Finite Group Databasebeta

← #27151560

Subgroups — #27151560

Order 12416.

398 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261362C211yesmaximal, characteristic, complemented
2#20261207C211yesmaximal, characteristic
3#20261038C211yesmaximal, characteristic, complemented
4#20261036C211yesmaximal, characteristic, complemented
5#20261035C211yesmaximal, characteristic, complemented
6#20261033C211yesmaximal, characteristic, complemented
7#20261023C211yesmaximal, characteristic, complemented
8#20261022C211yesmaximal, characteristic, complemented
9#20261014C211yesmaximal, characteristic, complemented
10#20260776C211yesmaximal, characteristic, complemented
11#20260772C211yesmaximal, characteristic, complemented
12#20260760C211yesmaximal, characteristic, complemented
13#20260738C211yesmaximal, characteristic
14#102×D97C211yesmaximal, characteristic
15#474×C97C211yesmaximal, characteristic, complemented
16C2×D4×D97C2211yescharacteristic
17C2×#13095494C2211yescharacteristic
18C2×C8×D97C2211yescharacteristic
19#15690809C2211yescharacteristic, complemented
20#15690808—21no—
21#15690802C2211yescharacteristic, complemented
22#15690801C2211yescharacteristic, complemented
23#15690800—21no—
24#15690788—21no—
25#15690786—21no—
26C4×D388C2211yescharacteristic
27D4×Dic97C2211yescharacteristic
28#15690772C2211yescharacteristic
29C4×C22⋊D97—21no—
30#15690770—21no—
31#15690768C2211yescharacteristic
32#15690768C2211yescharacteristic
33#15690767C2211yescharacteristic
34C4⋊C4×D97C2211yescharacteristic
35C2×#13095488—21no—
36C22⋊C4×D97—21no—
37#15690744C2211yescharacteristic, complemented
38#15690743C2211yescharacteristic, complemented
39#15690742C2211yescharacteristic, complemented
40#15690741C2211yescharacteristic, complemented
41#15690741C2211yescharacteristic, complemented
42#15690741C2211yescharacteristic, complemented
43C4⋊#11715895C2211yescharacteristic, complemented
44#15690739C2211yescharacteristic, complemented
45#15690739C2211yescharacteristic, complemented
46#15690732C2211yescharacteristic
47#15690732C2211yescharacteristic
48#15690731C2211yescharacteristic
49#15690731C2211yescharacteristic
50#15690730C2211yescharacteristic
51#15690730C2211yescharacteristic
52C4⋊Q8×C97C2211yescharacteristic, complemented
53C4×D4×C97C2211yescharacteristic
54#107×C97C2211yescharacteristic, complemented
55#107×C97C2211yescharacteristic, complemented
56C4⋊C8×C97C2211yescharacteristic, complemented
57#102×C97C2211yescharacteristic
58#102×C97C2211yescharacteristic
59C23×D97—21no—
60C2×Dic194—21no—
61C2×D388C2311yescharacteristic
62C2×C22⋊D97—21no—
63D4×D97—22no—
64D4×D97Q814yescomplemented
65C22×Dic97—21no—
66C2×C4×D97—21no—
67C2×C4×D97—21no—
68C2×C4×D97—21no—
69C2×C4×D97C2311yescharacteristic
70#13095494—22no—
71C2×#11715895C2311yescharacteristic
72C2×#11715895C2311yescharacteristic
73C8×D97—22no—
74C4⋊Dic97—21no—
75C4⋊Dic97C2311yescharacteristic
76C4⋊Dic97C2311yescharacteristic
77C4⋊Dic97C2311yescharacteristic
78#13095490—21no—
79#13095490—21no—
80#13095490—21no—
81#13095490—21no—
82#13095490—21no—
83#13095490D411yescharacteristic
84#13095490D411yescharacteristic
85C22⋊Dic97—21no—
86#13095488—21no—
87#13095488—21no—
88#13095488—21no—
89#13095488—21no—
