Finite Group Databasebeta

← #27141467

Subgroups — #27141467

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
1C2×#15690766C212yesmaximal
2C2×#15690766C211yesmaximal, characteristic
3#20260502C211yesmaximal, characteristic
4#20260501C212yesmaximal
5#342×C97C211yesmaximal, characteristic
6C23×Dic97C2211yescharacteristic
7#15690766—24no—
8#15690766—22no—
9#15690766—22no—
10#15690766C418yescomplemented
11C2×#13095490—22no—
12C2×#13095490C412yescomplemented
13C2×C22⋊Dic97C2211yescharacteristic
14C2×C22⋊Dic97C2212yes—
15C2×C4×Dic97—22no—
16C2×C4×Dic97C412yescomplemented
17C2×C22⋊C4×C97C2211yescharacteristic
18C2×C22⋊C4×C97C2212yes—
19C22×Dic97—22no—
20C22×Dic97C2×C412yes—
21C22×Dic97C2×C411yescharacteristic
22C22×Dic97C2×C411yescharacteristic
23C22×Dic97C2×C414yes—
24C22×Dic97D412yes—
25C22×Dic97D412yes—
26C22×Dic97D412yes—
27C22×Dic97Q812yes—
28#13095490—28no—
29#13095490—24no—
30C22⋊Dic97—22no—
31C22⋊Dic97—22no—
32C22⋊Dic97C2×C414yes—
33C4×Dic97—28no—
34C4×Dic97—22no—
35C4×Dic97—22no—
36C24×C97C2311yescharacteristic
37C22×C4×C97—22no—
38C22×C4×C97C2×C412yes—
39C22⋊C4×C97—22no—
40C22⋊C4×C97—22no—
41C22⋊C4×C97C2×C414yes—
42C23×C97—42no—
43C23×C97C22×C411yescharacteristic
44C23×C97C22×C412yes—
45C23×C97C2×D412yes—
46C23×C97C2×D411yescharacteristic
47C23×C97C2×Q811yescharacteristic
48C2×C4×C97—22no—
49C2×C4×C97—24no—
50C2×C4×C97—44no—
51C2×C4×C97—42no—
52C2×Dic97—22no—
53C2×Dic97—24no—
54C2×Dic97—44no—
55C2×Dic97—42no—
56C2×Dic97C4212yes—
57C2×Dic97C4212yes—
58C2×Dic97C22⋊C414yes—
59C2×Dic97C22⋊C411yescharacteristic
60C2×Dic97C22⋊C411yescharacteristic
61C2×Dic97C22⋊C414yes—
62C2×Dic97C22⋊C412yes—
63C2×Dic97C4⋊C412yes—
64C2×Dic97C4⋊C412yes—
65C2×Dic97C4⋊C414yes—
66C2×Dic97C4⋊C414yes—
67C22×C97—22no—
68C22×C97—24no—
69C22×C97—42no—
70C22×C97—41no—
71C22×C97—41no—
72C22×C97C2×C4211yescharacteristic
73C22×C97C2×C22⋊C412yes—
74C22×C97C2×C22⋊C411yescharacteristic
75C22×C97C2×C4⋊C411yescharacteristic
76C22×C97C2×C4⋊C412yes—
77C4×C97—44no—
78C4×C97—42no—
79Dic97—44no—
80Dic97—42no—
81Dic97#9512yes—
82Dic97#9514yes—
83Dic97#9512yes—
84C2×C97—22no—
85C2×C97—21no—
86C2×C97—21no—
87C2×C97—22no—
88C2×C97C2×#9511yescharacteristic
89C2×C97#41012yes—
90#1740—971nomaximal, Sylow
91C97#174011yescharacteristic, complemented, Sylow
92C2×#117—971no—
93C2×#117—972no—
94#342—972no—
95#342—971no—
96#342D9711yescharacteristic
97C23×C4—971no—
98#117—978no—
99#117—1942no—
100#117—1942no—
101#117—1944no—
102C2×C4⋊C4—972no—
103C2×C4⋊C4—1942no—
104C2×C22⋊C4—972no—
105C2×C22⋊C4—971no—
106C2×C22⋊C4C2×D9711yescharacteristic
107C2×C22⋊C4C2×D9712yes—
108C2×C42—972no—
109C2×C42—1942no—
110C24C22×D9711yescharacteristic
111C22×C4—22no—
112C22×C4—971no—
113C22×C4—971no—
114C22×C4—974no—
115C22×C4—972no—
116C22×C4—972no—
117C22×C4—972no—
118C22×C4—972no—
119C22×C4—972no—
120C22×C4—1942no—
121C22×C4C4×D9712yes—
122C4⋊C4—1944no—
123C4⋊C4—1948no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—974no—
127C22⋊C4—1942no—
128C22⋊C4—1942no—
129C22⋊C4C4×D9714yes—
130C42—1942no—
131C42—1942no—
132C42—1948no—
133C23—42no—
134C23C2×C4×D9711yescharacteristic
135C23C2×C4×D9712yes—
136C23D4×D9712yes—
137C23D4×D9711yescharacteristic
138C23Q8×D9711yescharacteristic
139C2×C4—24no—
140C2×C4—22no—
141C2×C4—44no—
142C2×C4—42no—
143C2×C4—972no—
144C2×C4—972no—
145C2×C4—974no—
146C2×C4—974no—
147C2×C4—971no—
148C2×C4—971no—
149C2×C4—972no—
150C2×C4—972no—
151C2×C4—972no—
152C2×C4—974no—
153C2×C4—974no—
154C2×C4—3884no—
155C2×C4—3882no—
156C2×C4—1944no—
157C2×C4—1942no—
158C22—22no—
159C22—24no—
160C22—42no—
161C22—41no—
162C22—41no—
163C22C42×D9711yescharacteristic
164C22C22⋊C4×D9712yes—
165C22C22⋊C4×D9711yescharacteristic
166C22C4⋊C4×D9711yescharacteristic
167C22C4⋊C4×D9712yes—
168C4—44no—
169C4—42no—
170C4—972no—
171C4—974no—
172C4—972no—
173C4—3884no—
174C4—3882no—
175C2—22no—
176C2—21no—
177C2—21no—
178C2—22no—
179C2#95×D9711yescharacteristic
180C2#2026070712yes—
subgroupG/H
centerC22C22⋊C4×D97
commutatorC22×C97C2×C42
FrattiniC24C22×D97
Fitting#342×C97C2
socleC22×C97C2×C22⋊C4
prime radical, cosocleC24×C97C23
hypercenter#342D97
hypocenterC97#1740
LCSG ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
10. #15690766C4semidirect
12. C2×#13095490C4semidirect
16. C2×C4×Dic97C4semidirect
91. C97#1740semidirect

Maximal subgroups

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

subgroupindex
C2×#156907662
#202605022
#202605012
#342×C972
#174097