Finite Group Databasebeta

← #27153277

Subgroups — #27153277

Order 12416.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690835C211yesmaximal, characteristic, complemented
2C2×#15690830C211yesmaximal, characteristic, complemented
3#20261381C211yesmaximal, characteristic, complemented
4#20261122C211yesmaximal, characteristic, complemented
5#20260979C211yesmaximal, characteristic, complemented
6#20260969C211yesmaximal, characteristic, complemented
7#20260957C211yesmaximal, characteristic, complemented
8#20260927C211yesmaximal, characteristic, complemented
9#20260897C211yesmaximal, characteristic, complemented
10#20260880C211yesmaximal, characteristic, complemented
11#20260878C211yesmaximal, characteristic, complemented
12#20260868C211yesmaximal, characteristic, complemented
13#20260833C211yesmaximal, characteristic, complemented
14#20260807C211yesmaximal, characteristic, complemented
15#550×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C22×D388C2211yescharacteristic
18#15690835—24no—
19#15690830—24no—
20C2×#13095501C2211yescharacteristic, complemented
21C2×#13095499C2211yescharacteristic, complemented
22C2×#13095499C2211yescharacteristic, complemented
23C2×#13095498C2211yescharacteristic
24C2×#13095498C2211yescharacteristic
25C2×#13095496C2211yescharacteristic
26C2×#13095496C2211yescharacteristic
27C2×#13095496C2211yescharacteristic
28C2×#13095496C2211yescharacteristic
29C2×#13095495C2211yescharacteristic
30C22×#11715895C2211yescharacteristic, complemented
31#15690798C2211yescharacteristic
32#15690795—21no—
33#15690789C2211yescharacteristic
34#15690784C2211yescharacteristic
35#15690783—21no—
36#15690778—21no—
37C4×D388C2211yescharacteristic
38C22⋊C4×D97—21no—
39#15690743C2211yescharacteristic, complemented
40C4⋊#11715895C2211yescharacteristic, complemented
41#15690734C2211yescharacteristic, complemented
42#15690733C2211yescharacteristic, complemented
43#15690731C2211yescharacteristic, complemented
44#15690731C2211yescharacteristic, complemented
45#15690730C2211yescharacteristic
46#15690730C2211yescharacteristic
47C22⋊#11715895C2211yescharacteristic
48C22⋊#11715895C2211yescharacteristic, complemented
49#15690721C2211yescharacteristic, complemented
50C2×#41×C97C2211yescharacteristic
51C2×#41×C97C2211yescharacteristic, complemented
52#127×C97—21no—
53#127×C97C2211yescharacteristic, complemented
54#121×C97—21no—
55#121×C97—21no—
56#121×C97C2211yescharacteristic
57#121×C97C2211yescharacteristic, complemented
58C4×Q8×C97C2211yescharacteristic, complemented
59C4×D4×C97—21no—
60C4×D4×C97C2211yescharacteristic
61C23×D97—21no—
62#13095511—22no—
63#13095511—41no—
64C2×Dic194C2311yescharacteristic
65C2×D388—22no—
66C2×D388—21no—
67C2×D388—41no—
68C2×D388C2311yescharacteristic
69C2×D388C2311yescharacteristic
70C2×C22⋊D97—21no—
71C2×C22⋊D97—21no—
72C2×C4×D97—21no—
73C2×C4×D97—21no—
74#13095501—22no—
75#13095499—22no—
76#13095499—41no—
77#13095498—22no—
78#13095498D414yescomplemented
79#13095496—22no—
80#13095496—22no—
81#13095496—22no—
82#13095496—41no—
83#13095495—22no—
84C2×#11715895—22no—
85C2×#11715895C2311yescharacteristic
86C2×#11715895C2311yescharacteristic
87C2×#11715895C2311yescharacteristic
88C2×#11715895C2311yescharacteristic
89C4⋊Dic97C2311yescharacteristic
90#13095490—21no—
91C22⋊Dic97—21no—
92#13095488—21no—
93#13095488—21no—
94#13095488—21no—
95#13095488—21no—
96#41×C97—22no—
97#41×C97—22no—
98#41×C97—41no—
99#41×C97—41no—
100C2×Q8×C97C2311yescharacteristic
101C2×D4×C97—21no—
102C2×D4×C97—21no—
103C2×D4×C97—21no—
104C2×D4×C97—21no—
105C2×D4×C97—21no—
106C2×D4×C97—21no—
107C2×D4×C97D411yescharacteristic
108C2×D4×C97C2311yescharacteristic
