Finite Group Databasebeta

← #29833352

Subgroups — #29833352

Order 15616.

696 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840196C211yesmaximal, characteristic
2#21454689C211yesmaximal, characteristic, complemented
3#21454666C211yesmaximal, characteristic, complemented
4#21447692C211yesmaximal, characteristic, complemented
5#21447685C211yesmaximal, characteristic, complemented
6#21447680C211yesmaximal, characteristic, complemented
7#21447670C211yesmaximal, characteristic, complemented
8C4×#14519721C211yesmaximal, characteristic, complemented
9D16×Dic61C211yesmaximal, characteristic, complemented
10C4×#14519718C211yesmaximal, characteristic, complemented
11#21447606C211yesmaximal, characteristic, complemented
12#21447523C211yesmaximal, characteristic, complemented
13#357×D61C211yesmaximal, characteristic, complemented
14#21447417C211yesmaximal, characteristic, complemented
15C61×#2177C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic
17C2×D8×D61C2211yescharacteristic
18#16840241—21no—
19#16840234—21no—
20#16840196C418yescomplemented
21C2×#14519722C2211yescharacteristic
22C2×#14519721C2211yescharacteristic
23C2×#14519718C2211yescharacteristic
24C2×#14519716C2211yescharacteristic
25C2×C16×D61C2211yescharacteristic
26#16839872—21no—
27#16839871—21no—
28#16839862—21no—
29#16839859—21no—
30C4×Dic244C2211yescharacteristic, complemented
31#16839856C2211yescharacteristic, complemented
32C4×D488C2211yescharacteristic
33D8×Dic61C2211yescharacteristic
34D8×Dic61C2211yescharacteristic, complemented
35#16839844C2211yescharacteristic
36#16839799C2211yescharacteristic
37#107×D61C2211yescharacteristic
38#16839765—21no—
39#16839760—21no—
40#16839716C2211yescharacteristic
41#16839589C2211yescharacteristic, complemented
42#16839588C2211yescharacteristic, complemented
43#16839587C2211yescharacteristic, complemented
44#16839567C2211yescharacteristic, complemented
45#16839565C2211yescharacteristic, complemented
46#16839564C2211yescharacteristic
47#16839563C2211yescharacteristic
48#16839563C2211yescharacteristic, complemented
49#16839562C2211yescharacteristic
50#16839537C2211yescharacteristic, complemented
51C4×#11758185C2211yescharacteristic, complemented
52C16×Dic61C2211yescharacteristic, complemented
53C2×D16×C61C2211yescharacteristic
54C4×D8×C61C2211yescharacteristic
55C4×D8×C61C2211yescharacteristic, complemented
56C61×#367C2211yescharacteristic, complemented
57C61×#357C2211yescharacteristic
58C61×#357C2211yescharacteristic, complemented
59C61×#346C2211yescharacteristic, complemented
60C2×#11758224—21no—
61C2×D4×D61—21no—
62#14519796—22no—
63#14519796C2×C414yescomplemented
64C2×Dic244C2311yescharacteristic
65C2×#11758200—21no—
66C2×D488C2311yescharacteristic
67C2×#11758199—21no—
68D8×D61—22no—
69D8×D61C2×C414yescomplemented
70C2×#11758189D411yescharacteristic
71C2×C8×D61C2311yescharacteristic
72C4×Dic122—21no—
73#14519742—21no—
74C4×D244—21no—
75D4×Dic61—21no—
76D4×Dic61—21no—
77#14519739—21no—
78C4×C22⋊D61—41no—
79C4×C22⋊D61—41no—
80#14519737—41no—
81#14519737—41no—
82#14519735—21no—
83#14519734D411yescharacteristic
84#14519733—41no—
85C2×#11758196—41no—
86C4⋊C4×D61—21no—
87C2×#11758198—41no—
88C22⋊C4×D61—41no—
89#14519722C2×C414yescomplemented
90#14519721C2×C414yescomplemented
91#14519718C2×C414yescomplemented
92#14519716C2×C414yescomplemented
93C2×#11758185C2311yescharacteristic
94C2×#11758185C2311yescharacteristic
95C16×D61—22no—
96#14519711C2311yescharacteristic
97#14519711C2311yescharacteristic
98#14519710—21no—
99#14519708—21no—
100#14519702—21no—
101#14519700—21no—
102#14519699—21no—
103#14519698—21no—
104#14519698—21no—
105#14519697—21no—
106#14519688D411yescharacteristic
107#14519687D411yescharacteristic
108C8×Dic61C2311yescharacteristic
109C2×D8×C61C2311yescharacteristic
110C2×D8×C61C2311yescharacteristic
111C4×D4×C61—21no—
112C4×D4×C61—21no—
113D16×C61C2×C414yescomplemented
114C2×C16×C61C2311yescharacteristic
115C2×C16×C61C2311yescharacteristic
116#107×C61C2311yescharacteristic
117#107×C61C2311yescharacteristic
118#102×C61—21no—
119#102×C61—21no—
