Finite Group Databasebeta

← #29725125

Subgroups — #29725125

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×#16840026C211yesmaximal, characteristic
2#21454680C211yesmaximal, characteristic, complemented
3#21447284C211yesmaximal, characteristic, complemented
4#21447277C211yesmaximal, characteristic, complemented
5#21447101C211yesmaximal, characteristic, complemented
6#21447094C211yesmaximal, characteristic
7#21446720C211yesmaximal, characteristic, complemented
8#21446312C211yesmaximal, characteristic, complemented
9#21446305C211yesmaximal, characteristic
10#21445944C211yesmaximal, characteristic, complemented
11#21445932C211yesmaximal, characteristic, complemented
12#21445929C211yesmaximal, characteristic, complemented
13#21445874C211yesmaximal, characteristic, complemented
14#21445620C211yesmaximal, characteristic, complemented
15C61×#1953C211yesmaximal, characteristic, complemented
16C2×#14519810C2211yescharacteristic
17C2×#14519805C2211yescharacteristic
18C22×#11758190C2211yescharacteristic
19C2×#14519783C2211yescharacteristic
20#16840239C2211yescharacteristic
21#16840236C2211yescharacteristic
22#16840026—22no—
23#16840026—22no—
24#16839864—22no—
25C4×#11758190—22no—
26#16839800C2211yescharacteristic
27#16839778C2211yescharacteristic
28#16839778C2211yescharacteristic
29#16839772C2211yescharacteristic
30#16839772C2211yescharacteristic
31#16839768C2211yescharacteristic
32#16839768C2211yescharacteristic
33#16839744C2211yescharacteristic
34C2×#14519695C2211yescharacteristic
35C2×#14519695C2211yescharacteristic
36C2×#14519693C2211yescharacteristic
37C2×#14519693C2211yescharacteristic
38#16839735C2211yescharacteristic
39#16839717C2211yescharacteristic
40#16839534C2211yescharacteristic
41#16839534C2211yescharacteristic
42#16839534C2211yescharacteristic
43#16839533C2211yescharacteristic
44#16839533C2211yescharacteristic
45#16839532C2211yescharacteristic
46#16839519C2211yescharacteristic
47#16839519C2211yescharacteristic
48C61×#430C2211yescharacteristic
49C61×#430C2211yescharacteristic
50C2×#108×C61C2211yescharacteristic
51C2×#108×C61C2211yescharacteristic
52C61×#405C2211yescharacteristic
53C61×#344C2211yescharacteristic
54C61×#344C2211yescharacteristic
55C2×#11758222C2311yescharacteristic
56C22×Dic122C2311yescharacteristic
57C22×D244C2311yescharacteristic
58#14519810—22no—
59#14519810—41no—
60#14519805—22no—
61#14519805—41no—
62C2×Dic244—21no—
63C2×#11758190—21no—
64C2×#11758190—21no—
65C2×#11758190—21no—
66C2×#11758190—21no—
67C2×#11758190—21no—
68C2×#11758190—21no—
69C2×#11758190D412yes—
70C2×#11758190D412yes—
71C2×D488—21no—
72#14519783—21no—
73#14519783—21no—
74#14519783—21no—
75#14519783—21no—
76C2×#11758194C2311yescharacteristic
77C2×#11758194C2311yescharacteristic
78C2×#11758189—21no—
79C2×C8×D61—21no—
80C4×Dic122—22no—
81#14519742—41no—
82C4×D244—22no—
83#14519739—41no—
84#14519736C2311yescharacteristic
85#14519736C2311yescharacteristic
86#14519736C2311yescharacteristic
87#14519733—22no—
88C22⋊C4×D61—22no—
89#14519711—21no—
90#14519709—21no—
91#14519703—41no—
