Finite Group Databasebeta

← #27157146

Subgroups — #27157146

Order 12416.

274 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 60 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#20261707C211yesmaximal, characteristic
2#20261678C211yesmaximal, characteristic
3#20261589C214yesmaximal, complemented
4C2×#15690786C211yesmaximal, characteristic
5C2×#13095520—21no—
6C2×C22⋊#87604—21no—
7#15690867—21no—
8#15690867C2212yes—
9(C2×#87604)⋊C4—21no—
10(C2×#87604)⋊C4C2212yes—
11C22×C4×D97C2211yescharacteristic
12#15690786—21no—
13#15690786—21no—
14#15690786C412yescomplemented
15#15690786C2212yes—
16C2×C4⋊Dic97C411yescharacteristic, complemented
17C2×#13095490—21no—
18C2×#13095488—21no—
19C22×Q8×C97C411yescharacteristic, complemented
20C22×#87604—22no—
21#13095520—42no—
22C2×#11715897—22no—
23C22⋊#87604—42no—
24C23×D97D411yescharacteristic
25C22×Dic97—21no—
26C22×Dic97D411yescharacteristic
27C2×C4×D97—42no—
28C2×C4×D97C2×C411yescharacteristic
29C2×C4×D97D411yescharacteristic
30C2×C4×D97D411yescharacteristic
31C2×C4×D97C2311yescharacteristic
32C4⋊Dic97—21no—
33C4⋊Dic97C2×C412yes—
34#13095490—22no—
35#13095490—22no—
36#13095488—22no—
37#13095488—22no—
38C2×Q8×C97—21no—
39C2×Q8×C97—42no—
40C2×Q8×C97C2×C412yes—
41C22×C4×C97—21no—
42C22×C4×C97C2×C411yescharacteristic
43C23×C97C22⋊C411yescharacteristic
44C22×D97—22no—
45C22×D97—22no—
46C22×D97—41no—
47C22×D97C22⋊C411yescharacteristic
48C22×D97C2×D411yescharacteristic
49C2×#87604—42no—
50C2×#87604—42no—
51C2×#87604—42no—
52Q8×C97—42no—
53Q8×C97—42no—
54C2×C4×C97—22no—
55C2×C4×C97—22no—
56C2×C4×C97—42no—
57C2×C4×C97C22⋊C411yescharacteristic
58C2×C4×C97C22×C411yescharacteristic
59C2×Dic97—22no—
60C2×Dic97—22no—
61C2×Dic97—42no—
62C2×Dic97C22⋊C411yescharacteristic
63C2×Dic97C2×D411yescharacteristic
64C4×D97—41no—
65C4×D97—41no—
66C4×D97—41no—
67C4×D97—41no—
68#11715897—42no—
69C22×C97—21no—
70C22×C97—21no—
71C22×C97#9912yes—
72C22×C97C2×C22⋊C411yescharacteristic
73C2×D97—21no—
74C2×D97—21no—
75C2×D97—21no—
76C2×D97—21no—
77C2×D97—21no—
78C2×D97—44no—
79C2×D97#13712yes—
80#87604—82no—
81C4×C97—21no—
82C4×C97—21no—
83C4×C97—42no—
84Dic97—21no—
85Dic97—21no—
86Dic97—42no—
87C2×C97—22no—
88C2×C97C2×#9911yescharacteristic
89C2×C97#41911yescharacteristic
90C2×C97#42111yescharacteristic
91D97—22no—
92D97—41no—
93#1511—971nomaximal, Sylow
94C97#151111yescharacteristic, complemented, Sylow
95C2×C22⋊Q8—971no—
96#407—971no—
97#381—971no—
98#328—974no—
99C22×Q8#8760411yescharacteristic, complemented
100C23×C4—971no—
101C2×M16—1941no—
102C22⋊Q8—972no—
103C22⋊Q8—972no—
104C22⋊Q8—1941no—
105C22⋊Q8—1941no—
106C2×C4⋊C4—971no—
107C2×C4⋊C4—1941no—
108C2×C22⋊C4—1941no—
109C2×C22⋊C4—1941no—
110C22⋊C8—972no—
111C22⋊C8—1941no—
112#95—972no—
113#95—1941no—
114C24—971no—
115C2×Q8—21no—
116C2×Q8—42no—
117C2×Q8C2×#8760412yes—
118C22×C4—21no—
119C22×C4—971no—
120C22×C4—971no—
121C22×C4—971no—
122C22×C4—971no—
123C22×C4—971no—
124C22×C4—3882no—
125C22×C4—1942no—
126C22×C4—1941no—
127C22×C4C2×#8760411yescharacteristic
128M16—3882no—
129C2×C8—1942no—
130C4⋊C4—972no—
131C4⋊C4—1942no—
132C4⋊C4—1942no—
133C4⋊C4—1941no—
134C22⋊C4—3882no—
135C22⋊C4—1942no—
136C22⋊C4—1942no—
137C23—971no—
138C23—971no—
139C23—3881no—
140C23—1942no—
141C23—1942no—
142C23C22⋊#8760411yescharacteristic
143Q8—42no—
144Q8—42no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—42no—
148C2×C4—971no—
149C2×C4—971no—
150C2×C4—3881no—
151C2×C4—3881no—
152C2×C4—3881no—
153C2×C4—3881no—
154C2×C4—3882no—
155C2×C4—3882no—
156C2×C4—3882no—
157C2×C4—3882no—
158C2×C4—1942no—
159C2×C4—1942no—
160C2×C4C22⋊#8760411yescharacteristic
161C2×C4C22×#8760411yescharacteristic
162C8—3882no—
163C22—21no—
164C22—21no—
165C22—972no—
166C22—3884no—
167C22—1941no—
168C22—1941no—
169C22—1941no—
170C22—1941no—
171C22—1941no—
172C22#1569086812yes—
173C22C2×C22⋊#8760411yescharacteristic
174C4—21no—
175C4—21no—
176C4—42no—
177C4—7762no—
178C4—3882no—
179C4—1941no—
180C4—1941no—
181C2—22no—
182C2—3881no—
183C2—1942no—
184C2C2×#1569086811yescharacteristic
185C2#2026172811yescharacteristic
186C2#2026173411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC22×Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC23×D97D4
cosocleC2×C4×D97C23
hypercenterC22×Q8#87604
hypocenterC97#1511
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
3. #20261589C2semidirect
14. #15690786C4semidirect
16. C2×C4⋊Dic97C4semidirect
19. C22×Q8×C97C4semidirect
94. C97#1511semidirect
99. C22×Q8#87604semidirect

Maximal subgroups

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

subgroupindex
#202617072
#202616782
#202615892
C2×#156907862
#151197