Finite Group Databasebeta

← #29836635

Subgroups — #29836635

Order 15616.

664 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840202C211yesmaximal, characteristic, complemented
2C2×#16840201C211yesmaximal, characteristic, complemented
3#21457568C211yesmaximal, characteristic, complemented
4#21457535C211yesmaximal, characteristic
5#21448070C211yesmaximal, characteristic, complemented
6#21447926C211yesmaximal, characteristic, complemented
7#21447925C211yesmaximal, characteristic, complemented
8#21447868C211yesmaximal, characteristic, complemented
9#21447862C211yesmaximal, characteristic, complemented
10#21447760C211yesmaximal, characteristic, complemented
11#21447755C211yesmaximal, characteristic, complemented
12#21447751C211yesmaximal, characteristic, complemented
13#21447674C211yesmaximal, characteristic, complemented
14#21447648C211yesmaximal, characteristic, complemented
15C61×#2189C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic
17C22×Dic244C2211yescharacteristic
18C2×Q16×D61C2211yescharacteristic
19#16840425—21no—
20#16840411—21no—
21#16840202—24no—
22#16840201—24no—
23C2×Dic488C2211yescharacteristic
24C2×Dic488C2211yescharacteristic
25C2×#14519723C2211yescharacteristic
26C2×#14519723C2211yescharacteristic
27C2×#14519723C2211yescharacteristic
28C2×#14519721C2211yescharacteristic
29C2×#14519721C2211yescharacteristic
30C2×#14519717C2211yescharacteristic
31C2×#14519716C2211yescharacteristic
32#16840140C2211yescharacteristic
33#16840140C2211yescharacteristic
34#16840004C2211yescharacteristic
35#16840003C2211yescharacteristic
36#16840000C2211yescharacteristic
37#16839971—21no—
38#16839965—21no—
39#16839924—21no—
40#16839921—21no—
41#16839919—21no—
42#16839917—21no—
43#16839856C2211yescharacteristic
44D8×Dic61C2211yescharacteristic
45C8×C22⋊D61C2211yescharacteristic
46#16839589C2211yescharacteristic
47#16839580C2211yescharacteristic
48#16839567C2211yescharacteristic
49#16839567C2211yescharacteristic
50#16839566C2211yescharacteristic
51#16839565C2211yescharacteristic
52#16839541C2211yescharacteristic
53C16×Dic61C2211yescharacteristic
54C2×#135×C61C2211yescharacteristic
55C2×Q32×C61C2211yescharacteristic
56C2×SD32×C61C2211yescharacteristic
57C61×#467C2211yescharacteristic
58C61×#358C2211yescharacteristic
59C61×#357C2211yescharacteristic
60C61×C22⋊C16C2211yescharacteristic
61C2×#11758224—21no—
62C22×Dic122—21no—
63C2×Q8×D61—21no—
64#14519796—22no—
65#14519796—41no—
66C2×Dic244—22no—
67C2×Dic244—21no—
68C2×Dic244—41no—
69C2×Dic244C2311yescharacteristic
70C2×Dic244C2311yescharacteristic
71C2×Dic244C2311yescharacteristic
72C2×#11758202—21no—
73C2×#11758202—21no—
74C2×#11758202—21no—
75Q16×D61—22no—
76Q16×D61—41no—
77C2×#11758200—21no—
78C2×C8×D61C2311yescharacteristic
79#14519776—41no—
80#14519775—21no—
81#14519775—21no—
82#14519760—41no—
83#14519758—41no—
84#14519757—21no—
85#14519756—21no—
86#14519755—41no—
87#14519754—41no—
88#14519753—21no—
89#14519749—41no—
90#14519742—21no—
91D4×Dic61—21no—
92C4×C22⋊D61D411yescharacteristic
93#14519737—41no—
94C2×#11758196—41no—
95Dic488—22no—
96Dic488—41no—
97#14519723—22no—
98#14519723—22no—
99#14519723—22no—
100#14519721—22no—
101#14519721D414yescomplemented
102#14519717—22no—
103#14519716—22no—
104C2×#11758185C2311yescharacteristic
