Finite Group Databasebeta

← #11721110

Subgroups — #11721110

Order 864.

304 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 60 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×#6229C211yesmaximal, characteristic, complemented
2C2×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
3C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4#88450C214yesmaximal, complemented
5#63864—31nomaximal
6#63863—31nomaximal
7C22×(C3⋊D3)⋊C3C2212yes—
8C2×C4×H(3,𝔽3)C2211yescharacteristic
9C2×#1084C2212yes—
10#6229—22no—
11C2×(C3⋊Dic3)⋊C3C2212yes—
12C4⋊((C3⋊D3)⋊C3)—22no—
13C2×C4⋊(C3⋊D3)D311yescharacteristic
14C2×C3×D12—31no—
15C2×C3×Dic6—31no—
16C2×C3×Dic6—31no—
17C2×C3×Dic6—61no—
18C4×C3⋊Dic3D311yescharacteristic, complemented
19C3×C4×Dic3—31no—
20#3722—34no—
21(C2×D3)⋊Dic3—34no—
22C22×H(3,𝔽3)C2311yescharacteristic
23C2×(C3⋊D3)⋊C3—22no—
24C2×(C3⋊D3)⋊C3—24no—
25C4×H(3,𝔽3)D412yes—
26#1084—22no—
27(C3⋊Dic3)⋊C3—22no—
28#947—91no—
29#936—91no—
30#875—91no—
31C22×C3⋊D3C2×D312yes—
32C22×C3×D3—32no—
33C2×C32×C4—21no—
34C2×C32×C4C2×D311yescharacteristic
35C2×C32×C4C2×D311yescharacteristic
36C2×C3⋊Dic3C2×D312yes—
37C4⋊(C3⋊D3)—22no—
38C2×C3×Dic3—32no—
39C2×C3×Dic3—32no—
40C2×C3×Dic3—32no—
41C2×C3×Dic3—62no—
42C3×D12—62no—
43C3×Dic6—64no—
44C3×Dic6—62no—
45C3×Dic6—62no—
46C2×H(3,𝔽3)C2×D411yescharacteristic
47C2×H(3,𝔽3)#4112yes—
48(C3⋊D3)⋊C3—42no—
49C2×C3×Q8—91no—
50C2×C3×D4—91no—
51C2×D12—31no—
52C2×D12—91no—
53C2×Dic6—31no—
54C2×Dic6—31no—
55C2×Dic6—61no—
56C3×C42—91no—
57#217—94no—
58(C2×D3)⋊C4—94no—
59(C2×D3)⋊C4—94no—
60C4×Dic3—31no—
61C4×Dic3—91no—
62C22×C32—21no—
63C22×C32C22×D311yescharacteristic
64C22×C32C22×D311yescharacteristic
65C2×C3⋊D3—24no—
66C2×C3⋊D3—22no—
67C2×C3×D3—62no—
68C2×C3×D3—64no—
69C32×C4—22no—
70C32×C4D1212yes—
71C32×C4#6712yes—
72C3⋊Dic3—22no—
73C3×Dic3—64no—
74C3×Dic3—62no—
75C3×Dic3—62no—
76C3×Dic3—62no—
77#124—271noSylow
78H(3,𝔽3)#12411yescharacteristic, complemented, Sylow
79C23×C3—92no—
80C22×D3—32no—
81C22×D3—92no—
82C3×Q8—182no—
83C3×D4—182no—
84C2×C3×C4—31no—
85C2×C3×C4—31no—
86C2×C3×C4—61no—
87C2×C3×C4—92no—
88C2×C3×C4—92no—
89C2×C3×C4D3211yescharacteristic
90C2×Dic3—32no—
91C2×Dic3—32no—
92C2×Dic3—32no—
93C2×Dic3—62no—
94C2×Dic3—92no—
95D12—62no—
96D12—182no—
97Dic6—62no—
98Dic6—62no—
99Dic6—64no—
100C2×C32—22no—
101C2×C32—21no—
102C2×C32C2×D1211yescharacteristic
103C2×C32#23512yes—
104C2×C32#23712yes—
105C2×C32C2×#6711yescharacteristic
106C3⋊D3—42no—
107C3×D3—122no—
108C2×Q8—91no—
109C2×D4—91no—
110C22⋊C4—274no—
111C42—91no—
112C22×C3—31no—
113C22×C3—31no—
114C22×C3—61no—
115C22×C3—182no—
116C22×C3—184no—
117C22×C3C2×D3211yescharacteristic
118C2×D3—64no—
119C2×D3—62no—
120C2×D3—182no—
121C2×D3—184no—
122C3×C4—32no—
123C3×C4—32no—
124C3×C4—62no—
125C3×C4—182no—
126C3×C4—182no—
127C3×C4#62612yes—
128Dic3—62no—
129Dic3—62no—
130Dic3—64no—
131Dic3—62no—
132Dic3—182no—
133C32—21no—
134C32#87511yescharacteristic, complemented
135C32#93611yescharacteristic, complemented
136C23—92no—
137Q8—182no—
138D4—182no—
139C2×C4—92no—
140C2×C4—92no—
141C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
142C2×C3—32no—
143C2×C3—32no—
144C2×C3—31no—
145C2×C3—31no—
146C2×C3—62no—
147C2×C3—61no—
148C2×C3—362no—
149C2×C3#379512yes—
150C2×C3C2×#62611yescharacteristic
151D3—122no—
152D3—362no—
153C22—184no—
154C22—182no—
155C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
156C4—182no—
157C4—182no—
158C4#620012yes—
159C3—31no—
160C3—31no—
161C3—61no—
162C3#6386011yescharacteristic
163C2—362no—
164C2#8864812yes—
165C2C2×#620011yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C23
Frattini, socleC22×C3C2×D32
FittingC2×C4×H(3,𝔽3)C22
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C2
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#124
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#6229C2semidirect
3. C4×(C3⋊Dic3)⋊C3C2semidirect
4. #88450C2semidirect
18. C4×C3⋊Dic3D3semidirect
78. H(3,𝔽3)#124semidirect
134. C32#875semidirect
135. C32#936semidirect

Maximal subgroups

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

subgroupindex
C2×#62292
C2×C4⋊((C3⋊D3)⋊C3)2
C4×(C3⋊Dic3)⋊C32
#884502
#638643
#638633