Finite Group Databasebeta

← #27157925

Subgroups — #27157925

Order 12416.

324 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 74 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#20261677C211yesmaximal, characteristic
2C4×C22⋊#87604C211yesmaximal, characteristic
3#20261613C214yesmaximal
4C2×#15690786C211yesmaximal, characteristic
5C2×C4⋊#87604—21no—
6C2×C22⋊#87604C2212yes—
7C2×C22⋊#87604C2212yes—
8C2×C4×#87604—21no—
9(C2×#87604)⋊C4—21no—
10(C2×#87604)⋊C4—21no—
11C22×C4×D97C2211yescharacteristic
12#15690786—21no—
13#15690786—21no—
14#15690786—21no—
15#15690786—21no—
16C2×C4⋊Dic97C411yescharacteristic, complemented
17C2×#13095490C412yescomplemented
18C2×#13095488C2212yes—
19C22×Q8×C97C411yescharacteristic, complemented
20C22×#87604—22no—
21C22×#87604—22no—
22C4⋊#87604—42no—
23C22⋊#87604—22no—
24C22⋊#87604—24no—
25C22⋊#87604Q814yescomplemented
26C4×#87604—42no—
27C23×D97C2311yescharacteristic
28C22×Dic97C2×C412yes—
29C22×Dic97C2×C411yescharacteristic
30C2×C4×D97—42no—
31C2×C4×D97D411yescharacteristic
32C2×C4×D97D411yescharacteristic
33C2×C4×D97D411yescharacteristic
34C2×C4×D97D411yescharacteristic
35C4⋊Dic97—21no—
36C4⋊Dic97—21no—
37#13095490—22no—
38#13095490—22no—
39#13095488—22no—
40#13095488—22no—
41C2×Q8×C97—21no—
42C2×Q8×C97—21no—
43C2×Q8×C97—42no—
44C22×C4×C97C2×C412yes—
45C22×C4×C97C2×C411yescharacteristic
46C23×C97C22×C411yescharacteristic
47C22×D97—41no—
48C22×D97—41no—
49C22×D97C2×D411yescharacteristic
50C22×D97C2×D411yescharacteristic
51C22×D97C2×Q812yes—
52C22×D97#4112yes—
53C2×#87604—24no—
54C2×#87604—42no—
55C2×#87604—42no—
56C2×#87604—42no—
57C2×#87604—42no—
58C2×#87604—42no—
59Q8×C97—42no—
60Q8×C97—42no—
61C2×C4×C97—22no—
62C2×C4×C97—22no—
63C2×C4×C97—42no—
64C2×C4×C97C22⋊C411yescharacteristic
65C2×C4×C97C22⋊C411yescharacteristic
66C2×Dic97—22no—
67C2×Dic97—41no—
68C2×Dic97—42no—
69C2×Dic97C22⋊C411yescharacteristic
70C2×Dic97C22⋊C411yescharacteristic
71C4×D97—22no—
72C4×D97—22no—
73C4×D97—41no—
74C4×D97—41no—
75C22×C97—21no—
76C22×C97—21no—
77C22×C97C2×C22⋊C411yescharacteristic
78C22×C97C4×Q812yes—
79C2×D97—21no—
80C2×D97—21no—
81C2×D97—21no—
82C2×D97—21no—
83C2×D97—42no—
84C2×D97—42no—
85C2×D97C22⋊Q812yes—
86C2×D97#12411yescharacteristic
87C2×D97C4⋊Q811yescharacteristic
88#87604—44no—
89#87604—82no—
90C4×C97—21no—
91C4×C97—42no—
92C4×C97#9912yes—
93Dic97—21no—
94Dic97—21no—
95Dic97—42no—
96C2×C97—22no—
97C2×C97#39111yescharacteristic
98C2×C97C2×#9911yescharacteristic
99C2×C97#41011yescharacteristic
100D97—22no—
101D97—21no—
102D97—21no—
103#1996—971nomaximal, Sylow
104C97#199611yescharacteristic, complemented, Sylow
105C2×C22⋊Q8—971no—
106#380—971no—
107C4×C22⋊C4—971no—
108#342—974no—
109C22×Q8#8760411yescharacteristic, complemented
110C23×C4—971no—
111C22⋊Q8—1941no—
112C22⋊Q8—1941no—
113C22⋊Q8—1941no—
114C22⋊Q8—1941no—
115C2×C4⋊C4—972no—
116C2×C4⋊C4—971no—
117C2×C4⋊C4—1941no—
118C2×C22⋊C4—972no—
119C2×C22⋊C4—972no—
120C2×C22⋊C4—972no—
121C2×C42—1941no—
122#95—1941no—
123#95—1941no—
124C24—971no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—42no—
128C22×C4—971no—
129C22×C4—971no—
130C22×C4—971no—
131C22×C4—971no—
132C22×C4—971no—
133C22×C4—972no—
134C22×C4—3882no—
135C22×C4—1942no—
136C22×C4—1942no—
137C22×C4C2×#8760412yes—
138C22×C4C2×#8760411yescharacteristic
139C4⋊C4—3882no—
140C4⋊C4—1942no—
141C4⋊C4—1942no—
142C4⋊C4—1941no—
143C4⋊C4—1941no—
144C22⋊C4—974no—
145C22⋊C4—1942no—
146C22⋊C4—1942no—
147C22⋊C4—1944no—
148C22⋊C4—1942no—
149C42—3882no—
150C23—972no—
151C23—972no—
152C23—971no—
153C23—971no—
154C23—3881no—
155C23—3881no—
156C23C22×#8760411yescharacteristic
157Q8—42no—
158Q8—42no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—42no—
162C2×C4—971no—
163C2×C4—971no—
164C2×C4—3881no—
165C2×C4—3881no—
166C2×C4—3882no—
167C2×C4—3882no—
168C2×C4—3882no—
169C2×C4—3882no—
170C2×C4—3882no—
171C2×C4—3882no—
172C2×C4—3881no—
173C2×C4—1944no—
174C2×C4—1942no—
175C2×C4—1942no—
176C2×C4—1942no—
177C2×C4C22⋊#8760411yescharacteristic
178C2×C4C22⋊#8760411yescharacteristic
179C22—21no—
180C22—21no—
181C22—971no—
182C22—972no—
183C22—971no—
184C22—3882no—
185C22—3882no—
186C22—1941no—
187C22—1941no—
188C22—1941no—
189C22—1941no—
190C22C2×C22⋊#8760411yescharacteristic
191C22Q8×#8760412yes—
192C4—21no—
193C4—42no—
194C4—7762no—
195C4—3884no—
196C4—3882no—
197C4—1941no—
198C4—1941no—
199C4#1569086812yes—
200C2—22no—
201C2—1941no—
202C2—1942no—
203C2—1941no—
204C2(C2×#87604)⋊Q811yescharacteristic
205C2C2×#1569086811yescharacteristic
206C2#2026171611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
FittingC22×Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radical, cosocleC23×D97C23
hypercenterC22×Q8#87604
hypocenterC97#1996
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
16. C2×C4⋊Dic97C4semidirect
17. C2×#13095490C4semidirect
19. C22×Q8×C97C4semidirect
25. C22⋊#87604Q8semidirect
104. C97#1996semidirect
109. C22×Q8#87604semidirect

Maximal subgroups

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

subgroupindex
#202616772
C4×C22⋊#876042
#202616132
C2×#156907862
#199697