Finite Group Databasebeta

← #13165771

Subgroups — #13165771

Order 1728.

242 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 48 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#11724364C211yesmaximal, characteristic
2#11724245C211yesmaximal, characteristic
3#11722807C211yesmaximal, characteristic
4#11722803C211yesmaximal, characteristic
5C3×#63746C211yesmaximal, characteristic
6#11722709C211yesmaximal, characteristic
7#11722708C211yesmaximal, characteristic
8#10593562—31nomaximal
9C2×C3⋊PSU(3,𝔽2)—21no—
10C3⋊D3×Dic6C2211yescharacteristic
11C4⋊(C3⋊#157)C2211yescharacteristic
12C4×C3⋊#157C2211yescharacteristic
13#88988C2211yescharacteristic
14C3×C4⋊#157C2211yescharacteristic
15C3×C4⋊#157C2211yescharacteristic
16C3×#3787C2211yescharacteristic
17#88945—21no—
18Dic3⋊#157—21no—
19Dic3×#157—21no—
20Q8×#157—31no—
21C4⋊PSU(3,𝔽2)—31no—
22#63784—31no—
23#63784—31no—
24#63746—31no—
25#63746D311yescharacteristic
26#63744—31no—
27C2×C3⋊#157—21no—
28C2×C3⋊#157D411yescharacteristic
29C2×C3⋊#157D411yescharacteristic
30C2×C3×#157—21no—
31C2×C3×#157—21no—
32C3⋊PSU(3,𝔽2)—42no—
33#6308Q811yescharacteristic, complemented
34C3×C4×C3⋊D3C2311yescharacteristic
35C32×Dic6Q811yescharacteristic, complemented
36#6293—21no—
37Dic3×C3⋊D3—21no—
38C3⋊#622—21no—
39C3×#622—21no—
40#4875—91nomaximal
41C2×PSU(3,𝔽2)—61no—
42Q8×C3⋊D3—31no—
43D3×Dic6—121no—
44C4⋊#157—31no—
45C4⋊#157—61no—
46C4⋊#157C2×D311yescharacteristic
47C4⋊#157C2×D311yescharacteristic
48C4×#157—31no—
49C4×#157—61no—
50#3787—31no—
51#3787C2×D311yescharacteristic
52#3777—61no—
53C2×C3×C3⋊D3C2×D411yescharacteristic
54C3⋊#157—21no—
55C3⋊#157—41no—
56C3⋊#157Q1612yes—
57C3×#157—41no—
58C3×#157—41no—
59C33×C4C2×Q811yescharacteristic
60C3⋊(C3⋊Dic3)—21no—
61C3×C3⋊Dic3#4111yescharacteristic
62C32×Dic3—21no—
63#888—91no—
64#887—91no—
65C3×#107—91no—
66#816—91no—
67Dic3⋊C8—91no—
68#810—91no—
69#807—91no—
70C2×#157—21no—
71C2×#157—21no—
72C2×#157—31no—
73C2×#157—31no—
74C2×#157—61no—
75PSU(3,𝔽2)—122no—
76C32×Q8—31no—
77C4×C3⋊D3—61no—
78C4×C3⋊D3C22×D311yescharacteristic
79#634—31no—
80#634—61no—
81#634—121no—
82#634—241no—
83C3×C4×D3—121no—
84C3×Dic6—41no—
85#627—122no—
86D3×Dic3—122no—
87#622—21no—
88#622—61no—
89#477—271noSylow
90C2×C33C22⋊Q811yescharacteristic
91C3×C3⋊D3C2×Q1611yescharacteristic
92C3×C3⋊D3AGL(1,ℤ/8)11yescharacteristic
93D3×Q8—361no—
94C2×Dic6—91no—
95C2×Dic6—181no—
96C2×C3×C8—91no—
97C3×C4⋊C4—91no—
98C3×C4⋊C4—91no—
99C4⋊Dic3—91no—
100Dic3⋊C4—181no—
101Dic3⋊C4—181no—
102C4×Dic3—91no—
103C4×Dic3—181no—
104C2×C3⋊C8—91no—
105C2×C3⋊D3D3×D411yescharacteristic
106C2×C3×D3—121no—
107#157—32no—
108#157—41no—
109#157—41no—
