Finite Group Databasebeta

← #13180432

Subgroups — #13180432

Order 1728.

242 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 51 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
1C4×C3⋊PSU(3,𝔽2)C211yesmaximal, characteristic
2#11724335C211yesmaximal, characteristic
3C3×C4⋊AGL(1,𝔽9)C211yesmaximal, characteristic
4AGL(1,𝔽9)⋊Dic3C212yesmaximal
5#11724105C212yesmaximal
6PSU(3,𝔽2)⋊Q8—31nomaximal
7C2×C3⋊PSU(3,𝔽2)C2211yescharacteristic
8C2×C3×AGL(1,𝔽9)C2212yes—
9#89040—21no—
10C4×C3⋊#157—21no—
11C3×C4×#157C2211yescharacteristic
12#88946C2211yescharacteristic
13#88945—21no—
14#88940—21no—
15#88940C2212yes—
16C4×PSU(3,𝔽2)—31no—
17#64222—31no—
18C4⋊AGL(1,𝔽9)D311yescharacteristic
19AGL(1,𝔽9)⋊C4—32no—
20PSU(3,𝔽2)⋊C4—32no—
21C2×C3⋊#157—21no—
22C2×C3⋊#157—21no—
23C2×C3×#157D411yescharacteristic
24C2×C3×#157C2311yescharacteristic
25C3⋊PSU(3,𝔽2)—21no—
26C3⋊PSU(3,𝔽2)Q812yescomplemented
27C3×AGL(1,𝔽9)—22no—
28C3×C4×C3⋊D3D411yescharacteristic
29C3×#634—41no—
30#6296—41no—
31#6294—22no—
32#4851—91nomaximal
33C2×PSU(3,𝔽2)—31no—
34C2×AGL(1,𝔽9)C2×D312yes—
35#3797—61no—
36D3×Dic6—121no—
37C4×#157—61no—
38C4×#157C2×D311yescharacteristic
39#3777—31no—
40#3777—61no—
41#3773—32no—
42#3773—61no—
43C2×C3×C3⋊D3C2×D411yescharacteristic
44C3⋊#157—22no—
45C3⋊#157—41no—
46C3×#157—21no—
47C3×#157C2×Q811yescharacteristic
48C3×#157#4111yescharacteristic
49C33×C4SD1611yescharacteristic
50C3×C3⋊Dic3—42no—
51C3×C3⋊Dic3SD1611yescharacteristic
52#869—91no—
53C4×Dic6—91no—
54C3×C4⋊C8—91no—
55#817—92no—
56#816—92no—
57C2×#157—61no—
58C2×#157—61no—
59C2×#157D1211yescharacteristic
60C2×#157C22×D311yescharacteristic
61PSU(3,𝔽2)—32no—
62PSU(3,𝔽2)—61no—
63AGL(1,𝔽9)—22no—
64C4×C3⋊D3D1211yescharacteristic
65#634—41no—
66C3×C4×D3—121no—
67C3×Dic6—121no—
68C3×Dic6—121no—
69C3×Dic6—241no—
70#627—122no—
71#627—121no—
72#624—62no—
73D3×Dic3—122no—
74#475—271noSylow
75C2×C33C2×SD1611yescharacteristic
76C3×C3⋊D3C22⋊Q811yescharacteristic
77C3×C3⋊D3#13711yescharacteristic
78D3×Q8—361no—
79C2×Dic6—91no—
80C2×Dic6—181no—
81C2×C3×C8—92no—
82C3×C42—91no—
83C4⋊Dic3—92no—
84C4⋊Dic3—91no—
85C4⋊Dic3—181no—
86Dic3⋊C4—181no—
87C4×Dic3—181no—
88C2×C3⋊D3C2×D1211yescharacteristic
89C2×C3×D3—121no—
90#157—21no—
91#157—62no—
92#157—121no—
93#157#23711yescharacteristic
94#157D3×Q811yescharacteristic
95C32×C4—41no—
96C32×C4—81no—
97C32×C4#20411yescharacteristic
98C3⋊Dic3—42no—
99C3⋊Dic3#20411yescharacteristic
100C3×Dic3—121no—
101C3×Dic3—122no—
102C3×Dic3—122no—
103C3×Dic3—242no—
104C4⋊Q8—271no—
105C4×Q8—271no—
106#106—272no—
107C4⋊C8—91no—
108#103—272no—
109C33#47511yescharacteristic, complemented, Sylow
110C3×Q8—361no—
111C2×C3×C4—91no—
112C2×C3×C4—91no—
113C2×C3×C4—91no—
114C2×Dic3—182no—
115C2×Dic3—181no—
116C2×Dic3—181no—
117C4×D3—121no—
118C4×D3—362no—
119Dic6—92no—
120Dic6—121no—
121Dic6—121no—
122Dic6—181no—
123Dic6—241no—
124Dic6—361no—
125Dic6—362no—
126C3×C8—182no—
127C2×C32—41no—
128C2×C32—81no—
129C2×C32C2×#20411yescharacteristic
130C3⋊D3#89711yescharacteristic
131C3⋊D3#90911yescharacteristic
132C3×D3—121no—
133C3×D3—121no—
134C2×Q8—271no—
135C2×Q8—541no—
136C2×C8—92no—
137C4⋊C4—272no—
138C4⋊C4—271no—
139C4⋊C4—541no—
140C4⋊C4—541no—
141C42—91no—
142C42—541no—
143C22×C3—91no—
144C2×D3—121no—
145C3×C4—41no—
146C3×C4—81no—
147C3×C4—91no—
148C3×C4—91no—
149C3×C4—91no—
150C3×C4—181no—
151C3×C4—362no—
152C3×C4AΓL(1,𝔽9)11yescharacteristic
153Dic3—121no—
154Dic3—122no—
155Dic3—122no—
156Dic3—182no—
157Dic3—242no—
158Dic3—361no—
159C32—41no—
160C32—81no—
161C32#485111yescharacteristic, complemented
162Q8—272no—
163Q8—361no—
164Q8—541no—
165Q8—1081no—
166C2×C4—91no—
167C2×C4—91no—
168C2×C4—91no—
169C2×C4—542no—
170C2×C4—541no—
171C2×C4—541no—
172C8—182no—
173C2×C3—41no—
174C2×C3—81no—
175C2×C3—91no—
176C2×C3—91no—
177C2×C3C2×AΓL(1,𝔽9)11yescharacteristic
178D3—121no—
179D3—121no—
180C22—91no—
181C4—91no—
182C4—91no—
183C4—91no—
184C4—181no—
185C4—362no—
186C4—542no—
187C4—1081no—
188C4AGL(1,𝔽9)⋊D311yescharacteristic
189C3—41no—
190C3—81no—
191C3PSU(3,𝔽2)⋊Q811yescharacteristic, complemented
192C2—91no—
193C2—91no—
194C2C2×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#475
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
26. C3⋊PSU(3,𝔽2)Q8semidirect
109. C33#475semidirect
161. C32#4851semidirect
191. C3PSU(3,𝔽2)⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C4×C3⋊PSU(3,𝔽2)2
#117243352
C3×C4⋊AGL(1,𝔽9)2
AGL(1,𝔽9)⋊Dic32
#117241052
PSU(3,𝔽2)⋊Q83
#48519