Finite Group Databasebeta

← #27151686

Subgroups — #27151686

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×#15690790C211yesmaximal, characteristic
2#117×D97C211yesmaximal, characteristic
3#20261079C212yesmaximal, complemented
4#20261078C212yesmaximal, complemented
5#20261072C212yesmaximal, complemented
6#20261067C212yesmaximal, complemented
7#20260775C211yesmaximal, characteristic, complemented
8#20260745C211yesmaximal, characteristic, complemented
9#20260743C211yesmaximal, characteristic, complemented
10#20260698C211yesmaximal, characteristic
11#428×C97C211yesmaximal, characteristic, complemented
12C22×Dic194C2211yescharacteristic
13C22×C4×D97C2211yescharacteristic
14C2×#13095495C2211yescharacteristic
15#15690790—21no—
16#15690790—21no—
17#15690790C2212yes—
18#15690790C2212yes—
19#15690769C2211yescharacteristic
20#15690768C2211yescharacteristic
21#15690768C2211yescharacteristic
22#15690767—21no—
23#15690766—21no—
24C2×C4⋊Dic97—21no—
25C4⋊C4×D97C2211yescharacteristic
26C4⋊C4×D97C2211yescharacteristic
27C2×#13095488—21no—
28C22⋊C4×D97—21no—
29C42×D97—21no—
30#15690743C2212yescomplemented
31#15690741C2212yescomplemented
32#15690735C2212yescomplemented
33#15690735C2212yescomplemented
34#15690733C2212yescomplemented
35#15690733C2212yescomplemented
36#15690722C2212yes—
37#15690722C2212yes—
38C2×M16×C97C2211yescharacteristic
39#117×C97C2211yescharacteristic
40C2×C4⋊C4×C97C2211yescharacteristic
41#107×C97C2212yescomplemented
42#106×C97C2212yescomplemented
43C23×D97D411yescharacteristic
44C2×Dic194—22no—
45C2×Dic194—22no—
46C2×Dic194—21no—
47C2×Dic194C2312yes—
48C22×Dic97—21no—
49C22×Dic97D411yescharacteristic
50C2×C4×D97—22no—
51C2×C4×D97—22no—
52C2×C4×D97—21no—
53C2×C4×D97—21no—
54C2×C4×D97D411yescharacteristic
55C2×C4×D97D411yescharacteristic
56C2×C4×D97C2311yescharacteristic
57C2×C4×D97C2311yescharacteristic
58#13095495—22no—
59C2×#11715895C2312yes—
60C4⋊Dic97—22no—
61C4⋊Dic97—22no—
62C4⋊Dic97C2311yescharacteristic
63C4⋊Dic97C2311yescharacteristic
64#13095490—21no—
65#13095490—21no—
66C22⋊Dic97—21no—
67#13095488—21no—
68#13095488—21no—
69#13095488—22no—
70#13095488—22no—
71C4×Dic97—21no—
72C4×Dic97—21no—
73C4×Dic97—21no—
74C22×C4×C97—21no—
75C22×C4×C97C2311yescharacteristic
76M16×C97—22no—
77C2×C8×C97C2312yes—
78C4⋊C4×C97—21no—
79C4⋊C4×C97C2311yescharacteristic
80C4⋊C4×C97C2311yescharacteristic
81C4⋊C4×C97C2312yes—
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
90C2×C4×C97—21no—
91C2×C4×C97—21no—
92C2×C4×C97—22no—
93C2×C4×C97—42no—
94C2×C4×C97C2×D411yescharacteristic
95C2×C4×C97#4112yes—
96C2×C4×C97#4112yes—
97C2×C4×C97C2411yescharacteristic
98C2×Dic97—22no—
99C2×Dic97—21no—
100C2×Dic97—21no—
101C2×Dic97—22no—
102C2×Dic97—42no—
103C2×Dic97C2×D411yescharacteristic
104C2×Dic97C2×D411yescharacteristic
105C4×D97—22no—
106C4×D97—22no—
107C4×D97—21no—
108C4×D97—21no—
109C4×D97—21no—
110C4×D97—21no—
111C4×D97—42no—
112C4×D97—42no—
113Dic194—22no—
114Dic194—22no—
115Dic194—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—41no—
130C4×C97—41no—
131C4×C97—42no—
132C4×C97#12312yes—
133C4×C97C2×#4111yescharacteristic
134C4×C97C2×#4111yescharacteristic
135Dic97—21no—
136Dic97—21no—
137Dic97—42no—
138Dic97—41no—
139Dic97—41no—
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#428D9711yescharacteristic, complemented
156#419—971no—
157#419—971no—
158#407—971no—
159C22×Q8—971no—
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—1941no—
176C2×C4⋊C4C2×D9711yescharacteristic
177C2×C22⋊C4—1941no—
178C2×C22⋊C4—1941no—
179C2×C42—1941no—
180#107—972no—
181#107C2×D9712yescomplemented
182#106—972no—
183#106C2×D9712yescomplemented
184#103—972no—
185#103—972no—
186#103—972no—
187#103—972no—
188C22⋊C8—972no—
189C22⋊C8—972no—
190C24—971no—
191C2×Q8—972no—
192C2×Q8—1942no—
193C2×Q8—1942no—
194C2×Q8—1941no—
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—21no—
212C4⋊C4—971no—
213C4⋊C4—971no—
214C4⋊C4—1941no—
215C4⋊C4—1941no—
216C4⋊C4—1942no—
217C4⋊C4—1942no—
218C4⋊C4C22×D9711yescharacteristic
219C4⋊C4C22×D9711yescharacteristic
220C4⋊C4C22×D9712yes—
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—3882no—
238Q8—1942no—
239Q8—1942no—
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#1309551112yes—
261C2×C4#1309551212yes—
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#1569079512yes—
275C22C2×D4×D9711yescharacteristic
276C4—41no—
277C4—42no—
278C4—41no—
279C4—3882no—
280C4—3881no—
281C4—3881no—
282C4—1941no—
283C4—1941no—
284C4#1569079512yes—
285C4C2×#1309551111yescharacteristic
286C4C2×#1309551211yescharacteristic
287C2—22no—
288C2—3881no—
289C2—1942no—
290C2C2×#1569079511yescharacteristic
291C2#137×D9711yescharacteristic
292C2#2026153111yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#428×C97C2
socleC22×C97C22×D4
prime radicalC23×D97D4
hypercenter#428D97
hypocenterC97#3095
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
3. #20261079C2semidirect
4. #20261078C2semidirect
5. #20261072C2semidirect
6. #20261067C2semidirect
7. #20260775C2semidirect
8. #20260745C2semidirect
9. #20260743C2semidirect
11. #428×C97C2semidirect
30. #15690743C22semidirect
31. #15690741C22semidirect
32. #15690735C22semidirect
33. #15690735C22semidirect
34. #15690733C22semidirect
35. #15690733C22semidirect
41. #107×C97C22semidirect
42. #106×C97C22semidirect
147. C97#3095semidirect
155. #428D97semidirect
181. #107C2×D97semidirect
183. #106C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907902
#117×D972
#202610792
#202610782
#202610722
#202610672
#202607752
#202607452
#202607432
#202606982
#428×C972
#309597