Finite Group Databasebeta

← #13141245

Subgroups — #13141245

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×#396C311yesmaximal, characteristic
3#396⋊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×#396—31no—
12C3×#396C3211yescharacteristic
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—61no—
33C2×C32×C4—61no—
34C2×C32×C4—61no—
35C2×C32×C4—61no—
36C2×C32×C4—61no—
37C2×C22⋊C9—46no—
38C2×C22⋊C9—43no—
39#3963−1+211yescharacteristic, complemented, Sylow
40C2×3−1+2—162no—
41C2×3−1+2—161no—
42C24×C3—31no—
43C24×C3C3×A411yescharacteristic
44C22×C3×C4—33no—
45C22×C3×C4—31no—
46C22×C3×C4—33no—
47C22×C3×C4—31no—
48C3×C4⋊C4—62no—
49C3×C4⋊C4—66no—
50C3×C22⋊C4—62no—
51C3×C22⋊C4—66no—
52C22×C32—31no—
53C22×C32—31no—
54C22×C32—41no—
55C22×C32—62no—
56C22×C32—121no—
57C22×C32C2×SL(2,𝔽3)11yescharacteristic
58C32×C4—62no—
59C32×C4—121no—
60C22×C9—163no—
61C22⋊C9—43no—
62C2×C4⋊C4—31no—
63C2×C22⋊C4—31no—
64#95C2×3−1+211yescharacteristic, complemented
653−1+2—161noSylow
66C23×C3—32no—
67C23×C3—31no—
68C23×C3—61no—
69C23×C3—61no—
70C23×C3—63no—
71C23×C3—63no—
72C23×C3C3×SL(2,𝔽3)12yes—
73C23×C3C2×C3×A411yescharacteristic
74C2×C3×C4—32no—
75C2×C3×C4—36no—
76C2×C3×C4—61no—
77C2×C3×C4—63no—
78C2×C3×C4—63no—
79C2×C3×C4—61no—
80C2×C3×C4—61no—
81C2×C3×C4—63no—
82C2×C3×C4—61no—
83C2×C3×C4—63no—
84C2×C3×C4—63no—
85C2×C3×C4—61no—
86C2×C32—31no—
87C2×C32—31no—
88C2×C32—42no—
89C2×C32#126⋊C311yescharacteristic
90C2×C9—166no—
91C2×C9—163no—
92C24#109411yescharacteristic
93C22×C4—31no—
94C22×C4—31no—
95C4⋊C4—62no—
96C22⋊C4—62no—
97C22×C3—31no—
98C22×C3—33no—
99C22×C3—33no—
100C22×C3—31no—
101C22×C3—31no—
102C22×C3—41no—
103C22×C3—62no—
104C22×C3—66no—
105C22×C3—121no—
106C22×C3—121no—
107C22×C3—123no—
108C22×C3C2×C3×SL(2,𝔽3)11yescharacteristic
109C3×C4—66no—
110C3×C4—62no—
111C3×C4—123no—
112C3×C4—121no—
113C32#396⋊C311yescharacteristic
114C9—163no—
115C23—61no—
116C23—61no—
117C23#620412yes—
118C23C2×#109411yescharacteristic
119C2×C4—32no—
120C2×C4—61no—
121C2×C4—61no—
122C2×C4—61no—
123C2×C4—61no—
124C2×C4—61no—
125C2×C3—33no—
126C2×C3—33no—
127C2×C3—31no—
128C2×C3—31no—
129C2×C3—31no—
130C2×C3—42no—
131C2×C3—122no—
132C2×C3C3×#126⋊C311yescharacteristic
133C22—31no—
134C22—31no—
135C22—41no—
136C22—62no—
137C22—121no—
138C22C2×#620411yescharacteristic
139C4—62no—
140C4—121no—
141C3—31no—
142C3C3×#396⋊C311yescharacteristic
143C2—31no—
144C2—31no—
145C2—42no—
146C2#1172135011yescharacteristic
subgroupG/H
centerC2×C3C3×#126⋊C3
commutator, cosocleC3×#95C2×C32
Frattini, socleC23×C3C2×C3×A4
FittingC32×#396C3
prime radicalC24×C32A4
hypercenterC2×C32#126⋊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. #396⋊C9C3semidirect
8. #95⋊C9C2×C3semidirect
39. #3963−1+2semidirect
64. #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×#3963
#396⋊C93
C22×#10944