Finite Group Databasebeta

← #13047980

Subgroups — #13047980

Order 1296.

324 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C3×Q8×H(3,𝔽3)C211yesmaximal, characteristic
2#10619956C211yesmaximal, characteristic
3C3⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic
4#88863C311yesmaximal, characteristic, complemented
5C3×#3756—31nomaximal
6#3756⋊C3—33nomaximal
7C3×C4×H(3,𝔽3)—21no—
8C3×C4×H(3,𝔽3)C2211yescharacteristic
9C3⋊((C3⋊Dic3)⋊C3)—21no—
10C33×Q8—21no—
11C33×Q8D311yescharacteristic
12C33×Q8C2×C311yescharacteristic
13#6308C2×C311yescharacteristic
14C3×#634—31no—
15C3⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic
16C3×C3⋊(C3⋊C8)—31no—
17Q8×H(3,𝔽3)—23no—
18Q8×H(3,𝔽3)D313yes—
19#634⋊C3—33no—
20(C3⋊(C3⋊C8))⋊C3—33no—
21C2×C3×H(3,𝔽3)D411yescharacteristic
22#3756—31no—
23#3756—33no—
24#3756—93no—
25C3×#216—93no—
26C3×#216—91no—
27C33×C4—22no—
28C33×C4—21no—
29C33×C4—21no—
30C33×C4—21no—
31C33×C4C2×D311yescharacteristic
32C33×C4C22×C311yescharacteristic
33C3⋊(C3⋊Dic3)—21no—
34C3×C3⋊Dic3—61no—
35C4×H(3,𝔽3)—26no—
36C4×H(3,𝔽3)—23no—
37C4×H(3,𝔽3)—23no—
38C4×H(3,𝔽3)C2×D313yes—
39(C3⋊Dic3)⋊C3—63no—
40C3×H(3,𝔽3)Q1611yescharacteristic, complemented, Sylow
41C32×Q8—21no—
42C32×Q8—23no—
43C32×Q8—33no—
44C32×Q8—33no—
45C32×Q8—63no—
46C32×Q8C3×D313yes—
47C32×Q8C3×D311yescharacteristic
48C32×Q8C3⋊D311yescharacteristic
49#634—31no—
50#634—33no—
51#634—93no—
52C3×Dic6—93no—
53C3×Dic6—91no—
54C3⋊(C3⋊C8)—31no—
55C3⋊(C3⋊C8)—33no—
56C3⋊(C3⋊C8)—93no—
57C3×C3⋊C8—93no—
58C3×C3⋊C8—91no—
59C2×C33—21no—
60C2×C33#6711yescharacteristic
61C2×C33C3×D411yescharacteristic
62C2×H(3,𝔽3)—23no—
63C2×H(3,𝔽3)#6713yes—
64C3×Q16—271no—
65#216—93no—
66#216—91no—
67#216—273no—
68C32×C4—21no—
69C32×C4—22no—
70C32×C4—26no—
71C32×C4—23no—
72C32×C4—21no—
73C32×C4—21no—
74C32×C4—23no—
75C32×C4—33no—
76C32×C4—33no—
77C32×C4—63no—
78C32×C4—63no—
79C32×C4—66no—
80C32×C4—63no—
81C32×C4C2×C3×D313yes—
82C32×C4C2×C3×D311yescharacteristic
83C32×C4C2×C3⋊D311yescharacteristic
84C3⋊Dic3—61no—
85C3⋊Dic3—63no—
86C3⋊Dic3—183no—
87C3×Dic3—183no—
88C3×Dic3—181no—
89C33—21no—
90C33#21611yescharacteristic, complemented
91C33C3×Q1611yescharacteristic, complemented
92H(3,𝔽3)—23no—
93H(3,𝔽3)#21613yescomplemented
94C3×Q8—33no—
95C3×Q8—31no—
96C3×Q8—61no—
97C3×Q8—63no—
98C3×Q8—271no—
99C3×Q8(C3⋊D3)⋊C313yes—
100C3×Q8C3×C3⋊D311yescharacteristic
101Dic6—93no—
102Dic6—91no—
103Dic6—273no—
104C3×C8—271no—
105C3⋊C8—93no—
106C3⋊C8—91no—
107C3⋊C8—273no—
108C2×C32—21no—
109C2×C32—23no—
110C2×C32—33no—
111C2×C32—33no—
112C2×C32—63no—
113C2×C32C3×#6713yes—
114C2×C32C3×#6711yescharacteristic
115C2×C32#63811yescharacteristic
116Q16—271noSylow
117C3×C4—23no—
118C3×C4—21no—
119C3×C4—33no—
120C3×C4—31no—
121C3×C4—63no—
122C3×C4—61no—
123C3×C4—62no—
124C3×C4—66no—
125C3×C4—61no—
126C3×C4—63no—
127C3×C4—541no—
128C3×C4C2×(C3⋊D3)⋊C313yes—
129C3×C4C2×C3×C3⋊D311yescharacteristic
130Dic3—183no—
131Dic3—181no—
132Dic3—543no—
133C32—21no—
134C32—23no—
135C32—33no—
136C32—33no—
137C32—63no—
138C32C3×#21611yescharacteristic
139C32C3×#21613yescomplemented
140C32#375611yescharacteristic, complemented
141Q8—271no—
142Q8C3⋊((C3⋊D3)⋊C3)11yescharacteristic
143C8—271no—
144C2×C3—33no—
145C2×C3—31no—
146C2×C3—61no—
147C2×C3—63no—
148C2×C3#638⋊C313yes—
149C2×C3C3×#63811yescharacteristic
150C4—21no—
151C4—541no—
152C4C2×C3⋊((C3⋊D3)⋊C3)11yescharacteristic
153C3—33no—
154C3—31no—
155C3—61no—
156C3—63no—
157C3#3756⋊C313yescomplemented
158C3C3×#375611yescharacteristic
159C2#1061997711yescharacteristic
subgroupG/H
centerC2#10619977
commutator, cosocleC33×C4C22×C3
FrattiniC3×C4C2×C3×C3⋊D3
FittingC3×Q8×H(3,𝔽3)C2
socleC2×C32C3×#67
prime radicalC2×C3×H(3,𝔽3)D4
hypercenterQ8C3⋊((C3⋊D3)⋊C3)
hypocenterC33C3×Q16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
4. #88863C3semidirect
40. C3×H(3,𝔽3)Q16semidirect
90. C33#216semidirect
91. C33C3×Q16semidirect
93. H(3,𝔽3)#216semidirect
139. C32C3×#216semidirect
140. C32#3756semidirect
157. C3#3756⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×H(3,𝔽3)2
#106199562
C3⋊((C3⋊(C3⋊C8))⋊C3)2
#888633
C3×#37563
#3756⋊C33