Finite Group Databasebeta

← #27151680

Subgroups — #27151680

Order 12416.

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

normal subgroups only

This group has 410 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
1C2×#15690769C211yesmaximal, characteristic
2#20261073C214yesmaximal, complemented
3#20261068C212yesmaximal, complemented
4#20261063C212yesmaximal, complemented
5#20260775C212yesmaximal, complemented
6#20260744C212yesmaximal, complemented
7#20260699C211yesmaximal, characteristic
8C2×C22⋊Q8×C97C211yesmaximal, characteristic
9C2×#13095495C2212yes—
10#15690769—21no—
11#15690769—21no—
12#15690769C2212yes—
13#15690769C2212yes—
14C2×C4⋊Dic97C2211yescharacteristic
15C2×C4⋊Dic97C2211yescharacteristic
16C2×C22⋊Dic97—21no—
17C2×C4×Dic97—21no—
18#15690743C2214yescomplemented
19#15690741C2214yescomplemented
20#15690734C2214yescomplemented
21#15690734C2214yescomplemented
22C22⋊#11715895C2212yes—
23C22⋊#11715895C2212yes—
24C22×Q8×C97C2211yescharacteristic
25C23×C4×C97C2211yescharacteristic
26C22⋊Q8×C97—21no—
27C22⋊Q8×C97—21no—
28C22⋊Q8×C97C2212yes—
29C22⋊Q8×C97C2212yes—
30C2×C4⋊C4×C97—21no—
31C2×C4⋊C4×C97C2211yescharacteristic
32C2×C22⋊C4×C97—21no—
33C22×Dic97—21no—
34C22×Dic97—21no—
35#13095495—24no—
36C2×#11715895C2314yes—
37C4⋊Dic97—21no—
38C4⋊Dic97—21no—
39C4⋊Dic97C2312yes—
40C4⋊Dic97C2312yes—
41C22⋊Dic97—22no—
42C22⋊Dic97—22no—
43C4×Dic97—22no—
44C4×Dic97—22no—
45C24×C97D411yescharacteristic
46C2×Q8×C97—21no—
47C2×Q8×C97—24no—
48C2×Q8×C97C2312yes—
49C22×C4×C97—24no—
50C22×C4×C97—21no—
51C22×C4×C97—21no—
52C22×C4×C97D411yescharacteristic
53C22×C4×C97D411yescharacteristic
54C22×C4×C97D411yescharacteristic
55C22×C4×C97C2311yescharacteristic
56C22×C4×C97C2311yescharacteristic
57C22×C4×C97C2311yescharacteristic
58C4⋊C4×C97—22no—
59C4⋊C4×C97—22no—
60C4⋊C4×C97—21no—
61C4⋊C4×C97C2312yes—
62C22⋊C4×C97—22no—
63C22⋊C4×C97—22no—
64C23×C97—22no—
65C23×C97—22no—
66C23×C97—41no—
67C23×C97C2×D411yescharacteristic
68C23×C97C2×D411yescharacteristic
69C23×C97C2×D411yescharacteristic
70Q8×C97—24no—
71Q8×C97—42no—
72C2×C4×C97—22no—
73C2×C4×C97—22no—
74C2×C4×C97—22no—
75C2×C4×C97—22no—
76C2×C4×C97—22no—
77C2×C4×C97—22no—
78C2×C4×C97—22no—
79C2×C4×C97—42no—
80C2×C4×C97—42no—
81C2×C4×C97C2×D411yescharacteristic
82C2×C4×C97C2×D411yescharacteristic
83C2×C4×C97C2×D411yescharacteristic
84C2×C4×C97#4114yes—
85C2×C4×C97C2411yescharacteristic
86C2×Dic97—22no—
87C2×Dic97—22no—
88C2×Dic97—42no—
89C2×Dic97—42no—
90#11715895—24no—
91C22×C97—21no—
92C22×C97—21no—
93C22×C97—21no—
94C22×C97—21no—
95C22×C97—21no—
96C22×C97—21no—
97C22×C97—21no—
98C22×C97—44no—
99C22×C97#12312yes—
100C22×C97#13712yes—
101C22×C97C22×D411yescharacteristic
102C4×C97—21no—
103C4×C97—21no—
104C4×C97—42no—
105C4×C97—42no—
106C4×C97#12312yes—
107C4×C97C2×#4112yes—
108Dic97—42no—
109Dic97—42no—
110C2×C97—22no—
111C2×C97—22no—
112C2×C97—41no—
113C2×C97C2×#12311yescharacteristic
114C2×C97C2×#13711yescharacteristic