92#14519703—41no—
93#14519702—22no—
94#14519702—41no—
95#14519699—22no—
96#14519699—41no—
97#14519695—22no—
98#14519695—21no—
99#14519695—41no—
100#14519693—22no—
101#14519693—21no—
102#14519693—41no—
103#14519691—22no—
104#14519687—21no—
105C8×Dic61—21no—
106C2×M16×C61C2311yescharacteristic
107C2×M16×C61C2311yescharacteristic
108C2×M16×C61C2311yescharacteristic
109C2×M16×C61C2311yescharacteristic
110C2×M16×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112#117×C61C2311yescharacteristic
113#108×C61—21no—
114#108×C61—21no—
115#108×C61—21no—
116#108×C61—21no—
117#108×C61—41no—
118#108×C61—41no—
119#97×C61—21no—
120C4×C8×C61—21no—
121C23×D61—21no—
122C22×C4×C61C2411yescharacteristic
123C2×C22⋊D61—21no—
124C22×Dic61—21no—
125#11758222—21no—
126#11758222—21no—
127#11758222—21no—
128#11758222—21no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244C2×D412yes—
135C2×D244#4111yescharacteristic
136C2×C4×D61—21no—
137C2×C4×D61—42no—
138C2×Dic122—21no—
139C2×Dic122—21no—
140C2×Dic122—21no—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122C2×D412yes—
144C2×Dic122#4111yescharacteristic
145M16×C61—21no—
146M16×C61—21no—
147M16×C61—21no—
148M16×C61—21no—
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×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61C2×D411yescharacteristic
161C2×C8×C61C2×D411yescharacteristic
162C2×C8×C61#4111yescharacteristic
163C2×C8×C61#4111yescharacteristic
164C2×C8×C61#4111yescharacteristic
165C2×C8×C61#4111yescharacteristic
166C4⋊C4×C61—21no—
167C22⋊C4×C61#4112yes—
168C42×C61—21no—
169C22⋊Dic61—21no—
170C22⋊Dic61—21no—
171C22⋊Dic61—21no—
172#11758198—42no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176#11758196—42no—
177C4×Dic61—21no—
178C4×Dic61—21no—
179C4×Dic61—21no—
180C4×Dic61—42no—
181#11758194—21no—
182#11758194—21no—
183#11758194—21no—
184#11758194—21no—
185C2×#92418—21no—
186C2×#92418—21no—
187Dic244—42no—
188D488—42no—
189#11758190—22no—
190#11758190—22no—
191#11758190—22no—
192#11758190—22no—
193#11758190—42no—
194#11758190—42no—
195#11758189—41no—
196#11758189—41no—
197C8×D61—41no—
198C8×D61—41no—
199C23×C61C2×#4111yescharacteristic
200C22×D61—21no—
201C22×D61—22no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—161no—
205C2×C4×C61—22no—
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C2×#4111yescharacteristic
208C2×C4×C61C2×#4111yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C61H(5,𝔽2)11yescharacteristic
211C2×C4×C61H(5,𝔽2)11yescharacteristic
212C22⋊D61—41no—
213C22⋊D61—41no—
214C2×Dic61—21no—
215C2×Dic61—22no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221D244—22no—
222D244—21no—
223D244—22no—
224D244—21no—
225D244—41no—
226D244—41no—
227C4×D61—41no—
228C4×D61—41no—
229C4×D61—162no—
230Dic122—22no—
231Dic122—21no—
232Dic122—22no—
233Dic122—21no—
234Dic122—41no—
235Dic122—41no—
236C8×C61—41no—
237C8×C61—41no—
238C8×C61—41no—
239C8×C61—41no—
240C8×C61—41no—
241C8×C61#12311yescharacteristic
242C8×C61#12311yescharacteristic
243C8×C61#12311yescharacteristic
244C8×C61#12311yescharacteristic
245#92418—41no—
