Finite Group Databasebeta

← #27151532

Subgroups — #27151532

Order 12416.

440 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261288C211yesmaximal, characteristic, complemented
2C2×#15690772C211yesmaximal, characteristic
3#20261038C212yesmaximal, complemented
4#20261034C212yesmaximal, complemented
5#20261024C212yesmaximal, complemented
6#20261014C212yesmaximal, complemented
7#20260775C211yesmaximal, characteristic, complemented
8#20260758C211yesmaximal, characteristic, complemented
9#20260741C211yesmaximal, characteristic
10#20260739C211yesmaximal, characteristic
11#428×C97C211yesmaximal, characteristic, complemented
12C22×D388C2211yescharacteristic
13C22×C4×D97—21no—
14C2×#13095495C2211yescharacteristic
15#15690809C2211yescharacteristic, complemented
16#15690809C2211yescharacteristic, complemented
17#15690801C2211yescharacteristic, complemented
18#15690801C2211yescharacteristic, complemented
19#15690787—21no—
20#15690787—21no—
21C4×Dic194—21no—
22C4×Dic194—21no—
23#15690772—21no—
24#15690772—21no—
25#15690772C2212yes—
26#15690772C2212yes—
27#15690769C2211yescharacteristic
28C2×#13095488—21no—
29C2×C4×Dic97C2211yescharacteristic
30#15690743C2212yescomplemented
31#15690741C2212yescomplemented
32#15690739C2212yescomplemented
33#15690739C2212yescomplemented
34#15690732C2212yes—
35#15690732C2212yes—
36#15690730C2212yes—
37#15690730C2212yes—
38C2×M16×C97C2211yescharacteristic
39#117×C97C2211yescharacteristic
40C2×C4⋊C4×C97C2211yescharacteristic
41#107×C97C2212yescomplemented
42#106×C97C2212yescomplemented
43C23×D97—21no—
44C2×Dic194—21no—
45C2×Dic194—21no—
46C2×D388—21no—
47C2×D388—22no—
48C2×D388Q814yescomplemented
49C2×D388C2312yes—
50C22×Dic97D411yescharacteristic
51C22×Dic97D411yescharacteristic
52C2×C4×D97—22no—
53C2×C4×D97—22no—
54C2×C4×D97—44no—
55#13095495—22no—
56C2×#11715895C2312yes—
57C4⋊Dic97—21no—
58C4⋊Dic97—21no—
59C4⋊Dic97C2311yescharacteristic
60C4⋊Dic97C2311yescharacteristic
61#13095490—21no—
62#13095490—21no—
63#13095490—21no—
64#13095490—21no—
65C22⋊Dic97—21no—
66#13095488—22no—
67#13095488—22no—
68C4×Dic97—21no—
69C4×Dic97D411yescharacteristic
70C4×Dic97D411yescharacteristic
71C4×Dic97C2311yescharacteristic
72C4×Dic97C2311yescharacteristic
73C22×C4×C97—21no—
74C22×C4×C97C2311yescharacteristic
75M16×C97—22no—
76C2×C8×C97C2312yes—
77C4⋊C4×C97—21no—
78C4⋊C4×C97C2312yes—
79C4⋊C4×C97C2311yescharacteristic
80C4⋊C4×C97C2311yescharacteristic
81C22⋊C4×C97—21no—
82C42×C97—21no—
83C23×C97C2×D411yescharacteristic
84C22×D97—24no—
85C22×D97—22no—
86C22×D97—42no—
87C22×D97—42no—
88C2×C4×C97—21no—
89C2×C4×C97—21no—
90C2×C4×C97—22no—
91C2×C4×C97—42no—
92C2×C4×C97C2×D411yescharacteristic
93C2×C4×C97C2×Q812yes—
94C2×C4×C97#4112yes—
95C2×C4×C97C2411yescharacteristic
96C2×Dic97—22no—
97C2×Dic97—21no—
98C2×Dic97—21no—
99C2×Dic97—41no—
100C2×Dic97C2×D411yescharacteristic
101C2×Dic97C2×D411yescharacteristic
102C2×Dic97C2×D411yescharacteristic
103C2×Dic97C2×D411yescharacteristic
104D388—24no—
105D388—22no—
106D388C2×Q814yes—
107C4×D97—44no—
108C4×D97—44no—
109Dic194—42no—
110Dic194—42no—
111C8×C97—22no—
112#11715895—22no—
113C22×C97—21no—
114C22×C97—21no—
115C22×C97C22⋊Q812yes—
116C22×C97C22×D411yescharacteristic
117C2×D97—24no—
118C2×D97—48no—
119C2×D97—42no—
120C2×D97—42no—
121C4×C97—41no—
122C4×C97—42no—
123C4×C97—41no—
124C4×C97C22⋊Q812yes—
125C4×C97C22×Q811yescharacteristic
126C4×C97C2×#4111yescharacteristic
127Dic97—21no—
128Dic97—21no—
129Dic97—21no—
130Dic97—41no—
131Dic97—41no—
132Dic97AGL(1,ℤ/8)12yes—
133C2×C97—22no—
134C2×C97C2×C22⋊Q811yescharacteristic
135C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
