Finite Group Databasebeta

← #11722483

Subgroups — #11722483

Order 864.

549 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 549 conjugacy classes of subgroups — too many to draw. Only its 114 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
2#3828⋊C3C212yesmaximal, complemented
3#88672C214yesmaximal, complemented
4#88648C214yesmaximal, complemented
5((C3⋊D3)⋊C3)⋊Q8C214yesmaximal, complemented
6#64297—32nomaximal
7C2×C4×H(3,𝔽3)C2211yescharacteristic
8C2×#1084C2212yes—
9#6229C2214yes—
10#638⋊C3C2214yes—
11C4⋊((C3⋊D3)⋊C3)C2212yes—
12C4×(C3⋊D3)⋊C3C2214yes—
13#634⋊C3C2212yes—
14#6198C2214yes—
15((C3⋊D3)⋊C3)⋊C4C2218yes—
16#6196C2214yes—
17#3828D312yescomplemented
18C3×#235—32no—
19C2×C3×Dic6—32no—
20C2×C3×Dic6—61no—
21#3804—38no—
22#3799—34no—
23#3797—34no—
24#3796—34no—
25D3×Dic6—34no—
26C22×H(3,𝔽3)C2311yescharacteristic
27C2×(C3⋊D3)⋊C3C2314yes—
28C4×H(3,𝔽3)C2312yes—
29#1084C2314yes—
30(C3⋊Dic3)⋊C3C2314yes—
31#1010—92no—
32#1007—91no—
33C2×C32×C4—21no—
34C2×C32×C4C2×D312yes—
35#638C2×D314yes—
36C4⋊(C3⋊D3)C2×D312yes—
37C4×C3⋊D3C2×D314yes—
38#634C2×D312yes—
39C3×#67—34no—
40C2×C3×Dic3—34no—
41C2×C3×Dic3—62no—
42C3×D12—32no—
43C3×C4×D3—34no—
44C3×Dic6—32no—
45C3×Dic6—34no—
46C3×Dic6—34no—
47C3×Dic6—64no—
48#627—38no—
49#626—38no—
50#624—38no—
51D3×Dic3—38no—
52C2×H(3,𝔽3)—21no—
53C2×H(3,𝔽3)C2411yescharacteristic
54(C3⋊D3)⋊C3—24no—
55C3×#41—92no—
56C2×C3×Q8—91no—
57#239—94no—
58D3×Q8—94no—
59D3×Q8—94no—
60#237—94no—
61#237—98no—
62#235—32no—
63#235—94no—
64#235—94no—
65#235—182no—
66C2×Dic6—32no—
67C2×Dic6—61no—
68C2×Dic6—94no—
69C22×C32—21no—
70C22×C32C22×D312yes—
71C2×C3⋊D3C22×D314yes—
72C2×C3×D3—34no—
73C32×C4—22no—
74C32×C4C22×D312yes—
75C32×C4C22×D312yes—
76C3⋊Dic3C22×D314yes—
77C3×Dic3—34no—
78C3×Dic3—38no—
79C3×Dic3—64no—
802−1+4—271noSylow
81H(3,𝔽3)2−1+411yescharacteristic, complemented, Sylow
82C3×Q8—92no—
83C3×Q8—94no—
84C3×D4—92no—
85C3×D4—94no—
86C2×C3×C4—62no—
87C2×C3×C4—94no—
88C2×C3×C4—92no—
89C2×C3×C4—121no—
90C2×C3×C4D3211yescharacteristic
91#67—34no—
92#67—98no—
93#67—184no—
94C2×Dic3—34no—
95C2×Dic3—62no—
96C2×Dic3—98no—
97D12—32no—
98D12—94no—
99D12—182no—
100C4×D3—34no—
101C4×D3—98no—
102C4×D3—98no—
103C4×D3—184no—
104Dic6—34no—
105Dic6—34no—
106Dic6—32no—
107Dic6—64no—
108Dic6—94no—
109Dic6—98no—
110Dic6—182no—
111C2×C32—22no—
112C2×C32—22no—
113C2×C32—21no—
114C2×C32C23×D312yes—
115C3⋊D3—24no—
116C3×D3—64no—
117#41—92no—
118#41—274no—
119#41—274no—
120C2×Q8—91no—
121C2×Q8—274no—
122C22×C3—62no—
123C22×C3—94no—
124C22×C3—121no—
125C22×C3C2×D3211yescharacteristic
126C2×D3—34no—
127C2×D3—184no—
128C3×C4—62no—
129C3×C4—62no—
130C3×C4—94no—
131C3×C4—94no—
132C3×C4—122no—
133C3×C4C2×D3212yes—
134Dic3—38no—
135Dic3—34no—
136Dic3—64no—
137Dic3—184no—
138C32—21no—
139C32#101012yescomplemented
140Q8—94no—
141Q8—92no—
142Q8—274no—
143D4—94no—
144D4—92no—
145D4—274no—
146C2×C4—92no—
147C2×C4—94no—
148C2×C4—278no—
149C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
150C2×C3—21no—
151C2×C3—62no—
152C2×C3—122no—
153C2×C3—122no—
154C2×C3—121no—
155C2×C3—184no—
156C2×C3C22×D3211yescharacteristic
157D3—64no—
158D3—364no—
159C22—94no—
160C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
161C4—94no—
162C4—94no—
163C4C2×((C3⋊D3)⋊C3)⋊C212yes—
164C3—62no—
165C3—121no—
166C3#6429811yescharacteristic
167C2—21no—
168C2—184no—
169C2C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC2C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC2×H(3,𝔽3)C24
Frattini, socleC2×C3C22×D32
FittingC2×C4×H(3,𝔽3)C22
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)2−1+4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#6229C2semidirect
2. #3828⋊C3C2semidirect
3. #88672C2semidirect
4. #88648C2semidirect
5. ((C3⋊D3)⋊C3)⋊Q8C2semidirect
17. #3828D3semidirect
81. H(3,𝔽3)2−1+4semidirect
139. C32#1010semidirect

Maximal subgroups

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

subgroupindex
C2×#62292
#3828⋊C32
#886722
#886482
((C3⋊D3)⋊C3)⋊Q82
#642973