Finite Group Databasebeta

← #29836546

Subgroups — #29836546

Order 15616.

670 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840201C211yesmaximal, characteristic, complemented
2#21457554C211yesmaximal, characteristic, complemented
3#21457535C211yesmaximal, characteristic, complemented
4#21448089C211yesmaximal, characteristic, complemented
5#21447929C211yesmaximal, characteristic, complemented
6#21447910C211yesmaximal, characteristic, complemented
7#21447872C211yesmaximal, characteristic, complemented
8#21447867C211yesmaximal, characteristic, complemented
9#21447759C211yesmaximal, characteristic, complemented
10#21447759C211yesmaximal, characteristic, complemented
11#21447751C211yesmaximal, characteristic, complemented
12#21447739C211yesmaximal, characteristic, complemented
13#21447684C211yesmaximal, characteristic, complemented
14#21447644C211yesmaximal, characteristic, complemented
15C2×C61×#508C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic
17C2×Q16×D61C2211yescharacteristic
18#16840419—21no—
19#16840411—21no—
20#16840201—24no—
21C2×Dic488C2211yescharacteristic
22C2×#14519723C2211yescharacteristic
23C2×#14519723C2211yescharacteristic
24C2×#14519721C2211yescharacteristic
25C2×#14519721C2211yescharacteristic
26C2×#14519716C2211yescharacteristic
27#16840145C2211yescharacteristic
28#16840140C2211yescharacteristic
29#16840003C2211yescharacteristic
30#16840003C2211yescharacteristic
31#16840000C2211yescharacteristic
32#16839995C2211yescharacteristic
33#16839982—21no—
34#16839965—21no—
35#16839924—21no—
36#16839924—21no—
37#16839921—21no—
38#16839899—21no—
39Q16×Dic61C2211yescharacteristic
40D8×Dic61C2211yescharacteristic
41C8×C22⋊D61C2211yescharacteristic
42#16839588C2211yescharacteristic
43#16839580C2211yescharacteristic
44#16839567C2211yescharacteristic
45#16839567C2211yescharacteristic
46#16839566C2211yescharacteristic
47#16839566C2211yescharacteristic
48#16839563C2211yescharacteristic
49#16839563C2211yescharacteristic
50C22⋊#11758185C2211yescharacteristic
51#16839541C2211yescharacteristic
52#16839537C2211yescharacteristic
53C2×#135×C61C2211yescharacteristic
54C2×#135×C61C2211yescharacteristic
55C61×#508—24no—
56C2×Q32×C61C2211yescharacteristic
57C2×SD32×C61C2211yescharacteristic
58C2×SD32×C61C2211yescharacteristic
59C2×D16×C61C2211yescharacteristic
60C22×C16×C61C2211yescharacteristic
61C2×#11758224—21no—
62C2×Q8×D61—21no—
63#14519796—22no—
64#14519796—41no—
65C2×Dic244C2311yescharacteristic
66C2×Dic244C2311yescharacteristic
67C2×#11758202—21no—
68C2×#11758202—21no—
69Q16×D61—22no—
70Q16×D61—41no—
71C2×#11758200—21no—
72C2×#11758200—21no—
73C2×C8×D61C2311yescharacteristic
74#14519775—21no—
75#14519764—21no—
76#14519764—41no—
77#14519760—41no—
78#14519758—41no—
79#14519756—21no—
80#14519756—21no—
81#14519756—41no—
82#14519753—21no—
83#14519749—41no—
84#14519748—21no—
85#14519748—41no—
86Q8×Dic61—21no—
87D4×Dic61—21no—
88C4×C22⋊D61—41no—
89C4×C22⋊D61D411yescharacteristic
90C2×#11758196—41no—
91Dic488—22no—
92#14519723—22no—
93#14519723—41no—
94#14519721—22no—
95#14519721—41no—
96#14519716—22no—
97C2×#11758185C2311yescharacteristic
98C2×#11758185C2311yescharacteristic
99#14519711C2311yescharacteristic
100#14519711C2311yescharacteristic
101#14519706D411yescharacteristic
102#14519702—21no—
103#14519702—21no—
