Finite Group Databasebeta

← #392⋊H(3,𝔽3) #13141237

Subgroups — #392⋊H(3,𝔽3) #13141237

Order 1728.

394 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 29 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×#392C311yesmaximal, characteristic, complemented
3C3×#392⋊C3C313yesmaximal, complemented
4C4×C22⋊H(3,𝔽3)—41nomaximal
5C22×C32×D4—31no—
6C2×C32×C22⋊C4—31no—
7C32×#95C2×C311yescharacteristic
8C3×#95⋊C3C2×C313yescomplemented
9C2×C22⋊H(3,𝔽3)—41no—
10C3×#392—31no—
11C3×#392C3211yescharacteristic
12#392⋊C3—33no—
13C24×C32—31no—
14C2×C32×D4—62no—
15C2×C32×D4—64no—
16C22×C32×C4—31no—
17C22×C32×C4A411yescharacteristic
18C3×C4×A4—43no—
19C32×C22⋊C4—62no—
20C22⋊H(3,𝔽3)—41no—
21C4×H(3,𝔽3)—161no—
22C22×C3×D4—31no—
23C22×C3×D4—33no—
24C2×C3×C22⋊C4—33no—
25C2×C3×C22⋊C4—31no—
26C3×#95—31no—
27C3×#95C2×C3211yescharacteristic
28#95⋊C3—33no—
29C23×C32—32no—
30C23×C32—61no—
31C23×C32—62no—
32C23×C32—62no—
33C23×C32—61no—
34C23×C32C2×A411yescharacteristic
35C2×C3×A4—43no—
36C32×D4—68no—
37C2×C32×C4—32no—
38C2×C32×C4—61no—
39C2×C32×C4—62no—
40#392H(3,𝔽3)11yescharacteristic, complemented, Sylow
41C2×H(3,𝔽3)—161no—
42C24×C3—33no—
43C24×C3—31no—
44C2×C3×D4—62no—
45C2×C3×D4—66no—
46C2×C3×D4—64no—
47C2×C3×D4—612no—
48C22×C3×C4—33no—
49C22×C3×C4—31no—
50C22×C3×C4—31no—
51C22×C3×C4C3×A411yescharacteristic
52C4×A4—123no—
53C3×C22⋊C4—62no—
54C3×C22⋊C4—66no—
55C22×C32—31no—
56C22×C32—31no—
57C22×C32—64no—
58C22×C32—62no—
59C22×C32—121no—
60C22×C32—122no—
61C22×C32—121no—
62C22×C32#41⋊C311yescharacteristic
63C3×A4—43no—
64C32×C4—41no—
65C32×C4—62no—
66C32×C4—163no—
67C22×D4—31no—
68C2×C22⋊C4—31no—
69#95C2×H(3,𝔽3)11yescharacteristic
70H(3,𝔽3)—161noSylow
71C23×C3—32no—
72C23×C3—36no—
73C23×C3—31no—
74C23×C3—63no—
75C23×C3—61no—
76C23×C3—62no—
77C23×C3—66no—
78C23×C3—66no—
79C23×C3—62no—
80C23×C3—63no—
81C23×C3—61no—
82C23×C3C2×C3×A411yescharacteristic
83C2×A4—123no—
84C3×D4—624no—
85C3×D4—68no—
86C2×C3×C4—36no—
87C2×C3×C4—32no—
88C2×C3×C4—61no—
89C2×C3×C4—62no—
90C2×C3×C4—66no—
91C2×C3×C4—63no—
92C2×C32—31no—
93C2×C32—31no—
94C2×C32—122no—
95C2×C32—163no—
96C2×C32#127⋊C311yescharacteristic
97C24—31no—
98C2×D4—62no—
99C2×D4—64no—
100C22×C4—31no—
101C22×C4C22⋊H(3,𝔽3)11yescharacteristic
102C22⋊C4—62no—
103C22×C3—31no—
104C22×C3—33no—
105C22×C3—33no—
106C22×C3—31no—
107C22×C3—31no—
108C22×C3—612no—
109C22×C3—64no—
110C22×C3—66no—
111C22×C3—62no—
112C22×C3—123no—
113C22×C3—126no—
114C22×C3—121no—
115C22×C3—122no—
116C22×C3—123no—
117C22×C3—121no—
118C22×C3C3×#41⋊C311yescharacteristic
119A4—123no—
120C3×C4—41no—
121C3×C4—66no—
122C3×C4—62no—
123C3×C4—121no—
124C3×C4—483no—
125C32—163no—
126C32#392⋊C311yescharacteristic, complemented
127C23—32no—
128C23—61no—
129C23—62no—
130C23—62no—
131C23—61no—
132C23C2×C22⋊H(3,𝔽3)11yescharacteristic
133D4—68no—
134C2×C4—32no—
135C2×C4—61no—
136C2×C4—62no—
137C2×C3—33no—
138C2×C3—33no—
139C2×C3—31no—
140C2×C3—31no—
141C2×C3—31no—
142C2×C3—126no—
143C2×C3—122no—
144C2×C3—483no—
145C2×C3C3×#127⋊C311yescharacteristic
146C22—31no—
147C22—31no—
148C22—64no—
149C22—62no—
150C22—121no—
151C22—122no—
152C22—121no—
153C22#41⋊H(3,𝔽3)11yescharacteristic
154C4—41no—
155C4—62no—
156C3—31no—
157C3—483no—
158C3C3×#392⋊C311yescharacteristic
159C2—31no—
160C2—31no—
161C2—122no—
162C2#127⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
centerC2×C3C3×#127⋊C3
commutator, cosocleC3×#95C2×C32
Frattini, socleC23×C3C2×C3×A4
FittingC32×#392C3
hypercenterC2×C32#127⋊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×#392C3semidirect
3. C3×#392⋊C3C3semidirect
8. C3×#95⋊C3C2×C3semidirect
40. #392H(3,𝔽3)semidirect
126. C32#392⋊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×#3923
C3×#392⋊C33
C4×C22⋊H(3,𝔽3)4