Finite Group Databasebeta

← C22×C3×2−1+4⋊C3 #11919878

Subgroups — C22×C3×2−1+4⋊C3 #11919878

Order 1152.

2944 conjugacy classes of subgroups; only the 558 normal subgroups are shown.

normal subgroups only

This group has 2944 conjugacy classes of subgroups — too many to draw. Only its 558 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
1C2×C3×2−1+4⋊C3C2112yesmaximal, direct product factor
2C22×C3×#41⋊C3C211yesmaximal, characteristic, complemented
3C23×C3×SL(2,𝔽3)C212yesmaximal, complemented
4C22×C3×2−1+4C311yesmaximal, characteristic, complemented
5C22×2−1+4⋊C3C313yesmaximal, direct product factor
6C22×C32×D4—41nomaximal
7C3×2−1+4⋊C3C22116yesdirect product factor
8C2×C3×#41⋊C3C2216yescomplemented
9C22×C3×SL(2,𝔽3)C2211yescharacteristic
10C22×C3×SL(2,𝔽3)C22112yescomplemented
11C2×C3×2−1+4—312no—
12C2×C3×2−1+4C2×C3112yescomplemented
13C22×C3×#41—31no—
14C22×C3×#41—32no—
15C22×C3×#41C2×C311yescharacteristic, complemented
16C23×C3×Q8—31no—
17C23×C3×Q8C2×C312yescomplemented
18C2×2−1+4⋊C3C2×C3136yesdirect product factor
19C22×#41⋊C3C2×C313yescomplemented
20C23×SL(2,𝔽3)C2×C316yescomplemented
21C24×C32—42no—
22C2×C32×D4—412no—
23C22×C32×C4—41no—
24C3×#41⋊C3C2314yescomplemented
25C2×C3×SL(2,𝔽3)C2313yes—
26C2×C3×SL(2,𝔽3)C2318yescomplemented
27C22×2−1+4C3211yescharacteristic, complemented, Sylow
28C3×2−1+4—348no—
29C3×2−1+4—332no—
30C3×2−1+4C22×C3116yescomplemented
31C2×C3×#41—324no—
32C2×C3×#41—312no—
33C2×C3×#41—312no—
34C2×C3×#41—324no—
35C2×C3×#41—36no—
36C2×C3×#41C22×C316yescomplemented
37C22×C3×Q8—312no—
38C22×C3×Q8—312no—
39C22×C3×Q8—31no—
40C22×C3×Q8—32no—
41C22×C3×Q8—312no—
42C22×C3×Q8C22×C311yescharacteristic
43C22×C3×Q8C22×C3112yescomplemented
44C22×C3×D4—31no—
45C22×C3×D4—32no—
46C22×C3×D4—43no—
47C22×C3×D4A411yescharacteristic
48C23×C3×C4—32no—
49C23×C3×C4—32no—
50C23×C3×C4—31no—
512−1+4⋊C3C22×C3148yesdirect product factor
52C2×#41⋊C3C22×C3118yescomplemented
53C22×SL(2,𝔽3)C22×C313yes—
54C22×SL(2,𝔽3)C22×C3136yescomplemented
55C23×C32—412no—
56C23×C32—41no—
57C23×C32—88no—
58C32×D4—416no—
59C2×C32×C4—46no—
60C3×SL(2,𝔽3)C2411yescharacteristic
61C2×2−1+4—312no—
62C2×2−1+4C2×C32112yescomplemented
63C22×#41—31no—
64C22×#41—32no—
65C22×#41C2×C3211yescharacteristic, complemented
66C23×Q8—31no—
67C23×Q8C2×C3212yescomplemented
68C24×C3—31no—
69C24×C3—46no—
70C24×C3C2×A412yes—
71C3×#41—396no—
72C3×#41—38no—
73C3×#41—348no—
74C3×#41—316no—
75C3×#41—332no—
76C3×#41—312no—
77C3×#41C23×C314yescomplemented
78C2×C3×Q8—316no—
79C2×C3×Q8—312no—
80C2×C3×Q8—348no—
