Finite Group Databasebeta

← #27151682

Subgroups — #27151682

Order 12416.

410 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#15690786C211yesmaximal, characteristic
2#117×D97C211yesmaximal, characteristic
3#20261075C212yesmaximal, complemented
4#20261074C212yesmaximal, complemented
5#20261069C212yesmaximal, complemented
6#20261064C212yesmaximal, complemented
7#20260778C211yesmaximal, characteristic, complemented
8#20260777C211yesmaximal, characteristic, complemented
9#20260744C211yesmaximal, characteristic, complemented
10#20260698C211yesmaximal, characteristic
11#419×C97C211yesmaximal, characteristic, complemented
12C22×C4×D97C2211yescharacteristic
13C2×#13095495C2211yescharacteristic
14#15690786—21no—
15#15690786—21no—
16#15690786C2212yes—
17#15690786C2212yes—
18#15690769C2211yescharacteristic
19#15690768C2211yescharacteristic
20#15690768C2211yescharacteristic
21#15690767—21no—
22#15690766—21no—
23C2×C4⋊Dic97C2211yescharacteristic
24C2×#13095490—21no—
25C4⋊C4×D97C2211yescharacteristic
26C4⋊C4×D97C2211yescharacteristic
27C2×#13095488—21no—
28C22⋊C4×D97—21no—
29C42×D97—21no—
30#15690744C2212yescomplemented
31#15690743C2212yescomplemented
32#15690742C2212yescomplemented
33#15690741C2212yescomplemented
34#15690734C2212yescomplemented
35#15690734C2212yescomplemented
36#15690722C2212yes—
37#15690722C2212yes—
38C22×Q8×C97C2211yescharacteristic
39C2×M16×C97C2211yescharacteristic
40#117×C97C2211yescharacteristic
41#103×C97C2212yescomplemented
42#103×C97C2212yescomplemented
43C23×D97D411yescharacteristic
44C22×Dic97—21no—
45C22×Dic97D411yescharacteristic
46C2×C4×D97—22no—
47C2×C4×D97—22no—
48C2×C4×D97—21no—
49C2×C4×D97—21no—
50C2×C4×D97D411yescharacteristic
51C2×C4×D97D411yescharacteristic
52C2×C4×D97C2311yescharacteristic
53C2×C4×D97C2311yescharacteristic
54#13095495—22no—
55C2×#11715895C2312yes—
56C4⋊Dic97—21no—
57C4⋊Dic97C2311yescharacteristic
58C4⋊Dic97C2311yescharacteristic
59C4⋊Dic97C2312yes—
60#13095490—21no—
61#13095490—21no—
62#13095490—22no—
63#13095490—22no—
64C22⋊Dic97—21no—
65#13095488—21no—
66#13095488—21no—
67#13095488—22no—
68#13095488—22no—
69C4×Dic97—21no—
70C4×Dic97—21no—
71C4×Dic97—21no—
72C2×Q8×C97—21no—
73C2×Q8×C97—22no—
74C2×Q8×C97—22no—
75C2×Q8×C97C2312yes—
76C22×C4×C97—21no—
77C22×C4×C97C2311yescharacteristic
78M16×C97—22no—
79C2×C8×C97C2312yes—
80C4⋊C4×C97C2311yescharacteristic
81C4⋊C4×C97C2311yescharacteristic
82C22⋊C4×C97—21no—
83C42×C97—21no—
84C23×C97C2×D411yescharacteristic
85C22×D97—22no—
86C22×D97—22no—
87C22×D97—41no—
88C22×D97C2×D411yescharacteristic
89C22×D97C2×D411yescharacteristic
90Q8×C97—22no—
91Q8×C97—22no—
92Q8×C97—42no—
93C2×C4×C97—21no—
94C2×C4×C97—21no—
95C2×C4×C97—22no—
96C2×C4×C97—42no—
97C2×C4×C97C2×D411yescharacteristic
98C2×C4×C97#4112yes—
99C2×C4×C97#4112yes—
100C2×C4×C97C2411yescharacteristic
101C2×Dic97—22no—
102C2×Dic97—21no—
103C2×Dic97—21no—
104C2×Dic97—22no—
105C2×Dic97—42no—
106C2×Dic97C2×D411yescharacteristic
107C2×Dic97C2×D411yescharacteristic
108C4×D97—22no—
109C4×D97—22no—
110C4×D97—21no—
111C4×D97—21no—
112C4×D97—21no—
113C4×D97—21no—
114C4×D97—42no—
115C4×D97—42no—
116C8×C97—22no—
117#11715895—22no—
118C22×C97—21no—
119C22×C97—21no—
120C22×C97#12312yes—
121C22×C97C22×D411yescharacteristic
122C2×D97—21no—
123C2×D97—21no—
124C2×D97—21no—
125C2×D97—21no—
126C2×D97—21no—
127C2×D97—44no—
128C2×D97#13712yes—
129C4×C97—42no—
130C4×C97—41no—
131C4×C97—41no—
132C4×C97#12312yes—
133C4×C97C2×#4111yescharacteristic
134C4×C97C2×#4111yescharacteristic
135Dic97—21no—
136Dic97—21no—
137Dic97—41no—
138Dic97—41no—
139Dic97—42no—
140C2×C97—22no—
141C2×C97C2×#12311yescharacteristic
142C2×C97C2×#13711yescharacteristic
