Finite Group Databasebeta

← #11724515

Subgroups — #11724515

Order 864.

236 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 57 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
1C3×#3830C211yesmaximal, characteristic
2#88993C212yesmaximal, complemented
3C2×C3⋊#622C212yesmaximal, complemented
4#88989C211yesmaximal, characteristic, complemented
5#88988C211yesmaximal, characteristic, complemented
6#64287—31nomaximal
7C33×D4C411yescharacteristic
8C3×#638C412yes—
9C2×C3×C3⋊Dic3C2212yes—
10C3×C4×C3⋊D3C2211yescharacteristic
11C3×#634C411yescharacteristic
12C3⋊#622C2211yescharacteristic
13C3⋊#622C2211yescharacteristic
14C3⋊#622C2212yes—
15#3830D311yescharacteristic
16C3×#237—62no—
17#3792—32no—
18C2×#622—32no—
19#3788—31no—
20#3787—31no—
21C22×C33C2×C412yes—
22C2×C3×C3⋊D3C2×C411yescharacteristic
23C33×C4C2×C411yescharacteristic
24C3×C3⋊Dic3C2×C412yes—
25C3×C3⋊Dic3C2311yescharacteristic
26#948—91nomaximal
27C32×D4—22no—
28C32×D4—42no—
29C32×D4Dic311yescharacteristic
30#638Dic312yes—
31C2×C3⋊Dic3C2×D312yes—
32C4×C3⋊D3C2×D311yescharacteristic
33#634Dic311yescharacteristic
34C3×#67—64no—
35C2×C3×Dic3—64no—
36C3×C4×D3—62no—
37C3×Dic6—62no—
38#622—31no—
39#622—31no—
40#622—32no—
41C2×C33—22no—
42C2×C33C22×C411yescharacteristic
43C3×C3⋊D3—21no—
44C3×#41—91no—
45#237—62no—
46#208—91no—
47#208—92no—
48C2×C3⋊C8—91no—
49C2×C3⋊C8—92no—
50C22×C32—24no—
51C22×C32—44no—
52C22×C32C2×Dic312yes—
53C2×C3⋊D3C2×Dic311yescharacteristic
54C2×C3×D3—62no—
55C32×C4—22no—
56C32×C4—42no—
57C32×C4C2×Dic311yescharacteristic
58C3⋊Dic3C2×Dic312yes—
59C3⋊Dic3C22×D311yescharacteristic
60C3×Dic3—64no—
61C3×Dic3—62no—
62#131—271noSylow
63C33#13111yescharacteristic, complemented, Sylow
64C3×Q8—91no—
65C3×D4—22no—
66C3×D4—42no—
67C3×D4—92no—
68C3×D4#15711yescharacteristic
69C2×C3×C4—92no—
70C2×C3×C4—91no—
71#67—64no—
72C2×Dic3—64no—
73C4×D3—62no—
74Dic6—62no—
75C3⋊C8—91no—
76C3⋊C8—91no—
77C3⋊C8—92no—
78C2×C32—22no—
79C2×C32—22no—
80C2×C32—42no—
81C2×C32—44no—
82C2×C32—84no—
83C2×C32C22×Dic311yescharacteristic
84C3⋊D3—21no—
85C3×D3—122no—
86#41—91no—
87M16—271no—
88M16—272no—
89C2×C8—271no—
90C2×C8—272no—
91C22×C3—24no—
92C22×C3—44no—
93C22×C3—91no—
94C22×C3C2×#15712yes—
95C2×D3—62no—
96C3×C4—22no—
97C3×C4—42no—
98C3×C4—91no—
99C3×C4—92no—
100C3×C4C2×#15711yescharacteristic
101Dic3—64no—
102Dic3—62no—
103C32—22no—
104C32—42no—
105C32#94811yescharacteristic, complemented
106Q8—91no—
107D4—92no—
108D4C3⋊#15711yescharacteristic
109C2×C4—92no—
110C2×C4—91no—
111C8—271no—
112C8—271no—
113C8—272no—
114C2×C3—22no—
115C2×C3—22no—
116C2×C3—44no—
117C2×C3—42no—
118C2×C3—84no—
119C2×C3—181no—
120C2×C3C22×#15711yescharacteristic
121D3—122no—
122C22—91no—
123C22C2×C3⋊#15712yes—
124C4—91no—
125C4—92no—
126C4C2×C3⋊#15711yescharacteristic
127C3—22no—
128C3—42no—
129C3#6428711yescharacteristic, complemented
130C2—22no—
131C2—181no—
132C2C22×C3⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC2C22×C3⋊#157
commutator, socleC2×C33C22×C4
FittingC33×D4C4
prime radicalC3×#3830C2
cosocleC3×C3⋊Dic3C23
hypercenterD4C3⋊#157
hypocenterC33#131
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
2. #88993C2semidirect
3. C2×C3⋊#622C2semidirect
4. #88989C2semidirect
5. #88988C2semidirect
63. C33#131semidirect
105. C32#948semidirect
129. C3#64287semidirect

Maximal subgroups

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

subgroupindex
C3×#38302
#889932
C2×C3⋊#6222
#889892
#889882
#642873
#9489