Finite Group Databasebeta

← #27156974

Subgroups — #27156974

Order 12416.

332 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 84 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×C22⋊#11715897C212yesmaximal, complemented
2#20261581C212yesmaximal, complemented
3#20261579C212yesmaximal, complemented
4C2×#15690770C211yesmaximal, characteristic
5C22×#11715897—22no—
6C22⋊#11715897—22no—
7C22⋊#11715897C2214yes—
8C22×C22⋊D97C411yescharacteristic
9C23×Dic97C2211yescharacteristic
10C22×C4×D97C411yescharacteristic
11#15690770—22no—
12#15690770—22no—
13C2×#13095490C2211yescharacteristic
14C2×#13095488C2211yescharacteristic
15C2×C4×Dic97C411yescharacteristic
16C2×C22⋊C4×C97C411yescharacteristic
17C2×#11715897—24no—
18C2×#11715897—44no—
19C23×D97C2×C411yescharacteristic
20C2×C22⋊D97—22no—
21C2×C22⋊D97—22no—
22C2×C22⋊D97C814yescomplemented
23C22×Dic97—42no—
24C22×Dic97C2×C411yescharacteristic
25C22×Dic97C2×C411yescharacteristic
26C22×Dic97D412yes—
27C22×Dic97D412yes—
28C22×Dic97C2311yescharacteristic
29C2×C4×D97—21no—
30C2×C4×D97—21no—
31C2×C4×D97—42no—
32#13095490—22no—
33#13095488—22no—
34C4×Dic97—21no—
35C4×Dic97—21no—
36C24×C97C2×C411yescharacteristic
37C22×C4×C97C2×C411yescharacteristic
38C22×C4×C97C2×C411yescharacteristic
39C22⋊C4×C97—22no—
40C23×C97—41no—
41C23×C97—41no—
42C23×C97C22⋊C412yes—
43C23×C97C2×C812yes—
44C23×C97M1612yes—
45C23×C97C22×C411yescharacteristic
46C22×D97—21no—
47C22×D97—21no—
48C22×D97—41no—
49C22×D97—41no—
50C22×D97C2×C812yescomplemented
51C2×C4×C97—21no—
52C2×C4×C97—21no—
53C2×C4×C97—41no—
54C2×C4×C97—41no—
55C22⋊D97—28no—
56C2×Dic97—21no—
57C2×Dic97—21no—
58C2×Dic97—22no—
59C2×Dic97—24no—
60C2×Dic97—42no—
61C2×Dic97—41no—
62C2×Dic97C22⋊C411yescharacteristic
63C2×Dic97C22⋊C411yescharacteristic
64C2×Dic97C2×C812yescomplemented
65C2×Dic97C2×D412yes—
66C4×D97—42no—
67C4×D97—42no—
68#11715897—44no—
69C22×C97—21no—
70C22×C97—22no—
71C22×C97—21no—
72C22×C97—22no—
73C22×C97—42no—
74C22×C97—42no—
75C22×C97C22⋊C814yes—
76C22×C97C2×C22⋊C411yescharacteristic
77C22×C97C22×C811yescharacteristic
78C22×C97C2×M1611yescharacteristic
79C2×D97—22no—
80C2×D97—42no—
81C2×D97—41no—
82C2×D97—41no—
83C2×D97—42no—
84C4×C97—41no—
85C4×C97—41no—
86Dic97—22no—
87Dic97—22no—
88Dic97—21no—
89Dic97#9912yes—
90C2×C97—21no—
91C2×C97—21no—
92C2×C97—24no—
93C2×C97C2×C22⋊C811yescharacteristic
94C2×C97C2×#9911yescharacteristic
95C2×C97#41311yescharacteristic
96D97—42no—
97#1459—971nomaximal, Sylow
98C97#145911yescharacteristic, complemented, Sylow
99C2×C4×D4—971no—
100C2×C22⋊C8—972no—
101#324—972no—
102#323—972no—
103C22×D4—971no—
104C23×C4—971no—
105C23×C4—971no—
106C22×C8—1942no—
107C4×D4—1942no—
108C4×D4—1942no—
109C2×C4⋊C4—971no—
110C2×C22⋊C4—971no—
111C2×C22⋊C4#8760411yescharacteristic
112C2×C42—971no—
113C22⋊C8—974no—
114C22⋊C8—1942no—
115C24—971no—
116C24C2×#8760411yescharacteristic
117C2×D4—974no—
118C2×D4—1942no—
119C2×D4—1942no—
120C22×C4—972no—
121C22×C4—971no—
122C22×C4—972no—
123C22×C4—971no—
124C22×C4—971no—
125C22×C4—3882no—
126C22×C4—3882no—
127C22×C4—1941no—
128C22×C4—1941no—
129C22×C4C2×#8760411yescharacteristic
130C22×C4C2×#8760411yescharacteristic
131C2×C8—3884no—
132C2×C8—1944no—
133C4⋊C4—1942no—
134C22⋊C4—22no—
135C22⋊C4—1942no—
136C42—1941no—
137C42—1941no—
138C23—41no—
139C23—41no—
140C23—972no—
141C23—3881no—
142C23—3881no—
143C23—1941no—
144C23—1941no—
145C23C22⋊#8760412yes—
146C23#1309551912yes—
147C23#1309552112yes—
148C23C22×#8760411yescharacteristic
149D4—1948no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—971no—
155C2×C4—971no—
156C2×C4—972no—
157C2×C4—972no—
158C2×C4—3882no—
159C2×C4—3882no—
160C2×C4—3881no—
161C2×C4—3882no—
162C2×C4—1944no—
163C2×C4—1942no—
164C2×C4—1941no—
165C2×C4—1941no—
166C8—3884no—
167C22—21no—
168C22—22no—
169C22—21no—
170C22—22no—
171C22—42no—
172C22—42no—
173C22—3882no—
174C22—3882no—
175C22—3881no—
176C22—3881no—
177C22—1942no—
178C22#1569086714yes—
179C22C2×C22⋊#8760411yescharacteristic
180C22C2×#1309551911yescharacteristic
181C22C2×#1309552111yescharacteristic
182C4—41no—
183C4—41no—
184C4—972no—
185C4—1942no—
186C4—1942no—
187C4—1941no—
188C2—21no—
189C2—21no—
190C2—24no—
191C2—3882no—
192C2C2×#1569086711yescharacteristic
193C2D97⋊#9911yescharacteristic
194C2#2026172211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC22×C97C22×C8
FrattiniC23C22×#87604
FittingC2×C22⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×C22⋊D97C4
cosocleC22×Dic97C23
hypercenterC2×C22⋊C4#87604
hypocenterC97#1459
LCSG ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C22⋊#11715897C2semidirect
2. #20261581C2semidirect
3. #20261579C2semidirect
22. C2×C22⋊D97C8semidirect
50. C22×D97C2×C8semidirect
64. C2×Dic97C2×C8semidirect
98. C97#1459semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊#117158972
#202615812
#202615792
C2×#156907702
#145997