Finite Group Databasebeta

← #27158038

Subgroups — #27158038

Order 12416.

230 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 54 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#20261721C212yesmaximal
2#20261645C214yesmaximal, complemented
3#20261309C211yesmaximal, characteristic
4D97⋊M16—22no—
5#15690871C2214yes—
6#15690870—22no—
7C2×#13095511C2211yescharacteristic
8#15690808C2212yes—
9#15690799C412yes—
10#15690793—22no—
11#15690786—22no—
12#15690769C411yescharacteristic
13C22×Q8×C97C411yescharacteristic
14#13095521—42no—
15#13095520—24no—
16#13095519—42no—
17#13095511—42no—
18C2×Dic194C2311yescharacteristic
19C2×D388C2×C411yescharacteristic
20C2×C22⋊D97D412yes—
21C2×C4×D97D412yes—
22C4⋊Dic97—21no—
23#13095490—24no—
24C22⋊Dic97—21no—
25#13095488—24no—
26C4×Dic97C2×C412yes—
27C2×Q8×C97—21no—
28C2×Q8×C97—42no—
29C2×Q8×C97C2×C412yes—
30C22×C4×C97—21no—
31C22×C4×C97C2×C411yescharacteristic
32C23×C97C22⋊C411yescharacteristic
33C22×D97C22⋊C412yes—
34Q8×C97—42no—
35Q8×C97—42no—
36C2×C4×C97—22no—
37C2×C4×C97—22no—
38C2×C4×C97—42no—
39C2×C4×C97C22⋊C411yescharacteristic
40C2×C4×C97C22×C411yescharacteristic
41C22⋊D97—42no—
42C2×Dic97—22no—
43C2×Dic97C2×D412yes—
44D388—41no—
45C4×D97—22no—
46C4×D97—22no—
47Dic194—41no—
48#11715897—44no—
49C22×C97—21no—
50C22×C97—41no—
51C22×C97C2×C22⋊C411yescharacteristic
52C2×D97—22no—
53C2×D97—42no—
54C4×C97—21no—
55C4×C97—42no—
56C4×C97#9912yes—
57Dic97—22no—
58Dic97—42no—
59C2×C97—21no—
60C2×C97—41no—
61C2×C97C2×#9911yescharacteristic
62D97—42no—
63#2135—971nomaximal, Sylow
64C97#213511yescharacteristic, complemented, Sylow
65#536—971no—
66#413—972no—
67#356—974no—
68C2×#41—971no—
69C22×Q8#8760411yescharacteristic
70C2×M16—1942no—
71C4⋊Q8—972no—
72#124—972no—
73#123—1942no—
74C22⋊Q8—1942no—
75#117—971no—
76#101—974no—
77#100—1942no—
78#41—3882no—
79C2×Q8—21no—
80C2×Q8—42no—
81C2×Q8—971no—
82C2×Q8C2×#8760412yes—
83C2×D4—972no—
84C2×D4—971no—
85C22×C4—21no—
86C22×C4—972no—
87C22×C4C2×#8760411yescharacteristic
88M16—3882no—
89M16—1944no—
90C2×C8—3882no—
91C4⋊C4—1944no—
92C4⋊C4—1941no—
93C22⋊C4—1944no—
94C22⋊C4—1941no—
95C42—972no—
96C23—972no—
97C23C22⋊#8760411yescharacteristic
98Q8—42no—
99Q8—42no—
100Q8—3881no—
101D4—3881no—
102D4—3882no—
103C2×C4—22no—
104C2×C4—22no—
105C2×C4—42no—
106C2×C4—972no—
107C2×C4—1942no—
108C2×C4—1942no—
109C2×C4—1942no—
110C2×C4C22⋊#8760411yescharacteristic
111C2×C4C22×#8760411yescharacteristic
112C8—3884no—
113C22—21no—
114C22—41no—
115C22—3882no—
116C22—1942no—
117C22C2×C22⋊#8760411yescharacteristic
118C4—21no—
119C4—42no—
120C4—3882no—
121C4—1942no—
122C4#1569086812yes—
123C2—21no—
124C2—41no—
125C2—3882no—
126C2C2×#1569086811yescharacteristic
subgroupG/H
centerC2C2×#15690868
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC22×Q8×C97C4
socleC2×C97C2×#99
prime radicalC2×#13095511C22
cosocleC2×Dic194C23
hypercenterC22×Q8#87604
hypocenterC97#2135
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. #20261645C2semidirect
64. C97#2135semidirect

Maximal subgroups

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

subgroupindex
#202617212
#202616452
#202613092
#213597