Finite Group Databasebeta

← #13167654

Subgroups — #13167654

Order 1728.

226 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 55 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#11724364C212yesmaximal
2#11723093C211yesmaximal, characteristic
3C8×C3⋊#157C211yesmaximal, characteristic
4C3×#63768C211yesmaximal, characteristic
5#11722805C212yesmaximal
6#10593535—31nomaximal
7C2×C3⋊PSU(3,𝔽2)—22no—
8C4⋊(C3⋊#157)C2212yes—
9C4×C3⋊#157C2211yescharacteristic
10#88988C2211yescharacteristic
11C3×C4⋊#157C2212yes—
12#88945—22no—
13C3×C8×C3⋊D3C2211yescharacteristic
14C4⋊PSU(3,𝔽2)—32no—
15#63768—31no—
16#63768D311yescharacteristic
17C8×#157—31no—
18#63745—32no—
19C2×C3⋊#157—22no—
20C2×C3⋊#157D412yes—
21C2×C3×#157—22no—
22C3⋊PSU(3,𝔽2)—44no—
23C3×C4×C3⋊D3C2311yescharacteristic
24C3⋊#622Q812yescomplemented
25C33×C8Q811yescharacteristic, complemented
26C3×C3⋊(C3⋊C8)Q811yescharacteristic, complemented
27#4855—91nomaximal
28C2×PSU(3,𝔽2)—62no—
29C4⋊#157—32no—
30C4⋊#157C2×D312yes—
31C4×#157—31no—
32#3787—31no—
33#3777—62no—
34C8×C3⋊D3C2×D311yescharacteristic
35C3×D3×C8—121no—
36C2×C3×C3⋊D3C2×D411yescharacteristic
37C3⋊#157—42no—
38C3⋊#157SD1614yes—
39C3×#157—42no—
40C33×C4C2×Q811yescharacteristic
41C3×C3⋊Dic3C2×Q811yescharacteristic
42#888—92no—
43C3×#106—91no—
44#817—91no—
45C8×Dic3—91no—
46#808—92no—
47C2×#157—22no—
48C2×#157—32no—
49C2×#157—62no—
50PSU(3,𝔽2)—124no—
51C4×C3⋊D3C22×D311yescharacteristic
52C3×C4×D3—121no—
53#622—32no—
54C32×C8—41no—
55C32×C8—81no—
56C32×C8Dic611yescharacteristic, complemented
57C3⋊(C3⋊C8)Dic611yescharacteristic, complemented
58C3×C3⋊C8—121no—
59#498—271noSylow
60C2×C33C4⋊Q811yescharacteristic
61C3×C3⋊D3C2×SD1612yes—
62C2×Dic6—182no—
63C2×C3×C8—91no—
64C3×C4⋊C4—92no—
65C4⋊Dic3—92no—
66Dic3⋊C4—182no—
67C4×Dic3—91no—
68C2×C3⋊C8—91no—
69D3×C8—121no—
70C2×C3⋊D3D3×D411yescharacteristic
71C2×C3×D3—121no—
72#157—34no—
73#157—42no—
74#157—122no—
75C32×C4—41no—
76C32×C4—81no—
77C32×C4C2×Dic611yescharacteristic
78C3⋊Dic3D3×Q811yescharacteristic
79C3×Dic3—121no—
80C4⋊Q8—272no—
81#106—91no—
82#106—272no—
83#106—271no—
84C4×C8—271no—
85C33#49811yescharacteristic, complemented, Sylow
86C2×C3×C4—91no—
87C2×C3×C4—182no—
88C2×Dic3—92no—
89C2×Dic3—182no—
90C4×D3—121no—
91Dic6—364no—
92C3×C8—41no—
93C3×C8—81no—
94C3×C8—91no—
95C3×C8PSU(3,𝔽2)11yescharacteristic, complemented
96C3⋊C8—92no—
97C3⋊C8—121no—
98C2×C32—41no—
99C2×C32—81no—
100C2×C32#88811yescharacteristic
101C3⋊D3D3×SD1612yes—
102C3×D3—122no—
103C2×Q8—542no—
104C2×C8—91no—
105C2×C8—271no—
106C4⋊C4—92no—
107C4⋊C4—272no—
108C4⋊C4—542no—
109C42—271no—
110C22×C3—91no—
111C2×D3—121no—
112C3×C4—41no—
113C3×C4—81no—
114C3×C4—91no—
115C3×C4—362no—
116C3×C4C2×PSU(3,𝔽2)11yescharacteristic
117Dic3—94no—
118Dic3—121no—
119Dic3—362no—
120C32—41no—
121C32—81no—
122C32#485511yescharacteristic, complemented
123Q8—1084no—
124C2×C4—91no—
125C2×C4—182no—
126C2×C4—272no—
127C2×C4—542no—
128C8—91no—
129C8—272no—
130C8C3⋊PSU(3,𝔽2)11yescharacteristic, complemented
131C2×C3—41no—
132C2×C3—81no—
133C2×C3—92no—
134C2×C3C4⋊PSU(3,𝔽2)11yescharacteristic
135D3—122no—
136C22—91no—
137C4—91no—
138C4—274no—
139C4—362no—
140C4—1082no—
141C4C2×C3⋊PSU(3,𝔽2)11yescharacteristic
142C3—41no—
143C3—81no—
144C3#1059353511yescharacteristic, complemented
145C2—92no—
146C2#1172436411yescharacteristic
subgroupG/H
centerC2#11724364
commutator, cosocleC3×C4×C3⋊D3C23
FrattiniC4C2×C3⋊PSU(3,𝔽2)
FittingC33×C8Q8
socleC2×C33C4⋊Q8
prime radicalC2×C3×C3⋊D3C2×D4
hypercenterC8C3⋊PSU(3,𝔽2)
hypocenterC33#498
LCSG ▷ C3×C4×C3⋊D3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
24. C3⋊#622Q8semidirect
25. C33×C8Q8semidirect
26. C3×C3⋊(C3⋊C8)Q8semidirect
56. C32×C8Dic6semidirect
57. C3⋊(C3⋊C8)Dic6semidirect
85. C33#498semidirect
95. C3×C8PSU(3,𝔽2)semidirect
122. C32#4855semidirect
130. C8C3⋊PSU(3,𝔽2)semidirect
144. C3#10593535semidirect

Maximal subgroups

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

subgroupindex
#117243642
#117230932
C8×C3⋊#1572
C3×#637682
#117228052
#105935353
#48559