Finite Group Databasebeta

← #27151525

Subgroups — #27151525

Order 12416.

388 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 131 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#20261290C211yesmaximal, characteristic, complemented
2C4×#13095510C211yesmaximal, characteristic
3#20261045C212yesmaximal, complemented
4#20261037C212yesmaximal, complemented
5#20261027C211yesmaximal, characteristic, complemented
6#20261021C211yesmaximal, characteristic, complemented
7#20261015C211yesmaximal, characteristic, complemented
8#20260770C212yesmaximal, complemented
9C4⋊C8×D97C211yesmaximal, characteristic, complemented
10#20260728C212yesmaximal
11#476×C97C211yesmaximal, characteristic, complemented
12C2×#13095510C2211yescharacteristic
13C2×C8×D97C2212yes—
14#15690810C2211yescharacteristic
15#15690801C2212yes—
16#15690790—21no—
17#15690786—21no—
18Q8×Dic97—21no—
19C4×Dic194C2211yescharacteristic
20#15690775—21no—
21#15690775C2211yescharacteristic
22D4×Dic97C2211yescharacteristic
23C4×C22⋊D97—21no—
24#15690770—21no—
25#15690768C2211yescharacteristic
26#15690768C2212yes—
27#15690766—21no—
28C2×C4×Dic97—21no—
29C42×D97C2211yescharacteristic
30#15690743C2212yes—
31#15690743C2212yes—
32#15690742C2212yes—
33C4⋊#11715895C2211yescharacteristic
34#15690739C2212yes—
35#15690735C2212yes—
36#15690733C2212yes—
37#15690731C2212yes—
38C4⋊Q8×C97C2211yescharacteristic
39C4×D4×C97C2211yescharacteristic
40#106×C97C2212yes—
41C4⋊C8×C97C2211yescharacteristic
42#102×C97C2212yes—
43#13095510—22no—
44#13095510Q814yescomplemented
45C2×Dic194—21no—
46C2×Dic194C2311yescharacteristic
47C2×C22⋊D97—21no—
48C22×Dic97—21no—
49C22×Dic97—21no—
50C2×C4×D97D411yescharacteristic
51C2×C4×D97D411yescharacteristic
52C2×C4×D97C2311yescharacteristic
53C2×#11715895C2312yes—
54C8×D97—24no—
55C4⋊Dic97—21no—
56C4⋊Dic97—22no—
57C4⋊Dic97C2311yescharacteristic
58C4⋊Dic97C2312yes—
59#13095490—21no—
60#13095490—21no—
61#13095490—22no—
62C22⋊Dic97—21no—
63#13095488—21no—
64#13095488—22no—
65C4×Dic97—21no—
66C4×Dic97—21no—
67C4×Dic97—21no—
68C4×Dic97—22no—
69C4×Dic97D411yescharacteristic
70C4×Dic97D411yescharacteristic
71C4×Dic97C2311yescharacteristic
72C2×Q8×C97—21no—
73C2×D4×C97C2311yescharacteristic
74C22×C4×C97—21no—
75C2×C8×C97C2312yes—
76C4⋊C4×C97—21no—
77C4⋊C4×C97C2311yescharacteristic
78C4⋊C4×C97C2312yes—
79C22⋊C4×C97—21no—
80C42×C97C2311yescharacteristic
81C23×C97—21no—
82C22×D97C2×D411yescharacteristic
83Q8×C97—42no—
84D4×C97—21no—
85D4×C97C2×Q812yes—
86C2×C4×C97—21no—
87C2×C4×C97—22no—
88C2×C4×C97—42no—
89C2×C4×C97C2×D411yescharacteristic
90C2×C4×C97C2×D411yescharacteristic
91C2×C4×C97C2411yescharacteristic
92C22⋊D97—24no—
93C2×Dic97—21no—
94C2×Dic97—24no—
95C2×Dic97—21no—
96C2×Dic97—22no—
97C2×Dic97—42no—
98C2×Dic97C2×D411yescharacteristic
99C2×Dic97C2×D411yescharacteristic
100C2×Dic97C2×D411yescharacteristic
101C4×D97—22no—
102C4×D97—41no—
103C4×D97C2×Q812yes—
104C4×D97#4112yes—
105Dic194—21no—
106Dic194—42no—
107Dic194C2×Q812yes—
108C8×C97—22no—
109#11715895—22no—
110C22×C97—22no—
111C22×C97—41no—
112C22×C97—41no—
113C22×C97C22×D411yescharacteristic
114C2×D97—21no—
115C2×D97—21no—
116C2×D97C22⋊Q812yes—
117C4×C97—21no—
118C4×C97—41no—
119C4×C97—42no—
120C4×C97#13512yes—
121C4×C97C22×Q811yescharacteristic
122C4×C97C2×#4111yescharacteristic
123Dic97—22no—
124Dic97—21no—
125Dic97—21no—
126Dic97—42no—
127Dic97—41no—
