Finite Group Databasebeta

← #13165759

Subgroups — #13165759

Order 1728.

218 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 58 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#11722803C211yesmaximal, characteristic
3#11722803C211yesmaximal, characteristic
4C3×#63744C211yesmaximal, characteristic
5#11722705C211yesmaximal, characteristic
6#11722704C211yesmaximal, characteristic
7#11722703C211yesmaximal, characteristic
8#10593532—31nomaximal
9C2×C3⋊PSU(3,𝔽2)C411yescharacteristic
10C4×C3⋊#157C2211yescharacteristic
11C4×C3⋊#157C2211yescharacteristic
12#88988C2211yescharacteristic
13C3×C4×#157C2211yescharacteristic
14C3×#3787C2211yescharacteristic
15C3×#3787C2211yescharacteristic
16#88946C411yescharacteristic
17#88945C411yescharacteristic
18#88945C411yescharacteristic
19C3⋊D3×C3⋊C8C2211yescharacteristic
20C4×PSU(3,𝔽2)—31no—
21#63767—31no—
22#63767—31no—
23C8×#157—31no—
24#63744—31no—
25#63744—31no—
26#63744D311yescharacteristic
27C2×C3⋊#157C2×C411yescharacteristic
28C2×C3⋊#157C2×C411yescharacteristic
29C2×C3⋊#157C2×C411yescharacteristic
30C2×C3⋊#157C2×C411yescharacteristic
31C2×C3×#157C2×C411yescharacteristic
32C2×C3×#157C2×C411yescharacteristic
33C3⋊PSU(3,𝔽2)—22no—
34C3×C4×C3⋊D3C2311yescharacteristic
35C3⋊#622—21no—
36C3×#622—21no—
37C3×#622—21no—
38C3⋊(C3⋊(C3⋊C8))Q811yescharacteristic, complemented
39C32×C3⋊C8Q811yescharacteristic, complemented
40#4828—91nomaximal
41C2×PSU(3,𝔽2)—31no—
42C4×#157—31no—
43C4×#157—31no—
44C4×#157C2×D311yescharacteristic
45#3787—31no—
46#3787C2×D311yescharacteristic
47#3787C2×D311yescharacteristic
48#3777—31no—
49#3777—31no—
50#3777—31no—
51C8×C3⋊D3—31no—
52D3×C3⋊C8—121no—
53C2×C3×C3⋊D3C22×C411yescharacteristic
54C3⋊#157—21no—
55C3⋊#157—21no—
56C3⋊#157—21no—
57C3⋊#157M1612yes—
58C3×#157—21no—
59C3×#157—21no—
60C33×C4C2×Q811yescharacteristic
61C3×C3⋊Dic3#4111yescharacteristic
62C4×Dic6—91no—
63C3×C4⋊C8—91no—
64#815—91no—
65Dic3⋊C8—91no—
66Dic3⋊C8—91no—
67C8×Dic3—91no—
68#803—91no—
69C2×#157—31no—
70C2×#157—31no—
71C2×#157—31no—
72C2×#157—31no—
73C2×#157C4×D311yescharacteristic
74C2×#157C4×D311yescharacteristic
75PSU(3,𝔽2)—62no—
76C4×C3⋊D3C22×D311yescharacteristic
77C3×C4×D3—121no—
78#622—21no—
79#622—21no—
80#622—61no—
81C32×C8—31no—
82C3⋊(C3⋊C8)—31no—
83C3⋊(C3⋊C8)—121no—
84C3⋊(C3⋊C8)—241no—
85C3×C3⋊C8—41no—
86#446—271noSylow
87C2×C33C4×Q811yescharacteristic
88C3×C3⋊D3C2×M1611yescharacteristic
89C3×C3⋊D3#13111yescharacteristic
90C2×Dic6—91no—
91C2×C3×C8—91no—
92C2×C3×C8—91no—
93C3×C42—91no—
94C4⋊Dic3—91no—
95Dic3⋊C4—91no—
96Dic3⋊C4—91no—
97C4×Dic3—91no—