120C4×C8×C61C2311yescharacteristic
121C23×D61—41no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C22×C4×C61—41no—
125C22×C4×C61—41no—
126C2×C22⋊D61—41no—
127C2×C22⋊D61—41no—
128C22×Dic61—41no—
129C22×Dic61—41no—
130C22×Dic61—41no—
131#11758224—24no—
132#11758224—42no—
133D4×D61—24no—
134D4×D61—42no—
135C2×D244—21no—
136C2×C4×D61—41no—
137C2×C4×D61—41no—
138C2×C4×D61—41no—
139C2×C4×D61C2×D411yescharacteristic
140C2×Dic122—21no—
141D8×C61—21no—
142D8×C61—21no—
143D8×C61C22×C412yes—
144D8×C61C22×C412yes—
145C2×C8×C61C2×D411yescharacteristic
146C2×C8×C61C2411yescharacteristic
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C22⋊C4×C61—41no—
150C22⋊C4×C61—41no—
151C42×C61C2×D411yescharacteristic
152C22⋊Dic61—41no—
153C22⋊Dic61—41no—
154#11758200—24no—
155#11758199—24no—
156#11758198—41no—
157#11758198—41no—
158#11758198—41no—
159#11758198D811yescharacteristic
160#11758198D811yescharacteristic
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163#11758196—41no—
164#11758196—41no—
165#11758196—41no—
166#11758196D811yescharacteristic
167#11758196D811yescharacteristic
168C4×Dic61—41no—
169C4×Dic61—41no—
170C4×Dic61—41no—
171C4×Dic61C2×D411yescharacteristic
172C2×#92418C2×D411yescharacteristic
173C2×#92418C2×D411yescharacteristic
174Dic244—21no—
175Dic244C22×C412yes—
176D488—21no—
177D488C22×C412yes—
178#11758189—22no—
179C8×D61C22×C412yes—
180C8×D61#4112yes—
181C16×C61—21no—
182C16×C61C22×C412yes—
183#11758185—21no—
184#11758185C22×C412yes—
185C23×C61—41no—
186C23×C61—41no—
187C22×D61—41no—
188C22×D61—44no—
189C22×D61—161no—
190C22×D61—161no—
191C22×D61C2×D811yescharacteristic
192D4×C61—22no—
193D4×C61—22no—
194D4×C61—41no—
195D4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—161no—
199C2×C4×C61—161no—
200C2×C4×C61C2×D811yescharacteristic
201C2×C4×C61C2×D811yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C22⋊D61—44no—
204C22⋊D61—44no—
205C2×Dic61—41no—
206C2×Dic61—44no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—161no—
210C2×Dic61—161no—
211C2×Dic61C2×D811yescharacteristic
212C2×Dic61C2×D811yescharacteristic
213C2×Dic61C2×D811yescharacteristic
214D244—22no—
215D244—41no—
216C4×D61—161no—
217C4×D61—161no—
218C4×D61—161no—
219C4×D61C4×D412yes—
220C4×D61#13512yes—
221Dic122—22no—
222Dic122—41no—
223C8×C61—21no—
224C8×C61C23×C411yescharacteristic
225C8×C61C2×#4111yescharacteristic
226#92418—21no—
227#92418C4×D412yes—
228#33454—611nomaximal, Sylow
229C22×C61—42no—
230C22×C61—42no—
231C22×C61—81no—
232C22×C61—81no—
233C22×C61—81no—
234C22×C61—81no—
235C22×C61C22×D811yescharacteristic
236C2×D61—21no—
237C2×D61—21no—
238C2×D61—42no—
239C2×D61—81no—
240C2×D61—81no—
241C2×D61—164no—
242C2×D61C4×D812yes—
243C4×C61—21no—
244C4×C61—21no—
245C4×C61—81no—
246C4×C61—81no—
247C4×C61C2×C4×D411yescharacteristic
248C4×C61C2×#13511yescharacteristic
249Dic61—21no—
250Dic61—21no—
251Dic61—42no—
252Dic61—81no—
253Dic61—81no—
254Dic61C4×D812yes—
255C2×#509—611no—
256#2951—611no—
257#2948—611no—
258#2177—611no—
259#2177D6111yescharacteristic, complemented
260#2175—611no—
261#2175—611no—
262C4×SD32—611no—
263C4×SD32—611no—
264C4×D16—611no—
265C4×D16—611no—
266#2159—611no—
267#2138—611no—
268C2×#357—611no—
269#2108—611no—
270C2×C61—82no—
271C2×C61—82no—
272C2×C61C2×C4×D811yescharacteristic
273C2×C61#341511yescharacteristic
274C2×C61#341611yescharacteristic
275D61—22no—
276D61—82no—
277C2×#135—611no—
278C22×D8—611no—
279#520—1221no—
280#518—1221no—
281#509—618no—
282C2×SD32—611no—
283C2×SD32—611no—
284C2×D16—611no—
285C2×D16C2×D6111yescharacteristic
286C2×M32—611no—
287C22×C16—611no—
288#442—1221no—
289#442—1221no—
290#440—1221no—
291#440—1221no—
292C4×Q16—611no—
293C4×Q16—611no—
294C4×D8—611no—
295C4×D8—611no—
296C4×D8—611no—
297C4×D8—611no—
298C4×D8C2×D6111yescharacteristic
299C4×D8C2×D6111yescharacteristic, complemented
300#427—611no—
301C2×#107—611no—
302#418—1221no—
303#417—1221no—
304#405—611no—
305#367—611no—
306#367C2×D6111yescharacteristic, complemented
307#366—611no—
308#366—611no—
309#358—611no—
310#358—611no—
311#357—611no—
312#357—611no—
313#357—611no—
314#357—611no—
