Finite Group Databasebeta

← #27151662

Subgroups — #27151662

Order 12416.

458 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 129 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#20261247C211yesmaximal, characteristic
2C2×C4⋊C4×D97C211yesmaximal, characteristic, complemented
3#20261072C212yesmaximal, complemented
4#20261067C212yesmaximal, complemented
5#20261055C212yesmaximal, complemented
6#20261050C212yesmaximal, complemented
7#20260778C211yesmaximal, characteristic, complemented
8#20260737C211yesmaximal, characteristic, complemented
9#20260734C211yesmaximal, characteristic, complemented
10#20260698C211yesmaximal, characteristic, complemented
11#428×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2211yescharacteristic
13C22×C4×D97—21no—
14C22×C4×D97C2211yescharacteristic
15C2×#13095495C2211yescharacteristic
16#15690794—21no—
17#15690790C2211yescharacteristic
18#15690790C2211yescharacteristic
19#15690785C2211yescharacteristic
20#15690785C2211yescharacteristic
21#15690780—21no—
22C4×D388—21no—
23C4×D388—21no—
24C2×C4⋊Dic97C2211yescharacteristic
25C2×#13095490—21no—
26C4⋊C4×D97—21no—
27C4⋊C4×D97—21no—
28C4⋊C4×D97C2212yes—
29C4⋊C4×D97C2212yes—
30#15690743C2212yescomplemented
31#15690741C2212yescomplemented
32#15690735C2212yescomplemented
33#15690733C2212yescomplemented
34#15690732C2212yescomplemented
35#15690730C2212yescomplemented
36#15690722C2212yes—
37#15690722C2212yes—
38C2×M16×C97C2211yescharacteristic
39#117×C97C2211yescharacteristic
40C2×C4⋊C4×C97C2211yescharacteristic
41#107×C97C2212yescomplemented
42#106×C97C2212yescomplemented
43C23×D97D411yescharacteristic
44#13095511—22no—
45#13095511—22no—
46C2×Dic194C2311yescharacteristic
47C2×D388—21no—
48C2×D388—21no—
49C2×D388C2311yescharacteristic
50C2×C22⋊D97—21no—
51C22×Dic97—21no—
52C22×Dic97D411yescharacteristic
53C2×C4×D97—22no—
54C2×C4×D97—22no—
55C2×C4×D97—22no—
56C2×C4×D97—22no—
57C2×C4×D97—21no—
58C2×C4×D97—44no—
59C2×C4×D97D411yescharacteristic
60C2×C4×D97D411yescharacteristic
61C2×C4×D97C2311yescharacteristic
62C2×C4×D97C2311yescharacteristic
63#13095495—22no—
64C2×#11715895C2312yes—
65C4⋊Dic97—21no—
66C4⋊Dic97—21no—
67C4⋊Dic97—21no—
68C4⋊Dic97C2312yes—
69#13095490—22no—
70#13095490—22no—
71#13095488—21no—
72#13095488—21no—
73#13095488—21no—
74#13095488—21no—
75C22×C4×C97—21no—
76C22×C4×C97C2311yescharacteristic
77M16×C97—22no—
78C2×C8×C97C2312yes—
79C4⋊C4×C97—21no—
80C4⋊C4×C97C2312yes—
81C4⋊C4×C97C2311yescharacteristic
82C4⋊C4×C97C2311yescharacteristic
83C22⋊C4×C97—21no—
84C42×C97—21no—
85C23×C97C2×D411yescharacteristic
86C22×D97—22no—
87C22×D97—22no—
88C22×D97—22no—
89C22×D97—21no—
90C22×D97C2×D411yescharacteristic
91C22×D97C2×D411yescharacteristic
92C2×C4×C97—21no—
93C2×C4×C97—21no—
94C2×C4×C97—22no—
95C2×C4×C97—42no—
96C2×C4×C97C2×D411yescharacteristic
97C2×C4×C97#4112yes—
98C2×C4×C97#4112yes—
99C2×C4×C97C2411yescharacteristic
100C22⋊D97—42no—
101C2×Dic97—22no—
102C2×Dic97—21no—
103C2×Dic97—22no—
104C2×Dic97—42no—
105C2×Dic97C2×D411yescharacteristic
106C2×Dic97C2×D411yescharacteristic
107D388—21no—
108D388—21no—
109D388—42no—
110D388—42no—
111C4×D97—22no—
112C4×D97—22no—
113C4×D97—21no—
114C4×D97—21no—
115C4×D97—21no—
116C4×D97—21no—
117C4×D97—44no—
118C4×D97—44no—
119C4×D97—42no—
120Dic194—21no—
121Dic194—21no—
122C8×C97—22no—
123#11715895—22no—
124C22×C97—21no—
125C22×C97—21no—
126C22×C97#12312yes—
127C22×C97C22×D411yescharacteristic
128C2×D97—28no—
129C2×D97—21no—
130C2×D97—21no—
131C2×D97—21no—
132C2×D97—21no—
133C2×D97—41no—
134C2×D97—41no—
135C2×D97—41no—
136C2×D97AGL(1,ℤ/8)12yes—
137C2×D97#13712yes—
138C4×C97—42no—
139C4×C97—41no—
140C4×C97—41no—
141C4×C97#12312yes—
142C4×C97C2×#4111yescharacteristic
143C4×C97C2×#4111yescharacteristic
144Dic97—21no—
145Dic97—21no—
146Dic97—41no—
147Dic97—42no—
148C2×C97—22no—
149C2×C97C2×#12311yescharacteristic
150C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C97C2×#13711yescharacteristic