98C4×Dic3—91no—
99C2×C3⋊C8—91no—
100C2×C3⋊C8—91no—
101D3×C8—361no—
102C2×C3⋊D3C2×C4×D311yescharacteristic
103C2×C3×D3—121no—
104#157—21no—
105#157—21no—
106#157—32no—
107#157—61no—
108#157—61no—
109#157—61no—
110C32×C4—41no—
111C32×C4—81no—
112C32×C4D3×Q811yescharacteristic
113C3⋊Dic3#23711yescharacteristic
114C3×Dic3—121no—
115C4×Q8—271no—
116C4⋊C8—91no—
117C4⋊C8—271no—
118C4⋊C8—271no—
119#97—271no—
120#97—271no—
121C4×C8—271no—
122C33#44611yescharacteristic, complemented, Sylow
123C2×C3×C4—91no—
124C2×C3×C4—91no—
125C2×C3×C4—91no—
126C2×Dic3—91no—
127C2×Dic3—91no—
128C2×Dic3—91no—
129C2×Dic3—91no—
130C4×D3—121no—
131Dic6—182no—
132C3×C8—121no—
133C3×C8—181no—
134C3×C8—181no—
135C3⋊C8—91no—
136C3⋊C8—181no—
137C3⋊C8—361no—
138C3⋊C8—721no—
139C3⋊C8PSU(3,𝔽2)11yescharacteristic, complemented
140C2×C32—41no—
141C2×C32—81no—
142C2×C32#88711yescharacteristic
143C3⋊D3#90111yescharacteristic
144C3⋊D3D3×M1611yescharacteristic
145C3×D3—121no—
146C3×D3—121no—
147C2×Q8—271no—
148C2×C8—91no—
149C2×C8—91no—
150C2×C8—271no—
151C2×C8—271no—
152C4⋊C4—271no—
153C4⋊C4—271no—
154C4⋊C4—271no—
155C42—91no—
156C42—271no—
157C42—271no—
158C22×C3—91no—
159C2×D3—121no—
160C3×C4—41no—
161C3×C4—81no—
162C3×C4—91no—
163C3×C4—181no—
164C3×C4—181no—
165C3×C4C2×PSU(3,𝔽2)11yescharacteristic
166Dic3—92no—
167Dic3—121no—
168Dic3—181no—
169Dic3—181no—
170Dic3—181no—
171C32—41no—
172C32—81no—
173C32#482811yescharacteristic, complemented
174Q8—542no—
175C2×C4—91no—
176C2×C4—91no—
177C2×C4—91no—
178C2×C4—271no—
179C2×C4—271no—
180C2×C4—271no—
181C2×C4—271no—
182C8—31no—
183C8—181no—
184C8—181no—
185C8—271no—
186C8—541no—
187C2×C3—41no—
188C2×C3—81no—
189C2×C3—91no—
190C2×C3—91no—
191C2×C3C4×PSU(3,𝔽2)11yescharacteristic
192D3—121no—
193D3—121no—
194C22—91no—
195C4—91no—
196C4—181no—
197C4—181no—
198C4—272no—
199C4—541no—
200C4—541no—
201C4—541no—
202C4D3×PSU(3,𝔽2)11yescharacteristic
203C3—41no—
204C3—81no—
205C3#1059353211yescharacteristic, complemented
206C2—91no—
207C2—91no—
208C2#1172414811yescharacteristic
subgroupG/H
center, Frattini, hypercenterC4D3×PSU(3,𝔽2)
commutator, prime radicalC2×C3×C3⋊D3C22×C4
FittingC33×C4C2×Q8
socleC2×C33C4×Q8
cosocleC3×C4×C3⋊D3C23
hypocenterC33#446
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
38. C3⋊(C3⋊(C3⋊C8))Q8semidirect
39. C32×C3⋊C8Q8semidirect
122. C33#446semidirect
139. C3⋊C8PSU(3,𝔽2)semidirect
173. C32#4828semidirect
205. C3#10593532semidirect

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
#117228032
C3×#637442
#117227052
#117227042
#117227032
#105935323
#48289