246#92418—41no—
247#24987—611nomaximal, Sylow
248C22×C61—41no—
249C22×C61#52911yescharacteristic
250C22×C61#54611yescharacteristic
251C22×C61#55511yescharacteristic
252C2×D61—41no—
253C2×D61—42no—
254C2×D61—82no—
255C2×D61—81no—
256C2×D61—161no—
257C2×D61—161no—
258C4×C61—42no—
259C4×C61C2×#12311yescharacteristic
260C4×C61#54011yescharacteristic
261C4×C61#54611yescharacteristic
262C4×C61#55311yescharacteristic
263Dic61—42no—
264Dic61—41no—
265Dic61—81no—
266Dic61—81no—
267Dic61—81no—
268C2×#471—611no—
269#2949—611no—
270#2082—611no—
271#2081—611no—
272#2049—611no—
273#2048—611no—
274#1977—611no—
275#1953D6111yescharacteristic, complemented
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#242011yescharacteristic
287D61—81no—
288D61—82no—
289C2×#137—611no—
290C2×AGL(1,ℤ/8)—611no—
291C22×SD16—611no—
292C2×#131—611no—
293#519—611no—
294#518—611no—
295#471—1222no—
296#471—1222no—
297#440—1222no—
298C4×SD16—1222no—
299#430C2×D6111yescharacteristic
300#430C2×D6111yescharacteristic
301C2×#108C2×D6111yescharacteristic
302C2×#108C2×D6111yescharacteristic
303#427—611no—
304#421—611no—
305#421—611no—
306#419—611no—
307#419—611no—
308#418—611no—
309#418—611no—
310#413—611no—
311C2×#101—611no—
312C2×#101—611no—
313C2×#100—611no—
314C2×#100—611no—
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#344C2×D6111yescharacteristic
325#344C2×D6111yescharacteristic
326H(3,ℤ/4)—611no—
327H(3,ℤ/4)—611no—
328C61#2498711yescharacteristic, complemented, Sylow
329C2×#41—611no—
330C22×Q8—611no—
331C22×D4—611no—
332#137—1222no—
333#137—2441no—
334AGL(1,ℤ/8)—1222no—
335AGL(1,ℤ/8)—2441no—
336C2×Q16—1221no—
337C2×SD16—612no—
338C2×SD16—612no—
339C2×SD16—1221no—
340C2×SD16—1221no—
341C2×SD16—1221no—
342C2×SD16—1221no—
343C2×SD16—1221no—
344C2×SD16—1221no—
345C2×D8—1221no—
346#131—1221no—
347#131—1221no—
348#131—1221no—
349#131—1221no—
350C2×M16—611no—
351C2×M16—611no—
352C2×M16—1221no—
353C2×M16C22×D6111yescharacteristic
354C2×M16C22×D6111yescharacteristic
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—21no—
373#108—21no—
374#108—41no—
375#108—41no—
376#107—1221no—
377#106—1221no—
378#104—2441no—
379#104—2441no—
380#103—1222no—
381#103—2441no—
382#102—1222no—
383#102—2441no—
384#101—1222no—
385#101—1221no—
386#101—2441no—
387#100—1222no—
388#100—1221no—
389#100—2441no—
390#99—1222no—
391#97—21no—
392#97—1221no—
393C4×C8—21no—
394C4×C8—1221no—
395C24—1221no—
396#41—1221no—
397#41—1221no—
398#41—1221no—
399#41—1221no—
400C2×Q8—612no—
401C2×Q8—611no—
402C2×Q8—1221no—
403C2×Q8—1221no—
404C2×Q8—1221no—
405C2×Q8—1221no—
406C2×Q8—1221no—
407C2×D4—612no—
408C2×D4—611no—
409C2×D4—1221no—
410C2×D4—1221no—
411C2×D4—1221no—
412C2×D4—1221no—
413C2×D4—1221no—
414C2×D4—1221no—
415C22×C4—1221no—
416C22×C4—1221no—
417C22×C4—2442no—
418C22×C4C23×D6111yescharacteristic
419Q16—2442no—
420SD16—1222no—
421SD16—1222no—
422SD16—1222no—
423SD16—1222no—
424SD16—2442no—
425SD16—2442no—
426D8—2442no—
427M16—21no—
428M16—21no—
429M16—21no—
430M16—21no—
431M16—21no—
432M16—21no—
433M16—21no—
434M16—21no—
435M16—21no—
436M16—21no—
