Finite Group Databasebeta

← #11721864

Subgroups — #11721864

Order 864.

418 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 88 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×Q8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×C4⋊#272C211yesmaximal, characteristic
3#88542C214yesmaximal, complemented
4C4×#1084C211yesmaximal, characteristic, complemented
5C3×#947—32nomaximal
6C3×#947—32nomaximal
7C22×#272C2212yes—
8Q8×H(3,𝔽3)—22no—
9C2×C4×H(3,𝔽3)C2212yes—
10C2×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#1084C2212yes—
12C4⋊#272—22no—
13C2×C32×Q8D312yescomplemented
14C2×C32×Q8D312yescomplemented
15C2×C3×D12—32no—
16C2×C3×D12—32no—
17C3×(C2×D3)⋊C4—38no—
18C3×(C2×D3)⋊C4—38no—
19C3×C4×Dic3—32no—
20C3×C4×Dic3—32no—
21C22×H(3,𝔽3)C2311yescharacteristic
22C2×#272—22no—
23C2×#272—24no—
24C4×H(3,𝔽3)—22no—
25C4×H(3,𝔽3)D412yes—
26#1084—22no—
27C3×#124—91no—
28#947—32no—
29#947—32no—
30C22×C3×D3—34no—
31C22×C3×D3—34no—
32C32×Q8—24no—
33C32×Q8—24no—
34C2×C32×C4C2×D314yes—
35C2×C32×C4C2×D312yes—
36C2×C32×C4C2×D314yes—
37C2×C32×C4C2×D312yes—
38C2×C3×Dic3—34no—
39C2×C3×Dic3—34no—
40C3×D12—64no—
41C3×D12—64no—
42C2×H(3,𝔽3)C2×D411yescharacteristic
43C2×H(3,𝔽3)#4112yes—
44#272—42no—
45C2×C3×Q8—32no—
46C2×C3×Q8—62no—
47C2×C3×Q8C3⋊D311yescharacteristic
48C2×C3×D4—91no—
49C2×D12—32no—
50C2×D12—32no—
51C3×C22⋊C4—94no—
52C3×C42—91no—
53(C2×D3)⋊C4—38no—
54(C2×D3)⋊C4—38no—
55C4×Dic3—32no—
56C4×Dic3—32no—
57C22×C32C22×D312yes—
58C22×C32C22×D312yes—
59C2×C3×D3—68no—
60C2×C3×D3—68no—
61C2×C3×D3—64no—
62C2×C3×D3—64no—
63C32×C4—24no—
64C32×C4—24no—
65C32×C4#6714yes—
66C32×C4#6714yes—
67C3×Dic3—64no—
68C3×Dic3—64no—
69#124—91noSylow
70H(3,𝔽3)#12411yescharacteristic, complemented, Sylow
71C23×C3—92no—
72C22×D3—34no—
73C22×D3—34no—
74C3×Q8—22no—
75C3×Q8—64no—
76C3×Q8—64no—
77C3×D4—182no—
78C2×C3×C4—32no—
79C2×C3×C4—34no—
80C2×C3×C4—62no—
81C2×C3×C4—64no—
82C2×C3×C4—92no—
83C2×C3×C4C2×C3⋊D312yes—
84C2×C3×C4C2×C3⋊D311yescharacteristic
85C2×Dic3—34no—
86C2×Dic3—34no—
87D12—64no—
88D12—64no—
89C2×C32#23914yes—
90C2×C32#23914yes—
91C2×C32C2×#6712yes—
92C2×C32C2×#6712yes—
93C3×D3—124no—
94C3×D3—124no—
95C2×Q8#27211yescharacteristic, complemented
96C2×D4—91no—
97C22⋊C4—94no—
98C42—91no—
99C22×C3—32no—
100C22×C3—62no—
101C22×C3—184no—
102C22×C3—182no—
103C22×C3C22×C3⋊D311yescharacteristic
104C2×D3—64no—
105C2×D3—68no—
106C2×D3—64no—
107C2×D3—68no—
108C3×C4—22no—
109C3×C4—34no—
110C3×C4—64no—
111C3×C4—64no—
112C3×C4—64no—
113C3×C4—182no—
114C3×C4#63812yes—
115Dic3—64no—
116Dic3—64no—
117C32#94712yescomplemented
118C32#94712yescomplemented
119C23—92no—
120Q8—22no—
121D4—182no—
122C2×C4—92no—
123C2×C4C2×#27212yes—
124C2×C4C2×#27211yescharacteristic
125C2×C3—34no—
126C2×C3—32no—
127C2×C3—64no—
128C2×C3—62no—
129C2×C3—362no—
130C2×C3#383212yes—
131C2×C3C2×#63811yescharacteristic
132D3—124no—
133D3—124no—
134C22—184no—
135C22—182no—
136C22C22×#27211yescharacteristic
137C4—22no—
138C4—182no—
139C4#623512yes—
140C3—32no—
141C3—62no—
142C3#6415611yescharacteristic
143C2—362no—
144C2#8874812yes—
145C2C2×#623511yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C23
FittingC2×Q8×H(3,𝔽3)C2
prime radicalC2×C4⋊#272C2
hypercenterC2×C3×Q8C3⋊D3
hypocenterH(3,𝔽3)#124
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×H(3,𝔽3)C2semidirect
3. #88542C2semidirect
4. C4×#1084C2semidirect
13. C2×C32×Q8D3semidirect
14. C2×C32×Q8D3semidirect
70. H(3,𝔽3)#124semidirect
95. C2×Q8#272semidirect
117. C32#947semidirect
118. C32#947semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×H(3,𝔽3)2
C2×C4⋊#2722
#885422
C4×#10842
C3×#9473