Finite Group Databasebeta

← #11921052

Subgroups — #11921052

Order 1152.

296 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 45 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×#557⋊C3C211yesmaximal, characteristic, complemented
2C4⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic
3C3⋊(SL(2,𝔽3)⋊C8)C211yesmaximal, characteristic, complemented
4#73096—33nomaximal
5#71935—31nomaximal
6C2×C3⋊GL(2,𝔽3)—21no—
7C3×C4×SL(2,𝔽3)—21no—
8C3×C4×SL(2,𝔽3)C2211yescharacteristic
9#63636—41nomaximal
10C3×#557D311yescharacteristic, complemented
11#557⋊C3D313yescomplemented
12C4⋊GL(2,𝔽3)—33no—
13#5088—31no—
14#5027—31no—
15#5027—31no—
16SL(2,𝔽3)⋊C8—33no—
17#4547—31no—
18#4535—31no—
19#4535—31no—
20C2×C4⋊(C3⋊D3)—41no—
21C2×C3×SL(2,𝔽3)D411yescharacteristic
22C3⋊GL(2,𝔽3)—41no—
23C32×C4⋊C4—41no—
24C2×C3⋊(C3⋊C8)—41no—
25#1626—91noSylow
26C2×GL(2,𝔽3)—63no—
27C3×C4⋊Q8—31no—
28C3×#125—61no—
29C3×C4×Q8—21no—
30C3×C4×Q8—31no—
31C3×C4×Q8C2×D311yescharacteristic
32C2×#215—61no—
33C2×#215—61no—
34#874—31no—
35C4×SL(2,𝔽3)—23no—
36C4×SL(2,𝔽3)C2×D313yes—
37#809—61no—
38#809—62no—
39#809—61no—
40#809—123no—
41C4⋊(C3⋊C8)—31no—
42C4⋊(C3⋊C8)—31no—
43C4×C3⋊C8—31no—
44C22×C3⋊D3—81no—
45C2×C32×C4—41no—
46C2×C32×C4—81no—
47C4⋊(C3⋊D3)—42no—
48C4⋊(C3⋊D3)—81no—
49C3×SL(2,𝔽3)SD1611yescharacteristic, complemented
50C3⋊(C3⋊C8)—81no—
51#557C3⋊D311yescharacteristic, complemented
52#486—91no—
53#460—91no—
54#460—91no—
55#332—91no—
56Q8⋊C8—91no—
57Q8⋊C8—91no—
58C2×C3×Q8—31no—
59C2×C3×Q8#6711yescharacteristic
60C2×D12—61no—
61C2×D12—61no—
62C2×D12—61no—
63C2×D12—123no—
64C2×SL(2,𝔽3)#6713yes—
65GL(2,𝔽3)—123no—
66C3×C4⋊C4—31no—
67C3×C4⋊C4—32no—
68C3×C4⋊C4—43no—
69C3×C4⋊C4—61no—
70C3×C4⋊C4—61no—
71C3×C4⋊C4S411yescharacteristic
72C3×C42—31no—
73C3×C42—61no—
74#215—121no—
75#215—122no—
76#215—121no—
77C2×C3⋊C8—61no—
78C2×C3⋊C8—61no—
79C2×C3⋊C8—61no—
80C2×C3⋊C8—123no—
81C22×C32—41no—
82C2×C3⋊D3—82no—
83C2×C3⋊D3—161no—
84C2×C3⋊D3—161no—
85C32×C4—41no—
86C32×C4—41no—
87C32×C4—161no—
88C2×SD16—181no—
89C2×SD16—181no—
90C4⋊Q8—31no—
91#127—91no—
92#125—61no—
93C4×Q8—21no—
94C4×Q8—31no—
95C4×Q8C2×C3⋊D311yescharacteristic
96C4⋊C8—91no—
97C4⋊C8—91no—
98#102—181no—
99#102—182no—
100#102—181no—
101C4×C8—91no—
102C22×D3—121no—
103C22×D3—243no—
104C3×Q8—31no—
105C3×Q8—61no—
106C3×Q8—61no—
107C3×Q8#21511yescharacteristic
108C2×C3×C4—21no—
109C2×C3×C4—31no—
110C2×C3×C4—31no—
111C2×C3×C4—43no—
112C2×C3×C4—61no—
