Finite Group Databasebeta

← #13141248

Subgroups — #13141248

Order 1728.

258 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 35 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
1#11720339C211yesmaximal, characteristic, complemented
2C32×#399C311yesmaximal, characteristic
3#399⋊C9C313yesmaximal, complemented
4C22×#1094—41nomaximal
5C2×C32×C4⋊C4—31no—
6C2×C32×C22⋊C4—31no—
7C32×#95C2×C311yescharacteristic
8#95⋊C9C2×C313yescomplemented
9C2×#1094—42no—
10C2×#1094—41no—
11C3×#399—31no—
12C3×#399C3211yescharacteristic
13C24×C32A411yescharacteristic
14C22×C32×C4—31no—
15C22×C32×C4—31no—
16C22×C22⋊C9—43no—
17C32×C4⋊C4—62no—
18C32×C22⋊C4—62no—
19C22×3−1+2—161no—
20#1094—41no—
21C2×C3×C4⋊C4—31no—
22C2×C3×C4⋊C4—33no—
23C2×C3×C22⋊C4—33no—
24C2×C3×C22⋊C4—31no—
25C3×#95—31no—
26C3×#95C2×C3211yescharacteristic
27C23×C32—61no—
28C23×C32—61no—
29C23×C32SL(2,𝔽3)12yes—
30C23×C32C2×A411yescharacteristic
31C2×C32×C4—32no—
32C2×C32×C4—62no—
33C2×C32×C4—61no—
34C2×C32×C4—62no—
35C2×C22⋊C9—46no—
36C2×C22⋊C9—43no—
37#3993−1+211yescharacteristic, complemented, Sylow
38C2×3−1+2—162no—
39C2×3−1+2—161no—
40C24×C3—31no—
41C24×C3C3×A411yescharacteristic
42C22×C3×C4—33no—
43C22×C3×C4—31no—
44C22×C3×C4—33no—
45C22×C3×C4—31no—
46C3×C4⋊C4—62no—
47C3×C4⋊C4—66no—
48C3×C22⋊C4—62no—
49C3×C22⋊C4—66no—
50C22×C32—31no—
51C22×C32—31no—
52C22×C32—41no—
53C22×C32—62no—
54C22×C32—121no—
55C22×C32C2×SL(2,𝔽3)11yescharacteristic
56C32×C4—62no—
57C32×C4—121no—
58C22×C9—163no—
59C22⋊C9—43no—
60C2×C4⋊C4—31no—
61C2×C22⋊C4—31no—
62#95C2×3−1+211yescharacteristic, complemented
633−1+2—161noSylow
64C23×C3—32no—
65C23×C3—31no—
66C23×C3—61no—
67C23×C3—61no—
68C23×C3—63no—
69C23×C3—63no—
70C23×C3C3×SL(2,𝔽3)12yes—
71C23×C3C2×C3×A411yescharacteristic
72C2×C3×C4—36no—
73C2×C3×C4—32no—
74C2×C3×C4—62no—
75C2×C3×C4—66no—
76C2×C3×C4—63no—
77C2×C3×C4—61no—
78C2×C3×C4—66no—
79C2×C3×C4—62no—
80C2×C32—31no—
81C2×C32—31no—
82C2×C32—42no—
83C2×C32#127⋊C311yescharacteristic
84C2×C9—166no—
85C2×C9—163no—
86C24#109411yescharacteristic
87C22×C4—31no—
88C22×C4—31no—
89C4⋊C4—62no—
90C22⋊C4—62no—
91C22×C3—31no—
92C22×C3—33no—
93C22×C3—33no—
94C22×C3—31no—
95C22×C3—31no—
96C22×C3—41no—
97C22×C3—62no—
98C22×C3—66no—
99C22×C3—121no—
100C22×C3—121no—
101C22×C3—123no—
102C22×C3C2×C3×SL(2,𝔽3)11yescharacteristic
103C3×C4—66no—
104C3×C4—62no—
105C3×C4—123no—
106C3×C4—121no—
107C32#399⋊C311yescharacteristic
108C9—163no—
109C23—61no—
110C23—61no—
111C23#620412yes—
112C23C2×#109411yescharacteristic
113C2×C4—32no—
114C2×C4—62no—
115C2×C4—61no—
116C2×C4—62no—
117C2×C3—33no—
118C2×C3—33no—
119C2×C3—31no—
120C2×C3—31no—
121C2×C3—31no—
122C2×C3—42no—
123C2×C3—122no—
124C2×C3C3×#127⋊C311yescharacteristic
125C22—31no—
126C22—31no—
127C22—41no—
128C22—62no—
129C22—121no—
130C22C2×#620411yescharacteristic
131C4—62no—
132C4—121no—
133C3—31no—
134C3C3×#399⋊C311yescharacteristic
135C2—31no—
136C2—31no—
137C2—42no—
138C2#1172135311yescharacteristic
subgroupG/H
centerC2×C3C3×#127⋊C3
commutator, cosocleC3×#95C2×C32
Frattini, socleC23×C3C2×C3×A4
FittingC32×#399C3
prime radicalC24×C32A4
hypercenterC2×C32#127⋊C3
hypocenter#95C2×3−1+2
LCSG ▷ C3×#95 ▷ #95
UCSC1 ◁ C2×C3 ◁ C2×C32

Complements

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

NH ≅ G/Ntype
1. #11720339C2semidirect
3. #399⋊C9C3semidirect
8. #95⋊C9C2×C3semidirect
37. #3993−1+2semidirect
62. #95C2×3−1+2semidirect

Maximal subgroups

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

subgroupindex
#117203392
C32×#3993
#399⋊C93
C22×#10944