—41no—
262C2×C61#310011yescharacteristic
263C2×C61#341511yescharacteristic
264C2×C61#341611yescharacteristic
265D61—82no—
266D61—82no—
267D61—161no—
268C22×D8—611no—
269#542—1221no—
270#518—1221no—
271C2×SD32—611no—
272C2×SD32—611no—
273C2×D16—611no—
274C2×D16—611no—
275#500—611no—
276#495—611no—
277#489—1221no—
278#468—1221no—
279#467C2×D6111yescharacteristic
280#466—611no—
281#442—1221no—
282#442—1221no—
283#440—1221no—
284C4×Q16—611no—
285C4×D8—611no—
286C4×D8—611no—
287C4×D8—611no—
288C4×D8—611no—
289C4×D8—611no—
290#427C2×D6111yescharacteristic
291#418—1221no—
292#405—611no—
293#367—611no—
294#367C2×D6111yescharacteristic, complemented
295#366—611no—
296#366C2×D6111yescharacteristic, complemented
297C4⋊C16—611no—
298C4⋊C16—611no—
299#358C2×D6111yescharacteristic, complemented
300#358C2×D6111yescharacteristic, complemented
301#357—611no—
302#357—611no—
303#357—611no—
304#357—611no—
305#357—611no—
306#357—611no—
307C22⋊C16—611no—
308C22⋊C16C2×D6111yescharacteristic
309#346—611no—
310C4×C16—611no—
311C61#3367811yescharacteristic, complemented, Sylow
312C22×D4—1221no—
313C2×Q16C22×D6111yescharacteristic
314C2×SD16—1221no—
315C2×D8—611no—
316C2×D8—611no—
317C2×D8—611no—
318C2×D8—1222no—
319C2×D8—1222no—
320C2×D8—1221no—
321C2×M16—611no—
322C22×C8C22×D6111yescharacteristic
323C4⋊Q8—1221no—
324#126—2441no—
325#124—1221no—
326#124—2441no—
327#123—2441no—
328C22⋊Q8—21no—
329C22⋊Q8—2441no—
330#121—1221no—
331C4×Q8—1221no—
332C4×D4—1221no—
333C4×D4—1221no—
334C4×D4—1221no—
335C4×D4—1221no—
336C4×D4—1221no—
337C4×D4—2441no—
338C4×D4—2441no—
339#117—21no—
340#117—611no—
341C2×C22⋊C4—2441no—
342C2×C22⋊C4—2441no—
343SD32—1222no—
344SD32—1222no—
345D16—1222no—
346D16—1222no—
347C2×C16—611no—
348C2×C16—611no—
349C2×C16C22×D6111yescharacteristic
350C2×C16C22×D6111yescharacteristic
351#107—611no—
352#107—1221no—
353#107C22×D6111yescharacteristic
354#107C22×D6111yescharacteristic
355#107C22×D6111yescharacteristic
356#106—21no—
357#103—21no—
358#103—1221no—
359#102—1221no—
360#102—1221no—
361#102—1221no—
362#102—1221no—
363#102—1221no—
364#102—1221no—
365#97—611no—
366#97—611no—
367C4×C8—611no—
368C4×C8—611no—
369C24—2441no—
370C2×Q8—21no—
371C2×Q8—2441no—
372C2×D4—1224no—
373C2×D4—1221no—
374C2×D4—1221no—
375C2×D4—1221no—
376C2×D4—2442no—
377C2×D4—2441no—
378C2×D4—2441no—
379C22×C4—2441no—
380C22×C4—2441no—
381C22×C4—2441no—
382C22×C4—2441no—
383C22×C4—2441no—
384C22×C4D4×D6111yescharacteristic
385Q16—21no—
386Q16—21no—
387SD16—2442no—
388D8—612no—
389D8—612no—
390D8—1224no—
391D8—1221no—
392D8—1221no—
393D8—1221no—
394D8—1221no—
395M16—2441no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—611no—
399C2×C8—611no—
400C2×C8D4×D6111yescharacteristic
401C2×C8C23×D6111yescharacteristic
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—611no—
407C4⋊C4—611no—
408C4⋊C4—1221no—
409C4⋊C4—2441no—
410C4⋊C4—2441no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—611no—
414C22⋊C4—611no—
415C22⋊C4—2441no—
416C22⋊C4—2441no—
417C22⋊C4—2441no—
418C22⋊C4—2441no—
419C22⋊C4—2441no—
420C22⋊C4—2441no—
421C22⋊C4—2441no—
422C22⋊C4—2441no—
423C42—41no—
424C42—611no—
425C42—611no—
426C42—2441no—
427C16—21no—
428C16—21no—
429C16—1221no—
430C16—1221no—
431C23—9761no—
432C23—9761no—
433C23—2444no—
434C23—2441no—
435C23—2441no—
436C23—2441no—
437C23D8×D6111yescharacteristic
438Q8—41no—
439Q8—41no—
440Q8—9761no—
441D4—1222no—
442D4—1222no—
443D4—9761no—
444D4—2444no—
445D4—2441no—
446D4—2441no—
447D4—2441no—
448D4—2441no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—611no—
456C2×C4—611no—
457C2×C4—611no—
458C2×C4—611no—
459C2×C4—9761no—
460C2×C4—9761no—
461C2×C4—9761no—
462C2×C4—9761no—
463C2×C4—9761no—
464C2×C4—2441no—
465C2×C4D8×D6111yescharacteristic
466C2×C4C2×D4×D6111yescharacteristic
467C8—21no—
468C8—1221no—
469C8—1221no—
470C8#41×D6111yescharacteristic
471C8#41×D6111yescharacteristic
472C22—21no—
473C22—21no—
474C22—21no—
475C22—9764no—
476C22—4881no—
477C22—4881no—
478C22—4881no—
479C22—4881no—
480C22—4881no—
481C22—4881no—
482C22—4881no—
483C22—2442no—
484C22—2442no—
485C22C2×D8×D6111yescharacteristic
486C4—21no—
487C4—81no—
488C4—81no—
489C4—81no—
490C4—81no—
491C4—1221no—
492C4—1221no—
493C4—1221no—
494C4—1221no—
495C4—4881no—
496C4#1684027011yescharacteristic
497C4#1684056111yescharacteristic
498C2—41no—
499C2—9761no—
500C2—4882no—
501C2—4882no—
502C2#2145581611yescharacteristic
503C2#2145895711yescharacteristic
504C2#2145896411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2237C2
socleC22×C61C22×D8
prime radical#21447733C2
hypercenter#2237D61
hypocenterC61#33678
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2237

