Finite Group Databasebeta

← #27151679

Subgroups — #27151679

Order 12416.

410 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C22⋊Q8×D97C211yesmaximal, characteristic
2C2×#15690768C211yesmaximal, characteristic
3#20261079C212yesmaximal, complemented
4#20261075C212yesmaximal, complemented
5#20261071C211yesmaximal, characteristic, complemented
6#20261068C211yesmaximal, characteristic, complemented
7#20261065C211yesmaximal, characteristic, complemented
8#20260774C212yesmaximal, complemented
9#20260742C212yesmaximal, complemented
10#20260697C211yesmaximal, characteristic
11#483×C97C211yesmaximal, characteristic, complemented
12C2×Q8×D97C2211yescharacteristic
13C22×C4×D97C2211yescharacteristic
14C2×#13095494C2212yes—
15#15690790C2211yescharacteristic
16#15690789C2211yescharacteristic
17#15690788—21no—
18#15690787—21no—
19#15690786C2211yescharacteristic
20#15690768—21no—
21#15690768—21no—
22#15690768C2212yes—
23#15690768C2212yes—
24C2×C4⋊Dic97C2211yescharacteristic
25C4⋊C4×D97—21no—
26C4⋊C4×D97C2211yescharacteristic
27C2×#13095488—21no—
28C22⋊C4×D97—21no—
29C2×C4×Dic97—21no—
30#15690744C2212yescomplemented
31#15690743C2212yescomplemented
32#15690741C2212yescomplemented
33#15690735C2212yescomplemented
34#15690734C2212yescomplemented
35#15690733C2212yescomplemented
36C22⋊#11715895C2211yescharacteristic
37#15690722C2212yes—
38C22⋊Q8×C97C2211yescharacteristic
39C2×C4⋊C4×C97C2211yescharacteristic
40#106×C97C2212yescomplemented
41#103×C97C2212yescomplemented
42C22⋊C8×C97C2211yescharacteristic
43C23×D97D411yescharacteristic
44C2×Dic194—21no—
45C2×Dic194C2311yescharacteristic
46Q8×D97—24no—
47C22×Dic97—21no—
48C22×Dic97D411yescharacteristic
49C2×C4×D97—24no—
50C2×C4×D97—21no—
51C2×C4×D97—21no—
52C2×C4×D97D411yescharacteristic
53C2×C4×D97D411yescharacteristic
54C2×C4×D97C2311yescharacteristic
55C2×C4×D97C2311yescharacteristic
56#13095494—24no—
57C2×#11715895C2312yes—
58C4⋊Dic97—21no—
59C4⋊Dic97—21no—
60C4⋊Dic97C2311yescharacteristic
61C4⋊Dic97C2312yes—
62#13095490—21no—
63#13095490—21no—
64#13095490—21no—
65C22⋊Dic97—21no—
66#13095488—21no—
67#13095488—21no—
68#13095488—22no—
69#13095488—22no—
70C4×Dic97—22no—
71C4×Dic97—22no—
72C2×Q8×C97C2311yescharacteristic
73C22×C4×C97—21no—
74C22×C4×C97C2311yescharacteristic
75C2×C8×C97C2312yes—
76C4⋊C4×C97—21no—
77C4⋊C4×C97—21no—
78C4⋊C4×C97C2311yescharacteristic
79C4⋊C4×C97C2312yes—
80C22⋊C4×C97—21no—
81C23×C97C2×D411yescharacteristic
82C22×D97—22no—
83C22×D97—22no—
84C22×D97—41no—
85C22×D97C2×D411yescharacteristic
86C22×D97C2×D411yescharacteristic
87Q8×C97—22no—
88C2×C4×C97—22no—
89C2×C4×C97—21no—
90C2×C4×C97—21no—
91C2×C4×C97—22no—
92C2×C4×C97—42no—
93C2×C4×C97C2×D411yescharacteristic
94C2×C4×C97C2411yescharacteristic
95C2×Dic97—22no—
96C2×Dic97—22no—
97C2×Dic97—21no—
98C2×Dic97—21no—
99C2×Dic97—42no—
100C2×Dic97C2×D411yescharacteristic
101C2×Dic97C2×D411yescharacteristic
102C4×D97—22no—
103C4×D97—22no—
104C4×D97—22no—
105C4×D97—42no—
106C4×D97—42no—
107C4×D97#4114yes—
108Dic194—22no—
109Dic194—42no—
110C8×C97—22no—
111#11715895—22no—
112C22×C97—21no—
113C22×C97—21no—
114C22×C97#13712yes—
115C22×C97C22×D411yescharacteristic
116C2×D97—21no—
117C2×D97—21no—
118C2×D97—21no—
119C2×D97—21no—
120C2×D97—21no—
121C2×D97—44no—
122C2×D97#12312yes—
123C4×C97—21no—
124C4×C97—41no—
125C4×C97—42no—
126C4×C97—41no—
127C4×C97C2×#4112yes—
128Dic97—21no—
129Dic97—41no—
130Dic97—41no—
131Dic97—42no—
132Dic97#12312yes—
133C2×C97—22no—
134C2×C97C2×#12311yescharacteristic
135C2×C97C2×#13711yescharacteristic
