Finite Group Databasebeta

← C22×#64267 #11923728

Subgroups — C22×#64267 #11923728

Order 1152.

2192 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 2192 conjugacy classes of subgroups — too many to draw. Only its 201 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
1C22×C3×#41⋊C3C211yesmaximal, characteristic, complemented
2C2×#64267C2112yesmaximal, direct product factor
3C22×C3⋊GL(2,𝔽3)C212yesmaximal, complemented
4C22×#986—33nomaximal
5C22×#949—31nomaximal
6C22×C4⋊(C3⋊D3)—41nomaximal
7C2×C3×#41⋊C3C2216yescomplemented
8C22×C3×SL(2,𝔽3)C2211yescharacteristic, complemented
9#64267C22116yesdirect product factor
10C2×C3⋊GL(2,𝔽3)C22112yescomplemented
11C22×C3×#41D311yescharacteristic, complemented
12C23×D12—31no—
13C22×#41⋊C3D313yescomplemented
14C2×#986—336no—
15C22×GL(2,𝔽3)—36no—
16C2×#949—312no—
17C2×#949—312no—
18C22×#215—32no—
19C22×#213—32no—
20C22×#208—31no—
21C23×C3⋊D3—42no—
22C22×C32×C4—41no—
23C2×C4⋊(C3⋊D3)—412no—
24C3×#41⋊C3C2314yescomplemented
25C2×C3×SL(2,𝔽3)C2313yescomplemented
26C3⋊GL(2,𝔽3)C2318yescomplemented
27C22×AGL(1,ℤ/8)—91noSylow
28C24×D3—61no—
29C2×C3×#41—36no—
30C2×C3×#41C2×D316yescomplemented
31C22×C3×Q8C2×D311yescharacteristic
32C22×C3×D4—31no—
33C23×C3×C4—31no—
34C22×D12—32no—
35C22×D12—312no—
36C22×D12—66no—
37C22×D12—61no—
38C22×D12—123no—
39C2×#41⋊C3C2×D3118yescomplemented
40C22×SL(2,𝔽3)C2×D313yescomplemented
41#986—348no—
42C2×GL(2,𝔽3)—336no—
43#949—348no—
44#949—316no—
45C2×#215—312no—
46C2×#215—312no—
47C2×#213—324no—
48C2×#208—312no—
49C22×C3⋊C8—32no—
50C23×C32—41no—
51C22×C3⋊D3—412no—
52C22×C3⋊D3—88no—
53C2×C32×C4—46no—
54C4⋊(C3⋊D3)—416no—
55C3×SL(2,𝔽3)C2411yescharacteristic, complemented
56C22×#41C3⋊D311yescharacteristic, complemented
57C23×D4—91no—
58C2×AGL(1,ℤ/8)—912no—
59C2×AGL(1,ℤ/8)—912no—
60C22×SD16—92no—
61C22×D8—92no—
62C22×M16—91no—
63C24×C3—31no—
64C23×D3—612no—
65C23×D3—62no—
66C23×D3—126no—
67C23×D3—128no—
68C3×#41—312no—
69C3×#41—64no—
70C3×#41C22×D314yescomplemented
71C2×C3×Q8—33no—
72C2×C3×Q8C22×D313yes—
73C2×C3×D4—36no—
74C2×C3×D4—63no—
75C22×C3×C4—31no—
76C22×C3×C4—36no—
77C22×C3×C4—36no—
78C22×C3×C4—43no—
79C22×C3×C4S411yescharacteristic
80C2×D12—312no—
81C2×D12—316no—
82C2×D12—66no—
83C2×D12—624no—
84C2×D12—612no—
85C2×D12—1236no—
86#41⋊C3C22×D3112yescomplemented
87C2×SL(2,𝔽3)C22×D319yescomplemented
88GL(2,𝔽3)—324no—
89#215—324no—
90#215—38no—
91#213—332no—
92#208—316no—
93C2×C3⋊C8—312no—
94C22×C32—44no—
95C22×C32—43no—
96C2×C3⋊D3—48no—
97C2×C3⋊D3—824no—
98C32×C4—44no—
99C25—181no—
100C2×#41—36no—
101C2×#41C2×C3⋊D316yescomplemented
102C22×Q8C2×C3⋊D311yescharacteristic
103C22×D4—31no—
104C22×D4—92no—
105C22×D4—912no—
106C22×D4—186no—
107C22×D4—181no—
108C23×C4—31no—
109AGL(1,ℤ/8)—948no—
110AGL(1,ℤ/8)—916no—
111C2×SD16—912no—
112C2×SD16—912no—
113C2×D8—924no—
114C2×M16—912no—
115C22×C8—92no—
116C23×C3—36no—
117C23×C3—43no—
118C23×C3—64no—
119C23×C3C2×S411yescharacteristic
120C22×D3—616no—
121C22×D3—612no—
122C22×D3—1236no—
123C22×D3—1248no—
124C22×D3—128no—
125C22×D3—2424no—
126C3×Q8—33no—