315#357C2×D6111yescharacteristic
316#357C2×D6111yescharacteristic, complemented
317#346—611no—
318#346C2×D6111yescharacteristic, complemented
319C4×C16—611no—
320C4×C16—611no—
321C61#3345411yescharacteristic, complemented, Sylow
322C2×#41—1221no—
323C22×D4—1221no—
324#135—614no—
325#135—1222no—
326C2×Q16—611no—
327C2×SD16—1221no—
328C2×D8—614no—
329C2×D8—611no—
330C2×D8—1222no—
331C2×D8—1221no—
332C2×D8C22×D6111yescharacteristic
333C2×D8C22×D6111yescharacteristic
334C2×M16—611no—
335C22×C8—611no—
336C4×Q8—1221no—
337C4×Q8—1221no—
338C4×D4—21no—
339C4×D4—21no—
340C4×D4—1221no—
341C4×D4—1221no—
342C4×D4—1221no—
343C4×D4—1221no—
344C4×D4—2441no—
345C4×D4—2441no—
346C4×D4—2441no—
347C4×D4—2441no—
348#117—611no—
349#117—1221no—
350#117—2441no—
351C2×C4⋊C4—1221no—
352C2×C4⋊C4—2441no—
353C2×C22⋊C4—2441no—
354C2×C22⋊C4—2441no—
355SD32—614no—
356SD32—614no—
357D16—614no—
358D16C4×D6114yescomplemented
359M32—614no—
360C2×C16—611no—
361C2×C16—611no—
362C2×C16—1222no—
363C2×C16C22×D6111yescharacteristic
364C2×C16C22×D6111yescharacteristic
365#107—611no—
366#107—611no—
367#107—1221no—
368#107C22×D6111yescharacteristic
369#107C22×D6111yescharacteristic
370#106—1221no—
371#103—1221no—
372#103—1221no—
373#102—21no—
374#102—21no—
375#102—1221no—
376#102—1221no—
377#102—1221no—
378#102—1221no—
379#97—611no—
380#97—611no—
381C4×C8—611no—
382C4×C8C22×D6111yescharacteristic
383C24—2441no—
384#41—1224no—
385#41—2442no—
386C2×Q8—1221no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—1224no—
390C2×D4—1221no—
391C2×D4—2442no—
392C2×D4—2441no—
393C2×D4—2441no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—611no—
397C22×C4—2441no—
398C22×C4—2441no—
399C22×C4—2441no—
400C22×C4—2441no—
401C22×C4—2441no—
402C22×C4—2441no—
403Q16—612no—
404Q16—1221no—
405SD16—1224no—
406D8—21no—
407D8—21no—
408D8—612no—
409D8—1224no—
410D8—1221no—
411D8C2×C4×D6112yes—
412D8C2×C4×D6112yes—
413M16—1222no—
414C2×C8—611no—
415C2×C8—612no—
416C2×C8—612no—
417C2×C8—611no—
418C2×C8D4×D6111yescharacteristic
419C2×C8C23×D6111yescharacteristic
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—611no—
423C4⋊C4—611no—
424C4⋊C4—1221no—
425C4⋊C4—1221no—
426C4⋊C4—2441no—
427C4⋊C4—2441no—
428C4⋊C4—2441no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—611no—
432C22⋊C4—611no—
433C22⋊C4—2441no—
434C22⋊C4—2441no—
435C22⋊C4—2441no—
436C22⋊C4—2441no—
437C22⋊C4—2441no—
438C42—611no—
439C42—2441no—
440C42—2441no—
441C42—2441no—
442C42D4×D6111yescharacteristic
443C16—21no—
444C16—612no—
445C16—1221no—
446C16C2×C4×D6112yes—
447C23—41no—
448C23—41no—
449C23—611no—
450C23—9761no—
451C23—9761no—
452C23—2444no—
453C23—2441no—
454Q8—1222no—
455Q8—2441no—
456D4—22no—
457D4—22no—
458D4—41no—
459D4—41no—
460D4—1222no—
461D4—2444no—
462D4—2444no—
463D4—2441no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—161no—
467C2×C4—161no—
468C2×C4—611no—
469C2×C4—612no—
470C2×C4—612no—
471C2×C4—611no—
472C2×C4—611no—
473C2×C4—9761no—
474C2×C4—9761no—
475C2×C4—9761no—
476C2×C4—9761no—
477C2×C4—9761no—
478C2×C4—2444no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4—2441no—
482C2×C4D8×D6111yescharacteristic
483C2×C4D8×D6111yescharacteristic
484C2×C4C2×D4×D6111yescharacteristic
485C8—21no—
486C8—612no—
487C8—1221no—
488C8C22×C4×D6111yescharacteristic
489C8#41×D6111yescharacteristic
490C22—42no—
491C22—42no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22—612no—
497C22—1221no—
498C22—1221no—
499C22—9764no—
500C22—4881no—
501C22—4881no—
502C22—2442no—
503C22C2×D8×D6111yescharacteristic
504C4—21no—
505C4—21no—
506C4—81no—
507C4—81no—
508C4—612no—
509C4—1221no—
510C4—1221no—
511C4—4881no—
512C4—4881no—
513C4—2442no—
514C4C4×D4×D6111yescharacteristic
515C4#135×D6111yescharacteristic
516C2—82no—
517C2—82no—
518C2—1222no—
519C2—4882no—
520C2C4×D8×D6111yescharacteristic
521C2#2145895911yescharacteristic
522C2#2145896511yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
FittingC61×#2177C2
socleC22×C61C22×D8
prime radicalC2×#16840196C2
cosocleC2×C8×C61C24
hypercenter#2177D61
hypocenterC61#33454
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2177

