Finite Group Databasebeta

← #29834403

Subgroups — #29834403

Order 15616.

696 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458953C211yesmaximal, characteristic, complemented
2#21456243C211yesmaximal, characteristic, complemented
3C2×#16839879C211yesmaximal, characteristic, complemented
4#21447898C212yesmaximal, complemented
5#21447893C212yesmaximal, complemented
6#21447890C212yesmaximal, complemented
7#21447886C212yesmaximal, complemented
8C2×#16839582C211yesmaximal, characteristic, complemented
9#21447569C211yesmaximal, characteristic, complemented
10#21447568C211yesmaximal, characteristic, complemented
11C61×#3414C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C2×#14519797—21no—
14C2×#14519784—21no—
15#16840308—21no—
16#16840203—21no—
17#16840202C2212yes—
18#16840198—21no—
19#16840197C2212yes—
20#16840192—22no—
21C2×#14519717C2211yescharacteristic
22#16840147C2211yescharacteristic, complemented
23#16840145C2211yescharacteristic, complemented
24#16840143—21no—
25#16840140C2211yescharacteristic, complemented
26#16840137C2211yescharacteristic, complemented
27#16840133—21no—
28#16839879—21no—
29#16839879—21no—
30#16839879C2212yes—
31#16839879C2212yes—
32C2×#14519713C2211yescharacteristic
33C2×#14519704—21no—
34C2×C8×Dic61C2211yescharacteristic
35#16839582C2212yescomplemented
36#16839582C2212yescomplemented
37#16839579C2212yescomplemented
38#16839578C2212yescomplemented
39#16839576C2212yescomplemented
40#16839575C2212yescomplemented
41C2×#135×C61C2211yescharacteristic
42C2×#135×C61C2211yescharacteristic
43C61×#510—21no—
44C61×#510C2212yes—
45C61×#509—21no—
46C61×#509C2212yes—
47C61×#508—22no—
48C2×M32×C61C2211yescharacteristic
49C2×#11758222—21no—
50#14519799—21no—
51#14519799—21no—
52#14519799—21no—
53#14519799—21no—
54#14519797—41no—
55#14519797—41no—
56#14519797—41no—
57#14519797—41no—
58C2×Dic244C2311yescharacteristic
59C2×#11758202—21no—
60C2×#11758190—21no—
61C2×#11758200—21no—
62C2×#11758201—21no—
63C2×D488C2311yescharacteristic
64C2×#11758199—21no—
65#14519784—22no—
66#14519784—22no—
67#14519784—41no—
68#14519784—41no—
69C2×#11758194—21no—
70C2×#11758194—41no—
71C22×#92418—41no—
72C22×#92418D411yescharacteristic
73#14519775—21no—
74#14519773—21no—
75#14519766—21no—
76#14519764—21no—
77#14519749—42no—
78C4×C22⋊D61—42no—
79C2×C4×Dic61D411yescharacteristic
80#14519723—22no—
81#14519721—22no—
82#14519720—22no—
83#14519718—22no—
84#14519717—21no—
85#14519717C2312yes—
86C2×#11758185—21no—
87#14519713—21no—
88#14519713C2312yes—
89#14519704—22no—
90#14519704—22no—
91C8×Dic61D411yescharacteristic
92C8×Dic61D411yescharacteristic
93C8×Dic61C2311yescharacteristic
94C8×Dic61C2311yescharacteristic
95C2×#41×C61—21no—
96C2×#41×C61—21no—
97#135×C61—21no—
98#135×C61—21no—
99#135×C61—21no—
100#135×C61—21no—
101#135×C61—21no—
102#135×C61—21no—
103#135×C61D412yescomplemented
104#135×C61C2312yes—
105C2×Q16×C61D411yescharacteristic, complemented
106C2×Q16×C61C2311yescharacteristic
107C2×SD16×C61—21no—
108C2×SD16×C61—21no—
109C2×D8×C61D411yescharacteristic, complemented
110C2×D8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112Q32×C61—22no—
113SD32×C61—22no—
114SD32×C61—22no—
115D16×C61—22no—
116M32×C61—21no—
117M32×C61C2312yes—
118C2×C16×C61—21no—
