Finite Group Databasebeta

← #27151531

Subgroups — #27151531

Order 12416.

440 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261291C211yesmaximal, characteristic, complemented
2C4×D4×D97C211yesmaximal, characteristic
3#20261036C211yesmaximal, characteristic, complemented
4#20261035C211yesmaximal, characteristic, complemented
5#20261032C211yesmaximal, characteristic, complemented
6#20261024C211yesmaximal, characteristic, complemented
7#20261023C211yesmaximal, characteristic, complemented
8#20261014C211yesmaximal, characteristic, complemented
9#20261013C211yesmaximal, characteristic, complemented
10#20260776C211yesmaximal, characteristic, complemented
11#20260774C211yesmaximal, characteristic, complemented
12#20260757C211yesmaximal, characteristic, complemented
13#20260738C211yesmaximal, characteristic
14#20260738C211yesmaximal, characteristic
15#478×C97C211yesmaximal, characteristic, complemented
16C2×D4×D97C2211yescharacteristic
17C22×C4×D97—21no—
18C2×#13095494C2211yescharacteristic
19C2×#13095494C2211yescharacteristic
20#15690809C2211yescharacteristic, complemented
21#15690809C2211yescharacteristic, complemented
22#15690802C2211yescharacteristic, complemented
23#15690788—21no—
24#15690788—21no—
25#15690775—21no—
26#15690775—21no—
27C4×D388C2211yescharacteristic
28D4×Dic97C2211yescharacteristic
29#15690772C2211yescharacteristic
30C4×C22⋊D97—21no—
31#15690770—21no—
32#15690768C2211yescharacteristic
33#15690768C2211yescharacteristic
34C4⋊C4×D97C2211yescharacteristic
35C22⋊C4×D97—21no—
36C42×D97C2211yescharacteristic
37#15690744C2211yescharacteristic, complemented
38#15690743C2211yescharacteristic, complemented
39#15690743C2211yescharacteristic, complemented
40#15690742C2211yescharacteristic, 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
52#125×C97C2211yescharacteristic, complemented
53C4×D4×C97C2211yescharacteristic
54#107×C97C2211yescharacteristic, complemented
55#106×C97C2211yescharacteristic, complemented
56C4⋊C8×C97C2211yescharacteristic, complemented
57#102×C97C2211yescharacteristic
58#102×C97C2211yescharacteristic
59C23×D97—21no—
60C2×Dic194—21no—
61C2×Dic194—21no—
62C2×D388C2311yescharacteristic
63C2×C22⋊D97—21no—
64D4×D97—22no—
65D4×D97Q814yescomplemented
66C22×Dic97—21no—
67C2×C4×D97—21no—
68C2×C4×D97—21no—
69C2×C4×D97—21no—
70C2×C4×D97—44no—
71C2×C4×D97D411yescharacteristic
72C2×C4×D97D411yescharacteristic
73C2×C4×D97C2311yescharacteristic
74#13095494—22no—
75#13095494—22no—
76C2×#11715895C2311yescharacteristic
77C2×#11715895C2311yescharacteristic
78C4⋊Dic97C2311yescharacteristic
79C4⋊Dic97C2311yescharacteristic
80C4⋊Dic97C2311yescharacteristic
81#13095490—21no—
82#13095490—21no—
83#13095490—21no—
84#13095490—21no—
85#13095490—21no—
86C22⋊Dic97—21no—
87#13095488—21no—
88#13095488—21no—
89#13095488—21no—
90#13095488—21no—
91C4×Dic97—21no—
92C4×Dic97—21no—
93C4×Dic97D411yescharacteristic
94C4×Dic97D411yescharacteristic
95C4×Dic97C2311yescharacteristic
96C2×D4×C97C2311yescharacteristic
97C22×C4×C97—21no—
98C2×C8×C97C2311yescharacteristic
