Finite Group Databasebeta

← C22×C3×#986 #11923697

Subgroups — C22×C3×#986 #11923697

Order 1152.

1865 conjugacy classes of subgroups; only the 220 normal subgroups are shown.

normal subgroups only

This group has 1865 conjugacy classes of subgroups — too many to draw. Only its 220 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×C3×#986C2112yesmaximal, direct product factor
3C22×C3×GL(2,𝔽3)C212yesmaximal, complemented
4C22×#986C311yesmaximal, characteristic, direct product factor
5C22×C3×AGL(1,ℤ/8)—31nomaximal
6C22×C3×D12—41nomaximal
7C2×C3×#41⋊C3C2216yescomplemented
8C22×C3×SL(2,𝔽3)C2211yescharacteristic, complemented
9C3×#986C22116yesdirect product factor
10C2×C3×GL(2,𝔽3)C22112yescomplemented
11C22×C3×#41D311yescharacteristic, complemented
12C23×C3×D4—31no—
13C22×#41⋊C3—21no—
14C22×#41⋊C3C2×C311yescharacteristic, complemented
15C2×#986C2×C3112yesdirect product factor
16C22×GL(2,𝔽3)C2×C312yescomplemented
17C2×C3×AGL(1,ℤ/8)—312no—
18C2×C3×AGL(1,ℤ/8)—312no—
19C22×C3×SD16—32no—
20C22×C3×D8—32no—
21C22×C3×M16—31no—
22C23×C3×D3—42no—
23C22×C32×C4—41no—
24C2×C3×D12—412no—
25C3×#41⋊C3C2314yescomplemented
26C2×C3×SL(2,𝔽3)C2313yescomplemented
27C3×GL(2,𝔽3)C2318yescomplemented
28C22×AGL(1,ℤ/8)—31noSylow
29C25×C3—61no—
30C2×C3×#41—36no—
31C2×C3×#41C2×D316yescomplemented
32C22×C3×Q8C2×D311yescharacteristic
33C22×C3×D4—31no—
34C22×C3×D4—32no—
35C22×C3×D4—312no—
36C22×C3×D4—66no—
37C22×C3×D4—61no—
38C23×C3×C4—31no—
39C22×D12—41no—
40C2×#41⋊C3—26no—
41C2×#41⋊C3C22×C316yescomplemented
42C22×SL(2,𝔽3)—21no—
43C22×SL(2,𝔽3)C22×C311yescharacteristic, complemented
44#986C22×C3116yesdirect product factor
45C2×GL(2,𝔽3)C22×C3112yescomplemented
46C3×AGL(1,ℤ/8)—348no—
47C3×AGL(1,ℤ/8)—316no—
48C2×C3×SD16—312no—
49C2×C3×SD16—312no—
50C2×C3×D8—324no—
51C2×C3×M16—312no—
52C22×C3×C8—32no—
53C23×C32—41no—
54C22×C3×D3—412no—
55C22×C3×D3—88no—
56C2×C32×C4—46no—
57C3×D12—416no—
58C3×SL(2,𝔽3)C2411yescharacteristic, complemented
59C22×#41C3×D311yescharacteristic, complemented
60C23×D4—31no—
61C2×AGL(1,ℤ/8)—312no—
62C2×AGL(1,ℤ/8)—312no—
63C22×SD16—32no—
64C22×D8—32no—
65C22×M16—31no—
66C24×C3—31no—
67C24×C3—612no—
68C24×C3—62no—
69C24×C3—128no—
70C23×D3—42no—
71C3×#41—312no—
72C3×#41—64no—
73C3×#41C22×D314yescomplemented
74C2×C3×Q8—33no—
75C2×C3×Q8C22×D313yes—
76C2×C3×D4—36no—
77C2×C3×D4—312no—
78C2×C3×D4—316no—
79C2×C3×D4—63no—
80C2×C3×D4—66no—
81C2×C3×D4—624no—
82C2×C3×D4—612no—
83C22×C3×C4—31no—
84C22×C3×C4—36no—
85C22×C3×C4—36no—
86C22×C3×C4—41no—
87C22×C3×C4—81no—
88C22×C3×C4S411yescharacteristic
89C2×D12—412no—
90#41⋊C3—24no—
91#41⋊C3C23×C314yescomplemented
92C2×SL(2,𝔽3)—23no—
93C2×SL(2,𝔽3)C23×C313yescomplemented
94GL(2,𝔽3)C23×C318yescomplemented
95C3×SD16—324no—
96C3×SD16—38no—
97C3×D8—332no—
98C3×M16—316no—
99C2×C3×C8—312no—
100C22×C32—44no—
101C22×C32—43no—
102C2×C3×D3—48no—
103C2×C3×D3—824no—
104C32×C4—44no—
105C25—61no—
106C2×#41—36no—
107C2×#41C2×C3×D316yescomplemented
108C22×Q8C2×C3×D311yescharacteristic
109C22×D4—31no—
110C22×D4—32no—
111C22×D4—312no—
112C22×D4—66no—
113C22×D4—61no—
114C23×C4—31no—
115AGL(1,ℤ/8)—348no—
116AGL(1,ℤ/8)—316no—
117C2×SD16—312no—
118C2×SD16—312no—
119C2×D8—324no—
120C2×M16—312no—
121C22×C8—32no—
122C23×C3—36no—
123C23×C3—41no—
124C23×C3—616no—
125C23×C3—64no—
126C23×C3—612no—
127C23×C3—81no—
128C23×C3—1248no—
129C23×C3—128no—
130C23×C3C2×S411yescharacteristic
131C22×D3—412no—
132C22×D3—88no—
