Finite Group Databasebeta

← #29834341

Subgroups — #29834341

Order 15616.

712 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×#16840201C211yesmaximal, characteristic, complemented
2C2×SD32×D61C211yesmaximal, characteristic, complemented
3#21456299C211yesmaximal, characteristic
4#21454682C211yesmaximal, characteristic, complemented
5#21447882C211yesmaximal, characteristic, complemented
6#21447881C211yesmaximal, characteristic, complemented
7#21447880C211yesmaximal, characteristic, complemented
8#21447879C211yesmaximal, characteristic, complemented
9#21447858C211yesmaximal, characteristic, complemented
10#21447857C211yesmaximal, characteristic, complemented
11#21447856C211yesmaximal, characteristic, complemented
12#21447576C211yesmaximal, characteristic, complemented
13#21447533C211yesmaximal, characteristic, complemented
14#21447523C211yesmaximal, characteristic, complemented
15C61×#2207C211yesmaximal, characteristic, complemented
16C2×#14519801—21no—
17C2×#14519796C2211yescharacteristic
18C2×Q16×D61C2211yescharacteristic
19C2×D8×D61C2211yescharacteristic
20#16840325—21no—
21#16840238—21no—
22#16840201—24no—
23C2×Dic488C2211yescharacteristic, complemented
24C2×#14519723C2211yescharacteristic, complemented
25C2×#14519722C2211yescharacteristic
26C2×#14519721C2211yescharacteristic
27C2×#14519721C2211yescharacteristic, complemented
28C2×#14519720C2211yescharacteristic
29SD32×D61—24no—
30C2×#14519716C2211yescharacteristic
31C2×C16×D61C2211yescharacteristic
32#16840150—21no—
33#16840149—21no—
34#16840146C2211yescharacteristic
35#16840145C2211yescharacteristic
36#16840137C2211yescharacteristic
37#16839867—21no—
38#16839866—21no—
39C4×Dic244C2211yescharacteristic, complemented
40Q16×Dic61C2211yescharacteristic, complemented
41#16839856C2211yescharacteristic, complemented
42#16839799C2211yescharacteristic
43#16839769—21no—
44#16839716C2211yescharacteristic
45C4⋊#11758185C2211yescharacteristic, complemented
46#16839580C2211yescharacteristic, complemented
47#16839580C2211yescharacteristic, complemented
48#16839567C2211yescharacteristic, complemented
49#16839565C2211yescharacteristic, complemented
50#16839564C2211yescharacteristic
51#16839563C2211yescharacteristic
52#16839563C2211yescharacteristic, complemented
53#16839562C2211yescharacteristic
54C2×SD32×C61C2211yescharacteristic
55C2×SD32×C61C2211yescharacteristic, complemented
56C61×#493C2211yescharacteristic
57C4×Q16×C61C2211yescharacteristic, complemented
58C61×C4⋊C16C2211yescharacteristic, complemented
59C61×#357C2211yescharacteristic
60C61×#357C2211yescharacteristic, complemented
61C2×#11758224—21no—
62C2×Q8×D61—21no—
63C2×D4×D61—21no—
64#14519801—44no—
65#14519796—22no—
66#14519796—22no—
67C2×Dic244C2311yescharacteristic
68C2×Dic244C2311yescharacteristic
69C2×#11758202—21no—
70Q16×D61—22no—
71Q16×D61D414yescomplemented
72C2×#11758190—21no—
73C2×#11758200—21no—
74C2×#11758200—21no—
75C2×D488C2311yescharacteristic
76C2×#11758199—21no—
77C2×#11758199—21no—
78D8×D61—22no—
79D8×D61—22no—
80C2×#11758189D411yescharacteristic
81C2×C8×D61C2311yescharacteristic
82#14519773—21no—
83#14519765—21no—
84#14519764—21no—
85#14519749—41no—
86#14519749—41no—
87#14519745—41no—
88#14519745—41no—
89Q8×Dic61—21no—
90C4×Dic122—21no—
91C4×Dic122—41no—
92#14519742—21no—
93#14519742—41no—
94#14519735—21no—
95#14519735—41no—
96#14519734—41no—
97#14519734D411yescharacteristic
98Dic488—22no—
99#14519723—22no—
100#14519722—22no—
101#14519721—22no—
102#14519721—22no—
103#14519720—22no—
104#14519716—22no—
105C2×#11758185C2311yescharacteristic
106C2×#11758185C2311yescharacteristic
107C16×D61—22no—
108#14519711C2311yescharacteristic
109#14519708—21no—
110#14519702—21no—
111#14519701—21no—
112#14519700—21no—
113#14519688D411yescharacteristic
114#14519687D411yescharacteristic
115C8×Dic61C2311yescharacteristic
116C2×Q16×C61C2311yescharacteristic
117C2×D8×C61—21no—
118C2×D8×C61C2311yescharacteristic
119C2×D8×C61C2311yescharacteristic
120#127×C61—21no—
121C4×Q8×C61—21no—
122SD32×C61—22no—
123SD32×C61—22no—
124C2×C16×C61C2311yescharacteristic
125C2×C16×C61C2311yescharacteristic
126#107×C61C2311yescharacteristic
127#103×C61—21no—
128C4×C8×C61C2311yescharacteristic
129C23×D61—41no—
