Finite Group Databasebeta

← C2×C3×#5465 #11919838

Subgroups — C2×C3×#5465 #11919838

Order 1152.

816 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 64 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×#5465C211yesmaximal, characteristic, direct product factor
2C2×C3×H(3,𝔽4)C311yesmaximal, characteristic, complemented
3C2×#5465C313yesmaximal, direct product factor
4C23×C3×A4—42nomaximal
5C22×C3×SL(2,𝔽3)—43nomaximal
6C3×H(3,𝔽4)—33no—
7C3×H(3,𝔽4)—32no—
8C3×H(3,𝔽4)C2×C311yescharacteristic, complemented
9C2×C3×#124—33no—
10C2×C3×#120—32no—
11#5465C2×C313yesdirect product factor
12C22×C3×A4—412no—
13C22×C3×A4—42no—
14C2×C3×SL(2,𝔽3)—46no—
15C2×C3×SL(2,𝔽3)—43no—
16C2×H(3,𝔽4)C3211yescharacteristic, complemented, Sylow
17C25×C3A412yescomplemented
18C23×A4—46no—
19C22×C3×Q8A413yescomplemented
20C22×C3×D4—33no—
21C22×SL(2,𝔽3)—49no—
22C3×#124—33no—
23C3×#124—312no—
24C3×#124—33no—
25C3×#124—33no—
26C3×#124—33no—
27C3×#120—36no—
28C3×#120—36no—
29C3×#120—32no—
30C3×#120—32no—
31C2×C3×C22⋊C4—36no—
32C2×C3×C42—31no—
33C23×C32—161no—
34C2×C3×A4—46no—
35C2×C3×A4—48no—
36C3×SL(2,𝔽3)—43no—
37H(3,𝔽4)—33no—
38H(3,𝔽4)—32no—
39H(3,𝔽4)C2×C3211yescharacteristic, complemented
40C2×#124—33no—
41C2×#120—32no—
42C24×C3—32no—
43C24×C3—32no—
44C24×C3—412no—
45C24×C3C2×A412yescomplemented
46C22×A4—436no—
47C22×A4—46no—
48C2×C3×Q8—33no—
49C2×C3×Q8—46no—
50C2×C3×Q8C2×A413yescomplemented
51C2×C3×D4—36no—
52C2×C3×D4—33no—
53C2×C3×D4—33no—
54C2×C3×D4—126no—
55C22×C3×C4—33no—
56C2×SL(2,𝔽3)—418no—
57C2×SL(2,𝔽3)—49no—
58C3×C22⋊C4—36no—
59C3×C22⋊C4—312no—
60C3×C22⋊C4—36no—
61C3×C42—33no—
62C3×C42—31no—
63C22×C32—166no—
64C22×C32—161no—
65C3×A4—42no—
66C25C3×A412yescomplemented
67C22×Q8C3×A413yescomplemented
68C22×D4—33no—
69#124—33no—
70#124—312no—
71#124—33no—
72#124—33no—
73#124—33no—
74#120—36no—
75#120—36no—
76#120—32no—
77#120—32no—
78C2×C22⋊C4—36no—
79C2×C42—31no—
80C23×C3—32no—
81C23×C3—32no—
82C23×C3—46no—
83C23×C3—48no—
84C23×C3—612no—
85C23×C3—126no—
86C23×C3—128no—
87C23×C3—163no—
88C23×C3C22⋊A411yescharacteristic
89C2×A4—418no—
90C2×A4—424no—
91C3×Q8—43no—
92C3×Q8—123no—
93C3×D4—123no—
94C3×D4—123no—
95C3×D4—126no—
96C2×C3×C4—33no—
97C2×C3×C4—33no—
98C2×C3×C4—66no—
99SL(2,𝔽3)—49no—
100C2×C32—163no—
101C2×C32—164no—
102C24—32no—
103C24—32no—
104C24—412no—
105C24C2×C3×A412yescomplemented
106C2×Q8—33no—
107C2×Q8—46no—
108C2×Q8C2×C3×A413yescomplemented
109C2×D4—33no—
110C2×D4—33no—
111C2×D4—36no—
112C2×D4—126no—
113C22×C4—33no—
114C22⋊C4—36no—
115C22⋊C4—312no—
116C22⋊C4—36no—
117C42—31no—
118C42—33no—
119C22×C3—42no—
120C22×C3—66no—
121C22×C3—66no—
122C22×C3—122no—
123C22×C3—128no—
124C22×C3—128no—
125C22×C3—1618no—
126C22×C3—163no—
127C22×C3#99716yes—
128C22×C3C2×C22⋊A411yescharacteristic
129A4—46no—
130C3×C4—63no—
131C3×C4—63no—
132C32—161noSylow
133C23—32no—
134C23—32no—
135C23—46no—
136C23—48no—
137C23—612no—
138C23—126no—
139C23—128no—
140C23C3×C22⋊A411yescharacteristic
141Q8—43no—
142Q8—123no—
143D4—123no—
144D4—123no—
145D4—126no—
146C2×C4—33no—
147C2×C4—33no—
148C2×C4—66no—
149C2×C3—122no—
150C2×C3—122no—
151C2×C3—169no—
152C2×C3—1612no—
153C2×C3#546514yesdirect product factor
154C2×C3C2×#99713yes—
155C22—42no—
156C22—66no—
157C22—66no—
158C22—122no—
159C22—128no—
160C22—128no—
161C22C3×#99716yes—
162C22C2×C3×C22⋊A411yescharacteristic
163C4—63no—
164C4—63no—
165C3—163no—
166C3C2×#546511yescharacteristic, direct product factor
167C2—122no—
168C2—122no—
169C2C3×#546514yesdirect product factor
170C2C2×C3×#99713yes—
subgroupG/H
center, socle, hypercenterC23×C3C22⋊A4
commutator, cosocle, hypocenterH(3,𝔽4)C2×C32
FrattiniC22C2×C3×C22⋊A4
FittingC2×C3×H(3,𝔽4)C3
LCSG ▷ H(3,𝔽4)
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C3×#5465C2direct (N × H)
2. C2×C3×H(3,𝔽4)C3semidirect
3. C2×#5465C3direct (N × H)
8. C3×H(3,𝔽4)C2×C3semidirect
11. #5465C2×C3direct (N × H)
16. C2×H(3,𝔽4)C32semidirect
17. C25×C3A4semidirect
19. C22×C3×Q8A4semidirect
39. H(3,𝔽4)C2×C32semidirect
45. C24×C3C2×A4semidirect
50. C2×C3×Q8C2×A4semidirect
66. C25C3×A4semidirect
67. C22×Q8C3×A4semidirect
105. C24C2×C3×A4semidirect
108. C2×Q8C2×C3×A4semidirect
153. C2×C3#5465direct (N × H)
166. C3C2×#5465direct (N × H)
169. C2C3×#5465direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#54652
C2×C3×H(3,𝔽4)3
C2×#54653
C23×C3×A44
C22×C3×SL(2,𝔽3)4