Complements

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

NH ≅ G/Ntype
1. #21456459C2semidirect
2. #21454670C2semidirect
3. #21448010C2semidirect
4. #21447965C2semidirect
5. #21447955C2semidirect
6. #21447877C2semidirect
7. #21447877C2semidirect
8. #21447747C2semidirect
10. #21447694C2semidirect
11. #21447686C2semidirect
12. #21447668C2semidirect
13. #21447665C2semidirect
14. #21447623C2semidirect
15. C61×#2237C2semidirect
23. #16840166C22semidirect
24. #16840147C22semidirect
31. Q16×Dic61C22semidirect
36. #16839844C22semidirect
39. #16839589C22semidirect
40. #16839587C22semidirect
41. #16839580C22semidirect
42. #16839580C22semidirect
47. #16839562C22semidirect
48. #16839562C22semidirect
50. #16839537C22semidirect
51. C16×Dic61C22semidirect
54. C61×#367C22semidirect
55. C61×#366C22semidirect
56. C61×#358C22semidirect
57. C61×#358C22semidirect
248. #2237D61semidirect
294. #367C2×D61semidirect
296. #366C2×D61semidirect
299. #358C2×D61semidirect
300. #358C2×D61semidirect
311. C61#33678semidirect

Maximal subgroups

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

subgroupindex
#214564592
#214546702
#214480102
#214479652
#214479552
#214478772
#214477472
#214477332
#214476942
#214476862
#214476682
#214476652
#214476232
C61×#22372
#3367861