110#157—61no—
111#157—121no—
112C32×C4—41no—
113C32×C4—61no—
114C32×C4—81no—
115C32×C4D3×Q811yescharacteristic
116C3⋊Dic3—61no—
117C3⋊Dic3—122no—
118C3⋊Dic3—242no—
119C3⋊Dic3#23511yescharacteristic
120C3×Dic3—42no—
121C3×Dic3—121no—
122C4⋊Q8—271no—
123C4×Q8—271no—
124#107—91no—
125#107—271no—
126C4⋊C8—271no—
127#103—271no—
128#103—271no—
129C33#47711yescharacteristic, complemented, Sylow
130C3×Q8—121no—
131C2×C3×C4—91no—
132C2×C3×C4—181no—
133C2×C3×C4—181no—
134C2×Dic3—91no—
135C2×Dic3—91no—
136C2×Dic3—181no—
137C2×Dic3—181no—
138C4×D3—121no—
139C4×D3—362no—
140Dic6—91no—
141Dic6—181no—
142Dic6—361no—
143Dic6—362no—
144Dic6—362no—
145Dic6—721no—
146Dic6PSU(3,𝔽2)11yescharacteristic, complemented
147C3×C8—181no—
148C3⋊C8—181no—
149C2×C32—41no—
150C2×C32—81no—
151C2×C32#89611yescharacteristic
152C3⋊D3#91111yescharacteristic
153C3⋊D3D3×Q1611yescharacteristic
154C3×D3—121no—
155C3×D3—121no—
156C2×Q8—271no—
157C2×Q8—541no—
158C2×C8—91no—
159C2×C8—271no—
160C4⋊C4—91no—
161C4⋊C4—91no—
162C4⋊C4—271no—
163C4⋊C4—541no—
164C4⋊C4—541no—
165C42—271no—
166C42—541no—
167C22×C3—91no—
168C2×D3—121no—
169C3×C4—41no—
170C3×C4—81no—
171C3×C4—91no—
172C3×C4—122no—
173C3×C4—361no—
174C3×C4—361no—
175C3×C4C2×PSU(3,𝔽2)11yescharacteristic
176Dic3—21no—
177Dic3—92no—
178Dic3—121no—
179Dic3—181no—
180Dic3—181no—
181Dic3—362no—
182Dic3—361no—
183Dic3—722no—
184C32—41no—
185C32—81no—
186C32#487511yescharacteristic, complemented
187Q8—31no—
188Q8—271no—
189Q8—541no—
190Q8—1082no—
191C2×C4—91no—
192C2×C4—181no—
193C2×C4—181no—
194C2×C4—271no—
195C2×C4—271no—
196C2×C4—541no—
197C2×C4—541no—
198C8—181no—
199C8—541no—
200C2×C3—41no—
201C2×C3—81no—
202C2×C3—91no—
203C2×C3—91no—
204C2×C3C22⋊PSU(3,𝔽2)11yescharacteristic
205D3—121no—
206D3—121no—
207C22—91no—
208C4—61no—
209C4—91no—
210C4—272no—
211C4—361no—
212C4—361no—
213C4—541no—
214C4—541no—
215C4—1081no—
216C4D3×PSU(3,𝔽2)11yescharacteristic
217C3—41no—
218C3—81no—
219C3#1059356211yescharacteristic, complemented
220C2—91no—
221C2—91no—
222C2#1172415111yescharacteristic
subgroupG/H
centerC2#11724151
commutator, cosocleC3×C4×C3⋊D3C23
Frattini, hypercenterC4D3×PSU(3,𝔽2)
FittingC33×C4C2×Q8
socleC2×C33C22⋊Q8
prime radicalC2×C3×C3⋊D3C2×D4
hypocenterC33#477
LCSG ▷ C3×C4×C3⋊D3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
33. #6308Q8semidirect
35. C32×Dic6Q8semidirect
129. C33#477semidirect
146. Dic6PSU(3,𝔽2)semidirect
186. C32#4875semidirect
219. C3#10593562semidirect

Maximal subgroups

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

subgroupindex
#117243642
#117242452
#117228072
#117228032
C3×#637462
#117227092
#117227082
#105935623
#48759