119#41×C61—22no—
120#41×C61—22no—
121#41×C61—41no—
122#41×C61—41no—
123#41×C61—41no—
124#41×C61—41no—
125#41×C61—41no—
126#41×C61—41no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—41no—
132C2×D4×C61—41no—
133C22×C4×C61—41no—
134C22×C4×C61—41no—
135C22×C4×C61C2×D411yescharacteristic
136C2×C22⋊D61—42no—
137C2×C22⋊D61—41no—
138C22×Dic61—21no—
139#11758222—41no—
140#11758222—41no—
141#11758222—41no—
142#11758222—41no—
143C2×D244—21no—
144C2×C4×D61—41no—
145C2×Dic122—21no—
146Q16×C61—21no—
147Q16×C61—21no—
148Q16×C61—21no—
149Q16×C61C2×D412yes—
150SD16×C61—22no—
151SD16×C61—22no—
152SD16×C61—41no—
153SD16×C61—41no—
154D8×C61—21no—
155D8×C61—21no—
156D8×C61—21no—
157D8×C61C2×D412yes—
158C2×C8×C61C2×D411yescharacteristic
159C2×C8×C61C2×D411yescharacteristic
160C2×C8×C61C2×D411yescharacteristic
161C2×C8×C61#4111yescharacteristic
162C2×C8×C61#4111yescharacteristic
163C2×C8×C61C2411yescharacteristic
164C22⋊Dic61—42no—
165#11758202—41no—
166#11758202—41no—
167#11758201—41no—
168#11758201—41no—
169#11758200—41no—
170#11758200—41no—
171#11758199—41no—
172#11758199—41no—
173#11758198—42no—
174#11758196—42no—
175C4×Dic61—21no—
176C4×Dic61C2×D411yescharacteristic
177C4×Dic61C2×D411yescharacteristic
178#11758194—22no—
179#11758194—41no—
180#11758194—42no—
181#11758194—42no—
182C2×#92418—41no—
183C2×#92418—42no—
184C2×#92418—42no—
185C2×#92418D811yescharacteristic
186C2×#92418D811yescharacteristic
187C2×#92418D811yescharacteristic
188C2×#92418D811yescharacteristic
189C2×#92418C2×D411yescharacteristic
190C2×#92418C2×D411yescharacteristic
191Dic244—21no—
192Dic244—21no—
193D488—21no—
194D488—21no—
195#11758190—41no—
196#11758190—41no—
197C16×C61—22no—
198#11758185—22no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61AGL(1,ℤ/8)11yescharacteristic
202C22×D61—41no—
203Q8×C61—22no—
204Q8×C61—41no—
205Q8×C61—41no—
206Q8×C61—41no—
207D4×C61—22no—
208D4×C61—42no—
209D4×C61—42no—
210D4×C61—41no—
211D4×C61—41no—
212D4×C61—41no—
213D4×C61—81no—
214D4×C61—81no—
215C2×C4×C61—41no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—41no—
219C2×C4×C61—81no—
220C2×C4×C61—81no—
221C2×C4×C61#12111yescharacteristic
222C2×C4×C61#12111yescharacteristic
223C2×C4×C61C2×D811yescharacteristic
224C2×C4×C61C2×D811yescharacteristic
225C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C22⋊D61—84no—
228C22⋊D61—81no—
229C22⋊D61—81no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—42no—
233C2×Dic61—41no—
234D244—41no—
235D244—41no—
236C4×D61—81no—
237C4×D61—81no—
238Dic122—41no—
239Dic122—41no—
240C8×C61#12111yescharacteristic
241C8×C61#12111yescharacteristic
242C8×C61C22×D411yescharacteristic
243C8×C61C2×#4111yescharacteristic
244#92418—42no—
245#92418C2×D811yescharacteristic
246#92418C2×D811yescharacteristic
247#92418C2×D811yescharacteristic
248#92418C2×D811yescharacteristic
249#33594—611nomaximal, Sylow
250C22×C61—41no—
251C22×C61—42no—
252C22×C61—82no—
253C22×C61—82no—
254C22×C61—81no—
255C22×C61#45911yescharacteristic
256C22×C61#45911yescharacteristic
257C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
258C2×D61—82no—
259C2×D61—81no—
260C4×C61—42no—
261C4×C61—81no—
262C4×C61#45911yescharacteristic
263C4×C61#45911yescharacteristic
264C4×C61C2×#12111yescharacteristic
