Finite Group Databasebeta

← #27151651

Subgroups — #27151651

Order 12416.

420 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 137 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#20261246C211yesmaximal, characteristic
2C2×#15690769C211yesmaximal, characteristic, complemented
3#20261077C212yesmaximal, complemented
4#20261071C211yesmaximal, characteristic, complemented
5#20261066C211yesmaximal, characteristic, complemented
6#20261060C212yesmaximal, complemented
7#20261054C211yesmaximal, characteristic, complemented
8#20261049C211yesmaximal, characteristic, complemented
9#20260773C212yesmaximal, complemented
10#20260732C212yesmaximal, complemented
11C2×C22⋊C8×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2211yescharacteristic
13#15690793—21no—
14#15690789C2211yescharacteristic
15#15690789C2211yescharacteristic
16#15690784C2211yescharacteristic
17#15690784C2211yescharacteristic
18#15690779—21no—
19C4×C22⋊D97—21no—
20C4×C22⋊D97—21no—
21#15690769—21no—
22#15690769—21no—
23#15690769C2212yes—
24#15690769C2212yes—
25#15690769C2211yescharacteristic
26C2×C4⋊Dic97C2211yescharacteristic
27C2×C4⋊Dic97C2211yescharacteristic
28C2×C22⋊Dic97—21no—
29C2×C4×Dic97—21no—
30#15690744C2212yescomplemented
31#15690744C2212yescomplemented
32#15690742C2212yescomplemented
33#15690742C2212yescomplemented
34#15690735C2212yescomplemented
35#15690735C2212yescomplemented
36#15690732C2212yescomplemented
37#15690732C2212yescomplemented
38C23×C4×C97C2211yescharacteristic
39C22×C8×C97C2212yes—
40C22⋊C8×C97C2211yescharacteristic
41C22⋊C8×C97C2211yescharacteristic
42C22⋊C8×C97C2212yes—
43#13095511—24no—
44C2×Dic194C2311yescharacteristic
45C2×D388C2311yescharacteristic
46C2×C22⋊D97—21no—
47C2×C22⋊D97—21no—
48C2×C22⋊D97—21no—
49C22×Dic97—21no—
50C22×Dic97—21no—
51C2×C4×D97—21no—
52C4⋊Dic97—21no—
53C4⋊Dic97—21no—
54C4⋊Dic97C2312yes—
55C4⋊Dic97C2312yes—
56C4⋊Dic97C2311yescharacteristic
57C4⋊Dic97C2311yescharacteristic
58#13095490—21no—
59#13095490—21no—
60C22⋊Dic97—22no—
61C22⋊Dic97—22no—
62C22⋊Dic97—21no—
63C22⋊Dic97—21no—
64C22⋊Dic97—21no—
65#13095488—21no—
66#13095488—21no—
67C4×Dic97—22no—
68C4×Dic97—22no—
69C4×Dic97—21no—
70C24×C97D411yescharacteristic
71C22×C4×C97—24no—
72C22×C4×C97D411yescharacteristic
73C22×C4×C97D411yescharacteristic
74C22×C4×C97D411yescharacteristic
75C22×C4×C97C2311yescharacteristic
76C22×C4×C97C2311yescharacteristic
77C22×C4×C97C2311yescharacteristic
78C2×C8×C97—24no—
79C2×C8×C97C2312yes—
80C2×C8×C97C2312yes—
81C23×C97—22no—
82C23×C97—22no—
83C23×C97—41no—
84C23×C97C2×D411yescharacteristic
85C23×C97C2×D411yescharacteristic
86C23×C97C2×D411yescharacteristic
87C22×D97—21no—
88C2×C4×C97—22no—
89C2×C4×C97—22no—
90C2×C4×C97—22no—
91C2×C4×C97—22no—
92C2×C4×C97—22no—
93C2×C4×C97C2×D411yescharacteristic
94C2×C4×C97C2×D411yescharacteristic
95C2×C4×C97C2×D411yescharacteristic
96C2×C4×C97#4114yes—
97C2×C4×C97C2411yescharacteristic
98C22⋊D97—42no—
99C22⋊D97—42no—
100C22⋊D97—42no—
101C2×Dic97—22no—
102C2×Dic97—22no—
103C2×Dic97—21no—
104C2×Dic97—21no—
105C2×Dic97—21no—
106C2×Dic97—42no—
107C2×Dic97—42no—
108D388—22no—
109C4×D97—42no—
110Dic194—22no—
111C8×C97—22no—
112C8×C97—22no—
113C22×C97—21no—
114C22×C97—21no—
115C22×C97—21no—
116C22×C97—21no—
117C22×C97—21no—
118C22×C97—21no—
119C22×C97—21no—
120C22×C97—44no—
121C22×C97#12312yes—
122C22×C97#13512yes—
123C22×C97C22×D411yescharacteristic
124C2×D97—41no—
125C2×D97—41no—
126C2×D97—41no—
127C4×C97—21no—
128C4×C97—21no—
129C4×C97#12312yes—
130C4×C97C2×#4112yes—
131Dic97—41no—
132Dic97—42no—
133Dic97—41no—
134Dic97—42no—
135Dic97—41no—
136C2×C97—22no—
137C2×C97—22no—
138C2×C97—41no—
139C2×C97C2×#12311yescharacteristic
140C2×C97C2×#13511yescharacteristic
141C2×C97#57511yescharacteristic
