Finite Group Databasebeta

← #27139681

Subgroups — #27139681

Order 12416.

134 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 37 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#20260604C211yesmaximal, characteristic
2#20260461C211yesmaximal, characteristic
3#324×C97C211yesmaximal, characteristic
4C2×C4⋊Dic97C411yescharacteristic
5C2×#13095490—21no—
6C2×#13095490—21no—
7C2×C4×Dic97C411yescharacteristic
8C22⋊#11715895—21no—
9C2×C4⋊C4×C97C2211yescharacteristic
10C22⋊C8×C97—21no—
11C22×Dic97—21no—
12C22×Dic97—21no—
13C2×#11715895—41no—
14C4⋊Dic97—41no—
15#13095490—42no—
16#13095490—42no—
17C4×Dic97—21no—
18C4×Dic97C812yescomplemented
19C22×C4×C97C2×C411yescharacteristic
20C22×C4×C97D411yescharacteristic
21C22×C4×C97D411yescharacteristic
22C2×C8×C97—41no—
23C4⋊C4×C97—41no—
24C23×C97C22⋊C411yescharacteristic
25C2×C4×C97—21no—
26C2×C4×C97—21no—
27C2×C4×C97—21no—
28C2×C4×C97—21no—
29C2×C4×C97—41no—
30C2×C4×C97—41no—
31C2×C4×C97C2×C811yescharacteristic
32C2×C4×C97M1611yescharacteristic
33C2×Dic97—22no—
34C2×Dic97—22no—
35C2×Dic97—22no—
36C2×Dic97—41no—
37C2×Dic97—41no—
38C2×Dic97—41no—
39C8×C97—81no—
40#11715895—81no—
41C22×C97—21no—
42C22×C97—21no—
43C22×C97C22⋊C811yescharacteristic
44C22×C97#9911yescharacteristic
45C22×C97#10111yescharacteristic
46C4×C97—21no—
47C4×C97—21no—
48C4×C97—41no—
49C4×C97—41no—
50Dic97—24no—
51Dic97—81no—
52C2×C97—21no—
53C2×C97—21no—
54C2×C97#32411yescharacteristic
55C2×C97#35311yescharacteristic
56C2×C97#35611yescharacteristic
57#1325—971nomaximal, Sylow
58C97#132511yescharacteristic, complemented, Sylow
59#384—971no—
60#324—971no—
61#324D9711yescharacteristic
62C2×C4⋊C4—971no—
63C2×C4⋊C4—1941no—
64C2×C4⋊C4—1941no—
65C2×C4⋊C4C2×D9711yescharacteristic
66C2×C42—971no—
67C22⋊C8—21no—
68C22⋊C8—1941no—
69C22×C4—1941no—
70C22×C4—1941no—
71C22×C4C4×D9711yescharacteristic
72C22×C4D38811yescharacteristic
73C22×C4C22⋊D9711yescharacteristic
74C2×C8—41no—
75C2×C8—3881no—
76C4⋊C4—41no—
77C4⋊C4—3881no—
78C4⋊C4—3882no—
79C4⋊C4—3882no—
80C42—972no—
81C42—1941no—
82C23#1309548811yescharacteristic
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—21no—
87C2×C4—41no—
88C2×C4—41no—
89C2×C4—3881no—
90C2×C4—3881no—
91C2×C4—3881no—
92C2×C4—1942no—
93C2×C4—1942no—
94C2×C4—1942no—
95C2×C4C8×D9711yescharacteristic
96C2×C4#1309549411yescharacteristic
97C8—81no—
98C8—7761no—
99C22—21no—
100C22—21no—
101C22#1569072211yescharacteristic
102C22#1569072411yescharacteristic
103C22#1569072811yescharacteristic
104C4—21no—
105C4—21no—
106C4—41no—
107C4—41no—
108C4—7761no—
109C4—1944no—
110C2—21no—
111C2—21no—
112C2#2026046011yescharacteristic
113C2#2026052711yescharacteristic
114C2#2026053311yescharacteristic
subgroupG/H
centerC22#15690724
commutatorC2×C4×C97C2×C8
FrattiniC2×C4⋊C4C2×D97
Fitting#324×C97C2
socleC22×C97#99
prime radicalC23×C97C22⋊C4
cosocleC2×C4⋊C4×C97C22
hypercenter#324D97
hypocenterC97#1325
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #324

Complements

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

NH ≅ G/Ntype
18. C4×Dic97C8semidirect
58. C97#1325semidirect

Maximal subgroups

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

subgroupindex
#202606042
#202604612
#324×C972
#132597