104#14519701—21no—
105#14519701—21no—
106#14519698—21no—
107#14519698—21no—
108C22⋊#92418D411yescharacteristic
109#14519689D411yescharacteristic
110C8×Dic61C2311yescharacteristic
111C2×#41×C61—21no—
112C2×#41×C61—21no—
113#135×C61—22no—
114#135×C61—22no—
115#135×C61—41no—
116#135×C61—41no—
117C2×Q16×C61C2311yescharacteristic
118C2×Q16×C61C2311yescharacteristic
119C2×SD16×C61—21no—
120C2×SD16×C61—21no—
121C2×D8×C61C2311yescharacteristic
122C2×D8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124Q32×C61—22no—
125SD32×C61—22no—
126SD32×C61D414yescomplemented
127D16×C61—22no—
128C2×C16×C61—22no—
129C2×C16×C61C2311yescharacteristic
130C2×C16×C61C2311yescharacteristic
131#41×C61—42no—
132#41×C61—42no—
133#41×C61—81no—
134#41×C61—81no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—41no—
140C2×D4×C61—41no—
141C22×C4×C61—41no—
142C22×C4×C61—41no—
143C22×C4×C61C2×D411yescharacteristic
144C2×C22⋊D61—41no—
145C2×C22⋊D61—41no—
146C2×C22⋊D61D811yescharacteristic
147C22×Dic61—41no—
148C22×Dic61—41no—
149Q8×D61—42no—
150Q8×D61—81no—
151#11758224—42no—
152#11758224—81no—
153C2×C4×D61—41no—
154C2×C4×D61C2×D411yescharacteristic
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—41no—
158Q16×C61—21no—
159Q16×C61—21no—
160Q16×C61—21no—
161Q16×C61C2×D412yes—
162SD16×C61—42no—
163SD16×C61—42no—
164D8×C61—21no—
165D8×C61—21no—
166D8×C61—21no—
167D8×C61C2×D412yes—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61C2×D411yescharacteristic
171C2×C8×C61C2411yescharacteristic
172C22⋊Dic61—41no—
173C22⋊Dic61—41no—
174C22⋊Dic61—41no—
175C22⋊Dic61—41no—
176C22⋊Dic61—41no—
177C22⋊Dic61—41no—
178C22⋊Dic61D811yescharacteristic
179#11758202—42no—
180#11758202—81no—
181#11758200—42no—
182#11758200—81no—
183#11758198—41no—
184#11758198D811yescharacteristic
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—41no—
188#11758196—41no—
189#11758196—41no—
190#11758196—41no—
191#11758196—41no—
192#11758196—41no—
193#11758196D811yescharacteristic
194C4×Dic61—41no—
195C4×Dic61C2×D411yescharacteristic
196C2×#92418C2×D411yescharacteristic
197C2×#92418C2×D411yescharacteristic
198Dic244—21no—
199Dic244—21no—
200Dic244—21no—
201Dic244—21no—
202C8×D61—21no—
203C8×D61—21no—
204C16×C61—21no—
205C16×C61C2×D412yes—
206#11758185—21no—
207#11758185—21no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61C2×D811yescharacteristic
211C22×D61C2×D811yescharacteristic
212Q8×C61—22no—
213Q8×C61—41no—
214Q8×C61—41no—
215Q8×C61—41no—
216D4×C61—22no—
217D4×C61—41no—
218D4×C61—41no—
219D4×C61—41no—
220D4×C61—82no—
221D4×C61—82no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—82no—
227C2×C4×C61—82no—
228C2×C4×C61C2×D811yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C22⋊D61—41no—
231C22⋊D61—82no—
232C22⋊D61—82no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—82no—
238C2×Dic61—82no—
239C2×Dic61C2×D811yescharacteristic
240C2×Dic61C2×D811yescharacteristic
241C2×Dic61C2×D811yescharacteristic
242C4×D61—21no—
243C4×D61—21no—
244C4×D61—82no—
245Dic122—41no—
246Dic122—41no—
247Dic122—41no—
248Dic122—41no—
249Dic122—82no—
250C8×C61—21no—
251C8×C61C22×D411yescharacteristic
252C8×C61H(5,𝔽2)11yescharacteristic
253#92418—21no—
254#92418—21no—
255#33835—611nomaximal, Sylow
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—42no—
260C22×C61—81no—