143C2×C97#57511yescharacteristic
144D97—22no—
145D97—41no—
146#3095—971nomaximal, Sylow
147C97#309511yescharacteristic, complemented, Sylow
148C2×C22⋊Q8—971no—
149C2×#117—971no—
150#485—972no—
151#485—972no—
152#484—972no—
153#483—972no—
154#428—971no—
155#428—971no—
156#419—971no—
157#419D9711yescharacteristic, complemented
158#407—971no—
159C22×Q8C2×D9711yescharacteristic
160C23×C4—971no—
161C2×M16—971no—
162C2×M16C2×D9711yescharacteristic
163C22⋊Q8—972no—
164C22⋊Q8—972no—
165C22⋊Q8—1941no—
166C22⋊Q8—1941no—
167#117—971no—
168#117—971no—
169#117—971no—
170#117—1941no—
171#117—1941no—
172#117C2×D9711yescharacteristic
173C2×C4⋊C4—971no—
174C2×C4⋊C4—971no—
175C2×C4⋊C4—971no—
176C2×C4⋊C4—1941no—
177C2×C22⋊C4—1941no—
178C2×C22⋊C4—1941no—
179C2×C42—1941no—
180#107—972no—
181#107—972no—
182#106—972no—
183#106—972no—
184#103—972no—
185#103—972no—
186#103C2×D9712yescomplemented
187#103C2×D9712yescomplemented
188C22⋊C8—972no—
189C22⋊C8—972no—
190C24—971no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8C22×D9712yes—
195C22×C4—21no—
196C22×C4—971no—
197C22×C4—971no—
198C22×C4—971no—
199C22×C4—971no—
200C22×C4—971no—
201C22×C4—1941no—
202C22×C4—1941no—
203C22×C4—1941no—
204C22×C4—1942no—
205C22×C4—1942no—
206C22×C4C22×D9711yescharacteristic
207M16—22no—
208M16—1942no—
209C2×C8—972no—
210C2×C8C22×D9712yes—
211C4⋊C4—971no—
212C4⋊C4—971no—
213C4⋊C4—972no—
214C4⋊C4—1941no—
215C4⋊C4—1941no—
216C4⋊C4—1942no—
217C4⋊C4—1942no—
218C4⋊C4—1941no—
219C4⋊C4C22×D9711yescharacteristic
220C4⋊C4C22×D9711yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—1941no—
223C22⋊C4—1941no—
224C22⋊C4—1942no—
225C22⋊C4—1942no—
226C22⋊C4—1941no—
227C42—21no—
228C42—1941no—
229C42—1941no—
230C42—1941no—
231C23—971no—
232C23—971no—
233C23—3881no—
234C23—1942no—
235C23—1942no—
236C23D4×D9711yescharacteristic
237Q8—22no—
238Q8—22no—
239Q8—42no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—971no—
245C2×C4—971no—
246C2×C4—3882no—
247C2×C4—3882no—
248C2×C4—3882no—
249C2×C4—1941no—
250C2×C4—1941no—
251C2×C4—1942no—
252C2×C4—1942no—
253C2×C4—1942no—
254C2×C4—1942no—
255C2×C4—1941no—
256C2×C4—1941no—
257C2×C4—1941no—
258C2×C4—1941no—
259C2×C4D4×D9711yescharacteristic
260C2×C4#1309551012yes—
261C2×C4#1309551112yes—
262C2×C4C23×D9711yescharacteristic
263C8—22no—
264C8—1942no—
265C22—21no—
266C22—21no—
267C22—972no—
268C22—3884no—
269C22—1941no—
270C22—1941no—
271C22—1941no—
272C22—1941no—
273C22—1941no—
274C22#1569079212yes—
275C22C2×D4×D9711yescharacteristic
276C4—42no—
277C4—41no—
278C4—41no—
279C4—3881no—
280C4—3881no—
281C4—3882no—
282C4—1941no—
283C4—1941no—
284C4#1569079212yes—
285C4C2×#1309551011yescharacteristic
286C4C2×#1309551111yescharacteristic
287C2—22no—
288C2—3881no—
289C2—1942no—
290C2C2×#1569079211yescharacteristic
291C2#137×D9711yescharacteristic
292C2#2026153411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#419×C97C2
socleC22×C97C22×D4
prime radicalC23×D97D4
hypercenter#419D97
hypocenterC97#3095
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #20261075C2semidirect
4. #20261074C2semidirect
5. #20261069C2semidirect
6. #20261064C2semidirect
7. #20260778C2semidirect
8. #20260777C2semidirect
9. #20260744C2semidirect
11. #419×C97C2semidirect
30. #15690744C22semidirect
31. #15690743C22semidirect
32. #15690742C22semidirect
33. #15690741C22semidirect
34. #15690734C22semidirect
35. #15690734C22semidirect
41. #103×C97C22semidirect
42. #103×C97C22semidirect
147. C97#3095semidirect
157. #419D97semidirect
186. #103C2×D97semidirect
187. #103C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907862
#117×D972
#202610752
#202610742
#202610692
#202610642
#202607782
#202607772
#202607442
#202606982
#419×C972
#309597