133C3×Q8—33no—
134C3×Q8—61no—
135C3×Q8C23×D311yescharacteristic
136C3×D4—34no—
137C3×D4—38no—
138C3×D4—66no—
139C3×D4—612no—
140C3×D4—616no—
141C2×C3×C4—33no—
142C2×C3×C4—33no—
143C2×C3×C4—34no—
144C2×C3×C4—312no—
145C2×C3×C4—46no—
146C2×C3×C4—86no—
147C2×C3×C4C2×S416yes—
148D12—416no—
149SL(2,𝔽3)—21no—
150SL(2,𝔽3)C24×C311yescharacteristic, complemented
151C3×C8—38no—
152C2×C32—46no—
153C2×C32—41no—
154C3×D3—88no—
155C24—31no—
156C24—612no—
157C24—62no—
158C24—128no—
159#41—312no—
160#41—64no—
161#41C22×C3×D314yescomplemented
162C2×Q8—33no—
163C2×Q8C22×C3×D313yes—
164C2×D4—36no—
165C2×D4—312no—
166C2×D4—316no—
167C2×D4—63no—
168C2×D4—66no—
169C2×D4—624no—
170C2×D4—612no—
171C22×C4—31no—
172C22×C4—36no—
173C22×C4—36no—
174C22×C4C3×S411yescharacteristic
175SD16—324no—
176SD16—38no—
177D8—332no—
178M16—316no—
179C2×C8—312no—
180C22×C3—34no—
181C22×C3—44no—
182C22×C3—43no—
183C22×C3—612no—
184C22×C3—68no—
185C22×C3—84no—
186C22×C3—83no—
187C22×C3—1232no—
188C22×C3—1224no—
189C22×C3#98614yesdirect product factor
190C22×C3C22×S413yes—
191C2×D3—48no—
192C2×D3—824no—
193C3×C4—31no—
194C3×C4—33no—
195C3×C4—44no—
196C3×C4—84no—
197C3×C4C22×S414yes—
198C32—41noSylow
199C23—36no—
200C23—616no—
201C23—64no—
202C23—612no—
203C23—1248no—
204C23—128no—
205C23C2×C3×S411yescharacteristic
206Q8—33no—
207Q8—61no—
208Q8C23×C3×D311yescharacteristic
209D4—34no—
210D4—38no—
211D4—66no—
212D4—612no—
213D4—616no—
214C2×C4—33no—
215C2×C4—33no—
216C2×C4—34no—
217C2×C4—312no—
218C2×C4C2×C3×S416yes—
219C8—38no—
220C2×C3—41no—
221C2×C3—46no—
222C2×C3—64no—
223C2×C3—81no—
224C2×C3—86no—
225C2×C3—128no—
226C2×C3C2×#98616yesdirect product factor
227C2×C3C23×S411yescharacteristic
228D3—88no—
229C22—34no—
230C22—612no—
231C22—68no—
232C22—1232no—
233C22—1224no—
234C22C3×#98614yesdirect product factor
235C22C22×C3×S413yes—
236C4—31no—
237C4—33no—
238C4C22×C3×S414yes—
239C3—41no—
240C3—81no—
241C3C22×#98611yescharacteristic, direct product factor
242C2—64no—
243C2—128no—
244C2C2×C3×#98616yesdirect product factor
245C2C23×C3×S411yescharacteristic
subgroupG/H
center, socleC23×C3C2×S4
commutator, cosocle, hypocenterSL(2,𝔽3)C24×C3
FrattiniC2C23×C3×S4
FittingC22×C3×#41D3
hypercenterC22×C3×C4S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×#41⋊C3C2semidirect
2. C2×C3×#986C2direct (N × H)
3. C22×C3×GL(2,𝔽3)C2semidirect
4. C22×#986C3direct (N × H)
7. C2×C3×#41⋊C3C22semidirect
8. C22×C3×SL(2,𝔽3)C22semidirect
9. C3×#986C22direct (N × H)
10. C2×C3×GL(2,𝔽3)C22semidirect
11. C22×C3×#41D3semidirect
14. C22×#41⋊C3C2×C3semidirect
15. C2×#986C2×C3direct (N × H)
16. C22×GL(2,𝔽3)C2×C3semidirect
25. C3×#41⋊C3C23semidirect
26. C2×C3×SL(2,𝔽3)C23semidirect
27. C3×GL(2,𝔽3)C23semidirect
31. C2×C3×#41C2×D3semidirect
41. C2×#41⋊C3C22×C3semidirect
43. C22×SL(2,𝔽3)C22×C3semidirect
44. #986C22×C3direct (N × H)
45. C2×GL(2,𝔽3)C22×C3semidirect
58. C3×SL(2,𝔽3)C24semidirect
59. C22×#41C3×D3semidirect
73. C3×#41C22×D3semidirect
91. #41⋊C3C23×C3semidirect
93. C2×SL(2,𝔽3)C23×C3semidirect
94. GL(2,𝔽3)C23×C3semidirect
107. C2×#41C2×C3×D3semidirect
150. SL(2,𝔽3)C24×C3semidirect
161. #41C22×C3×D3semidirect
189. C22×C3#986direct (N × H)
226. C2×C3C2×#986direct (N × H)
234. C22C3×#986direct (N × H)
241. C3C22×#986direct (N × H)
244. C2C2×C3×#986direct (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×C3×#9862
C22×C3×GL(2,𝔽3)2
C22×#9863
C22×C3×AGL(1,ℤ/8)3
C22×C3×D124