99C2×C8×C97C2311yescharacteristic
100C4⋊C4×C97—21no—
101C4⋊C4×C97—21no—
102C4⋊C4×C97C2311yescharacteristic
103C4⋊C4×C97C2311yescharacteristic
104C4⋊C4×C97C2311yescharacteristic
105C22⋊C4×C97—21no—
106C42×C97C2311yescharacteristic
107C23×C97—21no—
108C22×D97—21no—
109C22×D97—24no—
110C22×D97—42no—
111C22×D97—42no—
112C22×D97C2×D411yescharacteristic
113D4×C97—21no—
114D4×C97C2×Q812yes—
115C2×C4×C97—21no—
116C2×C4×C97—21no—
117C2×C4×C97—21no—
118C2×C4×C97—42no—
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—42no—
127C2×Dic97C2×D411yescharacteristic
128C2×Dic97C2×D411yescharacteristic
129C2×Dic97C2×D411yescharacteristic
130D388—21no—
131D388C2×Q812yes—
132C4×D97—22no—
133C4×D97—41no—
134C4×D97—42no—
135C4×D97—42no—
136C4×D97—42no—
137C4×D97C2×Q812yes—
138C4×D97#4112yes—
139Dic194—42no—
140Dic194—42no—
141C8×C97—21no—
142C8×C97—21no—
143#11715895—21no—
144#11715895—21no—
145C22×C97—22no—
146C22×C97—41no—
147C22×C97—41no—
148C22×C97C22×D411yescharacteristic
149C2×D97—21no—
150C2×D97—21no—
151C2×D97—22no—
152C2×D97—48no—
153C2×D97—41no—
154C2×D97—41no—
155C2×D97C22⋊Q812yes—
156C4×C97—21no—
157C4×C97—21no—
158C4×C97—41no—
159C4×C97—41no—
160C4×C97—41no—
161C4×C97C22×Q811yescharacteristic
162C4×C97C2×#4111yescharacteristic
163Dic97—21no—
164Dic97—41no—
165Dic97—41no—
166Dic97—41no—
167Dic97C22⋊Q812yes—
168Dic97AGL(1,ℤ/8)12yes—
169C2×C97—42no—
170C2×C97C2×C22⋊Q811yescharacteristic
171C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
172C2×C97#57511yescharacteristic
173D97—22no—
174D97—42no—
175#3077—971nomaximal, Sylow
176C97#307711yescharacteristic, complemented, Sylow
177#533—971no—
178C2×C4×D4—971no—
179#478—971no—
180#478—971no—
181#478—971no—
182#478D9711yescharacteristic, complemented
183#476—971no—
184#476—971no—
185#474—971no—
186#474—971no—
187#428—971no—
188#428—971no—
189#423—971no—
190#418—971no—
191#418—971no—
192C22×D4—971no—
193C23×C4—1941no—
194C2×M16—971no—
195C2×M16—971no—
196C4⋊Q8—971no—
197C4⋊Q8—971no—
198#125—971no—
199#125C2×D9711yescharacteristic, complemented
200C22⋊Q8—1941no—
201C22⋊Q8—1941no—
202C4×Q8—1941no—
203C4×Q8—1941no—
204C4×D4—971no—
205C4×D4—971no—
206C4×D4—971no—
207C4×D4—1941no—
208C4×D4—1941no—
209C4×D4C2×D9711yescharacteristic
210#117—971no—
211#117—971no—
212C2×C4⋊C4—971no—
213C2×C22⋊C4—1941no—
214C2×C42—971no—
215#107—971no—
216#107—971no—
217#107—971no—
218#107C2×D9711yescharacteristic, complemented
219#106—971no—
220#106—971no—
221#106—971no—
222#106C2×D9711yescharacteristic, complemented
223C4⋊C8—971no—
224C4⋊C8—971no—
225C4⋊C8—971no—
226C4⋊C8C2×D9711yescharacteristic, complemented
227#102—971no—
228#102—971no—
229#102—971no—
230#102—971no—
231#102—971no—
232#102—971no—
233#102C2×D9711yescharacteristic
234#102C2×D9711yescharacteristic
235C24—1941no—
236C2×Q8—1941no—
237C2×Q8—1941no—
238C2×D4—974no—
239C2×D4—971no—
240C2×D4—1942no—
241C2×D4—1941no—
242C2×D4C22×D9711yescharacteristic
243C22×C4—21no—
244C22×C4—971no—
245C22×C4—971no—
246C22×C4—971no—
247C22×C4—3884no—
248C22×C4—1941no—
249C22×C4—1941no—
250C22×C4—1941no—
251C22×C4—1941no—
252M16—1942no—