109C2×D4×C97C2311yescharacteristic
110C22×C4×C97—21no—
111C22×C4×C97—21no—
112C22×C4×C97C2311yescharacteristic
113C22×C4×C97C2311yescharacteristic
114C4⋊C4×C97—21no—
115C4⋊C4×C97D411yescharacteristic
116C4⋊C4×C97C2311yescharacteristic
117C22⋊C4×C97—21no—
118C22⋊C4×C97—21no—
119C22⋊C4×C97D411yescharacteristic
120C22⋊C4×C97D411yescharacteristic
121C42×C97—21no—
122C42×C97C2311yescharacteristic
123C23×C97—21no—
124C23×C97—21no—
125C23×C97C2×D411yescharacteristic
126C23×C97C2×D411yescharacteristic
127C22×D97—21no—
128C22×D97—21no—
129C22×D97—42no—
130C22×D97—42no—
131C22×D97—81no—
132Q8×C97—21no—
133Q8×C97C2×D412yes—
134D4×C97—21no—
135D4×C97—21no—
136D4×C97—21no—
137D4×C97—21no—
138D4×C97—41no—
139D4×C97—42no—
140D4×C97—42no—
141D4×C97—42no—
142D4×C97—81no—
143D4×C97—81no—
144D4×C97—81no—
145C2×C4×C97—21no—
146C2×C4×C97—21no—
147C2×C4×C97—21no—
148C2×C4×C97—21no—
149C2×C4×C97—21no—
150C2×C4×C97—21no—
151C2×C4×C97—42no—
152C2×C4×C97—42no—
153C2×C4×C97C2×D411yescharacteristic
154C2×C4×C97C2×D411yescharacteristic
155C2×C4×C97C2×D411yescharacteristic
156C2×C4×C97C2×D411yescharacteristic
157C2×C4×C97C2411yescharacteristic
158C22⋊D97—42no—
159C22⋊D97—81no—
160C2×Dic97—21no—
161C2×Dic97—21no—
162D388—21no—
163D388—21no—
164D388—21no—
165D388—42no—
166D388C2×D412yes—
167C4×D97—42no—
168C4×D97—81no—
169Dic194—21no—
170Dic194—21no—
171#11715895—21no—
172#11715895—21no—
173#11715895—21no—
174#11715895C2×D412yes—
175C22×C97—21no—
176C22×C97—21no—
177C22×C97—21no—
178C22×C97—21no—
179C22×C97—21no—
180C22×C97—21no—
181C22×C97—41no—
182C22×C97—41no—
183C22×C97—41no—
184C22×C97—41no—
185C22×C97—41no—
186C22×C97—41no—
187C22×C97C22×D411yescharacteristic
188C2×D97—22no—
189C2×D97—41no—
190C2×D97—41no—
191C2×D97—41no—
192C2×D97—41no—
193C2×D97—41no—
194C2×D97—84no—
195C4×C97—21no—
196C4×C97—21no—
197C4×C97—22no—
198C4×C97—21no—
199C4×C97—21no—
200C4×C97—41no—
201C4×C97C22×D411yescharacteristic
202C4×C97H(5,𝔽2)11yescharacteristic
203Dic97—41no—
204Dic97—41no—
205C2×C97—41no—
206C2×C97—41no—
207C2×C97—81no—
208C2×C97—81no—
209C2×C97D4211yescharacteristic
210C2×C97#57511yescharacteristic
211C2×C97#57611yescharacteristic
212D97—42no—
213D97—81no—
214#3268—971nomaximal, Sylow
215C97#326811yescharacteristic, complemented, Sylow
216C2×AGL(1,ℤ/8)—971no—
217C2×#135—971no—
218#550D9711yescharacteristic, complemented
219#546—971no—
220#496—971no—
221#468—971no—
222#466—971no—
223#464—971no—
224#459—971no—
225#452—971no—
226#449—971no—
227#449—971no—
228#447—971no—
229#440—971no—
230#435—971no—
231C2×#41—971no—
232C2×#41C2×D9711yescharacteristic
233C2×#41C2×D9711yescharacteristic, complemented
234C22×D4—971no—
235AGL(1,ℤ/8)—1944no—
236#135—1944no—
237C2×Q16—971no—
238C2×SD16—971no—
239C2×SD16—971no—
240C2×SD16—971no—
241C2×SD16—971no—
242C2×D8—971no—
243C2×D8—971no—
244C2×D8—971no—
245C2×D8—971no—
246C2×M16—971no—
247C22×C8—971no—
248#127—21no—
249#127C2×D9711yescharacteristic, complemented
250#124—971no—
251#123—1941no—
252C22⋊Q8—971no—
253#121—21no—
254#121—21no—
255#121—971no—
256#121—1941no—
257#121C2×D9711yescharacteristic
258#121C2×D9711yescharacteristic, complemented
259#120—1941no—
260C4×Q8C2×D9711yescharacteristic, complemented
261C4×D4—21no—
262C4×D4—971no—
263C4×D4C2×D9711yescharacteristic
264C2×C22⋊C4—1941no—
265#107—971no—
266C4⋊C8—971no—
267#103—971no—
268#103—971no—
269#102—971no—
270#102—971no—
271#102—971no—
272#102—971no—
273C22⋊C8—971no—
274C22⋊C8—971no—
275#97—971no—
276C24—1941no—
277#41—22no—
278#41—22no—
279#41—41no—
280#41—41no—
281#41—3881no—
282#41—1942no—
283C2×Q8—971no—
284C2×Q8C22×D9711yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—971no—