115C2×C97#57511yescharacteristic
116#3095—971nomaximal, Sylow
117C97#309511yescharacteristic, complemented, Sylow
118C2×C22⋊Q8D9711yescharacteristic
119C2×#117—971no—
120#485—974no—
121#484—972no—
122#483—972no—
123#428—972no—
124#419—972no—
125#407—971no—
126C22×Q8C2×D9711yescharacteristic
127C23×C4C2×D9711yescharacteristic
128C2×M16—972no—
129C22⋊Q8—21no—
130C22⋊Q8—21no—
131C22⋊Q8C2×D9712yes—
132C22⋊Q8C2×D9712yes—
133#117—972no—
134#117—972no—
135#117—1941no—
136#117—1941no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—971no—
139C2×C4⋊C4—971no—
140C2×C4⋊C4C2×D9711yescharacteristic
141C2×C22⋊C4—21no—
142C2×C22⋊C4—1941no—
143C2×C42—1941no—
144#107—974no—
145#106—974no—
146#103—974no—
147#103—974no—
148C22⋊C8—972no—
149C22⋊C8—972no—
150C24C22⋊D9711yescharacteristic
151C2×Q8—21no—
152C2×Q8—24no—
153C2×Q8C22×D9712yes—
154C22×C4—24no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—1941no—
158C22×C4—1941no—
159C22×C4C22⋊D9711yescharacteristic
160C22×C4C22⋊D9711yescharacteristic
161C22×C4C22⋊D9711yescharacteristic
162C22×C4C22×D9711yescharacteristic
163C22×C4C22×D9711yescharacteristic
164C22×C4C22×D9711yescharacteristic
165M16—1944no—
166C2×C8—974no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—21no—
170C4⋊C4—972no—
171C4⋊C4—972no—
172C4⋊C4—1941no—
173C4⋊C4—1941no—
174C4⋊C4C22×D9712yes—
175C22⋊C4—22no—
176C22⋊C4—22no—
177C22⋊C4—1942no—
178C22⋊C4—1942no—
179C42—1942no—
180C42—1942no—
181C23—22no—
182C23—22no—
183C23—41no—
184C23C2×C22⋊D9711yescharacteristic
185C23C2×C22⋊D9711yescharacteristic
186C23C2×C22⋊D9711yescharacteristic
187Q8—24no—
188Q8—42no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—3882no—
199C2×C4—3882no—
200C2×C4—1942no—
201C2×C4—1942no—
202C2×C4C2×C22⋊D9711yescharacteristic
203C2×C4C2×C22⋊D9711yescharacteristic
204C2×C4C2×C22⋊D9711yescharacteristic
205C2×C4#1309551014yes—
206C2×C4C23×D9711yescharacteristic
207C8—1944no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—44no—
216C22#1569079112yes—
217C22#1569084112yes—
218C22C22×C22⋊D9711yescharacteristic
219C4—21no—
220C4—21no—
221C4—42no—
222C4—42no—
223C4—3882no—
224C4—3882no—
225C4#1569079112yes—
226C4C2×#1309551012yes—
227C2—22no—
228C2—22no—
229C2—41no—
230C2C2×#1569079111yescharacteristic
231C2C2×#1569084111yescharacteristic
232C2#2026153311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×C22⋊Q8×C97C2
socleC22×C97C22×D4
prime radicalC24×C97D4
hypercenterC2×C22⋊Q8D97
hypocenterC97#3095
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #20261073C2semidirect
3. #20261068C2semidirect
4. #20261063C2semidirect
5. #20260775C2semidirect
6. #20260744C2semidirect
18. #15690743C22semidirect
19. #15690741C22semidirect
20. #15690734C22semidirect
21. #15690734C22semidirect
117. C97#3095semidirect

Maximal subgroups

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

subgroupindex
C2×#156907692
#202610732
#202610682
#202610632
#202607752
#202607442
#202606992
C2×C22⋊Q8×C972
#309597