Finite Group Databasebeta

← C2×C3×#66440 #14519345

Subgroups — C2×C3×#66440 #14519345

Order 1944.

1112 conjugacy classes of subgroups; only the 496 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 496 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×#66440C211yesmaximal, characteristic, direct product factor
2C23×C3×3−1+2C311yesmaximal, characteristic, complemented
3C2×C3×C22⋊H(3,𝔽3)C311yesmaximal, characteristic
4C2×C3×#1094C311yesmaximal, characteristic, complemented
5C2×C32×C22⋊C9C311yesmaximal, characteristic, complemented
6C2×#66440C3127yesmaximal, direct product factor
7C2×C3×C9⋊A4C316yesmaximal, complemented
8C2×C3×C9×A4C313yesmaximal, complemented
9C2×C3×#737—41nomaximal
10C22×C3×3−1+2—31no—
11C22×C3×3−1+2—31no—
12C22×C3×3−1+2C2×C311yescharacteristic, complemented
13C3×C22⋊H(3,𝔽3)C2×C311yescharacteristic
14C3×#1094C2×C311yescharacteristic, complemented
15C32×C22⋊C9C2×C311yescharacteristic, complemented
16#66440C2×C3127yesdirect product factor
17C3×C9⋊A4C2×C316yescomplemented
18C3×C9×A4C2×C313yescomplemented
19C3×#737—41noSylow
20C23×C33C3211yescharacteristic
21C2×C32×A4C3213yes—
22C23×3−1+2C3219yescomplemented
23C23×C3×C9C3213yes—
24C2×C22⋊H(3,𝔽3)C3219yes—
25C2×#1094C3219yescomplemented
26C2×C3×C22⋊C9C3211yescharacteristic
27C2×C3×C22⋊C9C3213yes—
28C2×C3×C22⋊C9C3212yes—
29C2×C3×C22⋊C9C3219yescomplemented
30C2×C9⋊A4C32154yescomplemented
31C2×C9×A4C32127yescomplemented
32C2×#737—427no—
33C2×C3×3−1+2—31no—
34C2×C3×3−1+2—31no—
35C2×C3×3−1+2—41no—
36C2×C3×3−1+2—46no—
37C2×C3×3−1+2A411yescharacteristic, complemented
38C2×C3×H(3,𝔽3)—41no—
39C2×C32×C9—41no—
40C2×C32×C9—43no—
41C22×C33—31no—
42C22×C33—31no—
43C22×C33C2×C3211yescharacteristic
44C32×A4C2×C3213yes—
45C22×3−1+2—39no—
46C22×3−1+2—39no—
47C22×3−1+2C2×C3219yescomplemented
48C22×C3×C9—33no—
49C22×C3×C9—33no—
50C22×C3×C9C2×C3213yes—
51C22⋊H(3,𝔽3)C2×C3219yes—
52#1094C2×C3219yescomplemented
53C3×C22⋊C9C2×C3211yescharacteristic
54C3×C22⋊C9C2×C3213yes—
55C3×C22⋊C9C2×C3212yes—
56C3×C22⋊C9C2×C3219yescomplemented
57C9⋊A4C2×C32154yescomplemented
58C9×A4C2×C32127yescomplemented
59#737—427no—
60C3×3−1+2—41no—
61C3×3−1+2—46no—
62C3×3−1+2C2×A411yescharacteristic, complemented
63C3×H(3,𝔽3)—41no—
64C32×C9—41no—
65C32×C9—43no—
66C23×C32—33no—
67C23×C32C3313yes—
68C23×C32C3311yescharacteristic
69C2×C3×A4—39no—
70C2×C3×A4C3319yes—
71C23×C9C3319yes—
72C2×C22⋊C9C3313yes—
73C2×C22⋊C9C3319yes—
74C2×C22⋊C9C3316yes—
75C2×C33—31no—
76C2×C33—31no—
77C2×C33—43no—
78C2×C33C3×A411yescharacteristic
79C2×3−1+2—39no—
80C2×3−1+2—39no—
81C2×3−1+2—49no—
82C2×3−1+2—454no—
83C2×3−1+2C3×A419yescomplemented
84C2×H(3,𝔽3)—49no—
85C2×C3×C9—33no—
86C2×C3×C9—33no—
87C2×C3×C9—49no—
88C2×C3×C9—41no—
89C2×C3×C9—42no—
90C2×C3×C9—43no—
91C2×C3×C9—427no—
92C2×C3×C9C3×A413yes—
93C22×C32—31no—
94C22×C32—31no—
95C22×C32—33no—
96C22×C32—33no—
97C22×C32—33no—
98C22×C32—93no—
99C22×C32—93no—
100C22×C32C2×C3313yes—
101C22×C32C2×C3311yescharacteristic
102C3×A4—39no—
103C3×A4C2×C3319yes—
104C22×C9—39no—
105C22×C9—39no—
106C22×C9C2×C3319yes—
107C22⋊C9C2×C3313yes—
108C22⋊C9C2×C3319yes—
109C22⋊C9C2×C3316yes—
110C33—43no—
111C33C2×C3×A411yescharacteristic
1123−1+2—49no—
1133−1+2—454no—
1143−1+2C2×C3×A419yescomplemented
115H(3,𝔽3)—49no—
116C3×C9—41no—
117C3×C9—43no—
118C3×C9—42no—
119C3×C9—49no—
120C3×C9—427no—
121C3×C9C2×C3×A413yes—
122C23×C3—33no—
123C23×C3#73713yescomplemented
124C23×C3C3411yescharacteristic
125C2×A4—39no—
126C2×C32—31no—
127C2×C32—31no—
128C2×C32—33no—
129C2×C32—33no—
130C2×C32—33no—
131C2×C32—49no—
132C2×C32—93no—
133C2×C32—93no—
134C2×C32—129no—
135C2×C32C32×A413yes—
136C2×C32C32×A411yescharacteristic
137C2×C9—39no—
138C2×C9—39no—
139C2×C9—43no—
140C2×C9—46no—
141C2×C9—49no—
142C2×C9C32×A419yes—
143C22×C3—33no—
144C22×C3—31no—
145C22×C3—33no—
146C22×C3—31no—
147C22×C3—33no—
148C22×C3—93no—
149C22×C3—93no—
150C22×C3C2×#73713yescomplemented
151C22×C3C2×C3411yescharacteristic
152A4—39no—
153C32—33no—
154C32—49no—
155C32—129no—
156C32C2×C32×A413yes—
157C32C2×C32×A411yescharacteristic
158C9—43no—
159C9—46no—
160C9—49no—
161C9C2×C32×A419yes—
162C23C3×#73711yescharacteristic, complemented, Sylow
163C2×C3—31no—
164C2×C3—31no—
165C2×C3—33no—
166C2×C3—33no—
167C2×C3—33no—
168C2×C3—93no—
169C2×C3—93no—
170C2×C3—129no—
171C2×C3#6644013yesdirect product factor
172C2×C3C33×A411yescharacteristic
173C22—31no—
174C22—31no—
175C22C2×C3×#73711yescharacteristic, complemented
176C3—33no—
177C3—129no—
178C3C2×#6644013yesdirect product factor
179C3C2×C33×A411yescharacteristic
180C2—31no—
181C2—31no—
182C2C3×#6644011yescharacteristic, direct product factor
subgroupG/H
centerC2×C32C32×A4
commutator, cosocleC22×C3C2×C34
FrattiniC3C2×C33×A4
FittingC23×C3×3−1+2C3
socleC23×C32C33
prime radicalC2×C3×C22⋊H(3,𝔽3)C3
hypercenterC2×C3×3−1+2A4
hypocenterC22C2×C3×#737
LCSG ▷ C22×C3 ▷ C22
UCSC1 ◁ C2×C32 ◁ C2×C3×3−1+2