136C2×C97#57511yescharacteristic
137D97—44no—
138#3077—971nomaximal, Sylow
139C97#307711yescharacteristic, complemented, Sylow
140#533—971no—
141C2×C4×D4—971no—
142#478—972no—
143#478—972no—
144#476—972no—
145#474—972no—
146#428—971no—
147#428D9711yescharacteristic, complemented
148#423—971no—
149#418—971no—
150#418—971no—
151C22×D4—971no—
152C23×C4—1941no—
153C2×M16—971no—
154C2×M16C2×D9711yescharacteristic
155C4⋊Q8—971no—
156C4⋊Q8—971no—
157#125—971no—
158#125—971no—
159C22⋊Q8—1941no—
160C22⋊Q8—1941no—
161C4×Q8—1941no—
162C4×Q8—1941no—
163C4×D4—972no—
164C4×D4—972no—
165C4×D4—1941no—
166C4×D4—1941no—
167#117—971no—
168#117C2×D9711yescharacteristic
169C2×C4⋊C4C2×D9711yescharacteristic
170C2×C22⋊C4—1941no—
171C2×C42—971no—
172#107—972no—
173#107C2×D9712yescomplemented
174#106—972no—
175#106C2×D9712yescomplemented
176C4⋊C8—972no—
177C4⋊C8—972no—
178#102—972no—
179#102—972no—
180#102—972no—
181#102—972no—
182C24—1941no—
183C2×Q8—1941no—
184C2×Q8—1941no—
185C2×D4—974no—
186C2×D4—972no—
187C2×D4—1941no—
188C2×D4—1942no—
189C22×C4—21no—
190C22×C4—971no—
191C22×C4—971no—
192C22×C4—3884no—
193C22×C4—1942no—
194C22×C4—1942no—
195C22×C4C22×D9711yescharacteristic
196M16—22no—
197M16—1942no—
198C2×C8—972no—
199C2×C8C22×D9712yes—
200C4⋊C4—21no—
201C4⋊C4—971no—
202C4⋊C4—971no—
203C4⋊C4—1941no—
204C4⋊C4—1941no—
205C4⋊C4—1941no—
206C4⋊C4—1941no—
207C4⋊C4—1941no—
208C4⋊C4—1941no—
209C4⋊C4C22×D9711yescharacteristic
210C4⋊C4C22×D9711yescharacteristic
211C4⋊C4C22×D9712yes—
212C22⋊C4—21no—
213C22⋊C4—1942no—
214C22⋊C4—1942no—
215C22⋊C4—1941no—
216C42—21no—
217C42—971no—
218C42—971no—
219C42—971no—
220C42—971no—
221C42—1941no—
222C23—3882no—
223C23—3882no—
224C23—1944no—
225C23—1942no—
226C23D4×D9711yescharacteristic
227Q8—3882no—
228Q8—3882no—
229D4—974no—
230D4—1944no—
231D4—1942no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—971no—
237C2×C4—971no—
238C2×C4—971no—
239C2×C4—971no—
240C2×C4—3884no—
241C2×C4—3884no—
242C2×C4—3881no—
243C2×C4—1942no—
244C2×C4—1941no—
245C2×C4—1941no—
246C2×C4D4×D9711yescharacteristic
247C2×C4Q8×D9712yes—
248C2×C4#1309551112yes—
249C2×C4C23×D9711yescharacteristic
250C8—22no—
251C8—1942no—
252C22—21no—
253C22—21no—
254C22—3888no—
255C22—3882no—
256C22—3882no—
257C22—1944no—
258C22#1569078812yes—
259C22C2×D4×D9711yescharacteristic
260C4—42no—
261C4—41no—
262C4—41no—
263C4—972no—
264C4—3881no—
265C4—3881no—
266C4—1941no—
267C4—1941no—
268C4—1941no—
269C4#1569078812yes—
270C4C2×Q8×D9711yescharacteristic
271C4C2×#1309551111yescharacteristic
272C2—22no—
273C2—3884no—
274C2C2×#1569078811yescharacteristic
275C2AGL(1,ℤ/8)×D9711yescharacteristic
276C2#2026153411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#428×C97C2
socleC22×C97C22×D4
prime radicalC22×D388C22
hypercenter#428D97
hypocenterC97#3077
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #20261288C2semidirect
3. #20261038C2semidirect
4. #20261034C2semidirect
5. #20261024C2semidirect
6. #20261014C2semidirect
7. #20260775C2semidirect
8. #20260758C2semidirect
11. #428×C97C2semidirect
15. #15690809C22semidirect
16. #15690809C22semidirect
17. #15690801C22semidirect
18. #15690801C22semidirect
30. #15690743C22semidirect
31. #15690741C22semidirect
32. #15690739C22semidirect
33. #15690739C22semidirect
41. #107×C97C22semidirect
42. #106×C97C22semidirect
48. C2×D388Q8semidirect
139. C97#3077semidirect
147. #428D97semidirect
173. #107C2×D97semidirect
175. #106C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202612882
C2×#156907722
#202610382
#202610342
#202610242
#202610142
#202607752
#202607582
#202607412
#202607392
#428×C972
#307797