Finite Group Databasebeta

← #13166270

Subgroups — #13166270

Order 1728.

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

normal subgroups only

This group has 368 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#11724250C212yesmaximal, complemented
2#11723339C211yesmaximal, characteristic
3#11723095C211yesmaximal, characteristic, complemented
4C3×#63767C211yesmaximal, characteristic, complemented
5#11722711C212yesmaximal, complemented
6#10593780—31nomaximal
7#89014C2212yes—
8#88989C2212yes—
9C3×C4×#157C2211yescharacteristic
10C3×#3787C2211yescharacteristic
11#88925—22no—
12D3×#622—22no—
13#88853C411yescharacteristic, complemented
14C3×C8×C3⋊D3C2211yescharacteristic
15C32×D24C411yescharacteristic, complemented
16#88794C412yescomplemented
17#64287—32no—
18#64121—31no—
19#63807—62no—
20#63783—32no—
21#63770—31no—
22#63767D311yescharacteristic, complemented
23C2×C3×#157—21no—
24#6308C2×C412yes—
25C3×C4×C3⋊D3C2311yescharacteristic
26C32×D12C2×C412yes—
27#6290—22no—
28D3×C3⋊Dic3—22no—
29C3⋊#622—22no—
30C3×#622D412yes—
31C33×C8C2×C411yescharacteristic
32C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
33#4717—91nomaximal
34#3830—32no—
35#3795—64no—
36#3792—62no—
37C2×#622—62no—
38C4×#157C2×D311yescharacteristic
39#3788—32no—
40#3787C2×D311yescharacteristic
41C32×D8—31no—
42#3754—32no—
43#3746—31no—
44#3746—62no—
45#3746—122no—
46C8×C3⋊D3C2×D311yescharacteristic
47C3×D24—22no—
48C3×D3×C8—62no—
49#3716—64no—
50C2×C3×C3⋊D3#4111yescharacteristic
51C2×C32×D3—22no—
52C3×#157—41no—
53C33×C4C22×C411yescharacteristic
54C3⋊(C3⋊Dic3)—22no—
55C3×C3⋊Dic3C2×D411yescharacteristic
56#912—182no—
57#904—91no—
58#901—92no—
59C3×#97—91no—
60#819—91no—
61#805—92no—
62C2×#157—21no—
63C32×D4—32no—
64#638—62no—
65C2×C3⋊Dic3—62no—
66C4×C3⋊D3C22×D311yescharacteristic
67#634—32no—
68#634—64no—
69#634—124no—
70C3×D12—24no—
71C3×C4×D3—62no—
72#625—124no—
73D3×Dic3—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—21no—
84C32×D3—42no—
85#237—184no—
86#235—92no—
87C3×D8—62no—
88C2×C3×C8—91no—
89C2×C3×C8—91no—
90C3×C42—91no—
91#214—184no—
92#208—92no—
93Dic12—91no—
94Dic12—182no—
95Dic12—362no—
96D24#15711yescharacteristic, complemented
97#204—92no—
98ΓL(1,𝔽25)—182no—
99D3×C8—62no—
100D3×C8—182no—
101C22×C32—62no—
102C2×C3⋊D3#23511yescharacteristic
103C2×C3×D3—44no—
104C2×C3×D3—62no—
105#157—41no—
106C32×C4—22no—
107C32×C4—42no—
108C32×C4C2×C4×D311yescharacteristic
109C3⋊Dic3—62no—
110C3⋊Dic3—124no—
111C3⋊Dic3—244no—
112C3⋊Dic3C2×D1211yescharacteristic
113C3×Dic3—62no—
114#135—271no—
115#131—272no—
116#108—271no—
117#104—272no—
118#97—91no—
119C33#44411yescharacteristic, complemented, Sylow
120C3×D4—64no—
121C2×C3×C4—91no—
122C2×C3×C4—181no—
123#67—182no—
124#67—364no—
125C2×Dic3—364no—
126D12C2×#15712yes—
127C4×D3—62no—
128C4×D3—182no—
129Dic6—92no—
130Dic6—184no—
131Dic6—364no—
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×C32—122no—
142C2×C32C4×D1211yescharacteristic
143C3⋊D3—21no—
144C3×D3—84no—
145C3×D3—122no—
146#41—272no—
147Q16—271no—
148SD16—272no—
149D8—31no—
150M16—272no—
151M16—542no—
152C2×C8—91no—
153C2×C8—91no—
154C2×C8—542no—
155C42—91no—
156C22×C3—91no—
157C22×C3—124no—
158C2×D3—22no—
159C2×D3—62no—
160C3×C4—22no—
161C3×C4—42no—
162C3×C4—91no—
163C3×C4—361no—
164C3×C4C22×#15711yescharacteristic
165Dic3—62no—
166Dic3—182no—
167Dic3—364no—
168Dic3—724no—
169C32—22no—
170C32—42no—
171C32#471711yescharacteristic, complemented
172Q8—272no—
173D4—32no—
174D4—542no—
175C2×C4—91no—
176C2×C4—181no—
177C2×C4—542no—
178C8—92no—
179C8—91no—
180C8—542no—
181C8D3×#15711yescharacteristic
182C2×C3—22no—
183C2×C3—42no—
184C2×C3—181no—
185C2×C3—244no—
186C2×C3D4×#15711yescharacteristic
187D3—42no—
188D3—122no—
189C22—62no—
190C22—91no—
191C4—91no—
192C4—361no—
193C4—542no—
194C4C2×D3×#15711yescharacteristic
195C3—22no—
196C3—42no—
197C3#1059378011yescharacteristic, complemented
198C2—122no—
199C2—181no—
200C2D12×#15711yescharacteristic
subgroupG/H
centerC2D12×#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC33×C8C2×C4
socleC2×C33C4×D4
prime radical#11723339C2
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. #11724250C2semidirect
3. #11723095C2semidirect
4. C3×#63767C2semidirect
5. #11722711C2semidirect
13. #88853C4semidirect
15. C32×D24C4semidirect
16. #88794C4semidirect
22. #63767D3semidirect
96. D24#157semidirect
119. C33#444semidirect
171. C32#4717semidirect
197. C3#10593780semidirect

Maximal subgroups

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

subgroupindex
#117242502
#117233392
#117230952
C3×#637672
#117227112
#105937803
#47179