Finite Group Databasebeta

← #27157159

Subgroups — #27157159

Order 12416.

208 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 56 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#20261680C211yesmaximal, characteristic
3#20261588C214yesmaximal
4#20261290C211yesmaximal, characteristic
5#15690877—21no—
6#15690867C2212yes—
7#15690865—21no—
8(C2×#87604)⋊C4—21no—
9(C2×#87604)⋊C4—21no—
10(C2×#87604)⋊C4C2212yes—
11#15690810C411yescharacteristic, complemented
12#15690801—21no—
13#15690790C2212yes—
14#15690786—21no—
15Q8×Dic97—21no—
16#15690775C412yescomplemented
17#15690768—21no—
18C42×D97C2211yescharacteristic
19C4⋊Q8×C97C411yescharacteristic, complemented
20C22×#87604—22no—
21C2×#11715897—22no—
22C2×Dic194C2×C412yes—
23C2×C4×D97D411yescharacteristic
24C2×C4×D97D411yescharacteristic
25C2×C4×D97C2311yescharacteristic
26C4⋊Dic97—21no—
27C4⋊Dic97—21no—
28C4⋊Dic97—22no—
29#13095490—22no—
30#13095488—22no—
31C4×Dic97—22no—
32C4×Dic97C2×C411yescharacteristic
33C4×Dic97D411yescharacteristic
34C4×Dic97D411yescharacteristic
35C2×Q8×C97—21no—
36C4⋊C4×C97—21no—
37C4⋊C4×C97C2×C412yes—
38C42×C97C2×C411yescharacteristic
39C22×D97C2×D411yescharacteristic
40C2×#87604—42no—
41C2×#87604—42no—
42C2×#87604—42no—
43Q8×C97—42no—
44C2×C4×C97—22no—
45C2×C4×C97C22⋊C411yescharacteristic
46C2×C4×C97C22⋊C411yescharacteristic
47C2×C4×C97C22×C411yescharacteristic
48C2×Dic97—22no—
49C2×Dic97C22⋊C411yescharacteristic
50C2×Dic97C22⋊C411yescharacteristic
51C2×Dic97C2×D411yescharacteristic
52C4×D97—22no—
53C4×D97—41no—
54C4×D97—41no—
55Dic194—42no—
56#11715897—42no—
57C22×C97C2×C22⋊C411yescharacteristic
58C2×D97—21no—
59C2×D97—21no—
60C2×D97#13712yes—
61#87604—82no—
62C4×C97—21no—
63C4×C97—21no—
64C4×C97—21no—
65C4×C97—42no—
66Dic97—21no—
67Dic97—21no—
68Dic97—21no—
69Dic97—42no—
70C2×C97#41011yescharacteristic
71C2×C97#41911yescharacteristic
72C2×C97#42111yescharacteristic
73D97—22no—
74#1515—971nomaximal, Sylow
75C97#151511yescharacteristic, complemented, Sylow
76#533—971no—
77#432—971no—
78#383—971no—
79#328—974no—
80C4⋊Q8—971no—
81C4⋊Q8#8760411yescharacteristic, complemented
82#125—1941no—
83C22⋊Q8—972no—
84C22⋊Q8—1941no—
85C4×Q8—972no—
86C4×Q8—1941no—
87#117—1941no—
88C2×C42—971no—
89C4⋊C8—1941no—
90C22⋊C8—972no—
91#97—1941no—
92#95—972no—
93#95—1941no—
94#95—1941no—
95C2×Q8—21no—
96C2×Q8—972no—
97C22×C4—971no—
98C22×C4—971no—
99C22×C4—971no—
100C22×C4—1942no—
101C2×C8—1942no—
102C4⋊C4—21no—
103C4⋊C4—1942no—
104C4⋊C4—1942no—
105C4⋊C4—1941no—
106C4⋊C4—1941no—
107C4⋊C4C2×#8760412yes—
108C22⋊C4—1942no—
109C42—971no—
110C42—971no—
111C42—971no—
112C42—1942no—
113C42C2×#8760411yescharacteristic
114C23—971no—
115Q8—42no—
116Q8—3882no—
117C2×C4—22no—
118C2×C4—971no—
119C2×C4—971no—
120C2×C4—971no—
121C2×C4—3881no—
122C2×C4—3881no—
123C2×C4—3882no—
124C2×C4—3882no—
125C2×C4—3882no—
126C2×C4—1942no—
127C2×C4—1942no—
128C2×C4C22⋊#8760411yescharacteristic
129C2×C4C22⋊#8760411yescharacteristic
130C2×C4C22×#8760411yescharacteristic
131C8—3882no—
132C22—972no—
133C22—1941no—
134C22—1941no—
135C22C2×C22⋊#8760411yescharacteristic
136C4—21no—
137C4—21no—
138C4—21no—
139C4—42no—
140C4—7762no—
141C4—3882no—
142C4—1941no—
143C4—1941no—
144C4—1941no—
145C2—1942no—
146C2#2026171611yescharacteristic
147C2#2026172811yescharacteristic
148C2#2026173311yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4⋊Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C2×D4
cosocleC2×C4×D97C23
hypercenterC4⋊Q8#87604
hypocenterC97#1515
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
11. #15690810C4semidirect
16. #15690775C4semidirect
19. C4⋊Q8×C97C4semidirect
75. C97#1515semidirect
81. C4⋊Q8#87604semidirect

Maximal subgroups

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

subgroupindex
#202617582
#202616802
#202615882
#202612902
#151597