128Dic97C22⋊Q812yes—
129C2×C97—42no—
130C2×C97C2×C22⋊Q811yescharacteristic
131C2×C97C2×#13511yescharacteristic
132C2×C97#57511yescharacteristic
133D97—22no—
134#3076—971nomaximal, Sylow
135C97#307611yescharacteristic, complemented, Sylow
136#533—971no—
137C4×#41—971no—
138#479—972no—
139#478—972no—
140#477—971no—
141#476D9711yescharacteristic, complemented
142#475—971no—
143#474—971no—
144#427—972no—
145C2×C4⋊C8—971no—
146#416—972no—
147C2×#41—971no—
148C22×C8—972no—
149C4⋊Q8—971no—
150C4⋊Q8C2×D9711yescharacteristic
151#125—972no—
152C22⋊Q8—1941no—
153C22⋊Q8—1941no—
154C4×Q8—971no—
155C4×Q8—971no—
156C4×Q8—1941no—
157C4×Q8—1941no—
158C4×D4—971no—
159C4×D4—1941no—
160C4×D4—1941no—
161C4×D4C2×D9711yescharacteristic
162#117—971no—
163#117—972no—
164#117—1941no—
165C2×C42—971no—
166C2×C42—1941no—
167#107—972no—
168#107—972no—
169#106—972no—
170#106C2×D9712yes—
171C4⋊C8—972no—
172C4⋊C8—971no—
173C4⋊C8C2×D9711yescharacteristic
174#103—972no—
175#103—972no—
176#102—972no—
177#102C2×D9712yes—
178#41—974no—
179#41—1942no—
180C2×Q8—21no—
181C2×Q8—971no—
182C2×Q8—1941no—
183C2×D4—1941no—
184C2×D4C22×D9711yescharacteristic
185C22×C4—21no—
186C22×C4—971no—
187C22×C4—971no—
188C22×C4—971no—
189C22×C4—1941no—
190C22×C4—1941no—
191C2×C8—972no—
192C2×C8—1944no—
193C2×C8C22×D9712yes—
194C4⋊C4—21no—
195C4⋊C4—972no—
196C4⋊C4—971no—
197C4⋊C4—1941no—
198C4⋊C4—1942no—
199C4⋊C4—1942no—
200C4⋊C4—1941no—
201C4⋊C4—1941no—
202C4⋊C4C22×D9712yes—
203C4⋊C4C22×D9711yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—1941no—
206C22⋊C4—1942no—
207C22⋊C4—1941no—
208C42—971no—
209C42—971no—
210C42—971no—
211C42—1941no—
212C42—1941no—
213C42—1942no—
214C42—1941no—
215C42C22×D9711yescharacteristic
216C23—21no—
217C23—971no—
218Q8—42no—
219Q8—972no—
220Q8—3882no—
221Q8—1941no—
222D4—21no—
223D4—1944no—
224D4Q8×D9712yes—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—971no—
229C2×C4—972no—
230C2×C4—972no—
231C2×C4—971no—
232C2×C4—971no—
233C2×C4—3881no—
234C2×C4—3882no—
235C2×C4—1944no—
236C2×C4—1941no—
237C2×C4—1942no—
238C2×C4—1941no—
239C2×C4—1942no—
240C2×C4D4×D9711yescharacteristic
241C2×C4D4×D9711yescharacteristic
242C2×C4C23×D9711yescharacteristic
243C8—22no—
244C8—1942no—
245C22—22no—
246C22—41no—
247C22—41no—
248C22—972no—
249C22—1941no—
250C22—1941no—
251C22C2×D4×D9711yescharacteristic
252C4—21no—
253C4—41no—
254C4—42no—
255C4—972no—
256C4—3882no—
257C4—3881no—
258C4—1942no—
259C4—1941no—
260C4—1941no—
261C4#1569083112yes—
262C4C2×Q8×D9711yescharacteristic
263C4#41×D9711yescharacteristic
264C2—42no—
265C2—1942no—
266C2C22⋊Q8×D9711yescharacteristic
267C2C2×#1569083111yescharacteristic
268C2#2026153111yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#476×C97C2
socleC22×C97C22×D4
prime radicalC2×#13095510C22
hypercenter#476D97
hypocenterC97#3076
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #20261290C2semidirect
3. #20261045C2semidirect
4. #20261037C2semidirect
5. #20261027C2semidirect
6. #20261021C2semidirect
7. #20261015C2semidirect
8. #20260770C2semidirect
9. C4⋊C8×D97C2semidirect
11. #476×C97C2semidirect
44. #13095510Q8semidirect
135. C97#3076semidirect
141. #476D97semidirect

Maximal subgroups

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

subgroupindex
#202612902
C4×#130955102
#202610452
#202610372
#202610272
#202610212
#202610152
#202607702
C4⋊C8×D972
#202607282
#476×C972
#307697