Finite Group Databasebeta

← #393⋊H(3,𝔽3) #13141240

Subgroups — #393⋊H(3,𝔽3) #13141240

Order 1728.

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

normal subgroups only

This group has 328 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#11722587C211yesmaximal, characteristic, complemented
2C32×#393C311yesmaximal, characteristic, complemented
3C3×#393⋊C3C313yesmaximal, complemented
4C22×C22⋊H(3,𝔽3)—41nomaximal
5C22×C32×Q8C2×C311yescharacteristic, complemented
6C3×C22⋊SL(2,𝔽3)C2×C313yescomplemented
7C2×C32×C22⋊C4—31no—
8C32×#95—31no—
9C2×C22⋊H(3,𝔽3)—42no—
10C2×C22⋊H(3,𝔽3)—41no—
11Q8⋊H(3,𝔽3)—82no—
12C3×#393—31no—
13C3×#393C3211yescharacteristic
14#393⋊C3—33no—
15C24×C32A411yescharacteristic
16C22×C3×A4—43no—
17C2×C32×Q8—62no—
18C22×C32×C4—31no—
19C22×C32×C4—31no—
20C32×C22⋊C4—62no—
21C22×H(3,𝔽3)—161no—
22C22⋊H(3,𝔽3)—41no—
23C22×C3×Q8—31no—
24C22×C3×Q8C2×C3211yescharacteristic
25C22⋊SL(2,𝔽3)—33no—
26C2×C3×C22⋊C4—33no—
27C2×C3×C22⋊C4—31no—
28C3×#95—31no—
29C3×#95—33no—
30C23×C32—61no—
31C23×C32—61no—
32C23×C32SL(2,𝔽3)12yescomplemented
33C23×C32C2×A411yescharacteristic
34C2×C3×A4—46no—
35C2×C3×A4—43no—
36C32×Q8—22no—
37C32×Q8—62no—
38C2×C32×C4—32no—
39C2×C32×C4—62no—
40C2×C32×C4—61no—
41C2×C32×C4—62no—
42C3×SL(2,𝔽3)—86no—
43#393H(3,𝔽3)11yescharacteristic, complemented, Sylow
44C2×H(3,𝔽3)—162no—
45C2×H(3,𝔽3)—161no—
46C24×C3—31no—
47C24×C3C3×A411yescharacteristic
48C22×A4—123no—
49C2×C3×Q8—62no—
50C2×C3×Q8—66no—
51C22×C3×C4—33no—
52C22×C3×C4—31no—
53C22×C3×C4—33no—
54C22×C3×C4—31no—
55C3×C22⋊C4—62no—
56C3×C22⋊C4—66no—
57C22×C32—31no—
58C22×C32—31no—
59C22×C32—41no—
60C22×C32—62no—
61C22×C32—121no—
62C22×C32—163no—
63C22×C32C2×SL(2,𝔽3)11yescharacteristic
64C3×A4—43no—
65C32×C4—62no—
66C32×C4—121no—
67C22×Q8C2×H(3,𝔽3)11yescharacteristic, complemented
68C2×C22⋊C4—31no—
69#95—31no—
70H(3,𝔽3)—161noSylow
71C23×C3—32no—
72C23×C3—31no—
73C23×C3—61no—
74C23×C3—61no—
75C23×C3—63no—
76C23×C3—63no—
77C23×C3C3×SL(2,𝔽3)12yes—
78C23×C3C2×C3×A411yescharacteristic
79C2×A4—126no—
80C2×A4—123no—
81C3×Q8—22no—
82C3×Q8—62no—
83C3×Q8—62no—
84C3×Q8—66no—
85C2×C3×C4—36no—
86C2×C3×C4—32no—
87C2×C3×C4—66no—
88C2×C3×C4—62no—
89C2×C3×C4—63no—
90C2×C3×C4—61no—
91C2×C3×C4—62no—
92C2×C3×C4—66no—
93SL(2,𝔽3)—246no—
94C2×C32—31no—
95C2×C32—31no—
96C2×C32—42no—
97C2×C32—166no—
98C2×C32—163no—
99C2×C32#120⋊C311yescharacteristic
100C24C22⋊H(3,𝔽3)11yescharacteristic
101C2×Q8—62no—
102C22×C4—31no—
103C22×C4—31no—
104C22⋊C4—62no—
105C22×C3—31no—
106C22×C3—33no—
107C22×C3—33no—
108C22×C3—31no—
109C22×C3—31no—
110C22×C3—41no—
111C22×C3—62no—
112C22×C3—66no—
113C22×C3—121no—
114C22×C3—121no—
115C22×C3—123no—
116C22×C3—483no—
117C22×C3C2×C3×SL(2,𝔽3)11yescharacteristic
118A4—123no—
119C3×C4—66no—
120C3×C4—62no—
121C3×C4—123no—
122C3×C4—121no—
123C32—163no—
124C32#393⋊C311yescharacteristic, complemented
125C23—61no—
126C23—61no—
127C23Q8⋊H(3,𝔽3)12yescomplemented
128C23C2×C22⋊H(3,𝔽3)11yescharacteristic
129Q8—22no—
130Q8—62no—
131C2×C4—32no—
132C2×C4—62no—
133C2×C4—61no—
134C2×C4—62no—
135C2×C3—33no—
136C2×C3—33no—
137C2×C3—31no—
138C2×C3—31no—
139C2×C3—31no—
140C2×C3—42no—
141C2×C3—122no—
142C2×C3—486no—
143C2×C3—483no—
144C2×C3C3×#120⋊C311yescharacteristic
145C22—31no—
146C22—31no—
147C22—41no—
148C22—62no—
149C22—121no—
150C22C2×Q8⋊H(3,𝔽3)11yescharacteristic
151C4—62no—
152C4—121no—
153C3—31no—
154C3—483no—
155C3C3×#393⋊C311yescharacteristic
156C2—31no—
157C2—31no—
158C2—42no—
159C2#120⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
centerC2×C3C3×#120⋊C3
commutator, cosocleC22×C3×Q8C2×C32
Frattini, socleC23×C3C2×C3×A4
FittingC32×#393C3
hypercenterC2×C32#120⋊C3
hypocenterC22×Q8C2×H(3,𝔽3)
LCSG ▷ C22×C3×Q8 ▷ C22×Q8
UCSC1 ◁ C2×C3 ◁ C2×C32

Complements

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

NH ≅ G/Ntype
1. #11722587C2semidirect
2. C32×#393C3semidirect
3. C3×#393⋊C3C3semidirect
5. C22×C32×Q8C2×C3semidirect
6. C3×C22⋊SL(2,𝔽3)C2×C3semidirect
32. C23×C32SL(2,𝔽3)semidirect
43. #393H(3,𝔽3)semidirect
67. C22×Q8C2×H(3,𝔽3)semidirect
124. C32#393⋊C3semidirect
127. C23Q8⋊H(3,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
#117225872
C32×#3933
C3×#393⋊C33
C22×C22⋊H(3,𝔽3)4