437M16—1221no—
438M16—1221no—
439M16—1221no—
440M16—1221no—
441M16—2441no—
442M16—2441no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—2441no—
451C2×C8—2441no—
452C2×C8#1175822211yescharacteristic
453C2×C8#1175822211yescharacteristic
454C2×C8D4×D6111yescharacteristic
455C2×C8D4×D6111yescharacteristic
456C2×C8#1175822611yescharacteristic
457C2×C8#1175822611yescharacteristic
458C4⋊C4—21no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—2442no—
463C22⋊C4—1221no—
464C22⋊C4—1221no—
465C22⋊C4—1221no—
466C22⋊C4—2442no—
467C22⋊C4#1175822612yes—
468C42—21no—
469C42—1221no—
470C42—1221no—
471C42—1221no—
472C42—2442no—
473C23—1221no—
474C23—1222no—
475C23—9761no—
476C23—2441no—
477C23—2441no—
478C23C2×#1175822611yescharacteristic
479Q8—1221no—
480Q8—1222no—
481Q8—1221no—
482Q8—1222no—
483Q8—2441no—
484Q8—2441no—
485D4—1221no—
486D4—1222no—
487D4—1221no—
488D4—1222no—
489D4—2441no—
490D4—2441no—
491D4—2441no—
492D4—2441no—
493C2×C4—22no—
494C2×C4—1221no—
495C2×C4—1222no—
496C2×C4—9762no—
497C2×C4—2441no—
498C2×C4—2441no—
499C2×C4—2441no—
500C2×C4—2441no—
501C2×C4—2441no—
502C2×C4—2441no—
503C2×C4—2441no—
504C2×C4C2×D4×D6111yescharacteristic
505C2×C4#41×D6111yescharacteristic
506C2×C4C2×#1175822211yescharacteristic
507C2×C4C2×#1175822611yescharacteristic
508C2×C4#1451982211yescharacteristic
509C2×C4#1451982311yescharacteristic
510C8—41no—
511C8—41no—
512C8—41no—
513C8—41no—
514C8—41no—
515C8—2441no—
516C8—2441no—
517C8#1451976211yescharacteristic
518C8#1451976211yescharacteristic
519C8#1451976211yescharacteristic
520C8#1451976211yescharacteristic
521C22—41no—
522C22—9761no—
523C22—9761no—
524C22—4882no—
525C22—4881no—
526C22—2441no—
527C22—2442no—
528C22#1684029911yescharacteristic
529C22#1684040311yescharacteristic
530C22#1684047811yescharacteristic
531C4—42no—
532C4—4881no—
533C4—4881no—
534C4—4881no—
535C4—2441no—
536C4—2442no—
537C4C2×#1451976211yescharacteristic
538C4#1684036311yescharacteristic
539C4#1684040911yescharacteristic
540C4#1684046511yescharacteristic
541C2—21no—
542C2—21no—
543C2—21no—
544C2—4882no—
545C2—4881no—
546C2#2144931211yescharacteristic
subgroupG/H
centerC2#21449312
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1953C2
socleC2×C61#2420
prime radicalC2×#14519805C22
hypercenter#1953D61
hypocenterC61#24987
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1953

Complements

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

NH ≅ G/Ntype
2. #21454680C2semidirect
3. #21447284C2semidirect
4. #21447277C2semidirect
5. #21447101C2semidirect
7. #21446720C2semidirect
8. #21446312C2semidirect
10. #21445944C2semidirect
11. #21445932C2semidirect
12. #21445929C2semidirect
13. #21445874C2semidirect
14. #21445620C2semidirect
15. C61×#1953C2semidirect
275. #1953D61semidirect
328. C61#24987semidirect

Maximal subgroups

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

subgroupindex
C2×#168400262
#214546802
#214472842
#214472772
#214471012
#214470942
#214467202
#214463122
#214463052
#214459442
#214459322
#214459292
#214458742
#214456202
C61×#19532
#2498761