Finite Group Databasebeta

← #11920556

Subgroups — #11920556

Order 1152.

271 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 271 conjugacy classes of subgroups — too many to draw. Only its 37 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
2C3×#5386C211yesmaximal, characteristic, complemented
3C3⋊C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#73021—31nomaximal
5#71116—31nomaximal
6C2×C3⋊GL(2,𝔽3)—21no—
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8C2×#3717—41nomaximal
9C3×#859—21no—
10C4⋊GL(2,𝔽3)—31no—
11C4⋊GL(2,𝔽3)—61no—
12#5386D311yescharacteristic, complemented
13C3×#475—31no—
14#5085—31no—
15#5027—31no—
16Q8×C3⋊C8D311yescharacteristic, complemented
17C8×SL(2,𝔽3)—31no—
18#4547—31no—
19#4485—31no—
20#4480—31no—
21C2×C4⋊(C3⋊D3)—41no—
22C2×C3×Dic6—41no—
23C2×C3×SL(2,𝔽3)D411yescharacteristic
24C3⋊GL(2,𝔽3)—41no—
25C2×C3×C3⋊C8—41no—
26#3717—44no—
27#1666—91noSylow
28C2×GL(2,𝔽3)—62no—
29C2×GL(2,𝔽3)—61no—
30C3×C4⋊Q8—31no—
31C3×C4×Q8C2×D311yescharacteristic
32C2×#215—61no—
33C2×#215—61no—
34C2×#215—61no—
35C2×#204—121no—
36#874—31no—
37C4×SL(2,𝔽3)—21no—
38C4×SL(2,𝔽3)C2×D311yescharacteristic
39#859—21no—
40C3×#106—61no—
41C3×C4⋊C8—31no—
42C3×#103—61no—
43#809—61no—
44C4⋊(C3⋊C8)—31no—
45C4⋊(C3⋊C8)—31no—
46C4×C3⋊C8—31no—
47C22×C3⋊D3—81no—
48C2×C32×C4—41no—
49C4⋊(C3⋊D3)—42no—
50C4⋊(C3⋊D3)—81no—
51C2×C3×Dic3—81no—
52C3×Dic6—42no—
53C3×Dic6—81no—
54C3×SL(2,𝔽3)SD1611yescharacteristic, complemented
55C3×C3⋊C8—42no—
56#492—91no—
57#475—31no—
58#460—91no—
59C8×Q8—31no—
60#334—91no—
61#332—91no—
62#331—91no—
63C2×C3×Q8—61no—
64C2×C3×Q8D1211yescharacteristic
65C2×D12—61no—
66C2×D12—61no—
67C2×D12—61no—
68C2×D12—121no—
69C2×D12—241no—
70C2×Dic6—41no—
71C2×SL(2,𝔽3)—21no—
72C2×SL(2,𝔽3)#6711yescharacteristic
73GL(2,𝔽3)—122no—
74GL(2,𝔽3)—121no—
75C2×C3×C8—61no—
76C2×C3×C8—121no—
77C3×C4⋊C4—31no—
78C3×C4⋊C4—61no—
79C3×C4⋊C4—61no—
80C3×C42—31no—
81#215—122no—
82#215—121no—
83#215—121no—
84#215—122no—
85C2×C3⋊C8—31no—
86C2×C3⋊C8—61no—
87C2×C3⋊C8S411yescharacteristic
88#204—124no—
89C22×C32—41no—
90C2×C3⋊D3—82no—
91C2×C3⋊D3—161no—
92C2×C3⋊D3—161no—
93C32×C4—41no—
94C32×C4—41no—
95C3×Dic3—82no—
96C2×SD16—181no—
97C2×SD16—181no—
98C2×SD16—181no—
99C4⋊Q8—31no—
100#127—91no—
101C4×Q8D3211yescharacteristic
102#106—61no—
103C4⋊C8—31no—
104C4⋊C8—91no—
105C4⋊C8—91no—
106#103—61no—
107#102—181no—
108C4×C8—91no—
109C22×D3—121no—
110C22×D3—241no—
111C22×D3—242no—
112C3×Q8—31no—
113C3×Q8—121no—
114C3×Q8—121no—
115C3×Q8#20411yescharacteristic, complemented
116C2×C3×C4—31no—
117C2×C3×C4—31no—
118C2×C3×C4—41no—
119C2×C3×C4—81no—
120C2×C3×C4—121no—
121C2×C3×C4C2×S411yescharacteristic