130C2×Q8×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—41no—
134C2×D4×C61—41no—
135C2×C22⋊D61—41no—
136C2×C22⋊D61—41no—
137C2×C22⋊D61—41no—
138C2×C22⋊D61—41no—
139C22×Dic61—41no—
140Q8×D61—24no—
141Q8×D61—42no—
142#11758224—42no—
143#11758224—42no—
144D4×D61—42no—
145D4×D61—42no—
146C2×D244—21no—
147C2×C4×D61—41no—
148C2×C4×D61C2×D411yescharacteristic
149C2×Dic122—21no—
150C2×Dic122—21no—
151C2×Dic122—41no—
152Q16×C61—21no—
153Q16×C61C2×D412yes—
154D8×C61—21no—
155D8×C61—21no—
156D8×C61—21no—
157D8×C61—21no—
158D8×C61—42no—
159C2×C8×C61C2×D411yescharacteristic
160C2×C8×C61C2411yescharacteristic
161C4⋊C4×C61—21no—
162C4⋊C4×C61—41no—
163C42×C61—41no—
164C42×C61C2×D411yescharacteristic
165C22⋊Dic61—41no—
166C22⋊Dic61—41no—
167#11758202—24no—
168#11758200—42no—
169#11758200—42no—
170#11758199—42no—
171#11758199—42no—
172#11758198—41no—
173#11758198—41no—
174#11758198—41no—
175#11758198D811yescharacteristic
176#11758198D811yescharacteristic
177C4⋊Dic61—21no—
178C4⋊Dic61—41no—
179#11758196—41no—
180#11758196—41no—
181#11758196D811yescharacteristic
182#11758196D811yescharacteristic
183C4×Dic61—41no—
184C4×Dic61—41no—
185C4×Dic61—41no—
186C4×Dic61—41no—
187C4×Dic61C2×D411yescharacteristic
188C2×#92418C2×D411yescharacteristic
189C2×#92418C2×D411yescharacteristic
190Dic244—21no—
191Dic244—21no—
192Dic244—21no—
193Dic244C2×D412yes—
194D488—21no—
195D488—21no—
196#11758190—42no—
197#11758189—22no—
198C8×D61C2×D412yes—
199C8×D61#4112yes—
200C16×C61—21no—
201C16×C61—21no—
202#11758185—21no—
203#11758185—21no—
204C23×C61—41no—
205C23×C61—41no—
206C22×D61—41no—
207C22×D61—82no—
208C22×D61—161no—
209C22×D61—161no—
210C22×D61C2×D811yescharacteristic
211Q8×C61—22no—
212Q8×C61—41no—
213D4×C61—41no—
214D4×C61—41no—
215D4×C61—41no—
216D4×C61—41no—
217D4×C61—161no—
218D4×C61—161no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61C2×D811yescharacteristic
222C2×C4×C61C2×D811yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C22⋊D61—82no—
225C22⋊D61—82no—
226C22⋊D61—161no—
227C22⋊D61—161no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—82no—
232C2×Dic61C2×D811yescharacteristic
233C2×Dic61C2×D811yescharacteristic
234C2×Dic61C2×D811yescharacteristic
235D244—41no—
236D244—41no—
237C4×D61—44no—
238C4×D61#12112yes—
239C4×D61AGL(1,ℤ/8)12yes—
240Dic122—22no—
241Dic122—41no—
242Dic122—44no—
243Dic122—41no—
244Dic122—41no—
245C8×C61—21no—
246C8×C61C22×D411yescharacteristic
247C8×C61C2×#4111yescharacteristic
248#92418—21no—
249#92418#12112yes—
250#33587—611nomaximal, Sylow
251C22×C61—81no—
252C22×C61—81no—
253C22×C61—81no—
254C22×C61—81no—
255C22×C61—81no—
256C22×C61—81no—
257C22×C61C22×D811yescharacteristic
258C2×D61—21no—
259C2×D61—21no—
260C2×D61—81no—
261C2×D61—81no—
262C2×D61—81no—
263C2×D61—164no—
264C2×D61#45912yes—
265C4×C61—21no—
266C4×C61—21no—
267C4×C61—42no—
268C4×C61—81no—
269C4×C61C2×#12111yescharacteristic
270C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
271Dic61—21no—
272Dic61—21no—
273Dic61—42no—
274Dic61—81no—
275Dic61—81no—
276Dic61#45912yes—
277C2×#510—611no—
278C22×SD32—611no—
279#3155—611no—
280#2950—611no—
281#2211—611no—
282#2211—611no—
283#2211—611no—
284#2211—611no—
285#2207—611no—
286#2207—611no—
287#2207—611no—
288#2207D6111yescharacteristic, complemented
289#2151—611no—
290#2140—611no—
291#2138—611no—
292C2×C61—161no—
293C2×C61—161no—
294C2×C61C2×#45911yescharacteristic
295C2×C61#341611yescharacteristic
296C2×C61#341611yescharacteristic
297D61—22no—
298D61—161no—
299C2×AGL(1,ℤ/8)—1221no—
300C2×#135—611no—
301C22×Q16—611no—
302C22×D8—611no—
303#535—1221no—
304#519—1221no—
305#510—1224no—
306C2×Q32—611no—
307C2×Q32—611no—
308C2×SD32—611no—
309C2×SD32—611no—
310C2×SD32—611no—
311C2×SD32—611no—
312C2×SD32—1224no—
313C2×SD32C2×D6111yescharacteristic
314C2×SD32C2×D6111yescharacteristic, complemented
315C2×M32—611no—
316C22×C16—611no—
317#496—1221no—
318#496—1221no—
319#495—611no—
320#495—611no—
321#493—611no—
322#493C2×D6111yescharacteristic
323#441—1221no—
324#441—1221no—
325C4×Q16—611no—
326C4×Q16—611no—
327C4×Q16—611no—
328C4×Q16C2×D6111yescharacteristic, complemented
329#427—611no—
330#419—1221no—
331#405—611no—
332C4⋊C16—611no—
333C4⋊C16—611no—
334C4⋊C16—611no—
335C4⋊C16C2×D6111yescharacteristic, complemented
336#358—611no—
