Finite Group Databasebeta

← #27156973

Subgroups — #27156973

Order 12416.

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

normal subgroups only

This group has 332 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#20261703C212yesmaximal, complemented
2#20261580C212yesmaximal, complemented
3#20261579C212yesmaximal, complemented
4C2×C4×D388C211yesmaximal, characteristic
5C2×#13095519—22no—
6#15690867—22no—
7C22⋊#11715897C2214yes—
8C22×D388C411yescharacteristic
9C22×C4×D97C411yescharacteristic
10C22×C4×D97C2211yescharacteristic
11C4×D388—22no—
12C4×D388—22no—
13C2×C4⋊Dic97C2211yescharacteristic
14C2×#13095488C411yescharacteristic
15C2×#13095488C2211yescharacteristic
16C2×C42×C97C411yescharacteristic
17#13095519—44no—
18C2×#11715897—24no—
19C23×D97C2×C411yescharacteristic
20C23×D97C2×C411yescharacteristic
21C2×D388—22no—
22C2×D388—22no—
23C2×D388C814yescomplemented
24C22×Dic97C2×C411yescharacteristic
25C22×Dic97C2311yescharacteristic
26C2×C4×D97—21no—
27C2×C4×D97—21no—
28C2×C4×D97—42no—
29C2×C4×D97—42no—
30C2×C4×D97D412yes—
31C2×C4×D97D412yes—
32C4⋊Dic97—22no—
33#13095488—22no—
34#13095488—22no—
35C22×C4×C97C2×C411yescharacteristic
36C22×C4×C97C2×C411yescharacteristic
37C22×C4×C97C2×C411yescharacteristic
38C42×C97—21no—
39C42×C97—21no—
40C23×C97C22×C411yescharacteristic
41C22×D97—21no—
42C22×D97—21no—
43C22×D97—41no—
44C22×D97—41no—
45C22×D97—41no—
46C22×D97—41no—
47C22×D97C22⋊C412yes—
48C22×D97C2×C812yes—
49C22×D97C2×C812yescomplemented
50C22×D97M1612yes—
51C2×C4×C97—21no—
52C2×C4×C97—21no—
53C2×C4×C97—22no—
54C2×C4×C97—21no—
55C2×C4×C97—41no—
56C2×C4×C97C22⋊C411yescharacteristic
57C2×C4×C97C22⋊C411yescharacteristic
58C2×C4×C97C2×C812yescomplemented
59C2×Dic97—21no—
60C2×Dic97—41no—
61C2×Dic97—41no—
62C2×Dic97C2×D412yes—
63D388—28no—
64C4×D97—24no—
65C4×D97—42no—
66C4×D97—42no—
67C4×D97—42no—
68#11715897—44no—
69C22×C97—21no—
70C22×C97—21no—
71C22×C97C2×C22⋊C411yescharacteristic
72C22×C97C22×C811yescharacteristic
73C22×C97C2×M1611yescharacteristic
74C2×D97—22no—
75C2×D97—22no—
76C2×D97—22no—
77C2×D97—42no—
78C2×D97—42no—
79C2×D97—42no—
80C2×D97—41no—
81C2×D97—41no—
82C2×D97—42no—
83C2×D97C22⋊C814yes—
84C4×C97—22no—
85C4×C97—21no—
86C4×C97—41no—
87C4×C97#9912yes—
88Dic97—22no—
89Dic97—41no—
90C2×C97—21no—
91C2×C97—21no—
92C2×C97C2×C22⋊C811yescharacteristic
93C2×C97C2×#9911yescharacteristic
94C2×C97#41311yescharacteristic
95D97—24no—
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—971no—
112C2×C42#8760411yescharacteristic
113C22⋊C8—974no—
114C22⋊C8—1942no—
115C24—971no—
116C24—971no—
117C2×D4—974no—
118C2×D4—1942no—
119C2×D4—1942no—
120C22×C4—971no—
121C22×C4—972no—
122C22×C4—972no—
123C22×C4—971no—
124C22×C4—3882no—
125C22×C4—3882no—
126C22×C4—1941no—
127C22×C4—1941no—
128C22×C4C2×#8760411yescharacteristic
129C22×C4C2×#8760411yescharacteristic
130C22×C4C2×#8760411yescharacteristic
131C2×C8—3884no—
132C2×C8—1944no—
133C4⋊C4—1942no—
134C22⋊C4—1942no—
135C22⋊C4—1942no—
136C42—21no—
137C42—21no—
138C23—972no—
139C23—972no—
140C23—972no—
141C23—972no—
142C23—3881no—
143C23—3881no—
144C23—3881no—
145C23—3881no—
146C23—1941no—
147C23—1941no—
148C23C22×#8760411yescharacteristic
149D4—1948no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—972no—
156C2×C4—3882no—
157C2×C4—3882no—
158C2×C4—3882no—
159C2×C4—3881no—
160C2×C4—3881no—
161C2×C4—1941no—
162C2×C4—1944no—
163C2×C4C22⋊#8760411yescharacteristic
164C2×C4C22⋊#8760411yescharacteristic
165C2×C4#1309551912yescomplemented
166C8—3884no—
167C22—21no—
168C22—21no—
169C22—974no—
170C22—3882no—
171C22—3882no—
172C22—3882no—
173C22—3882no—
174C22—3881no—
175C22—3881no—
176C22—1942no—
177C22—1942no—
178C22—1942no—
179C22C2×C22⋊#8760411yescharacteristic
180C22C2×#1309551911yescharacteristic
181C22D97⋊M1611yescharacteristic
182C4—22no—
183C4—21no—
184C4—41no—
185C4—3881no—
186C4—1942no—
187C4#1569086912yes—
188C2—21no—
189C2—21no—
190C2—3882no—
191C2—1944no—
192C2#2026170311yescharacteristic
193C2C2×#1569086911yescharacteristic
194C2#2026172111yescharacteristic
subgroupG/H
centerC2×C4C22⋊#87604
commutatorC22×C97C22×C8
FrattiniC23C22×#87604
FittingC2×C42×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D388C4
cosocleC22×Dic97C23
hypercenterC2×C42#87604
hypocenterC97#1459
LCSG ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. #20261703C2semidirect
2. #20261580C2semidirect
3. #20261579C2semidirect
23. C2×D388C8semidirect
49. C22×D97C2×C8semidirect
58. C2×C4×C97C2×C8semidirect
98. C97#1459semidirect
165. C2×C4#13095519semidirect

Maximal subgroups

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

subgroupindex
#202617032
#202615802
#202615792
C2×C4×D3882
#145997