Finite Group Databasebeta

← #13166272

Subgroups — #13166272

Order 1728.

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

normal subgroups only

This group has 376 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
1#11724252C212yesmaximal, complemented
2#11723341C211yesmaximal, characteristic
3#11723095C211yesmaximal, characteristic, complemented
4C3×#63767C211yesmaximal, characteristic, complemented
5#11722713C212yesmaximal, complemented
6#10593788—31nomaximal
7#89018C2212yes—
8#88989C2212yes—
9C3×C4×#157C2211yescharacteristic
10C3×#3787C2211yescharacteristic
11#88926—22no—
12#88924—22no—
13#88852C411yescharacteristic, complemented
14C3×C8×C3⋊D3C2211yescharacteristic
15C32×Dic12C411yescharacteristic, complemented
16#88795C412yescomplemented
17#64289—32no—
18#64128—31no—
19#63808—62no—
20#63785—32no—
21#63770—31no—
22#63767D311yescharacteristic, complemented
23C2×C3×#157—21no—
24C4⋊(C3⋊(C3⋊D3))C2×C412yes—
25C3×C4×C3⋊D3C2311yescharacteristic
26C32×Dic6C2×C412yes—
27#6292—22no—
28#6289—22no—
29C3⋊#622—22no—
30C3×#622D412yes—
31C33×C8C2×C411yescharacteristic
32C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
33#4717—91nomaximal
34#3832—32no—
35#3799—64no—
36C4×#157C2×D311yescharacteristic
37#3788—32no—
38#3788—62no—
39#3787—62no—
40#3787C2×D311yescharacteristic
41C32×Q16—31no—
42#3755—32no—
43#3745—31no—
44#3745—62no—
45#3745—122no—
46C8×C3⋊D3C2×D311yescharacteristic
47C3×Dic12—22no—
48C3×D3×C8—62no—
49#3717—64no—
50C2×C3⋊(C3⋊D3)—22no—
51C2×C3×C3⋊D3#4111yescharacteristic
52C3×#157—41no—
53C33×C4C22×C411yescharacteristic
54C3×C3⋊Dic3C2×D411yescharacteristic
55C32×Dic3—22no—
56#919—182no—
57#904—91no—
58#901—92no—
59C3×#97—91no—
60#819—91no—
61#805—92no—
62C2×#157—21no—
63C32×Q8—32no—
64C4⋊(C3⋊D3)—32no—
65C4⋊(C3⋊D3)—64no—
66C4⋊(C3⋊D3)—62no—
67C4⋊(C3⋊D3)—124no—
68C4×C3⋊D3—62no—
69C4×C3⋊D3C22×D311yescharacteristic
70C3×C4×D3—62no—
71C3×Dic6—24no—
72#626—124no—
73#624—124no—
74#622—62no—
75#622D1212yes—
76C32×C8—22no—
77C32×C8—42no—
78C32×C8C4×D311yescharacteristic
79C3⋊(C3⋊C8)C4×D311yescharacteristic
80C3×C3⋊C8—62no—
81#444—271noSylow
82C2×C33C4×D411yescharacteristic
83C3⋊(C3⋊D3)—42no—
84C3×C3⋊D3—21no—
85#239—184no—
86#235—92no—
87C3×Q16—62no—
88C2×C3×C8—91no—
89C2×C3×C8—91no—
90C3×C42—91no—
91#215—184no—
92#208—92no—
93Dic12#15711yescharacteristic, complemented
94D24—91no—
95D24—182no—
96D24—362no—
97#204—92no—
98ΓL(1,𝔽25)—182no—
99D3×C8—62no—
100D3×C8—182no—
101C2×C3⋊D3—62no—
102C2×C3⋊D3—124no—
103C2×C3⋊D3—244no—
104C2×C3⋊D3#23511yescharacteristic
105C2×C3×D3—62no—
106#157—41no—
107C32×C4—22no—
108C32×C4—42no—
109C32×C4—62no—
110C32×C4C2×C4×D311yescharacteristic
111C3⋊Dic3C2×D1211yescharacteristic
112C3×Dic3—44no—
113C3×Dic3—62no—
114#135—271no—
115#131—272no—
116#108—271no—
117#104—272no—
118#97—91no—
119C33#44411yescharacteristic, complemented, Sylow
120C3×Q8—64no—
121C2×C3×C4—91no—
122C2×C3×C4—181no—
123#67—182no—
124D12—92no—
125D12—184no—
126D12—364no—
127D12—364no—
128C4×D3—62no—
129C4×D3—182no—
130C4×D3—364no—
131Dic6C2×#15712yes—
132C3×C8—22no—
133C3×C8—42no—
134C3×C8—92no—
135C3×C8—91no—
136C3×C8C2×#15711yescharacteristic
137C3⋊C8—62no—
138C3⋊C8—182no—
139C2×C32—22no—
140C2×C32—42no—
141C2×C32C4×D1211yescharacteristic
142C3⋊D3—21no—
143C3⋊D3—122no—
144C3⋊D3—244no—
145C3⋊D3—484no—
146C3×D3—122no—
147#41—272no—
148Q16—31no—
149SD16—272no—
150D8—271no—
151M16—272no—
152M16—542no—
153C2×C8—91no—
154C2×C8—91no—
155C2×C8—542no—
156C42—91no—
157C22×C3—91no—
158C2×D3—62no—
159C2×D3—182no—
160C2×D3—364no—
161C2×D3—724no—
162C3×C4—22no—
163C3×C4—42no—
164C3×C4—91no—
165C3×C4—124no—
166C3×C4—361no—
167C3×C4C22×#15711yescharacteristic
168Dic3—22no—
169Dic3—62no—
170C32—22no—
171C32—42no—
172C32#471711yescharacteristic, complemented
173Q8—32no—
174D4—272no—
175D4—542no—
176C2×C4—91no—
177C2×C4—181no—
178C2×C4—542no—
179C8—92no—
180C8—91no—
181C8—542no—
182C8D3×#15711yescharacteristic
183C2×C3—22no—
184C2×C3—42no—
185C2×C3—181no—
186C2×C3D4×#15711yescharacteristic
187D3—122no—
188D3—362no—
189D3—724no—
190D3—1444no—
191C22—91no—
192C22—542no—
193C4—62no—
194C4—91no—
195C4—361no—
196C4C2×D3×#15711yescharacteristic
197C3—22no—
198C3—42no—
199C3#1059378811yescharacteristic, complemented
200C2—181no—
201C2—1082no—
202C2D12×#15711yescharacteristic
subgroupG/H
centerC2D12×#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC33×C8C2×C4
socleC2×C33C4×D4
prime radical#11723341C2
cosocleC3×C4×C3⋊D3C23
hypercenterC8D3×#157
hypocenterC33#444
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. #11724252C2semidirect
3. #11723095C2semidirect
4. C3×#63767C2semidirect
5. #11722713C2semidirect
13. #88852C4semidirect
15. C32×Dic12C4semidirect
16. #88795C4semidirect
22. #63767D3semidirect
93. Dic12#157semidirect
119. C33#444semidirect
172. C32#4717semidirect
199. C3#10593788semidirect

Maximal subgroups

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

subgroupindex
#117242522
#117233412
#117230952
C3×#637672
#117227132
#105937883
#47179