Finite Group Databasebeta

← #396⋊H(3,𝔽3) #13141246

Subgroups — #396⋊H(3,𝔽3) #13141246

Order 1728.

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

normal subgroups only

This group has 294 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#95⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C32×#396C311yesmaximal, characteristic, complemented
3C3×#396⋊C3C313yesmaximal, complemented
4C22×C22⋊H(3,𝔽3)—41nomaximal
5C2×C32×C4⋊C4—31no—
6C2×C32×C22⋊C4—31no—
7C32×#95C2×C311yescharacteristic, complemented
8C3×#95⋊C3C2×C313yescomplemented
9C2×C22⋊H(3,𝔽3)—42no—
10C2×C22⋊H(3,𝔽3)—41no—
11C3×#396—31no—
12C3×#396C3211yescharacteristic
13#396⋊C3—33no—
14C24×C32A411yescharacteristic
15C22×C3×A4—43no—
16C22×C32×C4—31no—
17C22×C32×C4—31no—
18C32×C4⋊C4—62no—
19C32×C22⋊C4—62no—
20C22×H(3,𝔽3)—161no—
21C22⋊H(3,𝔽3)—41no—
22C2×C3×C4⋊C4—31no—
23C2×C3×C4⋊C4—33no—
24C2×C3×C22⋊C4—33no—
25C2×C3×C22⋊C4—31no—
26C3×#95—31no—
27C3×#95C2×C3211yescharacteristic
28#95⋊C3—33no—
29C23×C32—61no—
30C23×C32—61no—
31C23×C32SL(2,𝔽3)12yes—
32C23×C32C2×A411yescharacteristic
33C2×C3×A4—46no—
34C2×C3×A4—43no—
35C2×C32×C4—32no—
36C2×C32×C4—61no—
37C2×C32×C4—61no—
38C2×C32×C4—61no—
39C2×C32×C4—61no—
40C2×C32×C4—61no—
41#396H(3,𝔽3)11yescharacteristic, complemented, Sylow
42C2×H(3,𝔽3)—162no—
43C2×H(3,𝔽3)—161no—
44C24×C3—31no—
45C24×C3C3×A411yescharacteristic
46C22×A4—123no—
47C22×C3×C4—33no—
48C22×C3×C4—31no—
49C22×C3×C4—33no—
50C22×C3×C4—31no—
51C3×C4⋊C4—62no—
52C3×C4⋊C4—66no—
53C3×C22⋊C4—62no—
54C3×C22⋊C4—66no—
55C22×C32—31no—
56C22×C32—31no—
57C22×C32—41no—
58C22×C32—62no—
59C22×C32—121no—
60C22×C32—163no—
61C22×C32C2×SL(2,𝔽3)11yescharacteristic
62C3×A4—43no—
63C32×C4—62no—
64C32×C4—121no—
65C2×C4⋊C4—31no—
66C2×C22⋊C4—31no—
67#95C2×H(3,𝔽3)11yescharacteristic, complemented
68H(3,𝔽3)—161noSylow
69C23×C3—32no—
70C23×C3—31no—
71C23×C3—61no—
72C23×C3—61no—
73C23×C3—63no—
74C23×C3—63no—
75C23×C3C3×SL(2,𝔽3)12yes—
76C23×C3C2×C3×A411yescharacteristic
77C2×A4—126no—
78C2×A4—123no—
79C2×C3×C4—32no—
80C2×C3×C4—36no—
81C2×C3×C4—61no—
82C2×C3×C4—63no—
83C2×C3×C4—63no—
84C2×C3×C4—61no—
85C2×C3×C4—61no—
86C2×C3×C4—63no—
87C2×C3×C4—61no—
88C2×C3×C4—63no—
89C2×C3×C4—63no—
90C2×C3×C4—61no—
91C2×C32—31no—
92C2×C32—31no—
93C2×C32—42no—
94C2×C32—166no—
95C2×C32—163no—
96C2×C32#126⋊C311yescharacteristic
97C24C22⋊H(3,𝔽3)11yescharacteristic
98C22×C4—31no—
99C22×C4—31no—
100C4⋊C4—62no—
101C22⋊C4—62no—
102C22×C3—31no—
103C22×C3—33no—
104C22×C3—33no—
105C22×C3—31no—
106C22×C3—31no—
107C22×C3—41no—
108C22×C3—62no—
109C22×C3—66no—
110C22×C3—121no—
111C22×C3—121no—
112C22×C3—123no—
113C22×C3—483no—
114C22×C3C2×C3×SL(2,𝔽3)11yescharacteristic
115A4—123no—
116C3×C4—66no—
117C3×C4—62no—
118C3×C4—123no—
119C3×C4—121no—
120C32—163no—
121C32#396⋊C311yescharacteristic, complemented
122C23—61no—
123C23—61no—
124C23Q8⋊H(3,𝔽3)12yes—
125C23C2×C22⋊H(3,𝔽3)11yescharacteristic
126C2×C4—32no—
127C2×C4—61no—
128C2×C4—61no—
129C2×C4—61no—
130C2×C4—61no—
131C2×C4—61no—
132C2×C3—33no—
133C2×C3—33no—
134C2×C3—31no—
135C2×C3—31no—
136C2×C3—31no—
137C2×C3—42no—
138C2×C3—122no—
139C2×C3—486no—
140C2×C3—483no—
141C2×C3C3×#126⋊C311yescharacteristic
142C22—31no—
143C22—31no—
144C22—41no—
145C22—62no—
146C22—121no—
147C22C2×Q8⋊H(3,𝔽3)11yescharacteristic
148C4—62no—
149C4—121no—
150C3—31no—
151C3—483no—
152C3C3×#396⋊C311yescharacteristic
153C2—31no—
154C2—31no—
155C2—42no—
156C2#126⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
centerC2×C3C3×#126⋊C3
commutator, cosocleC3×#95C2×C32
Frattini, socleC23×C3C2×C3×A4
FittingC32×#396C3
hypercenterC2×C32#126⋊C3
hypocenter#95C2×H(3,𝔽3)
LCSG ▷ C3×#95 ▷ #95
UCSC1 ◁ C2×C3 ◁ C2×C32

Complements

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

NH ≅ G/Ntype
1. #95⋊H(3,𝔽3)C2semidirect
2. C32×#396C3semidirect
3. C3×#396⋊C3C3semidirect
7. C32×#95C2×C3semidirect
8. C3×#95⋊C3C2×C3semidirect
41. #396H(3,𝔽3)semidirect
67. #95C2×H(3,𝔽3)semidirect
121. C32#396⋊C3semidirect

Maximal subgroups

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

subgroupindex
#95⋊H(3,𝔽3)2
C32×#3963
C3×#396⋊C33
C22×C22⋊H(3,𝔽3)4