105C2×#11758185C2311yescharacteristic
106#14519711C2311yescharacteristic
107#14519706D411yescharacteristic
108#14519702—21no—
109#14519701—21no—
110#14519700—21no—
111#14519700—21no—
112#14519698—21no—
113C22⋊#92418D411yescharacteristic
114#14519689D411yescharacteristic
115C8×Dic61C2311yescharacteristic
116C2×#41×C61—21no—
117#135×C61—22no—
118#135×C61—41no—
119C2×Q16×C61C2311yescharacteristic
120C2×Q16×C61C2311yescharacteristic
121C2×SD16×C61—21no—
122C2×D8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C22⋊Q8×C61—21no—
125Q32×C61—41no—
126SD32×C61—22no—
127C2×C16×C61C2311yescharacteristic
128C2×C16×C61C2311yescharacteristic
129#107×C61C2311yescharacteristic
130#103×C61—21no—
131#41×C61—42no—
132#41×C61—81no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—41no—
137C22×C4×C61—41no—
138C22×C4×C61C2×D411yescharacteristic
139C2×C22⋊D61—41no—
140C2×C22⋊D61D811yescharacteristic
141C22×Dic61—41no—
142C22×Dic61—41no—
143C22×Dic61—41no—
144Q8×D61—42no—
145Q8×D61—81no—
146#11758224—42no—
147#11758224—81no—
148C2×C4×D61—41no—
149C2×C4×D61C2×D411yescharacteristic
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—21no—
153C2×Dic122—42no—
154C2×Dic122—41no—
155C2×Dic122—41no—
156C2×Dic122—81no—
157Q16×C61—21no—
158Q16×C61—21no—
159Q16×C61—21no—
160Q16×C61—21no—
161SD16×C61—42no—
162D8×C61—21no—
163D8×C61C2×D412yes—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61C2×D411yescharacteristic
167C2×C8×C61C2411yescharacteristic
168C4⋊C4×C61—21no—
169C4⋊C4×C61—41no—
170C22⋊C4×C61—41no—
171C22⋊Dic61—41no—
172C22⋊Dic61—41no—
173C22⋊Dic61D811yescharacteristic
174#11758202—42no—
175#11758202—81no—
176#11758202—81no—
177#11758200—42no—
178#11758198—41no—
179#11758198—41no—
180#11758198D811yescharacteristic
181C4⋊Dic61—21no—
182C4⋊Dic61—41no—
183#11758196—41no—
184#11758196—41no—
185#11758196—41no—
186#11758196—41no—
187#11758196—41no—
188#11758196—41no—
189#11758196—41no—
190#11758196D811yescharacteristic
191C4×Dic61—41no—
192C4×Dic61C2×D411yescharacteristic
193C2×#92418C2×D411yescharacteristic
194C2×#92418C2×D411yescharacteristic
195Dic244—21no—
196Dic244—21no—
197Dic244—21no—
198Dic244—21no—
199Dic244—21no—
200Dic244—42no—
201Dic244C2×D412yes—
202C8×D61—21no—
203C8×D61—21no—
204C16×C61—21no—
205C16×C61—21no—
206#11758185—21no—
207#11758185C2×D412yes—
208C23×C61—41no—
209C23×C61C2×D811yescharacteristic
210C22×D61C2×D811yescharacteristic
211Q8×C61—41no—
212Q8×C61—41no—
213Q8×C61—41no—
214Q8×C61—41no—
215D4×C61—22no—
216D4×C61—41no—
217D4×C61—82no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—82no—
224C2×C4×C61C2×D811yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C22⋊D61—22no—
227C22⋊D61—82no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—82no—
233C2×Dic61—82no—
234C2×Dic61—82no—
235C2×Dic61C2×D811yescharacteristic
236C2×Dic61C2×D811yescharacteristic
237C2×Dic61C2×D811yescharacteristic
238C4×D61—21no—
239C4×D61—21no—
240C4×D61—82no—
241Dic122—22no—
242Dic122—41no—
243Dic122—41no—
244Dic122—41no—
245Dic122—41no—
246Dic122—41no—
247Dic122—82no—
248Dic122—82no—
249C8×C61—21no—
250C8×C61C22×D411yescharacteristic
251C8×C61H(5,𝔽2)11yescharacteristic
252#92418—21no—
253#92418—21no—
254#33842—611nomaximal, Sylow
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—42no—
259C22×C61—81no—
260C22×C61—81no—
