Finite Group Databasebeta

← #13158938

Subgroups — #13158938

Order 1728.

1012 conjugacy classes of subgroups; only the 240 normal subgroups are shown.

normal subgroups only

This group has 1012 conjugacy classes of subgroups — too many to draw. Only its 240 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
1C3×C42×C3⋊D3C211yesmaximal, characteristic
2C2×#88787C212yesmaximal
3#11722173C211yesmaximal, characteristic, complemented
4C32×#803C211yesmaximal, characteristic, complemented
5C3⋊C8×C3⋊Dic3C212yesmaximal, complemented
6#10592406—31nomaximal
7#10590334—34nomaximal
8C2×C3×C4×C3⋊D3C412yes—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C33×C42C2211yescharacteristic
11C2×C3⋊(C3⋊(C3⋊C8))C2212yes—
12C3×C4×C3⋊Dic3C412yes—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C2×C32×C3⋊C8C2212yes—
15#88787C418yescomplemented
16C2×#3743—32no—
17C42×C3⋊D3D311yescharacteristic
18C3×C42×D3—34no—
19C2×#3711—38no—
20C32×#97—31no—
21C8×C3⋊Dic3—32no—
22#63630—31no—
23#63630—34no—
24#63630—64no—
25C3×#803D314yescomplemented
26Dic3×C3⋊C8—38no—
27C22×C3×C3⋊D3C2×C411yescharacteristic
28C2×C33×C4C2×C412yes—
29C2×C33×C4C2311yescharacteristic
30C2×C3×C3⋊Dic3C2×C411yescharacteristic
31C2×C3×C3⋊Dic3C2×C412yes—
32C3×C4×C3⋊D3—24no—
33C3×C4×C3⋊D3C2×C414yes—
34C3⋊(C3⋊(C3⋊C8))C2×C414yescomplemented
35C32×C3⋊C8C2×C414yescomplemented
36C4×#208—91no—
37#4707—94no—
38C2×C4×C3⋊D3Dic312yes—
39C2×C4×C3⋊D3C2×D311yescharacteristic
40C2×C3×C4×D3—34no—
41C2×C3×C4×D3—38no—
42C2×C32×C8—32no—
43C32×C42—24no—
44C32×C42C2×D314yes—
45C32×C42C2×D311yescharacteristic
46C4×C3⋊Dic3Dic312yes—
47C4×C3⋊Dic3C2×D311yescharacteristic
48C2×C3⋊(C3⋊C8)—32no—
49C2×C3⋊(C3⋊C8)—38no—
50C2×C3⋊(C3⋊C8)—68no—
51#3743—38no—
52C3×C4×Dic3—38no—
53C3×C4×Dic3—34no—
54C2×C3×C3⋊C8C2×D318yes—
55#3711—332no—
56C22×C33C22×C411yescharacteristic
57C2×C3×C3⋊D3—22no—
58C2×C3×C3⋊D3C4212yes—
59C33×C4—22no—
60C33×C4C22×C412yes—
61C3×C3⋊Dic3C4212yes—
62C3×C3⋊Dic3M1614yescomplemented
63C2×C3×C42—91no—
64C2×#208—92no—
65C2×ΓL(1,𝔽25)—98no—
66C42×D3—34no—
67C3×#97—34no—
68C8×Dic3—98no—
69#803—94no—
70#803—91no—
71#803—184no—
72#803C3⋊D311yescharacteristic, complemented
73C4×C3⋊C8—92no—
74C22×C3⋊D3C2×Dic311yescharacteristic
75C22×C3×D3—34no—
76C2×C32×C4—28no—
77C2×C32×C4—24no—
78C2×C32×C4C4×D318yes—
79C2×C32×C4C2×Dic312yes—
80C2×C32×C4C22×D314yes—
81C2×C32×C4C22×D311yescharacteristic
82C2×C3⋊Dic3C2×Dic312yes—
83C2×C3⋊Dic3C2×Dic311yescharacteristic
84C4×C3⋊D3—24no—
85C4×C3⋊D3C4×D314yes—
86C2×C3×Dic3—34no—
87C2×C3×Dic3—38no—
88C3×C4×D3—316no—
89C3×C4×D3—616no—
90C32×C8—34no—
91C3⋊(C3⋊C8)—34no—
92C3⋊(C3⋊C8)—316no—
93C3⋊(C3⋊C8)—616no—
94C3×C3⋊C8C4×D3116yescomplemented
95C4×M16—271noSylow
96C2×C33C2×C4211yescharacteristic
97C2×C33C2×M1612yes—
98C3×C3⋊D3—22no—
99C22×C3×C4—91no—
100C22×C3×C4—92no—
101C2×C4×D3—34no—
102C2×C4×D3—38no—
103C2×C3×C8—38no—
104C3×C42—24no—
105C3×C42—92no—
106C3×C42—91no—
107C3×C42D3214yes—
108C3×C42C2×C3⋊D311yescharacteristic
109C4×Dic3—38no—
110C4×Dic3—34no—
111#208—98no—
112C2×C3⋊C8—98no—
113C2×C3⋊C8—92no—
114C2×C3⋊C8—188no—
115C2×C3⋊C8C2×C3⋊D312yes—
116ΓL(1,𝔽25)—932no—
117C22×C32—24no—