127C3×Q8—61no—
128C3×Q8C23×D311yescharacteristic
129C3×D4—34no—
130C3×D4—66no—
131C2×C3×C4—33no—
132C2×C3×C4—33no—
133C2×C3×C4—34no—
134C2×C3×C4—312no—
135C2×C3×C4—418no—
136C2×C3×C4C2×S416yes—
137D12—38no—
138D12—612no—
139D12—616no—
140D12—1248no—
141SL(2,𝔽3)C23×D313yescomplemented
142C3⋊C8—38no—
143C2×C32—46no—
144C2×C32—41no—
145C3⋊D3—88no—
146C24—31no—
147C24—1812no—
148C24—182no—
149C24—368no—
150#41—312no—
151#41—64no—
152#41C22×C3⋊D314yescomplemented
153C2×Q8—33no—
154C2×Q8C22×C3⋊D313yes—
155C2×D4—36no—
156C2×D4—63no—
157C2×D4—912no—
158C2×D4—916no—
159C2×D4—186no—
160C2×D4—1824no—
161C2×D4—1812no—
162C22×C4—31no—
163C22×C4—36no—
164C22×C4—36no—
165C22×C4C3⋊S411yescharacteristic
166SD16—924no—
167SD16—98no—
168D8—932no—
169M16—916no—
170C2×C8—912no—
171C22×C3—34no—
172C22×C3—412no—
173C22×C3—49no—
174C22×C3—612no—
175C22×C3#98614yescomplemented
176C22×C3C22×S413yes—
177C2×D3—68no—
178C2×D3—1224no—
179C2×D3—1232no—
180C2×D3—1224no—
181C2×D3—2472no—
182C3×C4—31no—
183C3×C4—33no—
184C3×C4—412no—
185C3×C4C22×S414yes—
186C32—41noSylow
187C23—36no—
188C23—64no—
189C23—1816no—
190C23—1812no—
191C23—3648no—
192C23—368no—
193C23C2×C3⋊S411yescharacteristic
194Q8—33no—
195Q8—61no—
196Q8C23×C3⋊D311yescharacteristic
197D4—34no—
198D4—66no—
199D4—98no—
200D4—1812no—
201D4—1816no—
202C2×C4—33no—
203C2×C4—33no—
204C2×C4—34no—
205C2×C4—312no—
206C2×C4C2×C3⋊S416yes—
207C8—98no—
208C2×C3—43no—
209C2×C3—418no—
210C2×C3—64no—
211C2×C3C2×#98616yescomplemented
212C2×C3C23×S411yescharacteristic
213D3—128no—
214D3—2424no—
215C22—34no—
216C22—612no—
217C22—188no—
218C22—3632no—
219C22—3624no—
220C22#6426714yesdirect product factor
221C22C22×C3⋊S413yes—
222C4—31no—
223C4—33no—
224C4C22×C3⋊S414yes—
225C3—43no—
226C3C22×#98611yescharacteristic, complemented
227C2—64no—
228C2—368no—
229C2C2×#6426716yesdirect product factor
230C2C23×C3⋊S411yescharacteristic
subgroupG/H
centerC23C2×C3⋊S4
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C24
FrattiniC2C23×C3⋊S4
FittingC22×C3×#41D3
socleC23×C3C2×S4
hypercenterC22×C4C3⋊S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×#41⋊C3C2semidirect
2. C2×#64267C2direct (N × H)
3. C22×C3⋊GL(2,𝔽3)C2semidirect
7. C2×C3×#41⋊C3C22semidirect
8. C22×C3×SL(2,𝔽3)C22semidirect
9. #64267C22direct (N × H)
10. C2×C3⋊GL(2,𝔽3)C22semidirect
11. C22×C3×#41D3semidirect
13. C22×#41⋊C3D3semidirect
24. C3×#41⋊C3C23semidirect
25. C2×C3×SL(2,𝔽3)C23semidirect
26. C3⋊GL(2,𝔽3)C23semidirect
30. C2×C3×#41C2×D3semidirect
39. C2×#41⋊C3C2×D3semidirect
40. C22×SL(2,𝔽3)C2×D3semidirect
55. C3×SL(2,𝔽3)C24semidirect
56. C22×#41C3⋊D3semidirect
70. C3×#41C22×D3semidirect
86. #41⋊C3C22×D3semidirect
87. C2×SL(2,𝔽3)C22×D3semidirect
101. C2×#41C2×C3⋊D3semidirect
141. SL(2,𝔽3)C23×D3semidirect
152. #41C22×C3⋊D3semidirect
175. C22×C3#986semidirect
211. C2×C3C2×#986semidirect
220. C22#64267direct (N × H)
226. C3C22×#986semidirect
229. C2C2×#64267direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
C2×#642672
C22×C3⋊GL(2,𝔽3)2
C22×#9863
C22×#9493
C22×C4⋊(C3⋊D3)4