Finite Group Databasebeta

← #27156975

Subgroups — #27156975

Order 12416.

312 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 80 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#20261752C212yesmaximal, complemented
2#20261579C212yesmaximal, complemented
3#20261578C212yesmaximal, complemented
4C2×C4×C22⋊D97C211yesmaximal, characteristic
5#15690877—22no—
6C22⋊#11715897C2214yes—
7C4×#11715897—22no—
8C22×C22⋊D97C411yescharacteristic
9C22×C4×D97C411yescharacteristic
10C4×C22⋊D97—22no—
11C4×C22⋊D97—22no—
12C2×#13095490C411yescharacteristic
13C2×C22⋊Dic97C2211yescharacteristic
14C2×#13095488C2211yescharacteristic
15C2×C4×Dic97C2211yescharacteristic
16C23×C4×C97C411yescharacteristic
17C2×#11715897—24no—
18C23×D97C2×C411yescharacteristic
19C2×C22⋊D97—22no—
20C2×C22⋊D97—22no—
21C2×C22⋊D97C814yescomplemented
22C22×Dic97C2×C411yescharacteristic
23C22×Dic97C2×C411yescharacteristic
24C22×Dic97C2311yescharacteristic
25C2×C4×D97—21no—
26C2×C4×D97—21no—
27C2×C4×D97—42no—
28#13095490—22no—
29C22⋊Dic97—22no—
30#13095488—22no—
31C4×Dic97D412yes—
32C4×Dic97D412yes—
33C24×C97C2×C411yescharacteristic
34C22×C4×C97—21no—
35C22×C4×C97—21no—
36C22×C4×C97—42no—
37C22×C4×C97C2×C411yescharacteristic
38C22×C4×C97C2×C411yescharacteristic
39C23×C97—21no—
40C23×C97—21no—
41C23×C97—41no—
42C23×C97—41no—
43C23×C97C2×C812yes—
44C23×C97C22×C411yescharacteristic
45C22×D97—21no—
46C22×D97—21no—
47C22×D97—41no—
48C22×D97—41no—
49C22×D97C2×C812yes—
50C2×C4×C97—22no—
51C2×C4×C97—21no—
52C2×C4×C97—42no—
53C2×C4×C97—42no—
54C2×C4×C97—41no—
55C2×C4×C97C22⋊C411yescharacteristic
56C2×C4×C97C22⋊C411yescharacteristic
57C22⋊D97—28no—
58C2×Dic97—21no—
59C2×Dic97—41no—
60C2×Dic97—41no—
61C2×Dic97C22⋊C412yes—
62C2×Dic97C2×C812yes—
63C2×Dic97M1612yes—
64C2×Dic97C2×D412yes—
65C4×D97—42no—
66C4×D97—42no—
67#11715897—44no—
68C22×C97—21no—
69C22×C97—21no—
70C22×C97—22no—
71C22×C97—42no—
72C22×C97—42no—
73C22×C97—41no—
74C22×C97—41no—
75C22×C97C2×C22⋊C411yescharacteristic
76C22×C97C22×C811yescharacteristic
77C22×C97C2×M1611yescharacteristic
78C2×D97—22no—
79C2×D97—42no—
80C2×D97—41no—
81C2×D97—41no—
82C2×D97—42no—
83C4×C97—21no—
84C4×C97—41no—
85C4×C97#10012yes—
86Dic97—22no—
87Dic97—41no—
88Dic97C22⋊C814yes—
89C2×C97—21no—
90C2×C97—21no—
91C2×C97—42no—
92C2×C97C2×C22⋊C811yescharacteristic
93C2×C97#41011yescharacteristic
94C2×C97C2×#10011yescharacteristic
95D97—42no—
96#1460—971nomaximal, Sylow
97C97#146011yescharacteristic, complemented, Sylow
98C2×C4×D4—971no—
99#431—972no—
100#323—972no—
101#323—972no—
102C22×D4—971no—
103C23×C4—971no—
104C23×C4#8760411yescharacteristic
105C4×D4—1942no—
106C4×D4—1942no—
107C2×C4⋊C4—971no—
108C2×C22⋊C4—971no—
109C2×C22⋊C4—971no—
110C2×C42—971no—
111C4⋊C8—1942no—
112C22⋊C8—974no—
113C4×C8—1942no—
114C24—971no—
115C24C2×#8760411yescharacteristic
116C2×D4—974no—
117C2×D4—1942no—
118C2×D4—1942no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—42no—
122C22×C4—971no—
123C22×C4—971no—
124C22×C4—971no—
125C22×C4—3882no—
126C22×C4—1941no—
127C22×C4—1941no—
128C22×C4C2×#8760411yescharacteristic
129C22×C4C2×#8760411yescharacteristic
130C2×C8—1944no—
131C4⋊C4—1942no—
132C22⋊C4—1942no—
133C22⋊C4—1942no—
134C42—972no—
135C42—972no—
136C23—21no—
137C23—21no—
138C23—41no—
139C23—41no—
140C23—972no—
141C23—3881no—
142C23—3881no—
143C23—1941no—
144C23—1941no—
145C23#1309551912yes—
146C23C22×#8760411yescharacteristic
147D4—1948no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—41no—
153C2×C4—972no—
154C2×C4—972no—
155C2×C4—972no—
156C2×C4—972no—
157C2×C4—3882no—
158C2×C4—3882no—
159C2×C4—3881no—
160C2×C4—3881no—
161C2×C4—1941no—
162C2×C4C22⋊#8760411yescharacteristic
163C2×C4C22⋊#8760411yescharacteristic
164C8—3884no—
165C22—21no—
166C22—21no—
167C22—22no—
168C22—42no—
169C22—42no—
170C22—41no—
171C22—41no—
172C22—3882no—
173C22—3882no—
174C22—3881no—
175C22—3881no—
176C22—1942no—
177C22C2×C22⋊#8760411yescharacteristic
178C22C2×#1309551911yescharacteristic
179C22D97⋊M1611yescharacteristic
180C4—21no—
181C4—41no—
182C4—974no—
183C4—3881no—
184C4—1942no—
185C4#1569087012yes—
186C2—21no—
187C2—21no—
188C2—42no—
189C2—3882no—
190C2#2026170311yescharacteristic
191C2#2026171411yescharacteristic
192C2C2×#1569087011yescharacteristic
subgroupG/H
centerC2×C4C22⋊#87604
commutatorC22×C97C22×C8
FrattiniC23C22×#87604
FittingC23×C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×C22⋊D97C4
cosocleC22×Dic97C23
hypercenterC23×C4#87604
hypocenterC97#1460
LCSG ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. #20261752C2semidirect
2. #20261579C2semidirect
3. #20261578C2semidirect
21. C2×C22⋊D97C8semidirect
97. C97#1460semidirect

Maximal subgroups

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

subgroupindex
#202617522
#202615792
#202615782
C2×C4×C22⋊D972
#146097