118C22×C32C2×C4×D314yes—
119C22×C32C22×Dic311yescharacteristic
120C2×C3⋊D3—22no—
121C2×C3⋊D3C4×Dic312yes—
122C2×C3×D3—38no—
123C2×C3×D3—68no—
124C32×C4—22no—
125C32×C4—28no—
126C32×C4—28no—
127C32×C4—48no—
128C32×C4C2×C4×D318yes—
129C32×C4C2×C4×D312yes—
130C3⋊Dic3#20814yescomplemented
131C3⋊Dic3C4×Dic312yes—
132C3×Dic3—38no—
133C3×Dic3—316no—
134C2×M16—272no—
135C2×C42—91no—
136#97—31no—
137#97—271no—
138C4×C8—272no—
139C33C4×M1611yescharacteristic, complemented, Sylow
140C23×C3—91no—
141C22×D3—34no—
142C2×C3×C4—28no—
143C2×C3×C4—24no—
144C2×C3×C4—91no—
145C2×C3×C4—92no—
146C2×C3×C4—94no—
147C2×C3×C4—184no—
148C2×C3×C4D3×Dic318yes—
149C2×C3×C4C4×C3⋊D312yes—
150C2×C3×C4C2×D3214yes—
151C2×C3×C4C22×C3⋊D311yescharacteristic
152C2×Dic3—38no—
153C2×Dic3—34no—
154C4×D3—316no—
155C4×D3—616no—
156C3×C8—316no—
157C3⋊C8—916no—
158C3⋊C8—94no—
159C3⋊C8—1816no—
160C3⋊C8C4×C3⋊D314yescomplemented
161C2×C32—28no—
162C2×C32—24no—
163C2×C32C42×D314yes—
164C2×C32D3×M1618yes—
165C2×C32C2×#20812yes—
166C2×C32C2×C4×Dic311yescharacteristic
167C3⋊D3—22no—
168C3×D3—68no—
169C22×C4—91no—
170C22×C4—92no—
171M16—278no—
172C2×C8—32no—
173C2×C8—272no—
174C42—92no—
175C42—91no—
176C42D3×C3⋊D311yescharacteristic
177C22×C3—24no—
178C22×C3—92no—
179C22×C3—182no—
180C22×C3C2×D3×Dic314yes—
181C22×C3C2×C4×C3⋊D311yescharacteristic
182C2×D3—38no—
183C2×D3—68no—
184C3×C4—28no—
185C3×C4—22no—
186C3×C4—28no—
187C3×C4—48no—
188C3×C4—92no—
189C3×C4—94no—
190C3×C4C4×D3218yes—
191C3×C4C2×C4×C3⋊D312yes—
192Dic3—316no—
193Dic3—38no—
194C32—24no—
195C32#470714yescomplemented
196C32C4×#20811yescharacteristic, complemented
197C23—91no—
198C2×C4—92no—
199C2×C4—91no—
200C2×C4—94no—
201C2×C4—184no—
202C2×C4Dic3×C3⋊D312yes—
203C2×C4C2×D3×C3⋊D311yescharacteristic
204C8—34no—
205C8—274no—
206C2×C3—28no—
207C2×C3—24no—
208C2×C3—182no—
209C2×C3D3×#20818yes—
210C2×C3C4×D3×Dic314yes—
211C2×C3C42×C3⋊D311yescharacteristic
212C2×C3M16×C3⋊D312yes—
213D3—68no—
214C22—92no—
215C22—182no—
216C22C2×Dic3×C3⋊D311yescharacteristic
217C4—22no—
218C4—94no—
219C4—92no—
220C4C4×D3×C3⋊D312yes—
221C3—24no—
222C3#1059033414yescomplemented
223C3#1059240611yescharacteristic, complemented
224C2—182no—
225C2C3⋊D3×#20812yes—
226C2C4×Dic3×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutatorC2×C33C2×C42
FittingC33×C42C22
socleC22×C33C22×C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C33×C4C23
hypercenterC42D3×C3⋊D3
hypocenterC33C4×M16
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
3. #11722173C2semidirect
4. C32×#803C2semidirect
5. C3⋊C8×C3⋊Dic3C2semidirect
15. #88787C4semidirect
25. C3×#803D3semidirect
34. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
35. C32×C3⋊C8C2×C4semidirect
62. C3×C3⋊Dic3M16semidirect
72. #803C3⋊D3semidirect
94. C3×C3⋊C8C4×D3semidirect
130. C3⋊Dic3#208semidirect
139. C33C4×M16semidirect
160. C3⋊C8C4×C3⋊D3semidirect
195. C32#4707semidirect
196. C32C4×#208semidirect
222. C3#10590334semidirect
223. C3#10592406semidirect

Maximal subgroups

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

subgroupindex
C3×C42×C3⋊D32
C2×#887872
#117221732
C32×#8032
C3⋊C8×C3⋊Dic32
#105924063
#105903343