261C22×C61—81no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61C22×D811yescharacteristic
266C2×D61—21no—
267C2×D61—21no—
268C2×D61—21no—
269C4×C61—21no—
270C4×C61—42no—
271C4×C61—81no—
272C4×C61D4211yescharacteristic
273C4×C61#57611yescharacteristic
274Dic61—21no—
275Dic61—21no—
276Dic61—21no—
277Dic61—81no—
278Dic61—81no—
279Dic61—81no—
280Dic61—81no—
281C2×#510—611no—
282C2×#508D6111yescharacteristic, complemented
283#3285—611no—
284#3283—611no—
285#2251—611no—
286#2220—611no—
287#2216—611no—
288#2210—611no—
289#2209—611no—
290#2189—611no—
291#2189—611no—
292#2188—611no—
293#2186—611no—
294#2175—611no—
295#2168—611no—
296C2×C61—41no—
297C2×C61—82no—
298C2×C61—161no—
299C2×C61D4×D811yescharacteristic
300C2×C61#341411yescharacteristic
301C2×C61#341711yescharacteristic
302D61—41no—
303C2×#135—611no—
304C2×#135C2×D6111yescharacteristic
305C2×#135C2×D6111yescharacteristic
306C22×Q16—611no—
307#548—1221no—
308#547—1221no—
309#510—1224no—
310#508—24no—
311C2×Q32—611no—
312C2×Q32—611no—
313C2×Q32—611no—
314C2×Q32C2×D6111yescharacteristic
315C2×SD32—611no—
316C2×SD32—611no—
317C2×SD32C2×D6111yescharacteristic
318C2×SD32C2×D6111yescharacteristic
319C2×D16C2×D6111yescharacteristic
320C2×M32—611no—
321C22×C16C2×D6111yescharacteristic
322#495—611no—
323#494—611no—
324#467—611no—
325#467—611no—
326#467—611no—
327#466—611no—
328#464—1221no—
329#461—1221no—
330#452—1221no—
331#452—1221no—
332#452—1221no—
333#448—1221no—
334C4×Q16—611no—
335C4×D8—611no—
336C8×D4—611no—
337#366—611no—
338C4⋊C16—611no—
339#358—611no—
340#358—611no—
341#358—611no—
342#358—611no—
343#357—611no—
344#357—611no—
345C22⋊C16—611no—
346C22⋊C16—611no—
347#346—611no—
348C61#3383511yescharacteristic, complemented, Sylow
349C2×#41—21no—
350C2×#41—21no—
351C2×#41—1221no—
352C22×Q8—1221no—
353#135—22no—
354#135—22no—
355#135—41no—
356#135—41no—
357#135—1222no—
358#135—2441no—
359C2×Q16—611no—
360C2×Q16—611no—
361C2×Q16—1221no—
362C2×Q16—1221no—
363C2×Q16—1222no—
364C2×Q16—2441no—
365C2×Q16C22×D6111yescharacteristic
366C2×Q16C22×D6111yescharacteristic
367C2×SD16—21no—
368C2×SD16—21no—
369C2×SD16—1221no—
370C2×SD16—1221no—
371C2×D8C22×D6111yescharacteristic
372C2×D8C22×D6111yescharacteristic
373C22×C8—611no—
374C22×C8C22×D6111yescharacteristic
375C4⋊Q8—1221no—
376#124—1221no—
377#124—2441no—
378#123—2441no—
379#123—2441no—
380C22⋊Q8—1221no—
381C22⋊Q8—1221no—
382C22⋊Q8—1221no—
383C22⋊Q8—2441no—
384#121—1221no—
385#121—2441no—
386#121—2441no—
387C4×Q8—1221no—
388C4×D4—611no—
389C4×D4—1221no—
390C4×D4—2441no—
391C2×C4⋊C4—2441no—
392Q32—22no—
393Q32—1222no—
394Q32—1222no—
395Q32—2441no—
396SD32—22no—
397SD32—1222no—
398SD32—2441no—
399SD32C22⋊D6114yescomplemented
400D16—22no—
401M32—1222no—
402C2×C16—22no—
403C2×C16—611no—
404C2×C16—611no—
405C2×C16C22×D6111yescharacteristic
406C2×C16C22×D6111yescharacteristic
407#107—611no—
408#107—611no—
409C4⋊C8—611no—
410#103—1221no—
411#103—1221no—
412#103—1221no—
413#103—1221no—
414#102—1221no—
415#102—1221no—
416C22⋊C8—611no—
417C22⋊C8—611no—
418C4×C8—611no—
419#41—42no—
420#41—42no—
421#41—81no—
422#41—81no—
423#41—4881no—
424#41—2442no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—1221no—
428C2×Q8—1221no—
429C2×Q8—4881no—
430C2×Q8—2441no—
431C2×Q8—2442no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—41no—
