Finite Group Databasebeta

← #27151678

Subgroups — #27151678

Order 12416.

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

normal subgroups only

This group has 462 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×#15690785C211yesmaximal, characteristic
2#117×D97C211yesmaximal, characteristic, complemented
3#20261062C212yesmaximal, complemented
4#20261061C212yesmaximal, complemented
5#20261055C212yesmaximal, complemented
6#20261050C212yesmaximal, complemented
7#20260775C211yesmaximal, characteristic, complemented
8#20260741C211yesmaximal, characteristic, complemented
9#20260739C211yesmaximal, characteristic, complemented
10#20260698C211yesmaximal, characteristic, complemented
11#428×C97C211yesmaximal, characteristic, complemented
12C22×D388—21no—
13C22×D388C2211yescharacteristic
14C22×C4×D97C2211yescharacteristic
15C2×#13095495C2211yescharacteristic
16#15690785—21no—
17#15690785—21no—
18#15690785C2212yes—
19#15690785C2212yes—
20#15690769C2211yescharacteristic
21#15690768C2211yescharacteristic, complemented
22#15690768C2211yescharacteristic, complemented
23#15690767—21no—
24#15690766—21no—
25C4⋊C4×D97C2211yescharacteristic, complemented
26C4⋊C4×D97C2211yescharacteristic, complemented
27C2×#13095488—21no—
28C22⋊C4×D97—21no—
29C42×D97—21no—
30#15690743C2212yescomplemented
31#15690741C2212yescomplemented
32#15690732C2212yescomplemented
33#15690732C2212yescomplemented
34#15690730C2212yescomplemented
35#15690730C2212yescomplemented
36#15690722C2212yescomplemented
37#15690722C2212yescomplemented
38C2×M16×C97C2211yescharacteristic
39#117×C97C2211yescharacteristic
40C2×C4⋊C4×C97C2211yescharacteristic
41#107×C97C2212yescomplemented
42#106×C97C2212yescomplemented
43C23×D97—21no—
44C23×D97D411yescharacteristic
45C2×D388—22no—
46C2×D388—22no—
47C2×D388—22no—
48C2×D388—22no—
49C2×D388—21no—
50C2×D388—44no—
51C2×D388C2312yes—
52C22×Dic97D411yescharacteristic
53C2×C4×D97—22no—
54C2×C4×D97—22no—
55C2×C4×D97—21no—
56C2×C4×D97—21no—
57C2×C4×D97D411yescharacteristic
58C2×C4×D97D411yescharacteristic
59C2×C4×D97C2311yescharacteristic
60C2×C4×D97C2311yescharacteristic
61#13095495—22no—
62C2×#11715895C2312yes—
63C4⋊Dic97C2311yescharacteristic
64C4⋊Dic97C2311yescharacteristic
65#13095490—21no—
66#13095490—21no—
67C22⋊Dic97—21no—
68#13095488—21no—
69#13095488—21no—
70#13095488—22no—
71#13095488—22no—
72C4×Dic97—21no—
73C4×Dic97—21no—
74C4×Dic97—21no—
75C22×C4×C97—21no—
76C22×C4×C97C2311yescharacteristic
77M16×C97—22no—
78C2×C8×C97C2312yes—
79C4⋊C4×C97—21no—
80C4⋊C4×C97C2311yescharacteristic
81C4⋊C4×C97C2311yescharacteristic
82C4⋊C4×C97C2312yes—
83C22⋊C4×C97—21no—
84C42×C97—21no—
85C23×C97C2×D411yescharacteristic
86C22×D97—22no—
87C22×D97—22no—
88C22×D97—22no—
89C22×D97—41no—
90C22×D97—42no—
91C22×D97—42no—
92C22×D97—42no—
93C22×D97C2×D411yescharacteristic
94C22×D97C2×D411yescharacteristic
95C2×C4×C97—21no—
96C2×C4×C97—21no—
97C2×C4×C97—22no—
98C2×C4×C97—42no—
99C2×C4×C97C2×D411yescharacteristic
100C2×C4×C97#4112yes—
101C2×C4×C97#4112yes—
102C2×C4×C97C2411yescharacteristic
103C2×Dic97—22no—
104C2×Dic97—21no—
105C2×Dic97—21no—
106C2×Dic97C2×D411yescharacteristic
107C2×Dic97C2×D411yescharacteristic
108D388—22no—
109D388—22no—
110D388—44no—
111D388—44no—
112D388—42no—
113C4×D97—22no—
114C4×D97—22no—
115C4×D97—21no—
116C4×D97—21no—
117C4×D97—21no—
118C4×D97—21no—
119C4×D97—42no—
120C4×D97—42no—
121C8×C97—22no—
122#11715895—22no—
123C22×C97—21no—
124C22×C97—21no—
125C22×C97#12312yes—
126C22×C97C22×D411yescharacteristic
127C2×D97—21no—
128C2×D97—21no—
129C2×D97—21no—
130C2×D97—21no—
131C2×D97—21no—
132C2×D97—44no—
133C2×D97—42no—
134C2×D97—42no—
135C2×D97—42no—
136C2×D97—84no—
137C2×D97AGL(1,ℤ/8)12yes—
138C4×C97—41no—
139C4×C97—42no—
140C4×C97—41no—
141C4×C97#12312yes—
142C4×C97C2×#4111yescharacteristic
143C4×C97C2×#4111yescharacteristic
144Dic97—21no—
145Dic97—21no—
146Dic97—41no—
147Dic97—41no—
148C2×C97—22no—
149C2×C97C2×#12311yescharacteristic
150C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C97#57511yescharacteristic
