Finite Group Databasebeta

← #27151649

Subgroups — #27151649

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
2#20261076C212yesmaximal, complemented
3#20261070C211yesmaximal, characteristic, complemented
4#20261065C211yesmaximal, characteristic, complemented
5#20261059C212yesmaximal, complemented
6#20261053C211yesmaximal, characteristic, complemented
7#20261048C211yesmaximal, characteristic, complemented
8#20260771C212yesmaximal, complemented
9#20260729C212yesmaximal, complemented
10C2×C22⋊#11715895C211yesmaximal, characteristic, complemented
11C2×#117×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2211yescharacteristic
13C22×#11715895C2212yes—
14#15690793—21no—
15#15690789C2211yescharacteristic
16#15690789C2211yescharacteristic
17#15690784C2211yescharacteristic
18#15690784C2211yescharacteristic
19#15690779—21no—
20C4×C22⋊D97—21no—
21C4×C22⋊D97—21no—
22#15690769C2211yescharacteristic
23#15690743C2212yescomplemented
24#15690743C2212yescomplemented
25#15690741C2212yescomplemented
26#15690741C2212yescomplemented
27#15690733C2212yescomplemented
28#15690733C2212yescomplemented
29#15690730C2212yescomplemented
30#15690730C2212yescomplemented
31C22⋊#11715895C2211yescharacteristic
32C22⋊#11715895C2211yescharacteristic
33C22⋊#11715895C2212yes—
34C23×C4×C97C2211yescharacteristic
35#117×C97—21no—
36#117×C97—21no—
37#117×C97C2212yes—
38#117×C97C2212yes—
39C2×C4⋊C4×C97C2211yescharacteristic
40C2×C4⋊C4×C97C2211yescharacteristic
41C2×C22⋊C4×C97—21no—
42C2×C42×C97—21no—
43#13095511—24no—
44C2×Dic194C2311yescharacteristic
45C2×D388C2311yescharacteristic
46C2×C22⋊D97—21no—
47C2×C22⋊D97—21no—
48C2×C22⋊D97—21no—
49C2×C4×D97—21no—
50C2×#11715895—24no—
51C2×#11715895C2312yes—
52C2×#11715895C2312yes—
53C4⋊Dic97C2311yescharacteristic
54C4⋊Dic97C2311yescharacteristic
55#13095490—21no—
56#13095490—21no—
57C22⋊Dic97—21no—
58C22⋊Dic97—21no—
59C22⋊Dic97—21no—
60#13095488—21no—
61#13095488—21no—
62C4×Dic97—21no—
63C24×C97D411yescharacteristic
64C22×C4×C97—24no—
65C22×C4×C97—21no—
66C22×C4×C97—21no—
67C22×C4×C97D411yescharacteristic
68C22×C4×C97D411yescharacteristic
69C22×C4×C97D411yescharacteristic
70C22×C4×C97C2311yescharacteristic
71C22×C4×C97C2311yescharacteristic
72C22×C4×C97C2311yescharacteristic
73C4⋊C4×C97—21no—
74C4⋊C4×C97—21no—
75C4⋊C4×C97C2312yes—
76C4⋊C4×C97C2312yes—
77C22⋊C4×C97—22no—
78C22⋊C4×C97—22no—
79C42×C97—22no—
80C42×C97—22no—
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×C97—22no—
94C2×C4×C97—22no—
95C2×C4×C97—42no—
96C2×C4×C97—42no—
97C2×C4×C97C2×D411yescharacteristic
98C2×C4×C97C2×D411yescharacteristic
99C2×C4×C97C2×D411yescharacteristic
100C2×C4×C97#4114yes—
101C2×C4×C97C2411yescharacteristic
102C22⋊D97—42no—
103C22⋊D97—42no—
104C22⋊D97—42no—
105C2×Dic97—21no—
106C2×Dic97—21no—
107C2×Dic97—21no—
108D388—22no—
109C4×D97—42no—
110Dic194—22no—
111#11715895—22no—
112#11715895—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—42no—
130C4×C97—42no—
131C4×C97#12312yes—
132C4×C97C2×#4112yes—
133Dic97—41no—
134Dic97—41no—
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×#117D9711yescharacteristic, complemented
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⋊C8—971no—
