Finite Group Databasebeta

← #11721298

Subgroups — #11721298

Order 864.

320 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C22×#1084C211yesmaximal, characteristic
2#3757⋊C3C212yesmaximal
3#88449C212yesmaximal, complemented
4#88447C212yesmaximal, complemented
5#63955—32nomaximal
6C23×H(3,𝔽3)C2211yescharacteristic
7C2×#1084C414yescomplemented
8C2×#1084C2211yescharacteristic
9C2×#1084C2211yescharacteristic
10C2×(C3⋊Dic3)⋊C3C2214yes—
11C22×C3×Dic3—32no—
12C22×C3×Dic3—61no—
13#3757D312yes—
14C3×#217—32no—
15Dic3⋊Dic3—34no—
16Dic32—34no—
17C22×H(3,𝔽3)—22no—
18C22×H(3,𝔽3)C2×C412yes—
19C22×H(3,𝔽3)C2311yescharacteristic
20#1084C2×C412yes—
21#1084C2×C412yes—
22(C3⋊Dic3)⋊C3—24no—
23#926—91no—
24#877—92no—
25C23×C32—21no—
26C23×C32C2×D312yes—
27C2×C3⋊Dic3C2×D314yes—
28C2×C3×Dic3—38no—
29C2×C3×Dic3—32no—
30C2×C3×Dic3—32no—
31C2×C3×Dic3—34no—
32C2×C3×Dic3—64no—
33C2×C3×Dic3—61no—
34C2×C3×Dic3—61no—
35C2×H(3,𝔽3)—22no—
36C2×H(3,𝔽3)C22×C411yescharacteristic
37C2×H(3,𝔽3)#4112yes—
38C22×C3×C4—91no—
39C22×Dic3—32no—
40C22×Dic3—61no—
41C3×C22⋊C4—92no—
42#217—32no—
43#217—92no—
44C4⋊Dic3—92no—
45Dic3⋊C4—94no—
46C4×Dic3—92no—
47C4×Dic3—94no—
48C22×C32—24no—
49C22×C32—24no—
50C22×C32—22no—
51C22×C32—21no—
52C22×C32C4×D314yes—
53C22×C32C22×D312yes—
54C3⋊Dic3—24no—
55C3×Dic3—34no—
56C3×Dic3—34no—
57C3×Dic3—62no—
58C3×Dic3—62no—
59C3×Dic3—64no—
60#117—271noSylow
61H(3,𝔽3)#11711yescharacteristic, complemented, Sylow
62C23×C3—32no—
63C23×C3—61no—
64C23×C3D3211yescharacteristic
65C2×C3×C4—94no—
66C2×C3×C4—91no—
67C2×C3×C4—91no—
68C2×C3×C4—94no—
69C2×Dic3—32no—
70C2×Dic3—32no—
71C2×Dic3—38no—
72C2×Dic3—34no—
73C2×Dic3—64no—
74C2×Dic3—61no—
75C2×Dic3—61no—
76C2×Dic3—94no—
77C2×C32—24no—
78C2×C32—24no—
79C2×C32—22no—
80C2×C32—21no—
81C2×C32C2×C4×D312yes—
82C2×C32#23714yes—
83C22×C4—91no—
84C4⋊C4—272no—
85C22⋊C4—92no—
86C42—272no—
87C22×C3—22no—
88C22×C3—34no—
89C22×C3—32no—
90C22×C3—64no—
91C22×C3—64no—
92C22×C3—62no—
93C22×C3—61no—
94C22×C3#62412yes—
95C22×C3C2×D3211yescharacteristic
96C3×C4—92no—
97C3×C4—92no—
98C3×C4—184no—
99Dic3—34no—
100Dic3—34no—
101Dic3—62no—
102Dic3—62no—
103Dic3—64no—
104Dic3—184no—
105C32—21no—
106C32#87712yescomplemented
107C23((C3⋊D3)⋊C3)⋊C211yescharacteristic
108C2×C4—94no—
109C2×C4—91no—
110C2×C4—91no—
111C2×C4—94no—
112C2×C3—22no—
113C2×C3—34no—
114C2×C3—32no—
115C2×C3—64no—
116C2×C3—64no—
117C2×C3—62no—
118C2×C3—61no—
119C2×C3#380512yes—
120C2×C3C2×#62411yescharacteristic
121C22—22no—
122C22#619912yes—
123C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
124C4—92no—
125C4—92no—
126C4—184no—
127C3—32no—
128C3—61no—
129C3#6395711yescharacteristic
130C2—22no—
131C2#8867212yes—
132C2C2×#619911yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C22×C4
Frattini, socleC22×C3C2×D32
Fitting, prime radicalC23×H(3,𝔽3)C22
cosocleC22×H(3,𝔽3)C23
hypercenterC23((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#117
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
3. #88449C2semidirect
4. #88447C2semidirect
7. C2×#1084C4semidirect
61. H(3,𝔽3)#117semidirect
106. C32#877semidirect

Maximal subgroups

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

subgroupindex
C22×#10842
#3757⋊C32
#884492
#884472
#639553