337#358—611no—
338#357—611no—
339#357—611no—
340#357—611no—
341#357—611no—
342#357C2×D6111yescharacteristic
343#357C2×D6111yescharacteristic, complemented
344C61#3358711yescharacteristic, complemented, Sylow
345C2×#41—1221no—
346C22×Q8—1221no—
347C22×D4—1221no—
348AGL(1,ℤ/8)—2444no—
349#135—1222no—
350#135—1222no—
351C2×Q16—614no—
352C2×Q16—611no—
353C2×Q16—611no—
354C2×Q16—1222no—
355C2×Q16—1221no—
356C2×Q16C22×D6111yescharacteristic
357C2×SD16—1221no—
358C2×SD16—1221no—
359C2×SD16—1221no—
360C2×D8—21no—
361C2×D8—611no—
362C2×D8—1222no—
363C2×D8—1222no—
364C2×D8—1221no—
365C2×D8—1221no—
366C2×D8C22×D6111yescharacteristic
367C2×D8C22×D6111yescharacteristic
368C2×M16—611no—
369C22×C8—611no—
370#127—21no—
371#127—1221no—
372#124—1221no—
373#124—1221no—
374#121—2441no—
375#121—2441no—
376#120—2441no—
377#120—2441no—
378C4×Q8—21no—
379C4×Q8—1221no—
380C4×Q8—1221no—
381C4×Q8—1221no—
382C4×Q8—2441no—
383C4×Q8—2441no—
384#117—611no—
385#117—1221no—
386#117—2441no—
387#117—2441no—
388Q32—1222no—
389Q32—1222no—
390SD32—22no—
391SD32—22no—
392SD32—1222no—
393SD32—1222no—
394SD32—1222no—
395SD32—1222no—
396M32—1222no—
397C2×C16—611no—
398C2×C16—611no—
399C2×C16—1222no—
400C2×C16C22×D6111yescharacteristic
401C2×C16C22×D6111yescharacteristic
402#107—611no—
403#107C22×D6111yescharacteristic
404#106—1221no—
405#103—21no—
406#103—1221no—
407#103—1221no—
408#103—1221no—
409#97—611no—
410#97—611no—
411C4×C8—611no—
412C4×C8C22×D6111yescharacteristic
413C24—2441no—
414#41—2442no—
415#41—2442no—
416C2×Q8—21no—
417C2×Q8—1224no—
418C2×Q8—1221no—
419C2×Q8—1221no—
420C2×Q8—2441no—
421C2×Q8—2442no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—41no—
425C2×D4—41no—
426C2×D4—1221no—
427C2×D4—2442no—
428C2×D4—2442no—
429C2×D4—2441no—
430C2×D4—2441no—
431C2×D4—2441no—
432C2×D4—2441no—
433C22×C4—611no—
434C22×C4—2441no—
435C22×C4—2441no—
436Q16—21no—
437Q16—612no—
438Q16—1224no—
439Q16—1221no—
440Q16—1221no—
441Q16—1221no—
442Q16D4×D6112yes—
443SD16—2442no—
444SD16—2442no—
445SD16—2442no—
446D8—21no—
447D8—21no—
448D8—21no—
449D8—21no—
450D8—42no—
451D8—1221no—
452D8—1221no—
453D8—2442no—
454D8—2442no—
455M16—1222no—
456C2×C8—611no—
457C2×C8—612no—
458C2×C8—612no—
459C2×C8—611no—
460C2×C8D4×D6111yescharacteristic
461C2×C8C23×D6111yescharacteristic
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—611no—
465C4⋊C4—611no—
466C4⋊C4—1221no—
467C4⋊C4—2441no—
468C4⋊C4—2441no—
469C4⋊C4—2441no—
470C22⋊C4—611no—
471C22⋊C4—611no—
472C22⋊C4—2441no—
473C22⋊C4—2441no—
474C22⋊C4—2441no—
475C22⋊C4—2441no—
476C22⋊C4—2441no—
477C42—41no—
478C42—611no—
479C42—2441no—
480C42—2441no—
481C42—2441no—
482C42—2441no—
483C42D4×D6111yescharacteristic
484C16—21no—
485C16—21no—
486C16—1221no—
487C16—1221no—
488C23—41no—
489C23—41no—
490C23—611no—
491C23—9761no—
492C23—9761no—
493C23—4882no—
494C23—2441no—
495Q8—22no—
496Q8—41no—
497Q8—1222no—
498Q8—2444no—
499Q8—2441no—
500Q8—2441no—
501Q8—2441no—
502D4—41no—
503D4—41no—
504D4—41no—
505D4—41no—
506D4—161no—
507D4—161no—
508D4—9761no—
509D4—9761no—
510D4—4882no—
511D4—4882no—
512D4—2441no—
513D4—2441no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—611no—
517C2×C4—612no—
518C2×C4—612no—
519C2×C4—611no—
520C2×C4—611no—
521C2×C4—4882no—
522C2×C4—2444no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4D8×D6111yescharacteristic
527C2×C4D8×D6111yescharacteristic
528C2×C4C2×D4×D6111yescharacteristic
529C8—21no—
530C8—612no—
531C8—1221no—
532C8C2×D4×D6111yescharacteristic
533C8#41×D6111yescharacteristic
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—612no—
541C22—1221no—
542C22—1221no—
543C22—9764no—
544C22—4881no—
545C22—4881no—
546C22—4881no—
547C22C2×D8×D6111yescharacteristic
548C4—21no—
549C4—21no—
550C4—42no—
551C4—81no—
552C4—612no—
553C4—1221no—
554C4—1221no—
555C4—4881no—
556C4—4881no—
557C4—2442no—
558C4#121×D6111yescharacteristic
559C4AGL(1,ℤ/8)×D6111yescharacteristic
560C2—161no—
561C2—161no—
562C2—1222no—
563C2—9761no—
564C2#459×D6111yescharacteristic
565C2#2145896111yescharacteristic
566C2#2145896511yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2207C2
socleC22×C61C22×D8
prime radical#21456299C2
hypercenter#2207D61
hypocenterC61#33587
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2207