81C2×C3×Q8—316no—
82C2×C3×Q8—312no—
83C2×C3×Q8—33no—
84C2×C3×Q8—312no—
85C2×C3×Q8—324no—
86C2×C3×Q8C23×C313yes—
87C2×C3×Q8C23×C318yescomplemented
88C2×C3×D4—36no—
89C2×C3×D4—36no—
90C2×C3×D4—324no—
91C2×C3×D4—436no—
92C2×C3×D4C2×A4112yes—
93C22×C3×C4—324no—
94C22×C3×C4—312no—
95C22×C3×C4—312no—
96C22×C3×C4—32no—
97C22×C3×C4—31no—
98C22×C3×C4—36no—
99C22×C3×C4—36no—
100C22×C3×C4—43no—
101C22×C3×C4C2×A411yescharacteristic
102#41⋊C3C23×C3112yescomplemented
103C2×SL(2,𝔽3)C23×C319yes—
104C2×SL(2,𝔽3)C23×C3124yescomplemented
105C22×C32—44no—
106C22×C32—43no—
107C22×C32—48no—
108C22×C32—824no—
109C32×C4—44no—
1102−1+4—348no—
1112−1+4—332no—
1122−1+4C22×C32116yescomplemented
113C2×#41—324no—
114C2×#41—312no—
115C2×#41—312no—
116C2×#41—324no—
117C2×#41—36no—
118C2×#41C22×C3216yescomplemented
119C22×Q8—312no—
120C22×Q8—312no—
121C22×Q8—31no—
122C22×Q8—32no—
123C22×Q8—312no—
124C22×Q8C22×C3211yescharacteristic
125C22×Q8C22×C32112yescomplemented
126C22×D4—31no—
127C22×D4—32no—
128C22×D4C3×A411yescharacteristic
129C23×C4—32no—
130C23×C4—32no—
131C23×C4—31no—
132C23×C3—28no—
133C23×C3—36no—
134C23×C3—43no—
135C23×C3—436no—
136C23×C3—64no—
137C23×C3—824no—
138C23×C3C22×A4112yes—
139C23×C3C22×A411yescharacteristic
140C3×Q8—33no—
141C3×Q8—38no—
142C3×Q8—324no—
143C3×Q8—32no—
144C3×Q8—316no—
145C3×Q8C24×C311yescharacteristic
146C3×D4—34no—
147C3×D4—312no—
148C3×D4—332no—
149C3×D4—448no—
150C3×D4C22×A4116yes—
151C2×C3×C4—332no—
152C2×C3×C4—312no—
153C2×C3×C4—33no—
154C2×C3×C4—33no—
155C2×C3×C4—38no—
156C2×C3×C4—324no—
157C2×C3×C4—312no—
158C2×C3×C4—34no—
159C2×C3×C4—418no—
160C2×C3×C4C22×A416yes—
161SL(2,𝔽3)C24×C313yes—
162C2×C32—46no—
163C2×C32—41no—
164C2×C32—88no—
165C24—31no—
166C24C2×C3×A412yes—
167#41—396no—
168#41—38no—
169#41—316no—
170#41—348no—
171#41—332no—
172#41—312no—
173#41C23×C3214yescomplemented
174C2×Q8—316no—
175C2×Q8—312no—
176C2×Q8—348no—
177C2×Q8—316no—
178C2×Q8—312no—
179C2×Q8—33no—
180C2×Q8—312no—
181C2×Q8—324no—
182C2×Q8C23×C3213yes—
183C2×Q8C23×C3218yescomplemented
184C2×D4—36no—
185C2×D4—36no—
186C2×D4—324no—
187C2×D4C2×C3×A4112yes—
188C22×C4—324no—
189C22×C4—31no—
190C22×C4—32no—
191C22×C4—312no—
192C22×C4—312no—
193C22×C4—36no—
194C22×C4—36no—
195C22×C4C2×C3×A411yescharacteristic
196C22×C3—224no—
197C22×C3—34no—
198C22×C3—412no—
199C22×C3—49no—
200C22×C3—424no—
201C22×C3—612no—
202C22×C3—872no—
203C22×C32−1+4⋊C314yesdirect product factor
204C22×C3C23×A418yes—
205C22×C3C23×A413yes—
206C3×C4—38no—
207C3×C4—33no—
208C3×C4—31no—
209C3×C4—412no—
210C3×C4C23×A414yes—