Complements

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

NH ≅ G/Ntype
2. #21454689C2semidirect
3. #21454666C2semidirect
4. #21447692C2semidirect
5. #21447685C2semidirect
6. #21447680C2semidirect
7. #21447670C2semidirect
8. C4×#14519721C2semidirect
9. D16×Dic61C2semidirect
10. C4×#14519718C2semidirect
11. #21447606C2semidirect
12. #21447523C2semidirect
13. #357×D61C2semidirect
14. #21447417C2semidirect
15. C61×#2177C2semidirect
20. #16840196C4semidirect
30. C4×Dic244C22semidirect
31. #16839856C22semidirect
34. D8×Dic61C22semidirect
41. #16839589C22semidirect
42. #16839588C22semidirect
43. #16839587C22semidirect
44. #16839567C22semidirect
45. #16839565C22semidirect
48. #16839563C22semidirect
50. #16839537C22semidirect
51. C4×#11758185C22semidirect
52. C16×Dic61C22semidirect
55. C4×D8×C61C22semidirect
56. C61×#367C22semidirect
58. C61×#357C22semidirect
59. C61×#346C22semidirect
63. #14519796C2×C4semidirect
69. D8×D61C2×C4semidirect
89. #14519722C2×C4semidirect
90. #14519721C2×C4semidirect
91. #14519718C2×C4semidirect
92. #14519716C2×C4semidirect
113. D16×C61C2×C4semidirect
259. #2177D61semidirect
299. C4×D8C2×D61semidirect
306. #367C2×D61semidirect
316. #357C2×D61semidirect
318. #346C2×D61semidirect
321. C61#33454semidirect
358. D16C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401962
#214546892
#214546662
#214476922
#214476852
#214476802
#214476702
C4×#145197212
D16×Dic612
C4×#145197182
#214476062
#214475232
#357×D612
#214474172
C61×#21772
#3345461