435C2×D4—41no—
436C2×D4—611no—
437C2×D4—2441no—
438C2×D4—2441no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—611no—
442C22×C4—2441no—
443C22×C4—2441no—
444C22×C4—2441no—
445C22×C4D4×D6111yescharacteristic
446Q16—21no—
447Q16—21no—
448Q16—21no—
449Q16—1221no—
450Q16—1221no—
451Q16—1221no—
452Q16—1221no—
453Q16—4881no—
454Q16—2442no—
455Q16C2×C22⋊D6112yes—
456SD16—42no—
457SD16—42no—
458SD16—4881no—
459SD16—2442no—
460D8—21no—
461D8—21no—
462D8—21no—
463D8C2×C22⋊D6112yes—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—611no—
467C2×C8—611no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8D4×D6111yescharacteristic
471C2×C8C23×D6111yescharacteristic
472C4⋊C4—611no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—2441no—
476C4⋊C4—2441no—
477C4⋊C4—2441no—
478C4⋊C4—2441no—
479C4⋊C4—2441no—
480C4⋊C4—2441no—
481C22⋊C4—611no—
482C22⋊C4—611no—
483C22⋊C4—2441no—
484C22⋊C4—2441no—
485C22⋊C4—2441no—
486C22⋊C4—2441no—
487C22⋊C4—2441no—
488C22⋊C4—2441no—
489C22⋊C4—2441no—
490C42—611no—
491C42—2441no—
492C16—21no—
493C16—1221no—
494C16—1221no—
495C16C2×C22⋊D6112yes—
496C23—41no—
497C23—41no—
498C23—611no—
499C23D8×D6111yescharacteristic
500Q8—22no—
501Q8—41no—
502Q8—41no—
503Q8—41no—
504Q8—4882no—
505Q8—2441no—
506Q8—2441no—
507Q8—2441no—
508Q8—2441no—
509D4—22no—
510D4—41no—
511D4—41no—
512D4—41no—
513D4—82no—
514D4—82no—
515D4—4882no—
516D4—4882no—
517D4—2441no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—611no—
525C2×C4—611no—
526C2×C4—611no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—4882no—
530C2×C4—4882no—
531C2×C4—4882no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4D8×D6111yescharacteristic
537C2×C4C2×D4×D6111yescharacteristic
538C8—21no—
539C8—1221no—
540C8—1221no—
541C8C22×C22⋊D6111yescharacteristic
542C8#1451982211yescharacteristic
543C22—21no—
544C22—21no—
545C22—21no—
546C22—42no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—1221no—
553C22—1221no—
554C22—1221no—
555C22C2×D8×D6111yescharacteristic
556C4—21no—
557C4—42no—
558C4—81no—
559C4—1221no—
560C4—1221no—
561C4—1221no—
562C4—4881no—
563C4—4881no—
564C4—4881no—
565C4—4881no—
566C4D4×C22⋊D6111yescharacteristic
567C4#1684056311yescharacteristic
568C2—41no—
569C2—82no—
570C2—161no—
571C2—2441no—
572C2D8×C22⋊D6111yescharacteristic
573C2#2145895211yescharacteristic
574C2#2145896911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#508C2
socleC22×C61C22×D8
hypercenterC2×#508D61
hypocenterC61#33835
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#508

Complements

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

NH ≅ G/Ntype
1. C2×#16840201C2semidirect
2. #21457554C2semidirect
3. #21457535C2semidirect
4. #21448089C2semidirect
5. #21447929C2semidirect
6. #21447910C2semidirect
7. #21447872C2semidirect
8. #21447867C2semidirect
9. #21447759C2semidirect
10. #21447759C2semidirect
11. #21447751C2semidirect
12. #21447739C2semidirect
13. #21447684C2semidirect
14. #21447644C2semidirect
15. C2×C61×#508C2semidirect
126. SD32×C61D4semidirect
282. C2×#508D61semidirect
348. C61#33835semidirect
399. SD32C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402012
#214575542
#214575352
#214480892
#214479292
#214479102
#214478722
#214478672
#214477592
#214477512
#214477392
#214476842
#214476442
C2×C61×#5082
#3383561