Finite Group Databasebeta

← #11921077

Subgroups — #11921077

Order 1152.

372 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C2×C3×#131⋊C3C211yesmaximal, characteristic
2#10593658C211yesmaximal, characteristic
3C2×C3⋊GU(2,𝔽3)C211yesmaximal, characteristic
4#73121—33nomaximal
5#72008—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7C3×#131⋊C3C412yescomplemented
8C3⋊GU(2,𝔽3)C412yescomplemented
9C3⋊#859C412yescomplemented
10C8×C3⋊Dic3—41nomaximal
11C2×C3×#131D311yescharacteristic
12C2×#131⋊C3D313yes—
13#5424—33no—
14C2×GU(2,𝔽3)—33no—
15C2×#837—31no—
16#5233—31no—
17#5101—31no—
18C2×C8×Dic3—31no—
19#4558—31no—
20#4555—31no—
21C3×#41⋊C3C2×C411yescharacteristic
22C3×#41⋊C3C2×C411yescharacteristic
23C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
24C2×C32×C8—41no—
25C4×C3⋊Dic3—41no—
26C2×C3⋊(C3⋊C8)—41no—
27#1764—91noSylow
28C2×C3×#41C2×D311yescharacteristic
29C2×#41⋊C3C2×D313yes—
30C3×#131—32no—
31C3×#131Dic312yescomplemented
32C2×C3×M16—31no—
33C22×C3×C8—31no—
34#926—31no—
35C2×C4×Dic3—31no—
36C2×#208—31no—
37C22×C3⋊C8—31no—
38#131⋊C3Dic316yescomplemented
39GU(2,𝔽3)—36no—
40#859—36no—
41#837—32no—
42#837—32no—
43#835—32no—
44#832—32no—
45#815—61no—
46C8×Dic3—62no—
47C8×Dic3—61no—
48C8×Dic3—123no—
49C2×C32×C4—41no—
50C2×C3⋊Dic3—42no—
51C3×SL(2,𝔽3)C4211yescharacteristic
52C32×C8—42no—
53C3⋊(C3⋊C8)—42no—
54C2×#131C3⋊D311yescharacteristic
55C2×#104—91no—
56#416—91no—
57#405—91no—
58C2×C4×C8—91no—
59#341—91no—
60#339—91no—
61C3×#41—31no—
62C3×#41—31no—
63C3×#41C4×D311yescharacteristic
64C3×#41C2×Dic311yescharacteristic
65C2×C3×Q8C4×D311yescharacteristic
66C2×C3×D4—31no—
67C22×C3×C4—31no—
68C22×Dic3—61no—
69#41⋊C3C4×D313yes—
70#41⋊C3C2×Dic313yes—
71C2×SL(2,𝔽3)C4×D313yes—
72C3×M16—32no—
73C3×M16—61no—
74C2×C3×C8—31no—
75C2×C3×C8—32no—
76C2×C3×C8—32no—
77C2×C3×C8—43no—
78C2×C3×C8S411yescharacteristic
79#217—61no—
80C4⋊Dic3—32no—
81C4×Dic3—32no—
82C4×Dic3—61no—
83C4×Dic3—61no—
84C4×Dic3—123no—
85#208—32no—
86#208—61no—
87C2×C3⋊C8—32no—
88C2×C3⋊C8—61no—
89C2×C3⋊C8—61no—
90C2×C3⋊C8—61no—
91C2×C3⋊C8—123no—
92C22×C32—41no—
93C32×C4—41no—
94C32×C4—41no—
95C3⋊Dic3—44no—
96C2×#41C2×C3⋊D311yescharacteristic
97#131—32no—
98#131C3⋊Dic312yescomplemented
99C2×M16—31no—
100C2×M16—91no—
101C22×C8—31no—
102C22×C8—91no—
103#117—91no—
104C2×C42—91no—
105#104—92no—
106#104—92no—
107#103—92no—
108#102—92no—
109#97—181no—
110C4×C8—182no—
111C4×C8—181no—
112C23×C3—31no—
113C3×Q8—31no—
114C3×Q8C4×Dic311yescharacteristic
