Finite Group Databasebeta

← C2×C3×#5464 #11919836

Subgroups — C2×C3×#5464 #11919836

Order 1152.

640 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 40 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×#5464C211yesmaximal, characteristic, direct product factor
2C2×C3×H(3,𝔽4)C311yesmaximal, characteristic, complemented
3C2×#5464C313yesmaximal, direct product factor
4C2×C3×C22⋊A4—42nomaximal
5C2×C3×C42⋊C3—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#5464C2×C313yesdirect product factor
12C3×C22⋊A4—42no—
13C3×C42⋊C3—43no—
14C2×H(3,𝔽4)C3211yescharacteristic, complemented, Sylow
15C25×C3A412yescomplemented
16C2×C22⋊A4—46no—
17C22×C3×Q8—31no—
18C22×C3×D4—33no—
19C3×#124—33no—
20C3×#124—312no—
21C3×#124—33no—
22C3×#124—33no—
23C3×#124—33no—
24C3×#120—36no—
25C3×#120—36no—
26C3×#120—32no—
27C3×#120—32no—
28C2×C3×C22⋊C4—36no—
29C2×C3×C42A413yescomplemented
30C2×C42⋊C3—49no—
31C2×C3×A4—168no—
32C2×C3×A4—161no—
33H(3,𝔽4)—33no—
34H(3,𝔽4)—32no—
35H(3,𝔽4)C2×C3211yescharacteristic, complemented
36C2×#124—33no—
37C2×#120—32no—
38C24×C3—32no—
39C24×C3—32no—
40C24×C3—122no—
41C24×C3—122no—
42C24×C3C2×A412yescomplemented
43C22⋊A4—46no—
44C2×C3×Q8—31no—
45C2×C3×Q8—33no—
46C2×C3×Q8—121no—
47C2×C3×Q8—121no—
48C2×C3×D4—33no—
49C2×C3×D4—33no—
50C2×C3×D4—36no—
51C2×C3×D4—123no—
52C2×C3×D4—123no—
53C22×C3×C4—33no—
54C3×C22⋊C4—36no—
55C3×C22⋊C4—312no—
56C3×C22⋊C4—36no—
57C3×C42—33no—
58C3×C42C2×A413yescomplemented
59C42⋊C3—49no—
60C3×A4—168no—
61C3×A4—161no—
62C25C3×A412yescomplemented
63C22×Q8—31no—
64C22×D4—33no—
65#124—33no—
66#124—312no—
67#124—33no—
68#124—33no—
69#124—33no—
70#120—36no—
71#120—36no—
72#120—32no—
73#120—32no—
74C2×C22⋊C4—36no—
75C2×C42C3×A413yescomplemented
76C23×C3—32no—
77C23×C3—32no—
78C23×C3—48no—
79C23×C3—66no—
80C23×C3—66no—
81C23×C3—122no—
82C23×C3—128no—
83C23×C3—126no—
84C23×C3C22⋊A411yescharacteristic
85C2×A4—1624no—
86C2×A4—163no—
87C3×Q8—121no—
88C3×Q8—123no—
89C3×D4—123no—
90C3×D4—123no—
91C3×D4—126no—
92C2×C3×C4—33no—
93C2×C3×C4—33no—
94C2×C3×C4—63no—
95C2×C3×C4—63no—
96C2×C32—641no—
97C24—32no—
98C24—32no—
99C24—122no—
100C24—122no—
101C24C2×C3×A412yescomplemented
102C2×Q8—31no—
103C2×Q8—33no—
104C2×Q8—121no—
105C2×Q8—121no—
106C2×D4—33no—
107C2×D4—33no—
108C2×D4—36no—
109C2×D4—123no—
110C2×D4—123no—
111C22×C4—33no—
112C22⋊C4—36no—
113C22⋊C4—312no—
114C22⋊C4—36no—
115C42—33no—
116C42C2×C3×A413yescomplemented
117C22×C3—31no—
118C22×C3—31no—
119C22×C3—48no—
120C22×C3—66no—
121C22×C3—66no—
122C22×C3—126no—
123C22×C3—128no—
124C22×C3—122no—
125C22×C3C2×C22⋊A411yescharacteristic
126A4—1624no—
127A4—163no—
128C3×C4—63no—
129C3×C4—63no—
130C32—641noSylow
131C23—32no—
132C23—32no—
133C23—48no—
134C23—66no—
135C23—66no—
136C23—122no—
137C23—128no—
138C23—126no—
139C23C3×C22⋊A411yescharacteristic
140Q8—121no—
141Q8—123no—
142D4—123no—
143D4—123no—
144D4—126no—
145C2×C4—33no—
146C2×C4—33no—
147C2×C4—63no—
148C2×C4—63no—
149C2×C3—31no—
150C2×C3—31no—
151C2×C3—122no—
152C2×C3—122no—
153C2×C3—643no—
154C2×C3#546411yescharacteristic, direct product factor
155C22—31no—
156C22—31no—
157C22—48no—
158C22—66no—
159C22—66no—
160C22—126no—
161C22—128no—
162C22—122no—
163C22C2×C3×C22⋊A411yescharacteristic
164C4—63no—
165C4—63no—
166C3—643no—
167C3C2×#546411yescharacteristic, direct product factor
168C2—31no—
169C2—31no—
170C2—122no—
171C2—122no—
172C2C3×#546411yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3#5464
commutator, cosocle, hypocenterH(3,𝔽4)C2×C32
FrattiniC22C2×C3×C22⋊A4
FittingC2×C3×H(3,𝔽4)C3
socleC23×C3C22⋊A4
LCSG ▷ H(3,𝔽4)
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C3×#5464C2direct (N × H)
2. C2×C3×H(3,𝔽4)C3semidirect
3. C2×#5464C3direct (N × H)
8. C3×H(3,𝔽4)C2×C3semidirect
11. #5464C2×C3direct (N × H)
14. C2×H(3,𝔽4)C32semidirect
15. C25×C3A4semidirect
29. C2×C3×C42A4semidirect
35. H(3,𝔽4)C2×C32semidirect
42. C24×C3C2×A4semidirect
58. C3×C42C2×A4semidirect
62. C25C3×A4semidirect
75. C2×C42C3×A4semidirect
101. C24C2×C3×A4semidirect
116. C42C2×C3×A4semidirect
154. C2×C3#5464direct (N × H)
167. C3C2×#5464direct (N × H)
172. C2C3×#5464direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#54642
C2×C3×H(3,𝔽4)3
C2×#54643
C2×C3×C22⋊A44
C2×C3×C42⋊C34