Finite Group Databasebeta

← #27157648

Subgroups — #27157648

Order 12416.

296 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 62 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
1C2×#15690869C212yesmaximal
2#20261612C214yesmaximal, complemented
3#20261209C211yesmaximal, characteristic
4C2×C22⋊#87604C2214yes—
5#15690869—22no—
6#15690869—22no—
7C22×D388C2211yescharacteristic
8C4×D388—22no—
9#15690772—22no—
10#15690769C411yescharacteristic, complemented
11C22⋊C4×D97C412yescomplemented
12C22⋊C4×D97C2212yes—
13#117×C97C411yescharacteristic, complemented
14C22×#87604—24no—
15C22⋊#87604—24no—
16C22⋊#87604—24no—
17C23×D97C2×C411yescharacteristic
18C23×D97C2311yescharacteristic
19C2×D388—24no—
20C2×D388D412yes—
21C2×D388D412yes—
22C2×C4×D97—22no—
23C2×C4×D97—22no—
24C4⋊Dic97—21no—
25C22⋊Dic97C2×C412yes—
26#13095488—22no—
27#13095488—22no—
28C4×Dic97—21no—
29C22×C4×C97C2×C411yescharacteristic
30C4⋊C4×C97—21no—
31C22⋊C4×C97C2×C412yes—
32C42×C97—21no—
33C23×C97C22×C411yescharacteristic
34C22×D97—22no—
35C22×D97—81no—
36C22×D97—81no—
37C22×D97C22⋊C412yes—
38C22×D97C2×D412yes—
39C22×D97#4114yes—
40C2×#87604—44no—
41C2×#87604—84no—
42C2×C4×C97—22no—
43C2×C4×C97—22no—
44C2×C4×C97C22⋊C411yescharacteristic
45C2×C4×C97C22⋊C411yescharacteristic
46C2×Dic97—22no—
47D388—24no—
48D388—24no—
49C4×D97—82no—
50C4×D97—82no—
51C22×C97—41no—
52C22×C97C2×C22⋊C411yescharacteristic
53C22×C97#11712yes—
54C2×D97—22no—
55C2×D97—42no—
56C2×D97—44no—
57C2×D97—42no—
58C2×D97—82no—
59C2×D97#12314yes—
60#87604—84no—
61C4×C97—21no—
62C4×C97—21no—
63C4×C97—42no—
64Dic97—42no—
65C2×C97—21no—
66C2×C97—22no—
67C2×C97#38111yescharacteristic
68D97—24no—
69D97—42no—
70#1828—971nomaximal, Sylow
71C97#182811yescharacteristic, complemented, Sylow
72#518—971no—
73C2×#99—972no—
74#342—974no—
75C22×D4—971no—
76C4×D4—1942no—
77C4×D4—1942no—
78#117—971no—
79#117#8760411yescharacteristic, complemented
80C2×C22⋊C4—972no—
81C2×C22⋊C4—972no—
82C2×C22⋊C4—974no—
83#99—1942no—
84#99—1942no—
85C24—971no—
86C24—971no—
87C2×D4—972no—
88C2×D4—972no—
89C2×D4—1944no—
90C22×C4—1942no—
91C22×C4—1942no—
92C22×C4—1944no—
93C22×C4C2×#8760411yescharacteristic
94C4⋊C4—21no—
95C4⋊C4—1941no—
96C22⋊C4—972no—
97C22⋊C4—1942no—
98C22⋊C4—1942no—
99C22⋊C4—1944no—
100C22⋊C4—1944no—
101C22⋊C4C2×#8760412yes—
102C42—21no—
103C42—1941no—
104C23—7761no—
105C23—7761no—
106C23—972no—
107C23—974no—
108C23—972no—
109C23—1942no—
110C23C22×#8760411yescharacteristic
111D4—1944no—
112D4—1944no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—7762no—
116C2×C4—7762no—
117C2×C4—7764no—
118C2×C4—3884no—
119C2×C4—1942no—
120C2×C4C22⋊#8760411yescharacteristic
121C2×C4C22⋊#8760411yescharacteristic
122C22—41no—
123C22—7762no—
124C22—974no—
125C22—3882no—
126C22—3882no—
127C22—3884no—
128C22—1942no—
129C22C2×C22⋊#8760411yescharacteristic
130C22#1569088912yes—
131C4—21no—
132C4—21no—
133C4—42no—
134C4—7764no—
135C4—3882no—
136C2—21no—
137C2—22no—
138C2—3882no—
139C2—1944no—
140C2#2026167811yescharacteristic
subgroupG/H
centerC2#20261678
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
Fitting#117×C97C4
socleC2×C97#381
prime radicalC22×D388C22
cosocleC23×D97C23
hypercenter#117#87604
hypocenterC97#1828
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C23 ◁ #117

Complements

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

NH ≅ G/Ntype
2. #20261612C2semidirect
10. #15690769C4semidirect
11. C22⋊C4×D97C4semidirect
13. #117×C97C4semidirect
71. C97#1828semidirect
79. #117#87604semidirect

Maximal subgroups

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

subgroupindex
C2×#156908692
#202616122
#202612092
#182897