156C2×#41—971no—
157C23×C4C2×D9711yescharacteristic
158C22×C8—972no—
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—21no—
168#117—21no—
169#117—971no—
170#117C2×D9712yes—
171#117C2×D9712yes—
172C2×C4⋊C4C2×D9711yescharacteristic
173C2×C4⋊C4C2×D9711yescharacteristic
174C2×C22⋊C4—21no—
175C2×C42—21no—
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⋊C8—971no—
185C22⋊C8—971no—
186C22⋊C8—972no—
187C24C22⋊D9711yescharacteristic
188#41—1944no—
189C2×Q8—971no—
190C2×D4—971no—
191C2×D4—1941no—
192C2×D4—1941no—
193C2×D4—1941no—
194C22×C4—24no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—1941no—
198C22×C4C22⋊D9711yescharacteristic
199C22×C4C22⋊D9711yescharacteristic
200C22×C4C22⋊D9711yescharacteristic
201C22×C4C22×D9711yescharacteristic
202C22×C4C22×D9711yescharacteristic
203C22×C4C22×D9711yescharacteristic
204C2×C8—972no—
205C2×C8—972no—
206C2×C8—1944no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—971no—
210C4⋊C4—971no—
211C4⋊C4—1941no—
212C4⋊C4—1941no—
213C4⋊C4C22×D9712yes—
214C4⋊C4C22×D9712yes—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—1941no—
218C22⋊C4—1941no—
219C22⋊C4—1941no—
220C22⋊C4—1941no—
221C22⋊C4—1941no—
222C42—22no—
223C42—22no—
224C42—1941no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—1941no—
229C23C2×C22⋊D9711yescharacteristic
230C23C2×C22⋊D9711yescharacteristic
231C23C2×C22⋊D9711yescharacteristic
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—22no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—3882no—
247C2×C4—1941no—
248C2×C4—1941no—
249C2×C4—1941no—
250C2×C4C2×C22⋊D9711yescharacteristic
251C2×C4C2×C22⋊D9711yescharacteristic
252C2×C4C2×C22⋊D9711yescharacteristic
253C2×C4#1309551114yes—
254C2×C4C23×D9711yescharacteristic
255C8—1942no—
256C8—1942no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22—3881no—
266C22—3881no—
267C22—3881no—
268C22#1569079312yes—
269C22#1569083012yes—
270C22C22×C22⋊D9711yescharacteristic
271C4—21no—
272C4—21no—
273C4—42no—
274C4—42no—
275C4—3881no—
276C4—3881no—
277C4—3881no—
278C4#1569079312yes—
279C4C2×#1309551112yes—
280C2—22no—
281C2—22no—
282C2—41no—
283C2—7761no—
284C2C2×#1569079311yescharacteristic
285C2C2×#1569083011yescharacteristic
286C2#2026153311yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#117×C97C2
socleC22×C97C22×D4
prime radical#20261246C2
hypercenterC2×#117D97
hypocenterC97#3091
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #20261076C2semidirect
3. #20261070C2semidirect
4. #20261065C2semidirect
5. #20261059C2semidirect
6. #20261053C2semidirect
7. #20261048C2semidirect
8. #20260771C2semidirect
9. #20260729C2semidirect
10. C2×C22⋊#11715895C2semidirect
11. C2×#117×C97C2semidirect
23. #15690743C22semidirect
24. #15690743C22semidirect
25. #15690741C22semidirect
26. #15690741C22semidirect
27. #15690733C22semidirect
28. #15690733C22semidirect
29. #15690730C22semidirect
30. #15690730C22semidirect
144. C97#3091semidirect
146. C2×#117D97semidirect

Maximal subgroups

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

subgroupindex
#202612462
#202610762
#202610702
#202610652
#202610592
#202610532
#202610482
#202607712
#202607292
C2×C22⋊#117158952
C2×#117×C972
#309197