Finite Group Databasebeta

← #29725121

Subgroups — #29725121

Order 15616.

638 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840022C211yesmaximal, characteristic
2#21454679C211yesmaximal, characteristic, complemented
3#21447283C211yesmaximal, characteristic, complemented
4#21447276C211yesmaximal, characteristic, complemented
5#21447098C211yesmaximal, characteristic, complemented
6#21447091C211yesmaximal, characteristic
7#21446610C211yesmaximal, characteristic, complemented
8#21446311C211yesmaximal, characteristic, complemented
9#21446304C211yesmaximal, characteristic
10#21445943C211yesmaximal, characteristic, complemented
11#21445931C211yesmaximal, characteristic, complemented
12#21445928C211yesmaximal, characteristic, complemented
13#21445874C211yesmaximal, characteristic, complemented
14#21445620C211yesmaximal, characteristic, complemented
15C61×#1977C211yesmaximal, characteristic, complemented
16C2×#14519808C2211yescharacteristic
17C2×#14519803C2211yescharacteristic
18C2×#14519783C2211yescharacteristic
19#16840240C2211yescharacteristic
20#16840237C2211yescharacteristic
21#16840022—22no—
22#16840022—22no—
23#16839863—22no—
24SD16×Dic61—22no—
25#16839811C2211yescharacteristic
26#16839811C2211yescharacteristic
27C2×#14519713C2211yescharacteristic
28#16839800C2211yescharacteristic
29#16839778C2211yescharacteristic
30#16839778C2211yescharacteristic
31#16839771C2211yescharacteristic
32#16839767C2211yescharacteristic
33#16839744C2211yescharacteristic
34C2×#14519696C2211yescharacteristic
35C2×#14519694C2211yescharacteristic
36#16839735C2211yescharacteristic
37#16839717C2211yescharacteristic
38#16839534C2211yescharacteristic
39#16839534C2211yescharacteristic
40#16839534C2211yescharacteristic
41#16839533C2211yescharacteristic
42#16839533C2211yescharacteristic
43#16839533C2211yescharacteristic
44#16839533C2211yescharacteristic
45#16839532C2211yescharacteristic
46#16839518C2211yescharacteristic
47#16839518C2211yescharacteristic
48C22×SD16×C61C2211yescharacteristic
49C2×#108×C61C2211yescharacteristic
50C61×#419C2211yescharacteristic
51C61×#418C2211yescharacteristic
52C2×#101×C61C2211yescharacteristic
53C2×#100×C61C2211yescharacteristic
54C61×#405C2211yescharacteristic
55C2×#11758222C2311yescharacteristic
56#14519808—22no—
57#14519808—41no—
58#14519803—22no—
59#14519803—41no—
60C2×#11758202—21no—
61C2×#11758200—21no—
62C2×#11758201—21no—
63C2×#11758199—21no—
64#14519783—21no—
65#14519783—21no—
66#14519783—21no—
67#14519783—21no—
68C2×#11758194C2311yescharacteristic
69C2×#11758194C2311yescharacteristic
70C2×#11758194C2311yescharacteristic
71C2×#11758194C2311yescharacteristic
72C2×#11758189—21no—
73C2×C8×D61—21no—
74Q8×Dic61—22no—
75Q8×Dic61—41no—
76D4×Dic61—22no—
77D4×Dic61—41no—
78#14519736—22no—
79#14519736C2311yescharacteristic
80#14519736C2311yescharacteristic
81#14519736C2311yescharacteristic
82C2×C22⋊Dic61—22no—
83#14519713—21no—
84#14519713—21no—
85#14519713—41no—
86#14519713—41no—
87#14519711—21no—
88#14519709—21no—
89#14519703—41no—
90#14519703—41no—
91#14519701—22no—
92#14519698—22no—
93#14519696—22no—
94#14519695—21no—
95#14519694—22no—
96#14519693—21no—