265C4×C61C22×D811yescharacteristic
266Dic61—22no—
267Dic61—41no—
268Dic61—81no—
269#3414—611no—
270#3414D6111yescharacteristic, complemented
271#3147—611no—
272C2×#443—611no—
273#2214—612no—
274#2213—612no—
275#2212—612no—
276#2212—612no—
277C2×#364—611no—
278#2148—611no—
279#2148—611no—
280C2×C61—21no—
281C2×C61—21no—
282C2×C61—21no—
283C2×C61—82no—
284C2×C61—161no—
285C2×C61C2×#45911yescharacteristic
286D61—161no—
287C2×#135—611no—
288C2×#135—1221no—
289C2×#135C2×D6111yescharacteristic
290C2×#135C2×D6111yescharacteristic
291C2×#131—1221no—
292#532—1221no—
293#510—21no—
294#510—612no—
295#510—1221no—
296#510C2×D6112yes—
297#509—21no—
298#509—612no—
299#509—1221no—
300#509C2×D6112yes—
301#508—22no—
302#508—1222no—
303C2×M32—611no—
304C2×M32C2×D6111yescharacteristic
305#495—611no—
306#495—611no—
307#495—1221no—
308#494—611no—
309#493—611no—
310#492—1221no—
311#443—612no—
312#443—612no—
313#443—1221no—
314#443—1221no—
315C2×#108—611no—
316C2×#104—1221no—
317C2×C4×C8—611no—
318#364—612no—
319#364—612no—
320#362—612no—
321#362—612no—
322#361—612no—
323#361—612no—
324C61#3359411yescharacteristic, complemented, Sylow
325C2×#41—21no—
326C2×#41—21no—
327C2×#41—1221no—
328#135—21no—
329#135—21no—
330#135—21no—
331#135—21no—
332#135—21no—
333#135—21no—
334#135—1221no—
335#135—1221no—
336#135—1221no—
337#135—1221no—
338#135—2441no—
339#135—2441no—
340#135—2441no—
341#135—2441no—
342#135C22⋊D6112yescomplemented
343#135C22×D6112yes—
344C2×Q16—611no—
345C2×Q16—1221no—
346C2×Q16C22⋊D6111yescharacteristic, complemented
347C2×Q16C22×D6111yescharacteristic
348C2×SD16—21no—
349C2×SD16—21no—
350C2×SD16—1221no—
351C2×SD16—1221no—
352C2×SD16—1221no—
353C2×D8—611no—
354C2×D8—1221no—
355C2×D8C22⋊D6111yescharacteristic, complemented
356C2×D8C22×D6111yescharacteristic
357#131—1222no—
358#131—1222no—
359#131—2441no—
360#131—2441no—
361C2×M16—1221no—
362C2×M16—2441no—
363C22×C8—611no—
364C22×C8—2441no—
365C22×C8C22×D6111yescharacteristic
366C4⋊Q8—1221no—
367#127—1221no—
368#124—1221no—
369#124—1221no—
370#121—2442no—
371C4×D4—2442no—
372C2×C42—611no—
373Q32—22no—
374Q32—1222no—
375SD32—22no—
376SD32—22no—
377SD32—1222no—
378SD32—1222no—
379D16—22no—
380D16—1222no—
381M32—21no—
382M32—612no—
383M32—1221no—
384M32C22×D6112yes—
385C2×C16—21no—
386C2×C16—1221no—
387#108—612no—
388#108—1221no—
389#104—1222no—
390#104—1222no—
391C4×C8—611no—
392C4×C8—611no—
393C4×C8—611no—
394C4×C8—611no—
395#41—22no—
396#41—22no—
397#41—41no—
398#41—41no—
399#41—41no—
400#41—41no—
401#41—41no—
402#41—41no—
403#41—2441no—
404#41—2441no—
405#41—2441no—
406#41—2441no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—1221no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—1221no—
415C2×D4—2442no—
416C2×D4—2441no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—1221no—
420C22×C4—2441no—
421C22×C4D4×D6111yescharacteristic
422Q16—21no—
423Q16—21no—
424Q16—21no—
425Q16—1221no—
426Q16—1221no—
427Q16—2441no—
428Q16—2441no—
429Q16C2×C22⋊D6112yes—
430SD16—22no—
431SD16—22no—
432SD16—41no—
433SD16—41no—
434SD16—2441no—
435SD16—2441no—
436SD16—2441no—
437SD16—2441no—
438SD16—2441no—
439SD16—2441no—
440D8—21no—
441D8—21no—
442D8—21no—
443D8—1221no—
444D8—1221no—
445D8—2441no—
446D8—2441no—