113C2×C3×C4—83no—
114C2×C3×C4C2×S411yescharacteristic
115D12—121no—
116D12—121no—
117D12—126no—
118D12—122no—
119D12—121no—
120D12—121no—
121D12—243no—
122SL(2,𝔽3)#21513yescomplemented
123C3⋊C8—121no—
124C3⋊C8—121no—
125C3⋊C8—121no—
126C3⋊C8—243no—
127C2×C32—41no—
128C2×C32—41no—
129C2×C32—41no—
130C3⋊D3—162no—
131C2×Q8—31no—
132C2×Q8#63811yescharacteristic
133C2×D4—181no—
134C2×D4—181no—
135C2×D4—181no—
136SD16—361no—
137SD16—362no—
138SD16—361no—
139C2×C8—181no—
140C2×C8—181no—
141C2×C8—181no—
142C4⋊C4—31no—
143C4⋊C4—32no—
144C4⋊C4—61no—
145C4⋊C4—61no—
146C4⋊C4C3⋊S411yescharacteristic
147C42—31no—
148C42—61no—
149C22×C3—43no—
150C22×C3GL(2,ℤ/4)11yescharacteristic
151C2×D3—246no—
152C2×D3—241no—
153C2×D3—241no—
154C2×D3—241no—
155C2×D3—483no—
156C2×D3—483no—
157C3×C4—31no—
158C3×C4—31no—
159C3×C4—41no—
160C3×C4—43no—
161C3×C4—43no—
162C3×C4—61no—
163C3×C4—121no—
164C3×C4—163no—
165C3×C4#98511yescharacteristic
166C3×C4#98611yescharacteristic
167C32—41noSylow
168C23—361no—
169Q8—31no—
170Q8—61no—
171Q8—61no—
172Q8#375511yescharacteristic
173D4—361no—
174D4—361no—
175D4—362no—
176D4—361no—
177D4—361no—
178C2×C4—21no—
179C2×C4—31no—
180C2×C4—31no—
181C2×C4—61no—
182C2×C4C2×C3⋊S411yescharacteristic
183C8—361no—
184C8—361no—
185C8—361no—
186C2×C3—43no—
187C2×C3—43no—
188C2×C3—43no—
189C2×C3#541711yescharacteristic
190C2×C3#542711yescharacteristic
191C2×C3#543111yescharacteristic
192D3—486no—
193D3—481no—
194C22—721no—
195C22—721no—
196C22—721no—
197C22#6426911yescharacteristic
198C4—31no—
199C4—31no—
200C4—41no—
201C4—61no—
202C4—121no—
203C4#6426611yescharacteristic
204C4#6426711yescharacteristic
205C3—43no—
206C3#7309611yescharacteristic, complemented
207C2—1441no—
208C2#1059365111yescharacteristic
209C2#1059366111yescharacteristic
210C2#1059366511yescharacteristic
subgroupG/H
centerC22#64269
commutator, cosocleC3×C4×SL(2,𝔽3)C22
FrattiniC2×C4C2×C3⋊S4
FittingC3×#557D3
socleC22×C3GL(2,ℤ/4)
prime radicalC4⋊(C3⋊GL(2,𝔽3))C2
hypercenterC4⋊C4C3⋊S4
hypocenterC3×SL(2,𝔽3)SD16
LCSG ▷ C3×C4×SL(2,𝔽3) ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#557⋊C3C2semidirect
3. C3⋊(SL(2,𝔽3)⋊C8)C2semidirect
10. C3×#557D3semidirect
11. #557⋊C3D3semidirect
49. C3×SL(2,𝔽3)SD16semidirect
51. #557C3⋊D3semidirect
122. SL(2,𝔽3)#215semidirect
206. C3#73096semidirect

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
C4⋊(C3⋊GL(2,𝔽3))2
C3⋊(SL(2,𝔽3)⋊C8)2
#730963
#719353
#636364