253M16—1942no—
254C2×C8—971no—
255C2×C8—971no—
256C2×C8C22×D9711yescharacteristic
257C2×C8C22×D9711yescharacteristic
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—971no—
261C4⋊C4—971no—
262C4⋊C4—971no—
263C4⋊C4—1941no—
264C4⋊C4—1941no—
265C4⋊C4—1941no—
266C4⋊C4—1941no—
267C4⋊C4—1941no—
268C4⋊C4C22×D9711yescharacteristic
269C4⋊C4C22×D9711yescharacteristic
270C4⋊C4C22×D9711yescharacteristic
271C22⋊C4—21no—
272C22⋊C4—1941no—
273C22⋊C4—1941no—
274C22⋊C4—1941no—
275C22⋊C4—1941no—
276C22⋊C4—1941no—
277C42—971no—
278C42—971no—
279C42—971no—
280C42—1941no—
281C42—1941no—
282C42C22×D9711yescharacteristic
283C23—21no—
284C23—971no—
285C23—3882no—
286C23—3882no—
287C23—1944no—
288C23—1941no—
289Q8—3882no—
290Q8—3882no—
291D4—21no—
292D4—972no—
293D4—1944no—
294D4—1941no—
295D4Q8×D9712yes—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—971no—
301C2×C4—971no—
302C2×C4—972no—
303C2×C4—972no—
304C2×C4—971no—
305C2×C4—3882no—
306C2×C4—3881no—
307C2×C4—3882no—
308C2×C4—3882no—
309C2×C4—3882no—
310C2×C4—1942no—
311C2×C4—1941no—
312C2×C4—1941no—
313C2×C4—1941no—
314C2×C4D4×D9711yescharacteristic
315C2×C4D4×D9711yescharacteristic
316C2×C4C23×D9711yescharacteristic
317C8—21no—
318C8—21no—
319C8—1941no—
320C8—1941no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—972no—
325C22—3888no—
326C22—3881no—
327C22—3881no—
328C22—1942no—
329C22—1941no—
330C22—1941no—
331C22C2×D4×D9711yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—972no—
338C4—972no—
339C4—3881no—
340C4—3881no—
341C4—3881no—
342C4—1941no—
343C4C2×Q8×D9711yescharacteristic
344C4#41×D9711yescharacteristic
345C2—42no—
346C2—3882no—
347C2—1942no—
348C2C22⋊Q8×D9711yescharacteristic
349C2AGL(1,ℤ/8)×D9711yescharacteristic
350C2#2026153211yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#478×C97C2
socleC22×C97C22×D4
prime radicalC2×D4×D97C22
hypercenter#478D97
hypocenterC97#3077
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #20261291C2semidirect
3. #20261036C2semidirect
4. #20261035C2semidirect
5. #20261032C2semidirect
6. #20261024C2semidirect
7. #20261023C2semidirect
8. #20261014C2semidirect
9. #20261013C2semidirect
10. #20260776C2semidirect
11. #20260774C2semidirect
12. #20260757C2semidirect
15. #478×C97C2semidirect
20. #15690809C22semidirect
21. #15690809C22semidirect
22. #15690802C22semidirect
37. #15690744C22semidirect
38. #15690743C22semidirect
39. #15690743C22semidirect
40. #15690742C22semidirect
41. #15690741C22semidirect
42. #15690741C22semidirect
43. C4⋊#11715895C22semidirect
44. #15690739C22semidirect
45. #15690739C22semidirect
52. #125×C97C22semidirect
54. #107×C97C22semidirect
55. #106×C97C22semidirect
56. C4⋊C8×C97C22semidirect
65. D4×D97Q8semidirect
176. C97#3077semidirect
182. #478D97semidirect
199. #125C2×D97semidirect
218. #107C2×D97semidirect
222. #106C2×D97semidirect
226. C4⋊C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202612912
C4×D4×D972
#202610362
#202610352
#202610322
#202610242
#202610232
#202610142
#202610132
#202607762
#202607742
#202607572
#202607382
#478×C972
#307797