Finite Group Databasebeta

← #11920977

Subgroups — #11920977

Order 1152.

324 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 49 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
1C4⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic
2#10593620C211yesmaximal, characteristic, complemented
3C3×C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#73021—33nomaximal
5#70938—31nomaximal
6C2×C3⋊GL(2,𝔽3)—21no—
7C2×#3744—41nomaximal
8C3×C4×SL(2,𝔽3)C2211yescharacteristic
9C3⋊#859—21no—
10C4⋊GL(2,𝔽3)—33no—
11#5386—33no—
12C3×C8×Q8D311yescharacteristic, complemented
13#5027—31no—
14#5020—31no—
15#4693—31no—
16C8×SL(2,𝔽3)D313yescomplemented
17#4485—31no—
18#4482—31no—
19#4457—31no—
20C2×C4⋊(C3⋊D3)—41no—
21C2×#634—41no—
22C2×C3×SL(2,𝔽3)D411yescharacteristic
23C3⋊GL(2,𝔽3)—41no—
24C2×C32×C8—41no—
25#3744—44no—
26#1666—91noSylow
27C2×GL(2,𝔽3)—63no—
28C3×C4×Q8C2×D311yescharacteristic
29C2×#215—61no—
30C2×#204—61no—
31C2×#204—61no—
32C2×#204—123no—
33#874—31no—
34#869—31no—
35C4×SL(2,𝔽3)C2×D313yes—
36#859—63no—
37C3×C4⋊C8—31no—
38C3×C4×C8—31no—
39#835—61no—
40#809—61no—
41#808—61no—
42C4⋊(C3⋊C8)—31no—
43C4⋊(C3⋊C8)—31no—
44C22×C3⋊D3—81no—
45C2×C32×C4—41no—
46C2×C3⋊Dic3—81no—
47C4⋊(C3⋊D3)—42no—
48C4⋊(C3⋊D3)—81no—
49#634—42no—
50#634—81no—
51C3×SL(2,𝔽3)SD1611yescharacteristic, complemented
52C32×C8—42no—
53#492—91no—
54#475—91no—
55#460—91no—
56C8×Q8C3⋊D311yescharacteristic, complemented
57#334—91no—
58#332—91no—
59#331—91no—
60C2×C3×Q8D1211yescharacteristic
61C2×D12—61no—
62C2×D12—61no—
63C2×D12—61no—
64C2×D12—123no—
65C2×Dic6—61no—
66C2×Dic6—123no—
67C2×SL(2,𝔽3)D1213yes—
68GL(2,𝔽3)—123no—
69C2×C3×C8—31no—
70C2×C3×C8—43no—
71C2×C3×C8S411yescharacteristic
72C3×C4⋊C4—31no—
73C3×C42—31no—
74#215—121no—
75#215—121no—
76C4⋊Dic3—61no—
77C4⋊Dic3—61no—
78C2×C3⋊C8—61no—
79C2×C3⋊C8—61no—
80#204—1212no—
81#204—122no—
82#204—122no—
83C22×C32—41no—
84C2×C3⋊D3—82no—
85C2×C3⋊D3—161no—
86C2×C3⋊D3—161no—
87C32×C4—41no—
88C32×C4—41no—
89C3⋊Dic3—82no—
90C2×SD16—181no—
91C2×SD16—181no—
92C2×SD16—181no—
93C4⋊Q8—91no—
94#127—91no—
95C4×Q8C2×C3⋊D311yescharacteristic
96#106—181no—
97C4⋊C8—31no—
98C4⋊C8—91no—
99C4⋊C8—91no—
100#103—181no—
101#102—181no—
102C4×C8—31no—
103C22×D3—121no—
104C22×D3—243no—
105C3×Q8—31no—
106C3×Q8#20411yescharacteristic, complemented
107C2×C3×C4—31no—
108C2×C3×C4—31no—
109C2×C3×C4—43no—
110C2×C3×C4C2×S411yescharacteristic
111C2×Dic3—121no—
112C2×Dic3—243no—
113D12—121no—
114D12—121no—
115D12—126no—
116D12—121no—
117D12—121no—
118D12—243no—
119D12—241no—
120Dic6—126no—