97#14519691—22no—
98#14519687—21no—
99C8×Dic61—21no—
100C22×Q8×C61C2311yescharacteristic
101C22×D4×C61C2311yescharacteristic
102C2×SD16×C61—21no—
103C2×SD16×C61—21no—
104C2×SD16×C61—21no—
105C2×SD16×C61—21no—
106C2×SD16×C61D412yes—
107C2×SD16×C61D412yes—
108C2×M16×C61C2311yescharacteristic
109C2×M16×C61C2311yescharacteristic
110C2×M16×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112#117×C61C2311yescharacteristic
113#108×C61—21no—
114#108×C61—21no—
115#103×C61—41no—
116#102×C61—41no—
117#101×C61—41no—
118#100×C61—41no—
119#97×C61—21no—
120C4×C8×C61—21no—
121C24×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61C2×D412yes—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61C2×D412yes—
134C22×C4×C61—21no—
135C22×C4×C61C2411yescharacteristic
136C2×C22⋊D61—21no—
137C22×Dic61—42no—
138#11758222—21no—
139#11758222—21no—
140#11758222—21no—
141#11758222—21no—
142C2×D244#4111yescharacteristic
143C2×C4×D61—21no—
144C2×Dic122#4111yescharacteristic
145SD16×C61—22no—
146SD16×C61—22no—
147SD16×C61—22no—
148SD16×C61—22no—
149M16×C61—21no—
150M16×C61—21no—
151M16×C61—21no—
152M16×C61—21no—
153M16×C61—21no—
154M16×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61C2×D411yescharacteristic
159C2×C8×C61C2×D411yescharacteristic
160C2×C8×C61#4111yescharacteristic
161C2×C8×C61#4111yescharacteristic
162C2×C8×C61#4111yescharacteristic
163C2×C8×C61#4111yescharacteristic
164C4⋊C4×C61—21no—
165C22⋊C4×C61—21no—
166C42×C61—21no—
167C22⋊Dic61—21no—
168C22⋊Dic61—21no—
169C22⋊Dic61—42no—
170C22⋊Dic61#4112yes—
171#11758202—42no—
172#11758201—42no—
173#11758200—42no—
174#11758199—42no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—42no—
179C4×Dic61—21no—
180C4×Dic61—21no—
181C4×Dic61—21no—
182C4×Dic61—42no—
183#11758194—21no—
184#11758194—21no—
185#11758194—21no—
186#11758194—21no—
187#11758194—21no—
188#11758194—21no—
189#11758194—21no—
190#11758194—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195#11758189—41no—
196#11758189—41no—
197C8×D61—41no—
198C8×D61—41no—
199C23×C61—22no—
200C23×C61—41no—
201C23×C61—41no—
202C23×C61—161no—
203C23×C61C2×#4111yescharacteristic
204C22×D61—21no—
205Q8×C61—22no—
206Q8×C61—22no—
207Q8×C61—41no—
208Q8×C61—41no—
209D4×C61—22no—
210D4×C61—22no—
211D4×C61—41no—
212D4×C61—41no—
213C2×C4×C61—22no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61H(5,𝔽2)11yescharacteristic
222C2×C4×C61H(5,𝔽2)11yescharacteristic
223C22⋊D61—41no—
224C22⋊D61—41no—
225C2×Dic61—21no—
226C2×Dic61—22no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—162no—
230D244—21no—
231D244—21no—
232C4×D61—41no—
233C4×D61—41no—
234Dic122—21no—
235Dic122—21no—
236C8×C61—41no—
237C8×C61—41no—
238C8×C61—41no—
239C8×C61#12311yescharacteristic
240C8×C61#12311yescharacteristic
241C8×C61#12311yescharacteristic
242C8×C61#12311yescharacteristic
243#92418—41no—
244#92418—41no—
245#92418—41no—
246#92418—41no—
247#24987—611nomaximal, Sylow
248C22×C61—41no—
249C22×C61—42no—
250C22×C61—82no—
251C22×C61—161no—
252C22×C61—161no—
253C22×C61#52911yescharacteristic
254C22×C61#54611yescharacteristic
255C22×C61#55511yescharacteristic
256C2×D61—41no—
257C2×D61—81no—
258C4×C61—42no—
259C4×C61—81no—