292C2×D4—971no—
293C2×D4—3881no—
294C2×D4—1942no—
295C2×D4—1941no—
296C2×D4—1941no—
297C2×D4—1941no—
298C2×D4C22⋊D9711yescharacteristic
299C2×D4C22×D9711yescharacteristic
300C2×D4C22×D9711yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—1941no—
304C22×C4—1941no—
305C22×C4C22×D9711yescharacteristic
306C22×C4C22×D9711yescharacteristic
307Q16—1942no—
308SD16—974no—
309SD16—3881no—
310SD16—1942no—
311SD16—1942no—
312D8—3881no—
313D8—1942no—
314D8—1942no—
315D8—1942no—
316M16—1942no—
317C2×C8—971no—
318C2×C8—971no—
319C2×C8—971no—
320C2×C8—971no—
321C2×C8—1942no—
322C4⋊C4—21no—
323C4⋊C4—971no—
324C4⋊C4—1941no—
325C4⋊C4C22⋊D9711yescharacteristic
326C4⋊C4C22×D9711yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—1941no—
330C22⋊C4—1941no—
331C22⋊C4—1941no—
332C22⋊C4—1941no—
333C22⋊C4—1941no—
334C22⋊C4C22⋊D9711yescharacteristic
335C22⋊C4C22⋊D9711yescharacteristic
336C42—21no—
337C42C22×D9711yescharacteristic
338C23—21no—
339C23—21no—
340C23—7761no—
341C23—3882no—
342C23—3882no—
343C23—1941no—
344C23—1941no—
345C23C2×C22⋊D9711yescharacteristic
346C23C2×C22⋊D9711yescharacteristic
347Q8—21no—
348Q8—1941no—
349Q8—1941no—
350Q8D4×D9712yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—41no—
356D4—42no—
357D4—42no—
358D4—42no—
359D4—81no—
360D4—81no—
361D4—81no—
362D4—7761no—
363D4—972no—
364D4—3882no—
365D4—3882no—
366D4—1941no—
367D4—1941no—
368D4—1941no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—7761no—
378C2×C4—3882no—
379C2×C4—1941no—
380C2×C4—1941no—
381C2×C4C2×C22⋊D9711yescharacteristic
382C2×C4C2×C22⋊D9711yescharacteristic
383C2×C4C2×C22⋊D9711yescharacteristic
384C2×C4C2×C22⋊D9711yescharacteristic
385C2×C4C23×D9711yescharacteristic
386C8—972no—
387C8—1941no—
388C8—1941no—
389C8—1941no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—7764no—
403C22—3881no—
404C22—3881no—
405C22—3881no—
406C22—3881no—
407C22—3881no—
408C22—1942no—
409C22C22×C22⋊D9711yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—22no—
415C4—41no—
416C4—3881no—
417C4—3881no—
418C4C2×D4×D9711yescharacteristic
419C4#1569085511yescharacteristic
420C2—41no—
421C2—41no—
422C2—81no—
423C2—81no—
424C2—7761no—
425C2—3882no—
426C2D4×C22⋊D9711yescharacteristic
427C2#2026153311yescharacteristic
428C2#2026153811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#550×C97C2
socleC22×C97C22×D4
hypercenter#550D97
hypocenterC97#3268
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#15690835C2semidirect
2. C2×#15690830C2semidirect
3. #20261381C2semidirect
4. #20261122C2semidirect
5. #20260979C2semidirect
6. #20260969C2semidirect
7. #20260957C2semidirect
8. #20260927C2semidirect
9. #20260897C2semidirect
10. #20260880C2semidirect
11. #20260878C2semidirect
12. #20260868C2semidirect
13. #20260833C2semidirect
14. #20260807C2semidirect
15. #550×C97C2semidirect
20. C2×#13095501C22semidirect
21. C2×#13095499C22semidirect
22. C2×#13095499C22semidirect
30. C22×#11715895C22semidirect
39. #15690743C22semidirect
40. C4⋊#11715895C22semidirect
41. #15690734C22semidirect
42. #15690733C22semidirect
43. #15690731C22semidirect
44. #15690731C22semidirect
48. C22⋊#11715895C22semidirect
49. #15690721C22semidirect
51. C2×#41×C97C22semidirect
53. #127×C97C22semidirect
57. #121×C97C22semidirect
58. C4×Q8×C97C22semidirect
78. #13095498D4semidirect
215. C97#3268semidirect
218. #550D97semidirect
233. C2×#41C2×D97semidirect
249. #127C2×D97semidirect
258. #121C2×D97semidirect
260. C4×Q8C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908352
C2×#156908302
#202613812
#202611222
#202609792
#202609692
#202609572
#202609272
#202608972
#202608802
#202608782
#202608682
#202608332
#202608072
#550×C972
#326897