211C32—41noSylow
212C23—28no—
213C23—36no—
214C23—64no—
215C23C22×C3×A4112yes—
216C23C22×C3×A411yescharacteristic
217Q8—33no—
218Q8—38no—
219Q8—324no—
220Q8—32no—
221Q8—316no—
222Q8C24×C3211yescharacteristic
223D4—34no—
224D4—312no—
225D4—332no—
226D4C22×C3×A4116yes—
227C2×C4—332no—
228C2×C4—33no—
229C2×C4—312no—
230C2×C4—33no—
231C2×C4—38no—
232C2×C4—324no—
233C2×C4—34no—
234C2×C4—312no—
235C2×C4C22×C3×A416yes—
236C2×C3—28no—
237C2×C3—418no—
238C2×C3—43no—
239C2×C3—64no—
240C2×C3—824no—
241C2×C3C2×2−1+4⋊C316yesdirect product factor
242C2×C3C24×A411yescharacteristic
243C22—224no—
244C22—34no—
245C22—612no—
246C22C3×2−1+4⋊C314yesdirect product factor
247C22C23×C3×A418yes—
248C22C23×C3×A413yes—
249C4—31no—
250C4—38no—
251C4—33no—
252C4C23×C3×A414yes—
253C3—43no—
254C3C22×2−1+4⋊C311yescharacteristic, direct product factor
255C2—28no—
256C2—64no—
257C2C2×C3×2−1+4⋊C316yesdirect product factor
258C2C24×C3×A411yescharacteristic
subgroupG/H
center, socleC23×C3C22×A4
commutator, cosocle, hypocenterQ8C24×C32
FrattiniC2C24×C3×A4
FittingC22×C3×2−1+4C3
hypercenterC22×C3×D4A4
LCSG ▷ Q8
UCSC1 ◁ C23×C3 ◁ C22×C3×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×2−1+4⋊C3C2direct (N × H)
2. C22×C3×#41⋊C3C2semidirect
3. C23×C3×SL(2,𝔽3)C2semidirect
4. C22×C3×2−1+4C3semidirect
5. C22×2−1+4⋊C3C3direct (N × H)
7. C3×2−1+4⋊C3C22direct (N × H)
8. C2×C3×#41⋊C3C22semidirect
10. C22×C3×SL(2,𝔽3)C22semidirect
12. C2×C3×2−1+4C2×C3semidirect
15. C22×C3×#41C2×C3semidirect
17. C23×C3×Q8C2×C3semidirect
18. C2×2−1+4⋊C3C2×C3direct (N × H)
19. C22×#41⋊C3C2×C3semidirect
20. C23×SL(2,𝔽3)C2×C3semidirect
24. C3×#41⋊C3C23semidirect
26. C2×C3×SL(2,𝔽3)C23semidirect
27. C22×2−1+4C32semidirect
30. C3×2−1+4C22×C3semidirect
36. C2×C3×#41C22×C3semidirect
43. C22×C3×Q8C22×C3semidirect
51. 2−1+4⋊C3C22×C3direct (N × H)
52. C2×#41⋊C3C22×C3semidirect
54. C22×SL(2,𝔽3)C22×C3semidirect
62. C2×2−1+4C2×C32semidirect
65. C22×#41C2×C32semidirect
67. C23×Q8C2×C32semidirect
77. C3×#41C23×C3semidirect
87. C2×C3×Q8C23×C3semidirect
102. #41⋊C3C23×C3semidirect
104. C2×SL(2,𝔽3)C23×C3semidirect
112. 2−1+4C22×C32semidirect
118. C2×#41C22×C32semidirect
125. C22×Q8C22×C32semidirect
173. #41C23×C32semidirect
183. C2×Q8C23×C32semidirect
203. C22×C32−1+4⋊C3direct (N × H)
241. C2×C3C2×2−1+4⋊C3direct (N × H)
246. C22C3×2−1+4⋊C3direct (N × H)
254. C3C22×2−1+4⋊C3direct (N × H)
257. C2C2×C3×2−1+4⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×2−1+4⋊C32
C22×C3×#41⋊C32
C23×C3×SL(2,𝔽3)2
C22×C3×2−1+43
C22×2−1+4⋊C33
C22×C32×D44