142D97—81no—
143#3091—971nomaximal, Sylow
144C97#309111yescharacteristic, complemented, Sylow
145#525—971no—
146C2×#117—971no—
147#485—972no—
148#484—971no—
149#483—971no—
150#482—972no—
151#481—971no—
152#480—971no—
153#427—972no—
154#416—972no—
155C2×C22⋊C8D9711yescharacteristic, complemented
156C2×#41—971no—
157C23×C4C2×D9711yescharacteristic
158C22×C8C2×D9712yes—
159#123—1941no—
160C22⋊Q8—971no—
161C22⋊Q8—971no—
162#121—971no—
163#121—971no—
164#120—1941no—
165C4×D4—1941no—
166C4×D4—1941no—
167#117—972no—
168#117—972no—
169#117—971no—
170#117—1941no—
171#117—1941no—
172C2×C4⋊C4—971no—
173C2×C4⋊C4—971no—
174C2×C22⋊C4—1941no—
175C2×C42—1941no—
176#107—972no—
177#107—972no—
178#106—972no—
179#106—972no—
180#103—972no—
181#103—972no—
182#102—972no—
183#102—972no—
184C22⋊C8C2×D9711yescharacteristic
185C22⋊C8C2×D9711yescharacteristic
186C22⋊C8C2×D9712yes—
187C24D38811yescharacteristic
188#41—1944no—
189C2×Q8—971no—
190C2×D4—971no—
191C2×D4—1941no—
192C2×D4—1941no—
193C2×D4—1941no—
194C22×C4—24no—
195C22×C4—1941no—
196C22×C4—1941no—
197C22×C4—1941no—
198C22×C4D38811yescharacteristic
199C22×C4D38811yescharacteristic
200C22×C4D38811yescharacteristic
201C22×C4C22×D9711yescharacteristic
202C22×C4C22×D9711yescharacteristic
203C22×C4C22×D9711yescharacteristic
204C2×C8—24no—
205C2×C8C22×D9712yes—
206C2×C8C22×D9712yes—
207C4⋊C4—972no—
208C4⋊C4—972no—
209C4⋊C4—971no—
210C4⋊C4—971no—
211C4⋊C4—1941no—
212C4⋊C4—1941no—
213C4⋊C4—1941no—
214C4⋊C4—1941no—
215C22⋊C4—1942no—
216C22⋊C4—1942no—
217C22⋊C4—1941no—
218C22⋊C4—1941no—
219C22⋊C4—1941no—
220C22⋊C4—1941no—
221C22⋊C4—1941no—
222C42—1942no—
223C42—1942no—
224C42—1941no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—1941no—
229C23C2×D38811yescharacteristic
230C23C2×D38811yescharacteristic
231C23C2×D38811yescharacteristic
232Q8—1942no—
233D4—3882no—
234D4—3882no—
235D4—3882no—
236D4—1942no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—3882no—
243C2×C4—3882no—
244C2×C4—3882no—
245C2×C4—1942no—
246C2×C4—1942no—
247C2×C4—1941no—
248C2×C4—1941no—
249C2×C4—1941no—
250C2×C4C2×D38811yescharacteristic
251C2×C4C2×D38811yescharacteristic
252C2×C4C2×D38811yescharacteristic
253C2×C4#1309551014yes—
254C2×C4C23×D9711yescharacteristic
255C8—22no—
256C8—22no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22—3881no—
266C22—3881no—
267C22—3881no—
268C22#1569079412yes—
269C22#1569083212yes—
270C22C22×D38811yescharacteristic
271C4—21no—
272C4—21no—
273C4—3881no—
274C4—3882no—
275C4—3881no—
276C4—3882no—
277C4—3881no—
278C4#1569079412yes—
279C4C2×#1309551012yes—
280C2—22no—
281C2—22no—
282C2—41no—
283C2—7761no—
284C2C2×#1569079411yescharacteristic
285C2C2×#1569083211yescharacteristic
286C2#2026153511yescharacteristic
subgroupG/H
centerC2×C4C2×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×C22⋊C8×C97C2
socleC22×C97C22×D4
prime radical#20261246C2
hypercenterC2×C22⋊C8D97
hypocenterC97#3091
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#15690769C2semidirect
3. #20261077C2semidirect
4. #20261071C2semidirect
5. #20261066C2semidirect
6. #20261060C2semidirect
7. #20261054C2semidirect
8. #20261049C2semidirect
9. #20260773C2semidirect
10. #20260732C2semidirect
11. C2×C22⋊C8×C97C2semidirect
30. #15690744C22semidirect
31. #15690744C22semidirect
32. #15690742C22semidirect
33. #15690742C22semidirect
34. #15690735C22semidirect
35. #15690735C22semidirect
36. #15690732C22semidirect
37. #15690732C22semidirect
144. C97#3091semidirect
155. C2×C22⋊C8D97semidirect

Maximal subgroups

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

subgroupindex
#202612462
C2×#156907692
#202610772
#202610712
#202610662
#202610602
#202610542
#202610492
#202607732
#202607322
C2×C22⋊C8×C972
#309197