447D8C2×C22⋊D6112yes—
448M16—1222no—
449M16—2442no—
450M16—2442no—
451M16—2441no—
452C2×C8—611no—
453C2×C8—611no—
454C2×C8—611no—
455C2×C8—611no—
456C2×C8—611no—
457C2×C8—611no—
458C2×C8—2442no—
459C2×C8—2442no—
460C2×C8—2441no—
461C2×C8D4×D6111yescharacteristic
462C2×C8#1175822411yescharacteristic
463C2×C8#1175822411yescharacteristic
464C2×C8C2×C22⋊D6111yescharacteristic
465C2×C8C2×C22⋊D6111yescharacteristic
466C2×C8C23×D6111yescharacteristic
467C4⋊C4—2442no—
468C22⋊C4—2442no—
469C22⋊C4—2442no—
470C42—611no—
471C42—611no—
472C42—1221no—
473C16—22no—
474C16—1222no—
475C23—41no—
476C23—41no—
477C23—2441no—
478C23#1451980111yescharacteristic
479Q8—22no—
480Q8—41no—
481Q8—41no—
482Q8—41no—
483Q8—2441no—
484Q8—2441no—
485D4—22no—
486D4—42no—
487D4—42no—
488D4—41no—
489D4—41no—
490D4—41no—
491D4—81no—
492D4—81no—
493D4—4884no—
494D4—4881no—
495D4—4881no—
496D4—2441no—
497D4—2441no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—1221no—
505C2×C4—1221no—
506C2×C4—4881no—
507C2×C4—4881no—
508C2×C4—2442no—
509C2×C4—2441no—
510C2×C4#1451974911yescharacteristic
511C2×C4#1451974911yescharacteristic
512C2×C4D8×D6111yescharacteristic
513C2×C4D8×D6111yescharacteristic
514C2×C4#1451980111yescharacteristic
515C2×C4C2×D4×D6111yescharacteristic
516C8—611no—
517C8—611no—
518C8—611no—
519C8—611no—
520C8—2442no—
521C8#1451974911yescharacteristic
522C8#1451974911yescharacteristic
523C8C22×C22⋊D6111yescharacteristic
524C8C2×#1175822411yescharacteristic
525C22—41no—
526C22—42no—
527C22—82no—
528C22—82no—
529C22—81no—
530C22—4882no—
531C22—4881no—
532C22#1683995511yescharacteristic
533C22#1683995511yescharacteristic
534C22C2×#1451980111yescharacteristic
535C4—42no—
536C4—81no—
537C4—1222no—
538C4—4881no—
539C4—2441no—
540C4#1683995511yescharacteristic
541C4#1683995511yescharacteristic
542C4C2×#1451974911yescharacteristic
543C4C2×D8×D6111yescharacteristic
544C2—21no—
545C2—21no—
546C2—21no—
547C2—82no—
548C2—161no—
549C2—9761no—
550C2C2×#1683995511yescharacteristic
subgroupG/H
centerC4#16839955
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3414C2
socleC2×C61C2×#459
hypercenter#3414D61
hypocenterC61#33594
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #3414

Complements

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

NH ≅ G/Ntype
1. #21458953C2semidirect
2. #21456243C2semidirect
3. C2×#16839879C2semidirect
4. #21447898C2semidirect
5. #21447893C2semidirect
6. #21447890C2semidirect
7. #21447886C2semidirect
8. C2×#16839582C2semidirect
9. #21447569C2semidirect
10. #21447568C2semidirect
11. C61×#3414C2semidirect
22. #16840147C22semidirect
23. #16840145C22semidirect
25. #16840140C22semidirect
26. #16840137C22semidirect
35. #16839582C22semidirect
36. #16839582C22semidirect
37. #16839579C22semidirect
38. #16839578C22semidirect
39. #16839576C22semidirect
40. #16839575C22semidirect
103. #135×C61D4semidirect
105. C2×Q16×C61D4semidirect
109. C2×D8×C61D4semidirect
270. #3414D61semidirect
324. C61#33594semidirect
342. #135C22⋊D61semidirect
346. C2×Q16C22⋊D61semidirect
355. C2×D8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214589532
#214562432
C2×#168398792
#214478982
#214478932
#214478902
#214478862
C2×#168395822
#214475692
#214475682
C61×#34142
#3359461