Finite Group Databasebeta

← #13180434

Subgroups — #13180434

Order 1728.

246 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 61 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#11724365C211yesmaximal, characteristic
2#11724335C211yesmaximal, characteristic
3C3×C4×AGL(1,𝔽9)C211yesmaximal, characteristic
4AGL(1,𝔽9)⋊Dic3C214yesmaximal
5AGL(1,𝔽9)⋊Q8—31nomaximal
6C2×C3⋊PSU(3,𝔽2)—21no—
7C2×C3×AGL(1,𝔽9)C2212yes—
8#89040—21no—
9C4⋊(C3⋊#157)—21no—
10C3×C4×#157C2211yescharacteristic
11#88946C2212yes—
12#88940—21no—
13#88940C2212yes—
14C4⋊PSU(3,𝔽2)—31no—
15#64222—31no—
16C4×AGL(1,𝔽9)D311yescharacteristic
17AGL(1,𝔽9)⋊C4—34no—
18C2×C3⋊#157—22no—
19C2×C3×#157D411yescharacteristic
20C2×C3×#157C2311yescharacteristic
21C3⋊PSU(3,𝔽2)—42no—
22C3×AGL(1,𝔽9)Q814yescomplemented
23C3×C4×C3⋊D3D411yescharacteristic
24C3×#634—41no—
25#6296—41no—
26#6294—22no—
27#4680—91nomaximal
28C2×PSU(3,𝔽2)—61no—
29C2×AGL(1,𝔽9)C2×D312yes—
30#3797—61no—
31D3×Dic6—121no—
32C4⋊#157—61no—
33C4×#157C2×D311yescharacteristic
34#3777—32no—
35#3773—32no—
36#3773—61no—
37C2×C3×C3⋊D3C2×D411yescharacteristic
38C3⋊#157—42no—
39C3×#157SD1612yes—
40C3×#157C2×Q812yes—
41C33×C4SD1611yescharacteristic
42C3×C3⋊Dic3—42no—
43C3×C3⋊Dic3SD1611yescharacteristic
44#869—91no—
45#869—91no—
46C3×C4×C8—91no—
47#817—94no—
48C2×#157—62no—
49C2×#157D1211yescharacteristic
50C2×#157C22×D311yescharacteristic
51PSU(3,𝔽2)—122no—
52AGL(1,𝔽9)Dic614yescomplemented
53C4×C3⋊D3D1211yescharacteristic
54#634—41no—
55C3×C4×D3—121no—
56C3×Dic6—121no—
57C3×Dic6—121no—
58C3×Dic6—241no—
59#627—122no—
60#627—121no—
61#624—62no—
62D3×Dic3—122no—
63#498—271noSylow
64C2×C33C2×SD1611yescharacteristic
65C3×C3⋊D3C4⋊Q811yescharacteristic
66C3×C3⋊D3C2×SD1611yescharacteristic
67D3×Q8—361no—
68C2×Dic6—181no—
69C2×Dic6—181no—
70C2×C3×C8—92no—
71C3×C42—91no—
72C4⋊Dic3—92no—
73C4⋊Dic3—92no—
74C4⋊Dic3—181no—
75C4⋊Dic3—181no—
76C2×C3⋊D3C2×D1211yescharacteristic
77C2×C3×D3—121no—
78#157—122no—
79#157#20412yes—
80#157C2×Dic612yes—
81C32×C4—41no—
82C32×C4—81no—
83C32×C4#20411yescharacteristic
84C3⋊Dic3—42no—
85C3⋊Dic3#20411yescharacteristic
86C3×Dic3—121no—
87C3×Dic3—122no—
88C3×Dic3—122no—
89C3×Dic3—242no—
90C4⋊Q8—271no—
91C4⋊Q8—271no—
92#106—274no—
93C4×C8—91no—
94C33#49811yescharacteristic, complemented, Sylow
95C3×Q8—361no—
96C2×C3×C4—91no—
97C2×C3×C4—91no—
98C2×C3×C4—91no—
99C2×Dic3—182no—
100C2×Dic3—182no—
101C4×D3—121no—
102C4×D3—362no—
103Dic6—121no—
104Dic6—121no—
105Dic6—241no—
106Dic6—361no—
107Dic6—362no—
108Dic6—362no—
109C3×C8—94no—
110C2×C32—41no—
111C2×C32—81no—
112C2×C32C2×#20411yescharacteristic
113C3⋊D3#86911yescharacteristic
114C3⋊D3C2×#20411yescharacteristic
115C3×D3—121no—
116C3×D3—121no—
117C2×Q8—541no—
118C2×Q8—541no—
119C2×C8—92no—
120C4⋊C4—272no—
121C4⋊C4—272no—
122C4⋊C4—541no—
123C4⋊C4—541no—
124C42—91no—
125C22×C3—91no—
126C2×D3—121no—
127C3×C4—41no—
128C3×C4—81no—
129C3×C4—92no—
130C3×C4—92no—
131C3×C4—91no—
132C3×C4—362no—
133C3×C4AΓL(1,𝔽9)11yescharacteristic
134Dic3—121no—
135Dic3—122no—
136Dic3—122no—
137Dic3—242no—
138Dic3—362no—
139C32—41no—
140C32—81no—
141C32#468011yescharacteristic, complemented
142Q8—361no—
143Q8—1081no—
144Q8—1082no—
145C2×C4—91no—
146C2×C4—91no—
147C2×C4—91no—
148C2×C4—542no—
149C2×C4—542no—
150C8—94no—
151C2×C3—41no—
152C2×C3—81no—
153C2×C3—91no—
154C2×C3—91no—
155C2×C3C2×AΓL(1,𝔽9)11yescharacteristic
156D3—121no—
157D3—121no—
158C22—91no—
159C4—92no—
160C4—92no—
161C4—91no—
162C4—362no—
163C4—1082no—
164C4AGL(1,𝔽9)⋊D311yescharacteristic
165C3—41no—
166C3—81no—
167C3AGL(1,𝔽9)⋊Q811yescharacteristic, complemented
168C2—91no—
169C2—91no—
170C2C2×AGL(1,𝔽9)⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C2×AGL(1,𝔽9)⋊D3
commutator, cosocleC2×C3×#157C23
FittingC33×C4SD16
socleC2×C33C2×SD16
prime radicalC2×C3×C3⋊D3C2×D4
hypercenterC4AGL(1,𝔽9)⋊D3
hypocenterC33#498
LCSG ▷ C2×C3×#157 ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
22. C3×AGL(1,𝔽9)Q8semidirect
52. AGL(1,𝔽9)Dic6semidirect
94. C33#498semidirect
141. C32#4680semidirect
167. C3AGL(1,𝔽9)⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#117243652
#117243352
C3×C4×AGL(1,𝔽9)2
AGL(1,𝔽9)⋊Dic32
AGL(1,𝔽9)⋊Q83
#46809