Finite Group Databasebeta

← #11721113

Subgroups — #11721113

Order 864.

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

normal subgroups only

This group has 296 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×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
2C2×#634⋊C3C211yesmaximal, characteristic, complemented
3C4×#1084C211yesmaximal, characteristic, complemented
4#88450C214yesmaximal, complemented
5#63864—31nomaximal
6#63860—31nomaximal
7C22×(C3⋊D3)⋊C3C2212yes—
8C2×C4×H(3,𝔽3)C2211yescharacteristic
9C2×#1084C2212yes—
10C2×(C3⋊Dic3)⋊C3C2212yes—
11C4⋊((C3⋊D3)⋊C3)—22no—
12#634⋊C3—22no—
13C2×C4⋊(C3⋊D3)D311yescharacteristic
14C2×#634D311yescharacteristic, complemented
15C2×C3×D12—31no—
16C2×C3×Dic6—31no—
17C3×C4×Dic3—31no—
18C3×C4×Dic3—31no—
19C3×C4×Dic3—61no—
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—
38#634—22no—
39C2×C3×Dic3—32no—
40C2×C3×Dic3—32no—
41C2×C3×Dic3—32no—
42C2×C3×Dic3—62no—
43C3×D12—62no—
44C3×Dic6—62no—
45C2×H(3,𝔽3)C2×D411yescharacteristic
46C2×H(3,𝔽3)#4112yes—
47(C3⋊D3)⋊C3—42no—
48C2×C3×Q8—91no—
49C2×C3×D4—91no—
50C2×D12—31no—
51C2×D12—181no—
52C2×Dic6—31no—
53C2×Dic6—91no—
54C3×C42—91no—
55#217—94no—
56(C2×D3)⋊C4—94no—
57(C2×D3)⋊C4—94no—
58C4×Dic3—31no—
59C4×Dic3—31no—
60C4×Dic3—61no—
61C22×C32—21no—
62C22×C32C22×D311yescharacteristic
63C22×C32C22×D311yescharacteristic
64C2×C3⋊D3—22no—
65C2×C3⋊D3—24no—
66C2×C3×D3—64no—
67C2×C3×D3—62no—
68C32×C4—22no—
69C32×C4#6712yes—
70C32×C4#6712yes—
71C3⋊Dic3—22no—
72C3×Dic3—64no—
73C3×Dic3—62no—
74C3×Dic3—62no—
75C3×Dic3—62no—
76#124—271noSylow
77H(3,𝔽3)#12411yescharacteristic, complemented, Sylow
78C23×C3—92no—
79C22×D3—32no—
80C22×D3—182no—
81C3×Q8—182no—
82C3×D4—182no—
83C2×C3×C4—31no—
84C2×C3×C4—61no—
85C2×C3×C4—92no—
86C2×C3×C4—92no—
87C2×C3×C4—121no—
88C2×C3×C4D3211yescharacteristic
89C2×Dic3—32no—
90C2×Dic3—32no—
91C2×Dic3—32no—
92C2×Dic3—62no—
93C2×Dic3—92no—
94D12—62no—
95D12—182no—
96Dic6—62no—
97Dic6—182no—
98C2×C32—22no—
99C2×C32—21no—
100C2×C32#23712yes—
101C2×C32#23912yes—
102C2×C32C2×#6711yescharacteristic
103C2×C32C2×#6711yescharacteristic
104C3⋊D3—42no—
105C3×D3—122no—
106C2×Q8—91no—
107C2×D4—91no—
108C22⋊C4—274no—
109C42—91no—
110C22×C3—31no—
111C22×C3—61no—
112C22×C3—121no—
113C22×C3—182no—
114C22×C3—184no—
115C22×C3C2×D3211yescharacteristic
116C2×D3—64no—
117C2×D3—62no—
118C2×D3—182no—
119C2×D3—364no—
120C3×C4—32no—
121C3×C4—62no—
122C3×C4—122no—
123C3×C4—182no—
124C3×C4—182no—
125C3×C4#62512yes—
126Dic3—62no—
127Dic3—62no—
128Dic3—64no—
129Dic3—62no—
130Dic3—182no—
131C32—21no—
132C32#93611yescharacteristic, complemented
133C32#94711yescharacteristic, complemented
134C23—92no—
135Q8—182no—
136D4—182no—
137C2×C4—92no—
138C2×C4—92no—
139C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
140C2×C3—32no—
141C2×C3—31no—
142C2×C3—62no—
143C2×C3—61no—
144C2×C3—122no—
145C2×C3—121no—
146C2×C3—362no—
147C2×C3#379612yes—
148C2×C3C2×#62511yescharacteristic
149D3—122no—
150D3—362no—
151C22—184no—
152C22—182no—
153C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
154C4—182no—
155C4—182no—
156C4#619812yes—
157C3—31no—
158C3—61no—
159C3—121no—
160C3#6386311yescharacteristic
161C2—362no—
162C2#8864912yes—
163C2C2×#619811yescharacteristic
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
2. C2×#634⋊C3C2semidirect
3. C4×#1084C2semidirect
4. #88450C2semidirect
14. C2×#634D3semidirect
77. H(3,𝔽3)#124semidirect
132. C32#936semidirect
133. C32#947semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊((C3⋊D3)⋊C3)2
C2×#634⋊C32
C4×#10842
#884502
#638643
#638603