121Dic6—121no—
122Dic6—121no—
123Dic6—243no—
124SL(2,𝔽3)#20413yescomplemented
125C3×C8—46no—
126C3×C8—61no—
127C3×C8GL(2,𝔽3)12yescomplemented
128C3⋊C8—121no—
129C3⋊C8—121no—
130C2×C32—41no—
131C2×C32—41no—
132C2×C32—41no—
133C3⋊D3—162no—
134C2×Q8—181no—
135C2×Q8C4⋊(C3⋊D3)11yescharacteristic
136C2×D4—181no—
137C2×D4—181no—
138C2×D4—181no—
139SD16—362no—
140SD16—361no—
141SD16—361no—
142SD16—362no—
143C2×C8—31no—
144C2×C8—181no—
145C2×C8—181no—
146C2×C8C3⋊S411yescharacteristic
147C4⋊C4—31no—
148C4⋊C4—181no—
149C4⋊C4—181no—
150C42—31no—
151C22×C3—43no—
152C22×C3C4⋊S411yescharacteristic
153C2×D3—246no—
154C2×D3—241no—
155C2×D3—241no—
156C2×D3—241no—
157C2×D3—483no—
158C2×D3—483no—
159C3×C4—31no—
160C3×C4—31no—
161C3×C4—43no—
162C3×C4—43no—
163C3×C4—61no—
164C3×C4C2×GL(2,𝔽3)11yescharacteristic
165C3×C4#98611yescharacteristic
166Dic3—246no—
167Dic3—241no—
168C32—41noSylow
169C23—361no—
170Q8—31no—
171Q8—361no—
172Q8—361no—
173Q8#374411yescharacteristic, complemented
174D4—361no—
175D4—361no—
176D4—361no—
177D4—361no—
178D4—721no—
179C2×C4—31no—
180C2×C4—31no—
181C2×C4—361no—
182C2×C4C2×C3⋊S411yescharacteristic
183C8—61no—
184C8—361no—
185C8—361no—
186C8C3⋊GL(2,𝔽3)12yescomplemented
187C2×C3—43no—
188C2×C3—43no—
189C2×C3—43no—
190C2×C3C4⋊GL(2,𝔽3)11yescharacteristic
191C2×C3#540111yescharacteristic
192C2×C3#540611yescharacteristic
193D3—486no—
194D3—481no—
195C22—721no—
196C22—721no—
197C22—721no—
198C22C4⋊(C3⋊S4)11yescharacteristic
199C4—31no—
200C4—31no—
201C4—61no—
202C4—721no—
203C4C2×C3⋊GL(2,𝔽3)11yescharacteristic
204C4#6426711yescharacteristic
205C3—43no—
206C3#7302111yescharacteristic, complemented
207C2—1441no—
208C2C4⋊(C3⋊GL(2,𝔽3))11yescharacteristic
209C2#1059363511yescharacteristic
210C2#1059364011yescharacteristic
subgroupG/H
centerC22C4⋊(C3⋊S4)
commutator, cosocleC3×C4×SL(2,𝔽3)C22
FrattiniC2×C4C2×C3⋊S4
FittingC3×C8×Q8D3
socleC22×C3C4⋊S4
prime radicalC4⋊(C3⋊GL(2,𝔽3))C2
hypercenterC2×C8C3⋊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 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #10593620C2semidirect
3. C3×C8×SL(2,𝔽3)C2semidirect
12. C3×C8×Q8D3semidirect
16. C8×SL(2,𝔽3)D3semidirect
51. C3×SL(2,𝔽3)SD16semidirect
56. C8×Q8C3⋊D3semidirect
106. C3×Q8#204semidirect
124. SL(2,𝔽3)#204semidirect
127. C3×C8GL(2,𝔽3)semidirect
173. Q8#3744semidirect
186. C8C3⋊GL(2,𝔽3)semidirect
206. C3#73021semidirect

Maximal subgroups

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

subgroupindex
C4⋊(C3⋊GL(2,𝔽3))2
#105936202
C3×C8×SL(2,𝔽3)2
#730213
#709383
C2×#37444