115C3×D4—31no—
116C3×D4—31no—
117C3×D4—61no—
118C2×C3×C4—31no—
119C2×C3×C4—31no—
120C2×C3×C4—31no—
121C2×C3×C4—31no—
122C2×C3×C4—31no—
123C2×C3×C4—43no—
124C2×C3×C4C2×S411yescharacteristic
125C2×Dic3—62no—
126C2×Dic3—64no—
127C2×Dic3—62no—
128C2×Dic3—126no—
129SL(2,𝔽3)C4×Dic313yes—
130C3×C8—32no—
131C3×C8—46no—
132C3×C8SL(2,ℤ/4)12yes—
133C3⋊C8—62no—
134C3⋊C8—62no—
135C3⋊C8—126no—
136C2×C32—42no—
137C2×C32—41no—
138#41—31no—
139#41—31no—
140#41C4×C3⋊D311yescharacteristic
141#41C2×C3⋊Dic311yescharacteristic
142C2×Q8C4×C3⋊D311yescharacteristic
143C2×D4—31no—
144C22×C4—31no—
145C22×C4—181no—
146M16—32no—
147M16—61no—
148M16—92no—
149M16—181no—
150C2×C8—31no—
151C2×C8—32no—
152C2×C8—32no—
153C2×C8—92no—
154C2×C8—181no—
155C2×C8—181no—
156C2×C8—181no—
157C2×C8C3⋊S411yescharacteristic
158C4⋊C4—92no—
159C22⋊C4—181no—
160C42—92no—
161C42—181no—
162C42—181no—
163C22×C3—31no—
164C22×C3—31no—
165C22×C3—43no—
166C22×C3—62no—
167C22×C3C4×S411yescharacteristic
168C3×C4—31no—
169C3×C4—31no—
170C3×C4—43no—
171C3×C4—43no—
172C3×C4C4×S411yescharacteristic
173C3×C4C2×SL(2,ℤ/4)11yescharacteristic
174Dic3—64no—
175Dic3—1212no—
176Dic3—122no—
177C32—41noSylow
178C23—31no—
179Q8—31no—
180Q8C4×C3⋊Dic311yescharacteristic
181D4—31no—
182D4—31no—
183D4—61no—
184C2×C4—31no—
185C2×C4—31no—
186C2×C4—31no—
187C2×C4—31no—
188C2×C4—31no—
189C2×C4—182no—
190C2×C4—184no—
191C2×C4—182no—
192C2×C4C2×C3⋊S411yescharacteristic
193C8—32no—
194C8—182no—
195C8—182no—
196C8C3⋊SL(2,ℤ/4)12yes—
197C2×C3—46no—
198C2×C3—43no—
199C2×C3—61no—
200C2×C3—61no—
201C2×C3#540412yes—
202C2×C3C4×SL(2,ℤ/4)11yescharacteristic
203C22—31no—
204C22—31no—
205C22—62no—
206C22C4×C3⋊S411yescharacteristic
207C4—31no—
208C4—31no—
209C4—184no—
210C4—362no—
211C4C4×C3⋊S411yescharacteristic
212C4C2×C3⋊SL(2,ℤ/4)11yescharacteristic
213C3—43no—
214C3#7312111yescharacteristic, complemented
215C2—61no—
216C2—61no—
217C2#1059363812yes—
218C2C4×C3⋊SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, hypercenterC2×C8C3⋊S4
commutator, hypocenterC3×SL(2,𝔽3)C42
FrattiniC2×C4C2×C3⋊S4
FittingC2×C3×#131D3
socleC22×C3C4×S4
prime radical, cosocleC2×C3×#41⋊C3C22
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
7. C3×#131⋊C3C4semidirect
8. C3⋊GU(2,𝔽3)C4semidirect
9. C3⋊#859C4semidirect
31. C3×#131Dic3semidirect
38. #131⋊C3Dic3semidirect
98. #131C3⋊Dic3semidirect
214. C3#73121semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#131⋊C32
#105936582
C2×C3⋊GU(2,𝔽3)2
#731213
#720083
C8×C3⋊Dic34