Complements

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

NH ≅ G/Ntype
1. C2×#16840201C2semidirect
2. C2×SD32×D61C2semidirect
4. #21454682C2semidirect
5. #21447882C2semidirect
6. #21447881C2semidirect
7. #21447880C2semidirect
8. #21447879C2semidirect
9. #21447858C2semidirect
10. #21447857C2semidirect
11. #21447856C2semidirect
12. #21447576C2semidirect
13. #21447533C2semidirect
14. #21447523C2semidirect
15. C61×#2207C2semidirect
23. C2×Dic488C22semidirect
24. C2×#14519723C22semidirect
27. C2×#14519721C22semidirect
39. C4×Dic244C22semidirect
40. Q16×Dic61C22semidirect
41. #16839856C22semidirect
45. C4⋊#11758185C22semidirect
46. #16839580C22semidirect
47. #16839580C22semidirect
48. #16839567C22semidirect
49. #16839565C22semidirect
52. #16839563C22semidirect
55. C2×SD32×C61C22semidirect
57. C4×Q16×C61C22semidirect
58. C61×C4⋊C16C22semidirect
60. C61×#357C22semidirect
71. Q16×D61D4semidirect
288. #2207D61semidirect
314. C2×SD32C2×D61semidirect
328. C4×Q16C2×D61semidirect
335. C4⋊C16C2×D61semidirect
343. #357C2×D61semidirect
344. C61#33587semidirect

Maximal subgroups

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

subgroupindex
C2×#168402012
C2×SD32×D612
#214562992
#214546822
#214478822
#214478812
#214478802
#214478792
#214478582
#214478572
#214478562
#214475762
#214475332
#214475232
C61×#22072
#3358761