122C2×Dic3—81no—
123D12—121no—
124D12—121no—
125D12—122no—
126D12—121no—
127D12—121no—
128D12—241no—
129D12—242no—
130D12—242no—
131D12—241no—
132Dic6—42no—
133Dic6—81no—
134SL(2,𝔽3)—21no—
135SL(2,𝔽3)#21511yescharacteristic, complemented
136C3×C8—122no—
137C3×C8—121no—
138C3⋊C8—61no—
139C3⋊C8—121no—
140C3⋊C8GL(2,𝔽3)12yescomplemented
141C2×C32—41no—
142C2×C32—41no—
143C2×C32—41no—
144C3⋊D3—162no—
145C2×Q8—61no—
146C2×Q8#62611yescharacteristic
147C2×D4—181no—
148C2×D4—181no—
149C2×D4—181no—
150SD16—362no—
151SD16—362no—
152SD16—361no—
153SD16—361no—
154C2×C8—31no—
155C2×C8—61no—
156C2×C8—91no—
157C2×C8—181no—
158C4⋊C4—31no—
159C4⋊C4—61no—
160C4⋊C4—61no—
161C42—31no—
162C22×C3—41no—
163C22×C3—81no—
164C22×C3C4⋊S411yescharacteristic
165C2×D3—242no—
166C2×D3—244no—
167C2×D3—241no—
168C2×D3—241no—
169C2×D3—241no—
170C2×D3—481no—
171C2×D3—481no—
172C2×D3—482no—
173C2×D3—482no—
174C3×C4—31no—
175C3×C4—31no—
176C3×C4—41no—
177C3×C4—41no—
178C3×C4—61no—
179C3×C4—81no—
180C3×C4—81no—
181C3×C4—241no—
182C3×C4C2×GL(2,𝔽3)11yescharacteristic
183C3×C4#98611yescharacteristic
184Dic3—82no—
185C32—41noSylow
186C23—361no—
187Q8—31no—
188Q8—121no—
189Q8—121no—
190Q8#371711yescharacteristic, complemented
191D4—361no—
192D4—361no—
193D4—361no—
194D4—361no—
195D4—721no—
196C2×C4—31no—
197C2×C4—31no—
198C2×C4—121no—
199C2×C4D3×S411yescharacteristic
200C8—32no—
201C8—121no—
202C8—181no—
203C8—361no—
204C2×C3—41no—
205C2×C3—41no—
206C2×C3—41no—
207C2×C3—81no—
208C2×C3—81no—
209C2×C3—81no—
210C2×C3C4⋊GL(2,𝔽3)11yescharacteristic
211C2×C3#540111yescharacteristic
212C2×C3#540611yescharacteristic
213D3—482no—
214D3—484no—
215D3—481no—
216C22—721no—
217C22—721no—
218C22—721no—
219C22#6420711yescharacteristic
220C4—31no—
221C4—31no—
222C4—61no—
223C4—241no—
224C4#6419911yescharacteristic
225C4D3×GL(2,𝔽3)11yescharacteristic
226C3—41no—
227C3—81no—
228C3#7302111yescharacteristic, complemented
229C2—1441no—
230C2#1059317811yescharacteristic
231C2#1059321511yescharacteristic
232C2#1059322411yescharacteristic
subgroupG/H
centerC22#64207
commutator, cosocleC3×C4×SL(2,𝔽3)C22
Frattini, hypercenterC2×C4D3×S4
FittingC3×C4×Q8C2×D3
socleC22×C3C4⋊S4
prime radicalC4⋊(C3⋊GL(2,𝔽3))C2
hypocenterC3×SL(2,𝔽3)SD16
LCSG ▷ C3×C4×SL(2,𝔽3) ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3×#5386C2semidirect
3. C3⋊C8×SL(2,𝔽3)C2semidirect
12. #5386D3semidirect
16. Q8×C3⋊C8D3semidirect
54. C3×SL(2,𝔽3)SD16semidirect
115. C3×Q8#204semidirect
135. SL(2,𝔽3)#215semidirect
140. C3⋊C8GL(2,𝔽3)semidirect
190. Q8#3717semidirect
228. 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
C3×#53862
C3⋊C8×SL(2,𝔽3)2
#730213
#711163
C2×#37174