Finite Group Databasebeta

← #27157272

Subgroups — #27157272

Order 12416.

202 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 58 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#20261755C211yesmaximal, characteristic
2#20261738C211yesmaximal, characteristic
3#20261595C212yesmaximal, complemented
4#20261594C212yesmaximal, complemented
5#20261287C211yesmaximal, characteristic
6C2×#13095520—21no—
7#15690877C2212yes—
8#15690877C2212yes—
9C4⋊#11715897—21no—
10C22⋊#11715897—21no—
11#15690865—21no—
12#15690809C2211yescharacteristic
13#15690808C2211yescharacteristic
14#15690800—21no—
15#15690789C411yescharacteristic
16#15690787—21no—
17Q8×Dic97C411yescharacteristic
18#15690775—21no—
19#15690775C411yescharacteristic
20#15690766—21no—
21C2×C4×Dic97C2211yescharacteristic
22C22⋊Q8×C97C411yescharacteristic
23#13095520—42no—
24C2×#11715897—22no—
25C2×#11715897—22no—
26C2×Dic194—21no—
27C2×Dic194C2×C411yescharacteristic
28C22×Dic97D411yescharacteristic
29C22×Dic97D411yescharacteristic
30C4⋊Dic97—21no—
31C4⋊Dic97C2×C411yescharacteristic
32#13095490—21no—
33#13095490—21no—
34#13095490—21no—
35#13095490—21no—
36#13095490—21no—
37C22⋊Dic97—21no—
38C4×Dic97—21no—
39C4×Dic97—21no—
40C4×Dic97C2×C411yescharacteristic
41C4×Dic97D411yescharacteristic
42C4×Dic97D411yescharacteristic
43C4×Dic97C2311yescharacteristic
44C2×Q8×C97C2×C411yescharacteristic
45C22×C4×C97C2×C411yescharacteristic
46C4⋊C4×C97—21no—
47C4⋊C4×C97C2×C411yescharacteristic
48C22⋊C4×C97—21no—
49C23×C97C22⋊C411yescharacteristic
50Q8×C97—41no—
51C2×C4×C97—21no—
52C2×C4×C97—21no—
53C2×C4×C97—41no—
54C2×C4×C97C22⋊C411yescharacteristic
55C2×C4×C97C22×C411yescharacteristic
56C2×Dic97—21no—
57C2×Dic97—21no—
58C2×Dic97—22no—
59C2×Dic97—21no—
60C2×Dic97—21no—
61C2×Dic97C22⋊C411yescharacteristic
62C2×Dic97C22⋊C411yescharacteristic
63C2×Dic97C2×D411yescharacteristic
64C2×Dic97C2×D411yescharacteristic
65Dic194—41no—
66Dic194—41no—
67Dic194—41no—
68#11715897—42no—
69#11715897—42no—
70C22×C97—21no—
71C22×C97—21no—
72C22×C97—21no—
73C22×C97C2×C22⋊C411yescharacteristic
74C4×C97—21no—
75C4×C97—21no—
76C4×C97—41no—
77C4×C97—41no—
78Dic97—21no—
79Dic97—21no—
80Dic97—41no—
81Dic97—41no—
82Dic97AGL(1,ℤ/8)11yescharacteristic
83Dic97#13711yescharacteristic
84Dic97#13712yes—
85C2×C97—41no—
86C2×C97#41311yescharacteristic
87C2×C97#41711yescharacteristic
88C2×C97#41911yescharacteristic
89#1559—971nomaximal, Sylow
90C97#155911yescharacteristic, complemented, Sylow
91#533—971no—
92#432—971no—
93#423—971no—
94#333—972no—
95#332—972no—
96C2×M16—1941no—
97C4⋊Q8—971no—
98C4⋊Q8—971no—
99#125—1941no—
100C22⋊Q8—971no—
101C22⋊Q8—1941no—
102C22⋊Q8#8760411yescharacteristic
103C4×Q8—971no—
104C4×Q8—971no—
105C4×Q8—1941no—
106#117—1941no—
107C2×C42—971no—
108C4⋊C8—972no—
109C4⋊C8—972no—
110C4⋊C8—1941no—
111C22⋊C8—1941no—
112#97—1941no—
113C2×Q8—971no—
114C2×Q8—1941no—
115C2×Q8C2×#8760411yescharacteristic
116C22×C4—971no—
117C22×C4—971no—
118C22×C4C2×#8760411yescharacteristic
119M16—3882no—
120C2×C8—1942no—
121C2×C8—1942no—
122C4⋊C4—21no—
123C4⋊C4—971no—
124C4⋊C4—1941no—
125C4⋊C4—1941no—
126C4⋊C4—1941no—
127C4⋊C4—1941no—
128C4⋊C4—1941no—
129C4⋊C4—1941no—
130C4⋊C4C2×#8760411yescharacteristic
131C22⋊C4—21no—
132C22⋊C4—1941no—
133C42—971no—
134C42—971no—
135C42—971no—
136C42—971no—
137C42—1941no—
138C42—1941no—
139C23C22⋊#8760411yescharacteristic
140Q8—41no—
141Q8—3881no—
142Q8—3881no—
143Q8—3881no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—41no—
147C2×C4—971no—
148C2×C4—971no—
149C2×C4—971no—
150C2×C4—971no—
151C2×C4—1941no—
152C2×C4—1941no—
153C2×C4—1942no—
154C2×C4—1941no—
155C2×C4—1941no—
156C2×C4C22⋊#8760411yescharacteristic
157C2×C4C22×#8760411yescharacteristic
158C8—3882no—
159C8—3882no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22C2×C22⋊#8760411yescharacteristic
164C4—21no—
165C4—21no—
166C4—41no—
167C4—41no—
168C4—971no—
169C4—971no—
170C4—972no—
171C4—3881no—
172C4—3881no—
173C4—1941no—
174C4—1941no—
175C2—41no—
176C2#2026172211yescharacteristic
177C2#2026172611yescharacteristic
178C2#2026172811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC22⋊Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC23×C97C22⋊C4
cosocleC4×Dic97C23
hypercenterC22⋊Q8#87604
hypocenterC97#1559
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. #20261595C2semidirect
4. #20261594C2semidirect
90. C97#1559semidirect

Maximal subgroups

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

subgroupindex
#202617552
#202617382
#202615952
#202615942
#202612872
#155997