Finite Group Databasebeta

← #27157262

Subgroups — #27157262

Order 12416.

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

normal subgroups only

This group has 198 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#20261752C211yesmaximal, characteristic
3#20261595C212yesmaximal, complemented
4#20261594C212yesmaximal, complemented
5#20261287C211yesmaximal, characteristic
6#15690877C2212yes—
7#15690877C2212yes—
8C22⋊#11715897—21no—
9C22⋊#11715897—21no—
10#15690865—21no—
11C4×#11715897—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
23C2×#11715897—22no—
24C2×#11715897—22no—
25C2×Dic194—21no—
26C2×Dic194C2×C411yescharacteristic
27C22×Dic97D411yescharacteristic
28C22×Dic97D411yescharacteristic
29C4⋊Dic97—21no—
30C4⋊Dic97C2×C411yescharacteristic
31#13095490—21no—
32#13095490—21no—
33#13095490—21no—
34#13095490—21no—
35#13095490—21no—
36C22⋊Dic97—21no—
37C4×Dic97—21no—
38C4×Dic97—21no—
39C4×Dic97C2×C411yescharacteristic
40C4×Dic97D411yescharacteristic
41C4×Dic97D411yescharacteristic
42C4×Dic97C2311yescharacteristic
43C2×Q8×C97C2×C411yescharacteristic
44C22×C4×C97C2×C411yescharacteristic
45C4⋊C4×C97—21no—
46C4⋊C4×C97C2×C411yescharacteristic
47C22⋊C4×C97—21no—
48C23×C97C22⋊C411yescharacteristic
49Q8×C97—41no—
50C2×C4×C97—21no—
51C2×C4×C97—21no—
52C2×C4×C97—41no—
53C2×C4×C97C22⋊C411yescharacteristic
54C2×C4×C97C22×C411yescharacteristic
55C2×Dic97—22no—
56C2×Dic97—22no—
57C2×Dic97—21no—
58C2×Dic97—21no—
59C2×Dic97C22⋊C411yescharacteristic
60C2×Dic97C22⋊C411yescharacteristic
61C2×Dic97C2×D411yescharacteristic
62C2×Dic97C2×D411yescharacteristic
63Dic194—42no—
64Dic194—41no—
65#11715897—42no—
66#11715897—42no—
67C22×C97—21no—
68C22×C97—21no—
69C22×C97—21no—
70C22×C97C2×C22⋊C411yescharacteristic
71C4×C97—21no—
72C4×C97—21no—
73C4×C97—41no—
74C4×C97—41no—
75Dic97—21no—
76Dic97—21no—
77Dic97—41no—
78Dic97—41no—
79Dic97#13512yes—
80Dic97#13712yes—
81C2×C97—41no—
82C2×C97#41311yescharacteristic
83C2×C97#41611yescharacteristic
84C2×C97#41911yescharacteristic
85#1554—971nomaximal, Sylow
86C97#155411yescharacteristic, complemented, Sylow
87#533—971no—
88#432—971no—
89#431—971no—
90#333—972no—
91#332—972no—
92C4⋊Q8—971no—
93C4⋊Q8—971no—
94#125—1941no—
95C22⋊Q8—971no—
96C22⋊Q8—1941no—
97C22⋊Q8#8760411yescharacteristic
98C4×Q8—971no—
99C4×Q8—971no—
100C4×Q8—1941no—
101#117—1941no—
102C2×C42—971no—
103C4⋊C8—972no—
104C4⋊C8—972no—
105C22⋊C8—1941no—
106C22⋊C8—1941no—
107#97—1941no—
108C4×C8—1941no—
109C2×Q8—971no—
110C2×Q8—1941no—
111C2×Q8C2×#8760411yescharacteristic
112C22×C4—971no—
113C22×C4—971no—
114C22×C4C2×#8760411yescharacteristic
115C2×C8—1942no—
116C2×C8—1942no—
117C4⋊C4—21no—
118C4⋊C4—971no—
119C4⋊C4—1941no—
120C4⋊C4—1941no—
121C4⋊C4—1941no—
122C4⋊C4—1941no—
123C4⋊C4—1941no—
124C4⋊C4—1941no—
125C4⋊C4C2×#8760411yescharacteristic
126C22⋊C4—21no—
127C22⋊C4—1941no—
128C42—971no—
129C42—971no—
130C42—971no—
131C42—971no—
132C42—1941no—
133C42—1941no—
134C23C22⋊#8760411yescharacteristic
135Q8—41no—
136Q8—3881no—
137Q8—3882no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—41no—
141C2×C4—971no—
142C2×C4—971no—
143C2×C4—971no—
144C2×C4—971no—
145C2×C4—1942no—
146C2×C4—1942no—
147C2×C4—1941no—
148C2×C4—1941no—
149C2×C4C22⋊#8760411yescharacteristic
150C2×C4C22×#8760411yescharacteristic
151C8—3882no—
152C8—3882no—
153C22—21no—
154C22—21no—
155C22—21no—
156C22C2×C22⋊#8760411yescharacteristic
157C4—21no—
158C4—21no—
159C4—41no—
160C4—41no—
161C4—972no—
162C4—972no—
163C4—3881no—
164C4—3881no—
165C4—1941no—
166C4—1941no—
167C2—41no—
168C2#2026172211yescharacteristic
169C2#2026172511yescharacteristic
170C2#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#1554
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
86. C97#1554semidirect

Maximal subgroups

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

subgroupindex
#202617552
#202617522
#202615952
#202615942
#202612872
#155497