260C4×C61C2×#12311yescharacteristic
261C4×C61#54011yescharacteristic
262C4×C61#54611yescharacteristic
263C4×C61#55311yescharacteristic
264Dic61—42no—
265Dic61—41no—
266Dic61—81no—
267Dic61—81no—
268C2×#471—611no—
269#2949—611no—
270#2082—611no—
271#2081—611no—
272#2049—611no—
273#2048—611no—
274#1977D6111yescharacteristic, complemented
275#1953—611no—
276#1905—611no—
277#1904—611no—
278#1833—611no—
279#1830—611no—
280#1829—611no—
281#1814—611no—
282#1757—611no—
283C2×C61—21no—
284C2×C61—21no—
285C2×C61—21no—
286C2×C61—82no—
287C2×C61#242011yescharacteristic
288D61—81no—
289C2×#137—611no—
290C2×AGL(1,ℤ/8)—611no—
291C22×SD16C2×D6111yescharacteristic
292C2×#131—611no—
293#519—611no—
294#518—611no—
295#471—1222no—
296#471—1222no—
297#440—1222no—
298C4×SD16—1222no—
299#430—611no—
300#430—611no—
301C2×#108—611no—
302C2×#108C2×D6111yescharacteristic
303#427—611no—
304#421—611no—
305#421—611no—
306#419—611no—
307#419C2×D6111yescharacteristic
308#418—611no—
309#418C2×D6111yescharacteristic
310#413—611no—
311C2×#101—611no—
312C2×#101C2×D6111yescharacteristic
313C2×#100—611no—
314C2×#100C2×D6111yescharacteristic
315#410—611no—
316#405—611no—
317#405C2×D6111yescharacteristic
318#344—611no—
319#344—611no—
320#344—611no—
321#344—611no—
322#344—611no—
323#344—611no—
324#344—611no—
325#344—611no—
326H(3,ℤ/4)—611no—
327H(3,ℤ/4)—611no—
328C61#2498711yescharacteristic, complemented, Sylow
329C2×#41—611no—
330C22×Q8C22×D6111yescharacteristic
331C22×D4C22×D6111yescharacteristic
332#137—1222no—
333#137—2441no—
334AGL(1,ℤ/8)—1222no—
335AGL(1,ℤ/8)—2441no—
336C2×Q16—1221no—
337C2×SD16—21no—
338C2×SD16—21no—
339C2×SD16—21no—
340C2×SD16—21no—
341C2×SD16—1221no—
342C2×SD16—1221no—
343C2×SD16D24412yes—
344C2×SD16D24412yes—
345C2×D8—1221no—
346#131—1221no—
347#131—1221no—
348#131—1221no—
349#131—1221no—
350C2×M16—611no—
351C2×M16—611no—
352C2×M16—611no—
353C2×M16—611no—
354C2×M16—1221no—
355C2×M16C22×D6111yescharacteristic
356C2×M16C22×D6111yescharacteristic
357C2×M16C22×D6111yescharacteristic
358C22×C8—1221no—
359C22×C8C22×D6111yescharacteristic
360C4×Q8—1222no—
361C4×Q8—2441no—
362C4×D4—1222no—
363C4×D4—2441no—
364#117—611no—
365#117—611no—
366#117—611no—
367#117—1222no—
368#117C22×D6111yescharacteristic
369C2×C22⋊C4—1222no—
370#108—21no—
371#108—21no—
372#108—1221no—
373#108—1221no—
374#108—2441no—
375#108—2441no—
376#107—1221no—
377#106—1221no—
378#104—2441no—
379#104—2441no—
380#103—41no—
381#103—1222no—
382#102—41no—
383#102—1222no—
384#101—41no—
385#101—1221no—
386#101—1222no—
387#100—41no—
388#100—1221no—
389#100—1222no—
390#99—1222no—
391#97—21no—
392#97—1221no—
393C4×C8—21no—
394C4×C8—1221no—
395C24—21no—
396#41—1221no—
397#41—1221no—
398#41—1221no—
399#41—1221no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—611no—
406C2×Q8C2×D24412yes—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—611no—
413C2×D4—1221no—
414C2×D4C2×D24412yes—
415C22×C4—21no—
416C22×C4—1221no—
417C22×C4—2442no—
418C22×C4C23×D6111yescharacteristic
419Q16—2442no—
420SD16—22no—
421SD16—22no—
422SD16—22no—
423SD16—22no—
424SD16—2442no—
425SD16—2442no—
426D8—2442no—
427M16—21no—
428M16—21no—
429M16—21no—
430M16—21no—