261C22×C61C22×D811yescharacteristic
262C2×D61—21no—
263C2×D61—21no—
264C2×D61—21no—
265C4×C61—21no—
266C4×C61—81no—
267C4×C61—81no—
268C4×C61—81no—
269C4×C61D4211yescharacteristic
270C4×C61#57611yescharacteristic
271Dic61—21no—
272Dic61—21no—
273Dic61—42no—
274Dic61—81no—
275Dic61—81no—
276Dic61—81no—
277Dic61#51012yescomplemented
278C2×#510—611no—
279C2×#510—611no—
280D4×Q16—611no—
281#3283—611no—
282#2247—611no—
283#2219—611no—
284#2219—611no—
285#2209—611no—
286#2208—611no—
287#2189—611no—
288#2189D6111yescharacteristic, complemented
289#2188—611no—
290#2188—611no—
291C4×SD32—611no—
292#2169—611no—
293C2×C61—41no—
294C2×C61—82no—
295C2×C61D4×D811yescharacteristic
296C2×C61C2×#51011yescharacteristic
297C2×C61#341711yescharacteristic
298D61—41no—
299C2×#135—611no—
300C2×#135C2×D6111yescharacteristic
301C22×Q16—611no—
302C22×Q16—611no—
303D4×Q8—1221no—
304#547—1221no—
305#510—1224no—
306#510—1224no—
307C2×Q32—611no—
308C2×Q32—611no—
309C2×Q32—611no—
310C2×Q32—611no—
311C2×Q32—611no—
312C2×Q32C2×D6111yescharacteristic
313C2×SD32—611no—
314C2×SD32—611no—
315C2×SD32C2×D6111yescharacteristic
316C2×M32—611no—
317C2×M32—611no—
318#494—611no—
319#494—611no—
320#467—611no—
321#467—611no—
322#467—611no—
323#467C2×D6111yescharacteristic
324#462—1221no—
325#461—1221no—
326#452—1221no—
327#452—1221no—
328#451—1221no—
329#451—1221no—
330C4×Q16—611no—
331C4×D8—611no—
332C8×D4—611no—
333#367—611no—
334C4⋊C16—611no—
335#358—611no—
336#358—611no—
337#358—611no—
338#358—611no—
339#358C2×D6111yescharacteristic
340#357C2×D6111yescharacteristic
341C22⋊C16—611no—
342C22⋊C16C2×D6111yescharacteristic
343C4×C16—611no—
344C61#3384211yescharacteristic, complemented, Sylow
345C2×#41—21no—
346C2×#41—1221no—
347C22×Q8—1221no—
348C22×Q8—1221no—
349#135—22no—
350#135—41no—
351#135—1222no—
352#135—2441no—
353C2×Q16—611no—
354C2×Q16—611no—
355C2×Q16—611no—
356C2×Q16—1222no—
357C2×Q16—1221no—
358C2×Q16—1221no—
359C2×Q16—1221no—
360C2×Q16—1222no—
361C2×Q16—1221no—
362C2×Q16—2441no—
363C2×Q16—2441no—
364C2×Q16C22×D6111yescharacteristic
365C2×Q16C22×D6111yescharacteristic
366C2×SD16—21no—
367C2×SD16—1221no—
368C2×D8C22×D6111yescharacteristic
369C22×C8—611no—
370C22×C8C22×D6111yescharacteristic
371C4⋊Q8—1221no—
372C4⋊Q8—1221no—
373C4⋊Q8—2441no—
374#123—2441no—
375#123—2441no—
376C22⋊Q8—21no—
377C22⋊Q8—1221no—
378C22⋊Q8—1221no—
379C22⋊Q8—1221no—
380C22⋊Q8—2441no—
381C22⋊Q8—2441no—
382#121—2441no—
383C4×Q8—1221no—
384C4×D4—611no—
385C4×D4—1221no—
386C4×D4—2441no—
387C2×C4⋊C4—2441no—
388Q32—41no—
389Q32—1222no—
390Q32—1222no—
391Q32—1222no—
392Q32—1222no—
393Q32—2441no—
394SD32—22no—
395SD32—614no—
396SD32—1222no—
397M32—1222no—
398M32—1222no—
399C2×C16—611no—
400C2×C16—611no—
401C2×C16C22×D6111yescharacteristic
402C2×C16C22×D6111yescharacteristic
403#107—611no—
404#107C22×D6111yescharacteristic
405C4⋊C8—611no—
406#103—21no—
407#103—1221no—
408#103—1221no—
409#103—1221no—
410#103—1221no—
411#102—1221no—
412C22⋊C8—611no—
413C22⋊C8—611no—
414C4×C8—611no—
415#41—42no—
416#41—81no—
417#41—4881no—
418#41—2442no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—1221no—
422C2×Q8—1221no—
423C2×Q8—1221no—
424C2×Q8—4881no—
425C2×Q8—4881no—
426C2×Q8—2442no—
427C2×Q8—2441no—
428C2×Q8—2442no—
429C2×Q8—2441no—
430C2×D4—21no—
431C2×D4—41no—