90#13095488D411yescharacteristic
91#13095488D411yescharacteristic
92C4×Dic97—21no—
93C4×Dic97—21no—
94C4×Dic97C2311yescharacteristic
95C2×Q8×C97—21no—
96C2×D4×C97C2311yescharacteristic
97C22×C4×C97—21no—
98C2×C8×C97C2311yescharacteristic
99C2×C8×C97C2311yescharacteristic
100C4⋊C4×C97—21no—
101C4⋊C4×C97C2311yescharacteristic
102C4⋊C4×C97C2311yescharacteristic
103C4⋊C4×C97C2311yescharacteristic
104C22⋊C4×C97—21no—
105C42×C97C2311yescharacteristic
106C23×C97—21no—
107C22×D97—21no—
108C22×D97—24no—
109C22×D97—81no—
110C22×D97—81no—
111C22×D97C2×D411yescharacteristic
112Q8×C97—81no—
113D4×C97—21no—
114D4×C97C2×Q812yes—
115C2×C4×C97—21no—
116C2×C4×C97—21no—
117C2×C4×C97—21no—
118C2×C4×C97—81no—
119C2×C4×C97C2×D411yescharacteristic
120C2×C4×C97C2×D411yescharacteristic
121C2×C4×C97C2411yescharacteristic
122C22⋊D97—24no—
123C2×Dic97—21no—
124C2×Dic97—21no—
125C2×Dic97—21no—
126C2×Dic97—81no—
127C2×Dic97C2×D411yescharacteristic
128C2×Dic97C2×D411yescharacteristic
129C2×Dic97C2×D411yescharacteristic
130D388—21no—
131D388C2×Q812yes—
132C4×D97—81no—
133C4×D97—81no—
134C4×D97—81no—
135C4×D97C2×Q812yes—
136C4×D97#4112yes—
137Dic194—81no—
138C8×C97—21no—
139C8×C97—21no—
140#11715895—21no—
141#11715895—21no—
142C22×C97—22no—
143C22×C97—41no—
144C22×C97—41no—
145C22×C97C22×D411yescharacteristic
146C2×D97—21no—
147C2×D97—21no—
148C2×D97—22no—
149C2×D97—41no—
150C2×D97—41no—
151C2×D97—84no—
152C2×D97C22⋊Q812yes—
153C4×C97—21no—
154C4×C97—21no—
155C4×C97—41no—
156C4×C97—41no—
157C4×C97—41no—
158C4×C97C22×Q811yescharacteristic
159C4×C97C2×#4111yescharacteristic
160Dic97—21no—
161Dic97—21no—
162Dic97—41no—
163Dic97—41no—
164Dic97—41no—
165Dic97C22⋊Q812yes—
166C2×C97—42no—
167C2×C97C2×C22⋊Q811yescharacteristic
168C2×C97#57611yescharacteristic
169C2×C97#57711yescharacteristic
170D97—22no—
171D97—42no—
172#3081—971nomaximal, Sylow
173C97#308111yescharacteristic, complemented, Sylow
174#544—971no—
175#518—971no—
176#478—971no—
177#478—971no—
178#478—971no—
179#478—971no—
180#476—971no—
181#476—971no—
182#474—971no—
183#474D9711yescharacteristic, complemented
184#428—971no—
185#427—971no—
186#424—971no—
187#418—971no—
188C2×#102—971no—
189C22×D4—971no—
190C2×M16—971no—
191C22×C8—971no—
192C4⋊Q8—971no—
193C4⋊Q8—1941no—
194C4⋊Q8C2×D9711yescharacteristic, complemented
195#125—971no—
196#125—971no—
197#125—1941no—
198C22⋊Q8—1941no—
199C22⋊Q8—1941no—
200C4×D4—971no—
201C4×D4—971no—
202C4×D4—971no—
203C4×D4—1941no—
204C4×D4—1941no—
205C4×D4C2×D9711yescharacteristic
206#117—971no—
207#117—971no—
208#117—971no—
209C2×C4⋊C4—971no—
210C2×C22⋊C4—1941no—
211C2×C22⋊C4—1941no—
212#107—971no—
213#107—971no—
214#107C2×D9711yescharacteristic, complemented
215#107C2×D9711yescharacteristic, complemented
216#106—971no—
217#106—971no—
218#106—971no—
219#106—971no—
220C4⋊C8—971no—
221C4⋊C8—971no—
222C4⋊C8—971no—
223C4⋊C8C2×D9711yescharacteristic, complemented
224#102—971no—
225#102—971no—
226#102—971no—
227#102—971no—
228#102—971no—
229#102—971no—
230#102C2×D9711yescharacteristic
231#102C2×D9711yescharacteristic
232C24—1941no—
233C2×Q8—21no—
234C2×Q8—1941no—
235C2×D4—974no—
236C2×D4—971no—
237C2×D4—1942no—
238C2×D4—1941no—
239C2×D4C22×D9711yescharacteristic
240C22×C4—21no—
