Finite Group Databasebeta

← #27158019

Subgroups — #27158019

Order 12416.

238 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 52 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#20261716C212yesmaximal
2#20261642C212yesmaximal, complemented
3#20261641C212yesmaximal, complemented
4#20261310C211yesmaximal, characteristic
5#15690889—22no—
6#15690869—22no—
7#15690869C2214yes—
8C2×#13095512C2211yescharacteristic
9C2×Q8×D97C411yescharacteristic, complemented
10#15690808C411yescharacteristic, complemented
11#15690799C2211yescharacteristic
12#15690798C2211yescharacteristic
13#15690795—22no—
14#15690788—22no—
15#15690767C411yescharacteristic, complemented
16C4⋊Q8×C97C411yescharacteristic, complemented
17C4⋊#87604—42no—
18C22⋊#87604—24no—
19C4×#87604—42no—
20#13095512—42no—
21C2×Dic194—21no—
22C2×Dic194C2×C411yescharacteristic
23Q8×D97—42no—
24C2×D388D412yes—
25C2×D388C2311yescharacteristic
26C2×C4×D97—21no—
27C2×C4×D97C2×C411yescharacteristic
28C2×C4×D97D412yes—
29#13095490—22no—
30#13095490—21no—
31#13095488—21no—
32#13095488—22no—
33#13095488—22no—
34C4×Dic97C2×C411yescharacteristic
35C2×Q8×C97C2×C411yescharacteristic
36C2×Q8×C97C2×C411yescharacteristic
37C4⋊C4×C97—22no—
38C42×C97C2×C411yescharacteristic
39C22×D97C22⋊C411yescharacteristic
40C22×D97C2×D412yes—
41C2×#87604—44no—
42Q8×C97—41no—
43Q8×C97—41no—
44C2×C4×C97—21no—
45C2×C4×C97—21no—
46C2×C4×C97C22⋊C412yes—
47C2×C4×C97C22×C411yescharacteristic
48C2×Dic97—21no—
49C2×Dic97—21no—
50C2×Dic97C22⋊C411yescharacteristic
51D388—42no—
52D388—41no—
53C4×D97—22no—
54C4×D97—22no—
55C4×D97—22no—
56C4×D97—42no—
57Dic194—42no—
58Dic194—41no—
59C22×C97C2×C22⋊C411yescharacteristic
60C2×D97—21no—
61C2×D97—22no—
62C2×D97—41no—
63C2×D97—42no—
64#87604—84no—
65C4×C97—21no—
66C4×C97—22no—
67C4×C97—41no—
68C4×C97—41no—
69Dic97—41no—
70Dic97—41no—
71Dic97#9912yes—
72C2×C97—21no—
73C2×C97C2×#9911yescharacteristic
74D97—42no—
75D97—41no—
76#2129—971nomaximal, Sylow
77C97#212911yescharacteristic, complemented, Sylow
78#536—971no—
79#410—972no—
80#354—972no—
81#354—972no—
82C2×#41—971no—
83C22×Q8—971no—
84C4⋊Q8—971no—
85C4⋊Q8#8760411yescharacteristic, complemented
86#124—971no—
87#124—971no—
88#123—1942no—
89C22⋊Q8—1942no—
90#117—971no—
91#117—1942no—
92#99—974no—
93#99—1942no—
94#41—3882no—
95C2×Q8—971no—
96C2×Q8—3882no—
97C2×Q8—1941no—
98C2×Q8C2×#8760411yescharacteristic
99C2×Q8C2×#8760411yescharacteristic
100C2×D4—972no—
101C2×D4—971no—
102C22×C4—972no—
103C22×C4—971no—
104C22×C4—1941no—
105C4⋊C4—22no—
106C4⋊C4—3882no—
107C4⋊C4—1941no—
108C4⋊C4—1942no—
109C22⋊C4—1941no—
110C22⋊C4—1942no—
111C22⋊C4—1942no—
112C22⋊C4—1944no—
113C42—971no—
114C42—3882no—
115C42C2×#8760411yescharacteristic
116C23—972no—
117C23—971no—
118Q8—41no—
119Q8—41no—
120Q8—3881no—
121Q8—3882no—
122D4—3882no—
123D4—3881no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—971no—
127C2×C4—3882no—
128C2×C4—3884no—
129C2×C4—1942no—
130C2×C4—1942no—
131C2×C4—1942no—
132C2×C4—1941no—
133C2×C4—1941no—
134C2×C4C22⋊#8760412yes—
135C2×C4C22×#8760411yescharacteristic
136C22—3882no—
137C22—3881no—
138C22—1942no—
139C22—1941no—
140C22C2×C22⋊#8760411yescharacteristic
141C4—22no—
142C4—21no—
143C4—41no—
144C4—41no—
145C4—7764no—
146C4—972no—
147C4—3881no—
148C4—3881no—
149C2—21no—
150C2—3882no—
151C2—3881no—
152C2D97⋊#9911yescharacteristic
subgroupG/H
centerC2D97⋊#99
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4⋊Q8×C97C4
socleC2×C97C2×#99
prime radicalC2×#13095512C22
cosocleC2×D388C23
hypercenterC4⋊Q8#87604
hypocenterC97#2129
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #20261642C2semidirect
3. #20261641C2semidirect
9. C2×Q8×D97C4semidirect
10. #15690808C4semidirect
15. #15690767C4semidirect
16. C4⋊Q8×C97C4semidirect
77. C97#2129semidirect
85. C4⋊Q8#87604semidirect

Maximal subgroups

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

subgroupindex
#202617162
#202616422
#202616412
#202613102
#212997