Finite Group Databasebeta

← #11724514

Subgroups — #11724514

Order 864.

292 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 61 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×C3×C4×C3⋊D3C211yesmaximal, characteristic
2#88994C212yesmaximal
3C4⋊(C3⋊#157)C212yesmaximal, complemented
4C4×C3⋊#157C212yesmaximal, complemented
5#64286—31nomaximal
6C22×C3×C3⋊D3C2211yescharacteristic
7C2×C3⋊#157C2214yes—
8C2×C33×C4C411yescharacteristic, complemented
9C2×C3×C3⋊Dic3C411yescharacteristic, complemented
10C3×C4×C3⋊D3C411yescharacteristic, complemented
11C3×C4×C3⋊D3C411yescharacteristic, complemented
12C3×C4×C3⋊D3C2211yescharacteristic
13C3×C4×C3⋊D3C2211yescharacteristic
14C2×C4×C3⋊D3D311yescharacteristic
15C2×C3×C4×D3—62no—
16C22⋊#157—32no—
17C4⋊#157—32no—
18C4×#157—32no—
19C22×C33C2×C411yescharacteristic
20C2×C3×C3⋊D3—22no—
21C2×C3×C3⋊D3C2×C411yescharacteristic
22C2×C3×C3⋊D3C2311yescharacteristic
23C3⋊#157—24no—
24C33×C4C2×C411yescharacteristic
25C33×C4C2×C411yescharacteristic
26C3×C3⋊Dic3C2×C411yescharacteristic
27C3×C3⋊Dic3C2×C411yescharacteristic
28#926—91nomaximal
29C22×C3⋊D3C2×D311yescharacteristic
30C22×C3×D3—62no—
31C2×#157—34no—
32C2×C32×C4—22no—
33C2×C32×C4—42no—
34C2×C32×C4Dic311yescharacteristic, complemented
35C2×C3⋊Dic3Dic311yescharacteristic, complemented
36C4×C3⋊D3Dic311yescharacteristic, complemented
37C4×C3⋊D3Dic311yescharacteristic, complemented
38C4×C3⋊D3C2×D311yescharacteristic
39C4×C3⋊D3C2×D311yescharacteristic
40C2×C3×Dic3—62no—
41C3×C4×D3—62no—
42C3×C4×D3—62no—
43C3×C4×D3—62no—
44C3×C4×D3—62no—
45C2×C33—21no—
46C2×C33C22×C411yescharacteristic
47C3×C3⋊D3—21no—
48C3×C3⋊D3#4112yes—
49C22×C3×C4—91no—
50C2×C4×D3—62no—
51#217—92no—
52C4⋊Dic3—92no—
53C4×Dic3—92no—
54C22×C32—22no—
55C22×C32—42no—
56C22×C32C2×Dic311yescharacteristic
57C2×C3⋊D3—22no—
58C2×C3⋊D3C2×Dic311yescharacteristic
59C2×C3⋊D3C22×D311yescharacteristic
60C2×C3×D3—62no—
61C2×C3×D3—62no—
62C2×C3×D3—68no—
63#157—64no—
64C32×C4—22no—
65C32×C4—22no—
66C32×C4—42no—
67C32×C4—42no—
68C32×C4C2×Dic311yescharacteristic
69C32×C4C2×Dic311yescharacteristic
70C3⋊Dic3C2×Dic311yescharacteristic
71C3⋊Dic3C2×Dic311yescharacteristic
72C3×Dic3—62no—
73C3×Dic3—62no—
74#117—271noSylow
75C33#11711yescharacteristic, complemented, Sylow
76C23×C3—91no—
77C22×D3—62no—
78C2×C3×C4—22no—
79C2×C3×C4—42no—
80C2×C3×C4—91no—
81C2×C3×C4—91no—
82C2×C3×C4—91no—
83C2×C3×C4—91no—
84C2×C3×C4—91no—
85C2×C3×C4#15711yescharacteristic, complemented
86C2×Dic3—62no—
87C2×Dic3—94no—
88C4×D3—62no—
89C4×D3—62no—
90C4×D3—62no—
91C4×D3—62no—
92C2×C32—21no—
93C2×C32—24no—
94C2×C32—22no—
95C2×C32—44no—
96C2×C32—42no—
97C2×C32C22×Dic311yescharacteristic
98C3⋊D3—21no—
99C3⋊D3#23512yes—
100C3×D3—64no—
101C3×D3—64no—
102C22×C4—91no—
103C4⋊C4—272no—
104C22⋊C4—272no—
105C42—272no—
106C22×C3—22no—
107C22×C3—42no—
108C22×C3—91no—
109C22×C3—91no—
110C22×C3—182no—
111C22×C3C2×#15711yescharacteristic
112C2×D3—62no—
113C2×D3—62no—
114C2×D3—68no—
115C3×C4—22no—
116C3×C4—22no—
117C3×C4—42no—
118C3×C4—42no—
119C3×C4—91no—
120C3×C4—91no—
121C3×C4C2×#15711yescharacteristic
122C3×C4C2×#15711yescharacteristic
123Dic3—62no—
124Dic3—62no—
125Dic3—184no—
126C32—22no—
127C32—42no—
128C32#92611yescharacteristic, complemented
129C23—91no—
130C2×C4—91no—
131C2×C4—91no—
132C2×C4—91no—
133C2×C4—91no—
134C2×C4—91no—
135C2×C4—274no—
136C2×C4C3⋊#15711yescharacteristic, complemented
137C2×C3—24no—
138C2×C3—21no—
139C2×C3—22no—
140C2×C3—44no—
141C2×C3—42no—
142C2×C3—92no—
143C2×C3—181no—
144C2×C3C22×#15711yescharacteristic
145D3—64no—
146D3—64no—
147C22—91no—
148C22—91no—
149C22—182no—
150C22C2×C3⋊#15711yescharacteristic
151C4—91no—
152C4—91no—
153C4—544no—
154C4C2×C3⋊#15711yescharacteristic
155C4C2×C3⋊#15711yescharacteristic
156C3—22no—
157C3—42no—
158C3#6428611yescharacteristic, complemented
159C2—21no—
160C2—92no—
161C2—181no—
162C2C22×C3⋊#15711yescharacteristic
subgroupG/H
centerC4C2×C3⋊#157
commutator, socleC2×C33C22×C4
FrattiniC2C22×C3⋊#157
FittingC2×C33×C4C4
prime radicalC22×C3×C3⋊D3C22
cosocleC2×C3×C3⋊D3C23
hypercenterC2×C4C3⋊#157
hypocenterC33#117
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. C4⋊(C3⋊#157)C2semidirect
4. C4×C3⋊#157C2semidirect
8. C2×C33×C4C4semidirect
9. C2×C3×C3⋊Dic3C4semidirect
10. C3×C4×C3⋊D3C4semidirect
11. C3×C4×C3⋊D3C4semidirect
34. C2×C32×C4Dic3semidirect
35. C2×C3⋊Dic3Dic3semidirect
36. C4×C3⋊D3Dic3semidirect
37. C4×C3⋊D3Dic3semidirect
75. C33#117semidirect
85. C2×C3×C4#157semidirect
128. C32#926semidirect
136. C2×C4C3⋊#157semidirect
158. C3#64286semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C3⋊D32
#889942
C4⋊(C3⋊#157)2
C4×C3⋊#1572
#642863
#9269