431M16—21no—
432M16—21no—
433M16—1221no—
434M16—1221no—
435M16—1221no—
436M16—1221no—
437M16—1221no—
438M16—1221no—
439M16—1221no—
440M16—1221no—
441M16—2441no—
442M16—2441no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—1221no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—2441no—
451C2×C8—2441no—
452C2×C8C2×D24411yescharacteristic
453C2×C8C2×D24411yescharacteristic
454C2×C8#1175822411yescharacteristic
455C2×C8#1175822411yescharacteristic
456C2×C8#1175822411yescharacteristic
457C2×C8#1175822411yescharacteristic
458C4⋊C4—21no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—2442no—
463C22⋊C4—21no—
464C22⋊C4—612no—
465C22⋊C4—1221no—
466C22⋊C4—1221no—
467C22⋊C4—2442no—
468C42—21no—
469C42—1221no—
470C42—1221no—
471C42—1221no—
472C42—2442no—
473C23—22no—
474C23—41no—
475C23—41no—
476C23—161no—
477C23—1221no—
478C23#41×D6111yescharacteristic
479Q8—22no—
480Q8—22no—
481Q8—41no—
482Q8—41no—
483Q8—1221no—
484Q8—1221no—
485D4—22no—
486D4—22no—
487D4—41no—
488D4—41no—
489D4—1221no—
490D4—1221no—
491D4—2441no—
492D4—2441no—
493C2×C4—22no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—1221no—
498C2×C4—1222no—
499C2×C4—9762no—
500C2×C4—2441no—
501C2×C4—2441no—
502C2×C4—2441no—
503C2×C4—2441no—
504C2×C4C22×D24411yescharacteristic
505C2×C4#41×D6111yescharacteristic
506C2×C4C2×#1175822411yescharacteristic
507C2×C4C2×#1175822411yescharacteristic
508C2×C4#1451982211yescharacteristic
509C2×C4#1451982211yescharacteristic
510C8—41no—
511C8—41no—
512C8—41no—
513C8—2441no—
514C8—2441no—
515C8—2441no—
516C8—2441no—
517C8#1451976111yescharacteristic
518C8#1451976111yescharacteristic
519C8#1451976111yescharacteristic
520C8#1451976111yescharacteristic
521C22—41no—
522C22—42no—
523C22—82no—
524C22—161no—
525C22—161no—
526C22—4881no—
527C22—2441no—
528C22#1684029111yescharacteristic
529C22#1684040711yescharacteristic
530C22#1684047611yescharacteristic
531C4—42no—
532C4—81no—
533C4—4881no—
534C4—4881no—
535C4—2442no—
536C4—2441no—
537C4C2×#1451976111yescharacteristic
538C4#1684036611yescharacteristic
539C4#1684040711yescharacteristic
540C4#1684045911yescharacteristic
541C2—21no—
542C2—21no—
543C2—21no—
544C2—82no—
545C2—4881no—
546C2#2144930911yescharacteristic
subgroupG/H
centerC2#21449309
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1977C2
socleC2×C61#2420
prime radicalC2×#14519803C22
hypercenter#1977D61
hypocenterC61#24987
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1977

Complements

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

NH ≅ G/Ntype
2. #21454679C2semidirect
3. #21447283C2semidirect
4. #21447276C2semidirect
5. #21447098C2semidirect
7. #21446610C2semidirect
8. #21446311C2semidirect
10. #21445943C2semidirect
11. #21445931C2semidirect
12. #21445928C2semidirect
13. #21445874C2semidirect
14. #21445620C2semidirect
15. C61×#1977C2semidirect
274. #1977D61semidirect
328. C61#24987semidirect

Maximal subgroups

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

subgroupindex
C2×#168400222
#214546792
#214472832
#214472762
#214470982
#214470912
#214466102
#214463112
#214463042
#214459432
#214459312
#214459282
#214458742
#214456202
C61×#19772
#2498761