Finite Group Databasebeta

← #11721114

Subgroups — #11721114

Order 864.

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

normal subgroups only

This group has 320 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⋊#272C211yesmaximal, characteristic
2C2×#634⋊C3C211yesmaximal, characteristic, complemented
3C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4#88451C214yesmaximal, complemented
5#63863—31nomaximal
6#63860—31nomaximal
7C22×#272C2212yes—
8C2×C4×H(3,𝔽3)C2211yescharacteristic
9C4⋊#272—22no—
10C2×(C3⋊Dic3)⋊C3C2212yes—
11C2×(C3⋊Dic3)⋊C3C2212yes—
12#634⋊C3—22no—
13C2×#634D311yescharacteristic, complemented
14C2×C3×D12—31no—
15C2×C3×D12—31no—
16C2×C3×D12—61no—
17C2×C3×Dic6—31no—
18C4×C3⋊Dic3D311yescharacteristic, complemented
19C3×C4×Dic3—31no—
20(C2×D3)⋊Dic3—34no—
21(C2×D3)⋊Dic3—34no—
22C22×H(3,𝔽3)C2311yescharacteristic
23C2×#272—22no—
24C2×#272—24no—
25C4×H(3,𝔽3)D412yes—
26(C3⋊Dic3)⋊C3—22no—
27(C3⋊Dic3)⋊C3—22no—
28#947—91no—
29#936—91no—
30#875—91no—
31C22×C3×D3—32no—
32C22×C3×D3—32no—
33C22×C3×D3—62no—
34C2×C32×C4—21no—
35C2×C32×C4C2×D311yescharacteristic
36C2×C32×C4C2×D311yescharacteristic
37C2×C3⋊Dic3C2×D312yes—
38C2×C3⋊Dic3C2×D312yes—
39#634—22no—
40C2×C3×Dic3—32no—
41C2×C3×Dic3—32no—
42C3×D12—64no—
43C3×D12—62no—
44C3×D12—62no—
45C3×Dic6—62no—
46C2×H(3,𝔽3)C2×D411yescharacteristic
47C2×H(3,𝔽3)#4112yes—
48#272—42no—
49C2×C3×Q8—91no—
50C2×C3×D4—91no—
51C2×D12—31no—
52C2×D12—31no—
53C2×D12—61no—
54C2×Dic6—31no—
55C2×Dic6—91no—
56C3×C42—91no—
57#217—94no—
58(C2×D3)⋊C4—94no—
59(C2×D3)⋊C4—94no—
60C4×Dic3—31no—
61C4×Dic3—181no—
62C22×C32—21no—
63C22×C32C22×D311yescharacteristic
64C22×C32C22×D311yescharacteristic
65C2×C3×D3—64no—
66C2×C3×D3—62no—
67C2×C3×D3—64no—
68C2×C3×D3—64no—
69C2×C3×D3—62no—
70C2×C3×D3—124no—
71C32×C4—22no—
72C32×C4D1212yes—
73C32×C4#6712yes—
74C3⋊Dic3—22no—
75C3⋊Dic3—22no—
76C3×Dic3—62no—
77C3×Dic3—62no—
78#124—271noSylow
79H(3,𝔽3)#12411yescharacteristic, complemented, Sylow
80C23×C3—92no—
81C22×D3—32no—
82C22×D3—32no—
83C22×D3—62no—
84C3×Q8—182no—
85C3×D4—182no—
86C2×C3×C4—31no—
87C2×C3×C4—61no—
88C2×C3×C4—92no—
89C2×C3×C4—92no—
90C2×C3×C4—121no—
91C2×C3×C4D3211yescharacteristic
92C2×Dic3—32no—
93C2×Dic3—32no—
94C2×Dic3—92no—
95C2×Dic3—182no—
96D12—62no—
97D12—62no—
98D12—64no—
99Dic6—62no—
100Dic6—182no—
101C2×C32—22no—
102C2×C32—21no—
103C2×C32C2×D1211yescharacteristic
104C2×C32#23512yes—
105C2×C32#23912yes—
106C2×C32C2×#6711yescharacteristic
107C3×D3—124no—
108C3×D3—122no—
109C3×D3—122no—
110C2×Q8—91no—
111C2×D4—91no—
112C22⋊C4—274no—
113C42—91no—
114C22×C3—31no—
115C22×C3—61no—
116C22×C3—121no—
117C22×C3—184no—
118C22×C3—182no—
119C22×C3C2×D3211yescharacteristic
120C2×D3—62no—
121C2×D3—64no—
122C2×D3—62no—
123C2×D3—64no—
124C2×D3—64no—
125C2×D3—124no—
126C3×C4—32no—
127C3×C4—62no—
128C3×C4—122no—
129C3×C4—182no—
130C3×C4—182no—
131C3×C4#62612yes—
132Dic3—62no—
133Dic3—62no—
134Dic3—182no—
135Dic3—182no—
136C32—21no—
137C32#87511yescharacteristic, complemented
138C32#94711yescharacteristic, complemented
139C23—92no—
140Q8—182no—
141D4—182no—
142C2×C4—92no—
143C2×C4—92no—
144C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
145C2×C3—32no—
146C2×C3—31no—
147C2×C3—62no—
148C2×C3—61no—
149C2×C3—122no—
150C2×C3—121no—
151C2×C3—362no—
152C2×C3#379912yes—
153C2×C3C2×#62611yescharacteristic
154D3—122no—
155D3—122no—
156D3—124no—
157C22—184no—
158C22—182no—
159C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
160C4—182no—
161C4—182no—
162C4#620012yes—
163C3—31no—
164C3—61no—
165C3—121no—
166C3#6386411yescharacteristic
167C2—362no—
168C2#8865212yes—
169C2C2×#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⋊#272C2
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×(C3⋊Dic3)⋊C3C2semidirect
4. #88451C2semidirect
13. C2×#634D3semidirect
18. C4×C3⋊Dic3D3semidirect
79. H(3,𝔽3)#124semidirect
137. C32#875semidirect
138. C32#947semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#2722
C2×#634⋊C32
C4×(C3⋊Dic3)⋊C32
#884512
#638633
#638603