Finite Group Databasebeta

← #11921048

Subgroups — #11921048

Order 1152.

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

normal subgroups only

This group has 332 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×C4⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
2C4⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
3C3⋊(SL(2,𝔽3)⋊C8)C211yesmaximal, characteristic, complemented
4#73092—33nomaximal
5#71918—31nomaximal
6C2×C3×#41⋊C3—21no—
7C2×C3⋊GL(2,𝔽3)—21no—
8C3×C4×SL(2,𝔽3)C2211yescharacteristic
9#63636—41nomaximal
10C3×#550D311yescharacteristic, complemented
11C4⋊(#41⋊C3)D313yescomplemented
12C4⋊GL(2,𝔽3)—33no—
13#5073—31no—
14#5027—31no—
15#5015—31no—
16SL(2,𝔽3)⋊C8—33no—
17#4544—31no—
18#4535—31no—
19D4⋊(C3⋊C8)—31no—
20C2×C4⋊(C3⋊D3)—41no—
21C3×#41⋊C3—41no—
22C2×C3×SL(2,𝔽3)D411yescharacteristic
23C3⋊GL(2,𝔽3)—41no—
24C32×C4⋊C4—41no—
25C2×C3⋊(C3⋊C8)—41no—
26#1621—91noSylow
27C2×C3×#41—21no—
28C2×#41⋊C3—23no—
29C2×GL(2,𝔽3)—63no—
30C3×#127—31no—
31C3×#121—61no—
32C3×C4×Q8C2×D311yescharacteristic
33C3×C4×D4—31no—
34C2×#215—61no—
35C2×#213—61no—
36C2×#213—62no—
37#874—31no—
38C4×SL(2,𝔽3)C2×D313yes—
39#809—61no—
40#809—61no—
41#809—123no—
42C4⋊(C3⋊C8)—31no—
43C4⋊(C3⋊C8)—31no—
44C4×C3⋊C8—31no—
45C22×C3⋊D3—81no—
46C2×C32×C4—41no—
47C2×C32×C4—81no—
48C4⋊(C3⋊D3)—42no—
49C4⋊(C3⋊D3)—81no—
50C3×SL(2,𝔽3)D811yescharacteristic, complemented
51C3⋊(C3⋊C8)—81no—
52#550C3⋊D311yescharacteristic, complemented
53#493—91no—
54#460—91no—
55#459—91no—
56#331—91no—
57Q8⋊C8—91no—
58D4⋊C8—91no—
59C3×#41—41no—
60C3×#41—121no—
61C2×C3×Q8#6711yescharacteristic
62C2×C3×D4—32no—
63C2×C3×D4—31no—
64C2×C3×D4—61no—
65C22×C3×C4—61no—
66C2×D12—61no—
67C2×D12—61no—
68C2×D12—61no—
69C2×D12—123no—
70#41⋊C3—43no—
71C2×SL(2,𝔽3)#6713yes—
72GL(2,𝔽3)—123no—
73C3×C4⋊C4—31no—
74C3×C4⋊C4—43no—
75C3×C4⋊C4S411yescharacteristic
76C3×C22⋊C4—61no—
77C3×C42—31no—
78#215—121no—
79#215—121no—
80#213—124no—
81#213—122no—
82C2×C3⋊C8—61no—
83C2×C3⋊C8—61no—
84C2×C3⋊C8—61no—
85C2×C3⋊C8—123no—
86C22×C32—41no—
87C2×C3⋊D3—82no—
88C2×C3⋊D3—161no—
89C2×C3⋊D3—161no—
90C32×C4—41no—
91C32×C4—41no—
92C32×C4—161no—
93C2×#41—21no—
94C2×SD16—181no—
95C2×D8—181no—
96C2×D8—182no—
97#127—31no—
98#127—91no—
99#121—61no—
100C4×Q8C2×C3⋊D311yescharacteristic
101C4×D4—31no—
102C4⋊C8—91no—
103C4⋊C8—91no—
104#102—181no—
105#102—181no—
106C4×C8—91no—
107C23×C3—61no—
108C22×D3—121no—
109C22×D3—243no—
110C3×Q8—31no—
111C3×Q8#21311yescharacteristic
112C3×D4—61no—
113C3×D4—61no—
114C3×D4—121no—
115C3×D4—121no—
116C3×D4—122no—
117C2×C3×C4—21no—
118C2×C3×C4—31no—
119C2×C3×C4—31no—
120C2×C3×C4—43no—
121C2×C3×C4—83no—
122C2×C3×C4—121no—
