Finite Group Databasebeta

← #27157266

Subgroups — #27157266

Order 12416.

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

normal subgroups only

This group has 228 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#20261758C211yesmaximal, characteristic
2#20261758C211yesmaximal, characteristic
3#20261594C214yesmaximal, complemented
4C4⋊Q8×D97C211yesmaximal, characteristic
5#15690877C2212yes—
6#15690877C2212yes—
7#15690867—21no—
8#15690867—21no—
9#15690865—21no—
10#15690865—21no—
11C2×Q8×D97—21no—
12#15690810C411yescharacteristic
13#15690809—21no—
14#15690809C2212yes—
15#15690808—21no—
16C4⋊C4×D97—21no—
17C4⋊C4×D97C412yes—
18C42×D97C2211yescharacteristic
19C4⋊Q8×C97C411yescharacteristic
20C2×#11715897—22no—
21C2×#11715897—22no—
22C2×Dic194—21no—
23C2×Dic194—22no—
24Q8×D97—44no—
25C2×C4×D97—22no—
26C2×C4×D97C2×C411yescharacteristic
27C2×C4×D97D411yescharacteristic
28C2×C4×D97D411yescharacteristic
29C4⋊Dic97—21no—
30C4⋊Dic97C2×C412yes—
31#13095490—22no—
32#13095490—22no—
33C4×Dic97D411yescharacteristic
34C4×Dic97D411yescharacteristic
35C4×Dic97C2311yescharacteristic
36C2×Q8×C97—21no—
37C4⋊C4×C97—21no—
38C4⋊C4×C97C2×C412yes—
39C42×C97C2×C411yescharacteristic
40C22×D97C22⋊C411yescharacteristic
41Q8×C97—42no—
42C2×C4×C97—22no—
43C2×C4×C97C22⋊C411yescharacteristic
44C2×C4×C97C22⋊C411yescharacteristic
45C2×C4×C97C22×C411yescharacteristic
46C2×Dic97—22no—
47C2×Dic97C22⋊C411yescharacteristic
48C2×Dic97C2×D411yescharacteristic
49C2×Dic97C2×D411yescharacteristic
50C4×D97—22no—
51C4×D97—22no—
52C4×D97—22no—
53C4×D97—44no—
54Dic194—42no—
55Dic194—44no—
56#11715897—42no—
57#11715897—42no—
58C22×C97C2×C22⋊C411yescharacteristic
59C2×D97—21no—
60C2×D97—21no—
61C2×D97#10112yes—
62C4×C97—21no—
63C4×C97—21no—
64C4×C97—21no—
65C4×C97—42no—
66Dic97—21no—
67Dic97—42no—
68Dic97AGL(1,ℤ/8)12yes—
69Dic97#13712yes—
70C2×C97C2×#10111yescharacteristic
71C2×C97#41811yescharacteristic
72C2×C97#41911yescharacteristic
73D97—22no—
74#1556—971nomaximal, Sylow
75C97#155611yescharacteristic, complemented, Sylow
76C2×C4⋊Q8—971no—
77#432—971no—
78#432—971no—
79#332—974no—
80C22×Q8—1941no—
81C4⋊Q8—972no—
82C4⋊Q8—971no—
83C4⋊Q8—1941no—
84C4⋊Q8—1941no—
85C4⋊Q8#8760411yescharacteristic
86C2×C4⋊C4—972no—
87C2×C4⋊C4—1941no—
88C2×C42—971no—
89C4⋊C8—972no—
90C4⋊C8—972no—
91C22⋊C8—1941no—
92C22⋊C8—1941no—
93#97—1941no—
94#97—1941no—
95C2×Q8—21no—
96C2×Q8—3884no—
97C2×Q8—1942no—
98C2×Q8—1941no—
99C22×C4—971no—
100C22×C4—971no—
101C22×C4—971no—
102C22×C4—1942no—
103C2×C8—1942no—
104C2×C8—1942no—
105C4⋊C4—21no—
106C4⋊C4—972no—
107C4⋊C4—1942no—
108C4⋊C4—1942no—
109C4⋊C4—1941no—
110C4⋊C4C2×#8760412yes—
111C42—971no—
112C42—971no—
113C42—971no—
114C42C2×#8760411yescharacteristic
115C23—971no—
116Q8—42no—
117Q8—3882no—
118Q8—3884no—
119C2×C4—22no—
120C2×C4—971no—
121C2×C4—971no—
122C2×C4—971no—
123C2×C4—3884no—
124C2×C4—1942no—
125C2×C4—1942no—
126C2×C4—1942no—
127C2×C4—1942no—
128C2×C4C22⋊#8760411yescharacteristic
129C2×C4C22⋊#8760411yescharacteristic
130C2×C4C22×#8760411yescharacteristic
131C8—3882no—
132C8—3882no—
133C22—972no—
134C22—1941no—
135C22—1941no—
136C22C2×C22⋊#8760411yescharacteristic
137C4—21no—
138C4—21no—
139C4—21no—
140C4—42no—
141C4—972no—
142C4—972no—
143C4—3882no—
144C4—1941no—
145C2—1942no—
146C2D97⋊#10111yescharacteristic
147C2#2026172711yescharacteristic
148C2#2026172811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4⋊Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C22⋊C4
cosocleC4×Dic97C23
hypercenterC4⋊Q8#87604
hypocenterC97#1556
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20261594C2semidirect
75. C97#1556semidirect

Maximal subgroups

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

subgroupindex
#202617582
#202615942
C4⋊Q8×D972
#155697