241C22×C4—971no—
242C22×C4—1941no—
243C22×C4—1941no—
244C22×C4—1941no—
245C22×C4—1941no—
246M16—1942no—
247C2×C8—971no—
248C2×C8—971no—
249C2×C8—1942no—
250C2×C8C22×D9711yescharacteristic
251C2×C8C22×D9711yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—971no—
254C4⋊C4—971no—
255C4⋊C4—971no—
256C4⋊C4—971no—
257C4⋊C4—971no—
258C4⋊C4—1941no—
259C4⋊C4—1941no—
260C4⋊C4—1941no—
261C4⋊C4—1941no—
262C4⋊C4—1941no—
263C4⋊C4—1941no—
264C4⋊C4C22×D9711yescharacteristic
265C4⋊C4C22×D9711yescharacteristic
266C4⋊C4C22×D9711yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—971no—
269C22⋊C4—971no—
270C22⋊C4—1941no—
271C22⋊C4—1941no—
272C22⋊C4—1941no—
273C22⋊C4—1941no—
274C22⋊C4—1941no—
275C42—971no—
276C42—1941no—
277C42—1941no—
278C42C22×D9711yescharacteristic
279C23—21no—
280C23—7761no—
281C23—7761no—
282C23—971no—
283C23—1944no—
284C23—1941no—
285Q8—81no—
286Q8—7761no—
287D4—21no—
288D4—972no—
289D4—1944no—
290D4—1941no—
291D4Q8×D9712yes—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—81no—
296C2×C4—7761no—
297C2×C4—7761no—
298C2×C4—7761no—
299C2×C4—7761no—
300C2×C4—971no—
301C2×C4—972no—
302C2×C4—972no—
303C2×C4—971no—
304C2×C4—971no—
305C2×C4—1941no—
306C2×C4—1941no—
307C2×C4—1941no—
308C2×C4D4×D9711yescharacteristic
309C2×C4D4×D9711yescharacteristic
310C2×C4C23×D9711yescharacteristic
311C8—21no—
312C8—21no—
313C8—1941no—
314C8—1941no—
315C22—22no—
316C22—41no—
317C22—41no—
318C22—7764no—
319C22—972no—
320C22—3881no—
321C22—3881no—
322C22—1942no—
323C22—1941no—
324C22—1941no—
325C22C2×D4×D9711yescharacteristic
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—972no—
332C4—3881no—
333C4—3881no—
334C4—3881no—
335C4—1941no—
336C4—1941no—
337C4C2×Q8×D9711yescharacteristic
338C4#41×D9711yescharacteristic
339C2—42no—
340C2—3882no—
341C2—1942no—
342C2C22⋊Q8×D9711yescharacteristic
343C2#2026153611yescharacteristic
344C2#2026154411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#474×C97C2
socleC22×C97C22×D4
prime radicalC2×D4×D97C22
hypercenter#474D97
hypocenterC97#3081
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #20261362C2semidirect
3. #20261038C2semidirect
4. #20261036C2semidirect
5. #20261035C2semidirect
6. #20261033C2semidirect
7. #20261023C2semidirect
8. #20261022C2semidirect
9. #20261014C2semidirect
10. #20260776C2semidirect
11. #20260772C2semidirect
12. #20260760C2semidirect
15. #474×C97C2semidirect
19. #15690809C22semidirect
21. #15690802C22semidirect
22. #15690801C22semidirect
37. #15690744C22semidirect
38. #15690743C22semidirect
39. #15690742C22semidirect
40. #15690741C22semidirect
41. #15690741C22semidirect
42. #15690741C22semidirect
43. C4⋊#11715895C22semidirect
44. #15690739C22semidirect
45. #15690739C22semidirect
52. C4⋊Q8×C97C22semidirect
54. #107×C97C22semidirect
55. #107×C97C22semidirect
56. C4⋊C8×C97C22semidirect
64. D4×D97Q8semidirect
173. C97#3081semidirect
183. #474D97semidirect
194. C4⋊Q8C2×D97semidirect
214. #107C2×D97semidirect
215. #107C2×D97semidirect
223. C4⋊C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202613622
#202612072
#202610382
#202610362
#202610352
#202610332
#202610232
#202610222
#202610142
#202607762
#202607722
#202607602
#202607382
#102×D972
#474×C972
#308197