123C2×C3×C4—121no—
124C2×C3×C4C2×S411yescharacteristic
125D12—121no—
126D12—121no—
127D12—126no—
128D12—122no—
129D12—121no—
130D12—121no—
131D12—243no—
132SL(2,𝔽3)#21313yescomplemented
133C3⋊C8—62no—
134C3⋊C8—121no—
135C3⋊C8—121no—
136C3⋊C8—243no—
137C2×C32—41no—
138C2×C32—41no—
139C2×C32—41no—
140C3⋊D3—162no—
141#41—41no—
142#41—121no—
143C2×Q8#63811yescharacteristic
144C2×D4—32no—
145C2×D4—31no—
146C2×D4—61no—
147C2×D4—181no—
148C2×D4—181no—
149C2×D4—181no—
150C22×C4—61no—
151SD16—361no—
152SD16—361no—
153D8—364no—
154D8—362no—
155C2×C8—181no—
156C2×C8—181no—
157C2×C8—181no—
158C4⋊C4—31no—
159C4⋊C4C3⋊S411yescharacteristic
160C22⋊C4—61no—
161C42—31no—
162C22×C3—43no—
163C22×C3—121no—
164C22×C3—121no—
165C22×C3—121no—
166C22×C3GL(2,ℤ/4)11yescharacteristic
167C2×D3—246no—
168C2×D3—241no—
169C2×D3—241no—
170C2×D3—241no—
171C2×D3—483no—
172C2×D3—483no—
173C3×C4—31no—
174C3×C4—31no—
175C3×C4—41no—
176C3×C4—43no—
177C3×C4—43no—
178C3×C4—61no—
179C3×C4—163no—
180C3×C4#98511yescharacteristic
181C3×C4#98611yescharacteristic
182C32—41noSylow
183C23—61no—
184C23—361no—
185Q8—31no—
186Q8#375311yescharacteristic
187D4—61no—
188D4—61no—
189D4—121no—
190D4—121no—
191D4—122no—
192D4—361no—
193D4—361no—
194D4—362no—
195D4—361no—
196D4—361no—
197C2×C4—21no—
198C2×C4—31no—
199C2×C4—31no—
200C2×C4—121no—
201C2×C4—121no—
202C2×C4C2×C3⋊S411yescharacteristic
203C8—182no—
204C8—361no—
205C8—361no—
206C2×C3—43no—
207C2×C3—43no—
208C2×C3—43no—
209C2×C3—241no—
210C2×C3#541511yescharacteristic
211C2×C3#542711yescharacteristic
212C2×C3#542911yescharacteristic
213D3—486no—
214D3—481no—
215C22—121no—
216C22—121no—
217C22—121no—
218C22—721no—
219C22—721no—
220C22—721no—
221C22#6426911yescharacteristic
222C4—31no—
223C4—31no—
224C4—41no—
225C4—61no—
226C4#6426611yescharacteristic
227C4#6426711yescharacteristic
228C3—43no—
229C3#7309211yescharacteristic, complemented
230C2—241no—
231C2—1441no—
232C2#1059364911yescharacteristic
233C2#1059366111yescharacteristic
234C2#1059366311yescharacteristic
subgroupG/H
centerC22#64269
commutator, cosocleC3×C4×SL(2,𝔽3)C22
FrattiniC2×C4C2×C3⋊S4
FittingC3×#550D3
socleC22×C3GL(2,ℤ/4)
hypercenterC4⋊C4C3⋊S4
hypocenterC3×SL(2,𝔽3)D8
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×C4⋊(#41⋊C3)C2semidirect
2. C4⋊(C3⋊GL(2,𝔽3))C2semidirect
3. C3⋊(SL(2,𝔽3)⋊C8)C2semidirect
10. C3×#550D3semidirect
11. C4⋊(#41⋊C3)D3semidirect
50. C3×SL(2,𝔽3)D8semidirect
52. #550C3⋊D3semidirect
132. SL(2,𝔽3)#213semidirect
229. C3#73092semidirect

Maximal subgroups

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

subgroupindex
C3×C4⋊(#41⋊C3)2
C4⋊(C3⋊GL(2,𝔽3))2
C3⋊(SL(2,𝔽3)⋊C8)2
#730923
#719183
#636364