136C2×C97#57511yescharacteristic
137D97—22no—
138D97—41no—
139#3095—971nomaximal, Sylow
140C97#309511yescharacteristic, complemented, Sylow
141C2×C22⋊Q8—971no—
142C2×#117—971no—
143#485—972no—
144#485—972no—
145#484—971no—
146#484—971no—
147#483—971no—
148#483D9711yescharacteristic, complemented
149#428—972no—
150#419—972no—
151#407—971no—
152C22×Q8—971no—
153C23×C4—971no—
154C2×M16—972no—
155C22⋊Q8—971no—
156C22⋊Q8—971no—
157C22⋊Q8—971no—
158C22⋊Q8—1941no—
159C22⋊Q8—1941no—
160C22⋊Q8C2×D9711yescharacteristic
161#117—972no—
162#117—972no—
163#117—1941no—
164#117—1941no—
165C2×C4⋊C4—971no—
166C2×C4⋊C4—971no—
167C2×C4⋊C4—1941no—
168C2×C4⋊C4C2×D9711yescharacteristic
169C2×C22⋊C4—1941no—
170C2×C22⋊C4—1941no—
171C2×C42—1941no—
172#107—972no—
173#107—972no—
174#106—972no—
175#106C2×D9712yescomplemented
176#103—972no—
177#103—972no—
178#103—972no—
179#103C2×D9712yescomplemented
180C22⋊C8—971no—
181C22⋊C8—972no—
182C22⋊C8C2×D9711yescharacteristic
183C24—971no—
184C2×Q8—971no—
185C2×Q8—1941no—
186C2×Q8—1944no—
187C2×Q8C22×D9711yescharacteristic
188C22×C4—21no—
189C22×C4—971no—
190C22×C4—971no—
191C22×C4—971no—
192C22×C4—971no—
193C22×C4—971no—
194C22×C4—1944no—
195C22×C4—1941no—
196C22×C4—1941no—
197C22×C4—1941no—
198C22×C4C22×D9711yescharacteristic
199M16—1944no—
200C2×C8—972no—
201C2×C8C22×D9712yes—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—972no—
205C4⋊C4—971no—
206C4⋊C4—1941no—
207C4⋊C4—1941no—
208C4⋊C4—1941no—
209C4⋊C4—1941no—
210C4⋊C4—1941no—
211C4⋊C4C22×D9712yes—
212C4⋊C4C22×D9711yescharacteristic
213C22⋊C4—21no—
214C22⋊C4—1942no—
215C22⋊C4—1942no—
216C22⋊C4—1941no—
217C22⋊C4—1941no—
218C22⋊C4—1941no—
219C42—1942no—
220C42—1942no—
221C23—971no—
222C23—971no—
223C23—3881no—
224C23—1942no—
225C23—1942no—
226C23D4×D9711yescharacteristic
227Q8—22no—
228Q8—3882no—
229Q8—1942no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—971no—
236C2×C4—974no—
237C2×C4—971no—
238C2×C4—3882no—
239C2×C4—3882no—
240C2×C4—3882no—
241C2×C4—1942no—
242C2×C4—1942no—
243C2×C4—1942no—
244C2×C4—1942no—
245C2×C4—1941no—
246C2×C4—1941no—
247C2×C4—1942no—
248C2×C4D4×D9711yescharacteristic
249C2×C4C23×D9711yescharacteristic
250C8—22no—
251C8—1942no—
252C22—21no—
253C22—21no—
254C22—972no—
255C22—3884no—
256C22—1941no—
257C22—1941no—
258C22—1941no—
259C22—1941no—
260C22—1941no—
261C22#1569083912yes—
262C22C2×D4×D9711yescharacteristic
263C4—21no—
264C4—41no—
265C4—42no—
266C4—41no—
267C4—972no—
268C4—3881no—
269C4—3881no—
270C4—3882no—
271C4—1941no—
272C4#41×D9712yes—
273C2—22no—
274C2—3881no—
275C2—1942no—
276C2#123×D9711yescharacteristic
277C2C2×#1569083911yescharacteristic
278C2#2026153411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#483×C97C2
socleC22×C97C22×D4
prime radicalC23×D97D4
hypercenter#483D97
hypocenterC97#3095
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
3. #20261079C2semidirect
4. #20261075C2semidirect
5. #20261071C2semidirect
6. #20261068C2semidirect
7. #20261065C2semidirect
8. #20260774C2semidirect
9. #20260742C2semidirect
11. #483×C97C2semidirect
30. #15690744C22semidirect
31. #15690743C22semidirect
32. #15690741C22semidirect
33. #15690735C22semidirect
34. #15690734C22semidirect
35. #15690733C22semidirect
40. #106×C97C22semidirect
41. #103×C97C22semidirect
140. C97#3095semidirect
148. #483D97semidirect
175. #106C2×D97semidirect
179. #103C2×D97semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D972
C2×#156907682
#202610792
#202610752
#202610712
#202610682
#202610652
#202607742
#202607422
#202606972
#483×C972
#309597