152D97—22no—
153D97—22no—
154D97—81no—
155#3092—971nomaximal, Sylow
156C97#309211yescharacteristic, complemented, Sylow
157#525—971no—
158C22×C4⋊C4—971no—
159#484—972no—
160#483—972no—
161#481—972no—
162#480—972no—
163#428—971no—
164#428D9711yescharacteristic, complemented
165#417—971no—
166#417—971no—
167#407—971no—
168C2×#41—971no—
169C23×C4—971no—
170C23×C4—1941no—
171C2×M16—971no—
172C2×M16C2×D9711yescharacteristic
173#123—1941no—
174C22⋊Q8—971no—
175C22⋊Q8—971no—
176#121—971no—
177#121—971no—
178#120—1941no—
179C4×D4—1941no—
180C4×D4—1941no—
181#117C2×D9711yescharacteristic
182C2×C4⋊C4—971no—
183C2×C4⋊C4—972no—
184C2×C4⋊C4—972no—
185C2×C4⋊C4—1941no—
186C2×C4⋊C4—1941no—
187C2×C4⋊C4—1941no—
188C2×C4⋊C4C2×D9711yescharacteristic
189#107—972no—
190#107C2×D9712yescomplemented
191#106—972no—
192#106C2×D9712yescomplemented
193#103—972no—
194#103—972no—
195#102—972no—
196#102—972no—
197C22⋊C8—972no—
198C22⋊C8—972no—
199C24—971no—
200#41—1942no—
201#41—1942no—
202C2×Q8—971no—
203C2×D4—971no—
204C2×D4—1941no—
205C2×D4—1941no—
206C2×D4—1941no—
207C22×C4—21no—
208C22×C4—971no—
209C22×C4—971no—
210C22×C4—971no—
211C22×C4—971no—
212C22×C4—971no—
213C22×C4—3884no—
214C22×C4—1942no—
215C22×C4—1942no—
216C22×C4—1941no—
217C22×C4—1941no—
218C22×C4—1942no—
219C22×C4—1942no—
220C22×C4C22×D9711yescharacteristic
221M16—22no—
222M16—1942no—
223C2×C8—972no—
224C2×C8C22×D9712yes—
225C4⋊C4—21no—
226C4⋊C4—972no—
227C4⋊C4—1942no—
228C4⋊C4—1942no—
229C4⋊C4—1941no—
230C4⋊C4—1941no—
231C4⋊C4—1941no—
232C4⋊C4C22×D9712yes—
233C4⋊C4C22×D9711yescharacteristic
234C4⋊C4C22×D9711yescharacteristic
235C22⋊C4—21no—
236C22⋊C4—1941no—
237C22⋊C4—1941no—
238C22⋊C4—1941no—
239C22⋊C4—1941no—
240C42—21no—
241C23—971no—
242C23—971no—
243C23—1942no—
244C23—1942no—
245C23—1941no—
246C23—1942no—
247C23D4×D9711yescharacteristic
248Q8—1941no—
249Q8—1941no—
250D4—3882no—
251D4—3882no—
252D4—3882no—
253D4—1941no—
254D4—1941no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—971no—
260C2×C4—971no—
261C2×C4—3884no—
262C2×C4—3884no—
263C2×C4—3882no—
264C2×C4—3882no—
265C2×C4—1941no—
266C2×C4—1942no—
267C2×C4—1942no—
268C2×C4—1942no—
269C2×C4—1942no—
270C2×C4—1941no—
271C2×C4—1941no—
272C2×C4—1941no—
273C2×C4—1941no—
274C2×C4D4×D9711yescharacteristic
275C2×C4#1309551112yes—
276C2×C4#1309551212yes—
277C2×C4C23×D9711yescharacteristic
278C8—22no—
279C8—1942no—
280C22—21no—
281C22—21no—
282C22—972no—
283C22—972no—
284C22—3881no—
285C22—3881no—
286C22—3881no—
287C22—1948no—
288C22—1941no—
289C22—1941no—
290C22—1941no—
291C22—1941no—
292C22#1569079512yes—
293C22C2×D4×D9711yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—3881no—
298C4—3882no—
299C4—1941no—
300C4—1941no—
301C4#1569079512yes—
302C4C2×#1309551111yescharacteristic
303C4C2×#1309551211yescharacteristic
304C2—22no—
305C2—7761no—
306C2—1942no—
307C2—1942no—
308C2C2×#1569079511yescharacteristic
309C2AGL(1,ℤ/8)×D9711yescharacteristic
310C2#137×D9711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#428×C97C2
socleC22×C97C22×D4
prime radical#20261247C2
hypercenter#428D97
hypocenterC97#3092
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
2. C2×C4⋊C4×D97C2semidirect
3. #20261072C2semidirect
4. #20261067C2semidirect
5. #20261055C2semidirect
6. #20261050C2semidirect
7. #20260778C2semidirect
8. #20260737C2semidirect
9. #20260734C2semidirect
10. #20260698C2semidirect
11. #428×C97C2semidirect
30. #15690743C22semidirect
31. #15690741C22semidirect
32. #15690735C22semidirect
33. #15690733C22semidirect
34. #15690732C22semidirect
35. #15690730C22semidirect
41. #107×C97C22semidirect
42. #106×C97C22semidirect
156. C97#3092semidirect
164. #428D97semidirect
190. #107C2×D97semidirect
192. #106C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202612472
C2×C4⋊C4×D972
#202610722
#202610672
#202610552
#202610502
#202607782
#202607372
#202607342
#202606982
#428×C972
#309297