432C2×D4—611no—
433C2×D4—2441no—
434C22×C4—41no—
435C22×C4—611no—
436C22×C4—2441no—
437C22×C4—2441no—
438C22×C4—2441no—
439C22×C4—2441no—
440C22×C4D4×D6111yescharacteristic
441Q16—21no—
442Q16—21no—
443Q16—21no—
444Q16—21no—
445Q16—612no—
446Q16—1221no—
447Q16—1221no—
448Q16—1221no—
449Q16—1221no—
450Q16—1221no—
451Q16—4881no—
452Q16—4881no—
453Q16—2442no—
454Q16—2442no—
455SD16—42no—
456SD16—2442no—
457D8—21no—
458D8D4×D6112yes—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—611no—
462C2×C8—611no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8D4×D6111yescharacteristic
466C2×C8C23×D6111yescharacteristic
467C4⋊C4—21no—
468C4⋊C4—41no—
469C4⋊C4—611no—
470C4⋊C4—1221no—
471C4⋊C4—2441no—
472C4⋊C4—2441no—
473C4⋊C4—2441no—
474C4⋊C4—2441no—
475C4⋊C4—2441no—
476C4⋊C4—2441no—
477C4⋊C4—2441no—
478C4⋊C4—2441no—
479C22⋊C4—41no—
480C22⋊C4—611no—
481C22⋊C4—611no—
482C22⋊C4—2441no—
483C22⋊C4—2441no—
484C22⋊C4—2441no—
485C22⋊C4—2441no—
486C42—611no—
487C42—2441no—
488C16—21no—
489C16—21no—
490C16—612no—
491C16—1221no—
492C23—41no—
493C23—611no—
494C23D8×D6111yescharacteristic
495Q8—41no—
496Q8—41no—
497Q8—41no—
498Q8—41no—
499Q8—1222no—
500Q8—4882no—
501Q8—4882no—
502Q8—2441no—
503Q8—2441no—
504Q8—2441no—
505Q8—2441no—
506Q8—2441no—
507D4—22no—
508D4—41no—
509D4—82no—
510D4—1222no—
511D4—4882no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—82no—
518C2×C4—611no—
519C2×C4—611no—
520C2×C4—611no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—4882no—
524C2×C4—4882no—
525C2×C4—4882no—
526C2×C4—4882no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4D8×D6111yescharacteristic
532C2×C4C2×D4×D6111yescharacteristic
533C8—21no—
534C8—1221no—
535C8—1221no—
536C8C2×D4×D6111yescharacteristic
537C8#1451982211yescharacteristic
538C22—21no—
539C22—21no—
540C22—21no—
541C22—42no—
542C22—81no—
543C22—81no—
544C22—1221no—
545C22—1221no—
546C22—1221no—
547C22C2×D8×D6111yescharacteristic
548C4—21no—
549C4—81no—
550C4—81no—
551C4—81no—
552C4—612no—
553C4—1221no—
554C4—1221no—
555C4—4881no—
556C4—4881no—
557C4—4881no—
558C4—2442no—
559C4#1684039411yescharacteristic
560C4#1684056611yescharacteristic
561C2—41no—
562C2—82no—
563C2—2441no—
564C2#2145748911yescharacteristic
565C2#510×D6111yescharacteristic
566C2#2145896911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2189C2
socleC22×C61C22×D8
prime radical#21457535C2
hypercenter#2189D61
hypocenterC61#33842
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2189

Complements

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

NH ≅ G/Ntype
1. C2×#16840202C2semidirect
2. C2×#16840201C2semidirect
3. #21457568C2semidirect
5. #21448070C2semidirect
6. #21447926C2semidirect
7. #21447925C2semidirect
8. #21447868C2semidirect
9. #21447862C2semidirect
10. #21447760C2semidirect
11. #21447755C2semidirect
12. #21447751C2semidirect
13. #21447674C2semidirect
14. #21447648C2semidirect
15. C61×#2189C2semidirect
101. #14519721D4semidirect
277. Dic61#510semidirect
288. #2189D61semidirect
344. C61#33842semidirect

Maximal subgroups

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

subgroupindex
C2×#168402022
C2×#168402012
#214575682
#214575352
#214480702
#214479262
#214479252
#214478682
#214478622
#214477602
#214477552
#214477512
#214476742
#214476482
C61×#21892
#3384261