152D97—22no—
153D97—41no—
154D97—82no—
155#3094—971nomaximal, Sylow
156C97#309411yescharacteristic, complemented, Sylow
157C2×#121—971no—
158C2×#117—971no—
159#482—972no—
160#482—972no—
161#481—972no—
162#480—972no—
163#428—971no—
164#428D9711yescharacteristic, complemented
165#418—971no—
166#418—971no—
167#407—971no—
168C22×D4—971no—
169C22×D4—1941no—
170C23×C4—971no—
171C2×M16—971no—
172C2×M16C2×D9711yescharacteristic
173#121—972no—
174#121—972no—
175#121—1941no—
176#121—1941no—
177#117—971no—
178#117—971no—
179#117—971no—
180#117—1941no—
181#117—1941no—
182#117C2×D9711yescharacteristic
183C2×C4⋊C4—971no—
184C2×C4⋊C4—971no—
185C2×C4⋊C4C2×D9711yescharacteristic
186C2×C22⋊C4—1941no—
187C2×C22⋊C4—1941no—
188C2×C42—1941no—
189#107—972no—
190#107C2×D9712yescomplemented
191#106—972no—
192#106C2×D9712yescomplemented
193#102—972no—
194#102—972no—
195#102—972no—
196#102—972no—
197C22⋊C8—972no—
198C22⋊C8—972no—
199C24—971no—
200C24—1941no—
201C2×D4—972no—
202C2×D4—3884no—
203C2×D4—1942no—
204C2×D4—1942no—
205C2×D4—1941no—
206C2×D4—1942no—
207C2×D4—1942no—
208C22×C4—21no—
209C22×C4—971no—
210C22×C4—971no—
211C22×C4—971no—
212C22×C4—971no—
213C22×C4—971no—
214C22×C4—1941no—
215C22×C4—1941no—
216C22×C4—1942no—
217C22×C4—1942no—
218C22×C4C22×D9711yescharacteristic
219M16—22no—
220M16—1942no—
221C2×C8—972no—
222C2×C8C22×D9712yes—
223C4⋊C4—21no—
224C4⋊C4—971no—
225C4⋊C4—971no—
226C4⋊C4—1941no—
227C4⋊C4—1941no—
228C4⋊C4C22×D9711yescharacteristic
229C4⋊C4C22×D9711yescharacteristic
230C4⋊C4C22×D9712yes—
231C22⋊C4—21no—
232C22⋊C4—1942no—
233C22⋊C4—1942no—
234C22⋊C4—1941no—
235C22⋊C4—1941no—
236C22⋊C4—1941no—
237C42—21no—
238C42—1941no—
239C42—1941no—
240C42—1941no—
241C23—971no—
242C23—971no—
243C23—3882no—
244C23—3882no—
245C23—3881no—
246C23—3882no—
247C23—1942no—
248C23—1942no—
249C23—1942no—
250C23D4×D9711yescharacteristic
251D4—3884no—
252D4—3884no—
253D4—3882no—
254D4—1942no—
255D4—1942no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—971no—
261C2×C4—971no—
262C2×C4—3882no—
263C2×C4—3882no—
264C2×C4—1941no—
265C2×C4—1941no—
266C2×C4—1942no—
267C2×C4—1942no—
268C2×C4—1942no—
269C2×C4—1941no—
270C2×C4—1941no—
271C2×C4—1941no—
272C2×C4—1941no—
273C2×C4D4×D9711yescharacteristic
274C2×C4#1309551112yes—
275C2×C4#1309551212yes—
276C2×C4C23×D9711yescharacteristic
277C8—22no—
278C8—1942no—
279C22—21no—
280C22—21no—
281C22—7764no—
282C22—972no—
283C22—3884no—
284C22—3882no—
285C22—3882no—
286C22—3882no—
287C22—1941no—
288C22—1941no—
289C22—1941no—
290C22—1941no—
291C22—1941no—
292C22#1569079512yes—
293C22C2×D4×D9711yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—3881no—
298C4—3881no—
299C4—1941no—
300C4—1941no—
301C4#1569079512yes—
302C4C2×#1309551111yescharacteristic
303C4C2×#1309551211yescharacteristic
304C2—22no—
305C2—7762no—
306C2—3881no—
307C2—1942no—
308C2C2×#1569079511yescharacteristic
309C2AGL(1,ℤ/8)×D9711yescharacteristic
310C2#2026153411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#428×C97C2
socleC22×C97C22×D4
prime radicalC2×#15690785C2
hypercenter#428D97
hypocenterC97#3094
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. #117×D97C2semidirect
3. #20261062C2semidirect
4. #20261061C2semidirect
5. #20261055C2semidirect
6. #20261050C2semidirect
7. #20260775C2semidirect
8. #20260741C2semidirect
9. #20260739C2semidirect
10. #20260698C2semidirect
11. #428×C97C2semidirect
21. #15690768C22semidirect
22. #15690768C22semidirect
25. C4⋊C4×D97C22semidirect
26. C4⋊C4×D97C22semidirect
30. #15690743C22semidirect
31. #15690741C22semidirect
32. #15690732C22semidirect
33. #15690732C22semidirect
34. #15690730C22semidirect
35. #15690730C22semidirect
36. #15690722C22semidirect
37. #15690722C22semidirect
41. #107×C97C22semidirect
42. #106×C97C22semidirect
156. C97#3094semidirect
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
C2×#156907852
#117×D972
#202610622
#202610612
#202610552
#202610502
#202607752
#202607412
#202607392
#202606982
#428×C972
#309497