Finite Group Databasebeta

← #27157439

Subgroups — #27157439

Order 12416.

432 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 138 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×D97⋊M16C212yesmaximal
2#20261615C214yesmaximal, complemented
3C2×C4⋊C4×D97C211yesmaximal, characteristic
4C2×#13095521—22no—
5D97⋊M16—24no—
6D97⋊M16C418yescomplemented
7C2×#13095520C2214yes—
8C2×#13095519—22no—
9C22×C4×D97C412yes—
10C22×C4×D97C2211yescharacteristic
11C2×C4⋊Dic97C411yescharacteristic
12C2×#13095490C2212yes—
13C4⋊C4×D97—22no—
14C4⋊C4×D97—22no—
15C2×C4⋊C4×C97C411yescharacteristic
16#13095521—28no—
17#13095520—24no—
18#13095520C2×C418yescomplemented
19#13095519—28no—
20C2×#11715897—24no—
21C23×D97C2×C411yescharacteristic
22C22×Dic97C2×C412yes—
23C22×Dic97C2311yescharacteristic
24C2×C4×D97—22no—
25C2×C4×D97—22no—
26C2×C4×D97—44no—
27C2×C4×D97C2×C414yes—
28C2×C4×D97D412yes—
29C2×C4×D97D412yes—
30C2×C4×D97D412yes—
31C2×C4×D97Q812yes—
32C4⋊Dic97—22no—
33#13095490—24no—
34C22×C4×C97C2×C412yes—
35C22×C4×C97C2×C411yescharacteristic
36C4⋊C4×C97—22no—
37C23×C97C22×C411yescharacteristic
38C22×D97—22no—
39C22×D97—22no—
40C22×D97C4212yes—
41C22×D97C22⋊C411yescharacteristic
42C22×D97C22⋊C411yescharacteristic
43C22×D97C4⋊C412yes—
44C2×C4×C97—22no—
45C2×C4×C97—42no—
46C2×C4×C97C4212yes—
47C2×C4×C97C22⋊C412yes—
48C2×C4×C97C4⋊C412yes—
49C2×Dic97—22no—
50C2×Dic97—42no—
51C2×Dic97C22×C412yes—
52C2×Dic97C2×D411yescharacteristic
53C2×Dic97C2×D412yes—
54C2×Dic97C2×Q811yescharacteristic
55C4×D97—44no—
56C4×D97—44no—
57C4×D97C22⋊C414yes—
58C4×D97C22⋊C414yes—
59C4×D97C4⋊C414yes—
60C4×D97C4⋊C414yes—
61#11715897—28no—
62C22×C97—21no—
63C22×C97—21no—
64C22×C97C2×C4211yescharacteristic
65C22×C97C2×C22⋊C411yescharacteristic
66C22×C97C2×C4⋊C411yescharacteristic
67C2×D97—21no—
68C2×D97—21no—
69C2×D97—24no—
70C2×D97—24no—
71C2×D97#9512yes—
72C2×D97#9512yes—
73C2×D97#10012yes—
74C2×D97#10112yes—
75C4×C97—42no—
76C4×C97#9514yes—
77Dic97—42no—
78Dic97C2×C22⋊C412yes—
79Dic97C2×C4⋊C412yes—
80C2×C97—21no—
81C2×C97—21no—
82C2×C97C2×#9511yescharacteristic
83C2×C97C2×#10011yescharacteristic
84C2×C97C2×#10111yescharacteristic
85D97—22no—
86D97#34314yescomplemented
87C2×#343—971nomaximal, Sylow
88C97C2×#34311yescharacteristic, complemented, Sylow
89C22×M16—972no—
90C22×C4⋊C4—971no—
91#343—974no—
92C23×C4—971no—
93C23×C4—972no—
94C2×M16—974no—
95C2×M16—978no—
96C2×M16—1944no—
97C2×M16—1942no—
98C22×C8—1942no—
99C2×C4⋊C4—972no—
100C2×C4⋊C4—971no—
101C2×C4⋊C4—1942no—
102C2×C4⋊C4—1942no—
103C2×C4⋊C4#8760411yescharacteristic
104C24—971no—
105C22×C4—971no—
106C22×C4—974no—
107C22×C4—972no—
108C22×C4—972no—
109C22×C4—972no—
110C22×C4—972no—
111C22×C4—972no—
112C22×C4—3884no—
113C22×C4—1942no—
114C22×C4—1942no—
115C22×C4C2×#8760412yes—
116C22×C4C2×#8760411yescharacteristic
117M16—978no—
118M16—1944no—
119M16—1948no—
120C2×C8—1944no—
121C2×C8—1948no—
122C4⋊C4—22no—
123C4⋊C4—1944no—
124C4⋊C4—1942no—
125C23—971no—
126C23—972no—
127C23—971no—
128C23—972no—
129C23—1942no—
130C23—1942no—
131C23C22×#8760411yescharacteristic
132C2×C4—22no—
133C2×C4—42no—
134C2×C4—972no—
135C2×C4—972no—
136C2×C4—971no—
137C2×C4—971no—
138C2×C4—974no—
139C2×C4—974no—
140C2×C4—974no—
141C2×C4—974no—
142C2×C4—3884no—
143C2×C4—3884no—
144C2×C4—3882no—
145C2×C4—1942no—
146C2×C4C4×#8760412yes—
147C2×C4C22⋊#8760412yes—
148C2×C4C4⋊#8760412yes—
149C8—1948no—
150C22—21no—
151C22—21no—
152C22—972no—
153C22—972no—
154C22—972no—
155C22—972no—
156C22—1944no—
157C22—1944no—
158C22—1941no—
159C22—1941no—
160C22C2×C4×#8760411yescharacteristic
161C22C2×C22⋊#8760411yescharacteristic
162C22C2×C4⋊#8760411yescharacteristic
163C4—42no—
164C4—972no—
165C4—972no—
166C4—3882no—
167C4(C2×#87604)⋊C414yes—
168C2—21no—
169C2—21no—
170C2—974no—
171C2—1942no—
172C2C2×(C2×#87604)⋊C411yescharacteristic
173C2D97⋊#10011yescharacteristic
174C2D97⋊#10111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC22×C97C2×C42
FrattiniC23C22×#87604
FittingC2×C4⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC23×D97C2×C4
cosocleC22×Dic97C23
hypercenterC2×C4⋊C4#87604
hypocenterC97C2×#343
LCSG ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #20261615C2semidirect
6. D97⋊M16C4semidirect
18. #13095520C2×C4semidirect
86. D97#343semidirect
88. C97C2×#343semidirect

Maximal subgroups

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

subgroupindex
C2×D97⋊M162
#202616152
C2×C4⋊C4×D972
C2×#34397