Finite Group Databasebeta

← #27152278

Subgroups — #27152278

Order 12416.

380 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 157 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#20261292C211yesmaximal, characteristic
2#20261141C214yesmaximal, complemented
3#20261138C212yesmaximal, complemented
4#20261131C212yesmaximal, complemented
5#20260775C214yesmaximal
6C4×#13095495C211yesmaximal, characteristic
7C2×C4⋊Q8×C97C211yesmaximal, characteristic
8C2×#13095495C2212yes—
9#15690810C2211yescharacteristic
10#15690810C2211yescharacteristic
11#15690802C2212yes—
12#15690789—21no—
13#15690789—21no—
14C4×Dic194—21no—
15C4×Dic194—21no—
16#15690769C2212yes—
17#15690743C2218yes—
18#15690741C2218yes—
19#15690721C2212yes—
20C4×#11715895C2212yes—
21C22×Q8×C97—21no—
22C4⋊Q8×C97—21no—
23C4⋊Q8×C97—21no—
24C4⋊Q8×C97C2212yes—
25C4⋊Q8×C97C2212yes—
26C2×C4⋊C4×C97—21no—
27C2×C4⋊C4×C97C2212yes—
28C2×C42×C97C2211yescharacteristic
29C2×Dic194—21no—
30C2×Dic194—21no—
31#13095495Q818yescomplemented
32C2×#11715895C2314yes—
33C4⋊Dic97—21no—
34C4⋊Dic97—21no—
35C4⋊Dic97C2312yes—
36C4⋊Dic97C2312yes—
37#13095490—22no—
38#13095490—22no—
39C22⋊Dic97—22no—
40C4×Dic97—22no—
41C2×Q8×C97—22no—
42C2×Q8×C97—22no—
43C2×Q8×C97—44no—
44C22×C4×C97—22no—
45C22×C4×C97D411yescharacteristic
46C22×C4×C97D411yescharacteristic
47C22×C4×C97C2311yescharacteristic
48C4⋊C4×C97—22no—
49C4⋊C4×C97—22no—
50C4⋊C4×C97—22no—
51C4⋊C4×C97C2314yes—
52C42×C97D411yescharacteristic
53C42×C97D411yescharacteristic
54C42×C97C2311yescharacteristic
55C42×C97C2311yescharacteristic
56C23×C97C2×D411yescharacteristic
57Q8×C97—44no—
58Q8×C97—44no—
59C2×C4×C97—22no—
60C2×C4×C97—22no—
61C2×C4×C97—24no—
62C2×C4×C97—44no—
63C2×C4×C97C2×D411yescharacteristic
64C2×C4×C97C2×D411yescharacteristic
65C2×C4×C97C2×D411yescharacteristic
66C2×C4×C97C2×D411yescharacteristic
67C2×C4×C97C2×D411yescharacteristic
68C2×C4×C97C2×Q814yes—
69C2×C4×C97C2411yescharacteristic
70C2×Dic97—22no—
71C2×Dic97—22no—
72Dic194—42no—
73Dic194—42no—
74#11715895C2×Q818yescomplemented
75C22×C97—21no—
76C22×C97—21no—
77C22×C97C4⋊Q812yes—
78C22×C97C22×D411yescharacteristic
79C4×C97—21no—
80C4×C97—21no—
81C4×C97—44no—
82C4×C97C4⋊Q812yes—
83C4×C97AGL(1,ℤ/8)12yes—
84C4×C97#13712yes—
85C4×C97C22×Q812yes—
86Dic97—42no—
87Dic97—42no—
88C2×C97—22no—
89C2×C97C2×C4⋊Q811yescharacteristic
90C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
91C2×C97C2×#13711yescharacteristic
92#3165—971nomaximal, Sylow
93C97#316511yescharacteristic, complemented, Sylow
94#533—971no—
95C2×C4⋊Q8D9711yescharacteristic
96#501—974no—
97#500—972no—
98#498—972no—
99#428—974no—
100C4×M16—971no—
101C22×Q8—21no—
102C2×M16—972no—
103C4⋊Q8—21no—
104C4⋊Q8—21no—
105C4⋊Q8—971no—
106C4⋊Q8—971no—
107C4⋊Q8C2×D9712yes—
108C4⋊Q8C2×D9712yes—
109#125—972no—
110C22⋊Q8—1941no—
111C22⋊Q8—1941no—
112C4×Q8—1941no—
113C4×Q8—1941no—
114#117—972no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4C2×D9712yes—
117C2×C42C2×D9711yescharacteristic
118#107—978no—
119#106—978no—
120#97—972no—
121C4×C8—972no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—44no—
125C2×Q8—1941no—
126C2×Q8—1941no—
127C22×C4—22no—
128C22×C4C22⋊D9711yescharacteristic
129C22×C4C22⋊D9711yescharacteristic
130C22×C4C22×D9711yescharacteristic
131M16—978no—
132C2×C8—974no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—972no—
137C4⋊C4—972no—
138C4⋊C4—1942no—
139C4⋊C4—1942no—
140C4⋊C4—1941no—
141C4⋊C4—1941no—
142C4⋊C4C22×D9714yes—
143C22⋊C4—1942no—
144C42—1942no—
145C42C22⋊D9711yescharacteristic
146C42C22⋊D9711yescharacteristic
147C42C22×D9711yescharacteristic
148C42C22×D9711yescharacteristic
149C23C2×C22⋊D9711yescharacteristic
150Q8—44no—
151Q8—44no—
152Q8—3882no—
153Q8—3882no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—44no—
158C2×C4—1942no—
159C2×C4—1942no—
160C2×C4C2×C22⋊D9711yescharacteristic
161C2×C4C2×C22⋊D9711yescharacteristic
162C2×C4C2×C22⋊D9711yescharacteristic
163C2×C4C2×C22⋊D9711yescharacteristic
164C2×C4C2×C22⋊D9711yescharacteristic
165C2×C4Q8×D9714yes—
166C2×C4C23×D9711yescharacteristic
167C8—978no—
168C22—21no—
169C22—21no—
170C22#1569080812yes—
171C22C22×C22⋊D9711yescharacteristic
172C4—21no—
173C4—21no—
174C4—44no—
175C4—3882no—
176C4—3882no—
177C4#1569080812yes—
178C4#1569083612yes—
179C4#1569084112yes—
180C4C2×Q8×D9712yes—
181C2—22no—
182C2C2×#1569080811yescharacteristic
183C2C2×#1569083611yescharacteristic
184C2C2×#1569084111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×C4⋊Q8×C97C2
socleC22×C97C22×D4
prime radicalC23×C97C2×D4
hypercenterC2×C4⋊Q8D97
hypocenterC97#3165
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #20261141C2semidirect
3. #20261138C2semidirect
4. #20261131C2semidirect
31. #13095495Q8semidirect
74. #11715895C2×Q8semidirect
93. C97#3165semidirect

Maximal subgroups

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

subgroupindex
#202612922
#202611412
#202611382
#202611312
#202607752
C4×#130954952
C2×C4⋊Q8×C972
#316597