Complements

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

NH ≅ G/Ntype
1. C3×#66440C2direct (N × H)
2. C23×C3×3−1+2C3semidirect
4. C2×C3×#1094C3semidirect
5. C2×C32×C22⋊C9C3semidirect
6. C2×#66440C3direct (N × H)
7. C2×C3×C9⋊A4C3semidirect
8. C2×C3×C9×A4C3semidirect
12. C22×C3×3−1+2C2×C3semidirect
14. C3×#1094C2×C3semidirect
15. C32×C22⋊C9C2×C3semidirect
16. #66440C2×C3direct (N × H)
17. C3×C9⋊A4C2×C3semidirect
18. C3×C9×A4C2×C3semidirect
22. C23×3−1+2C32semidirect
25. C2×#1094C32semidirect
29. C2×C3×C22⋊C9C32semidirect
30. C2×C9⋊A4C32semidirect
31. C2×C9×A4C32semidirect
37. C2×C3×3−1+2A4semidirect
47. C22×3−1+2C2×C32semidirect
52. #1094C2×C32semidirect
56. C3×C22⋊C9C2×C32semidirect
57. C9⋊A4C2×C32semidirect
58. C9×A4C2×C32semidirect
62. C3×3−1+2C2×A4semidirect
83. C2×3−1+2C3×A4semidirect
114. 3−1+2C2×C3×A4semidirect
123. C23×C3#737semidirect
150. C22×C3C2×#737semidirect
162. C23C3×#737semidirect
171. C2×C3#66440direct (N × H)
175. C22C2×C3×#737semidirect
178. C3C2×#66440direct (N × H)
182. C2C3×#66440direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#664402
C23×C3×3−1+23
C2×C3×C22⋊H(3,𝔽3)3
C2×C3×#10943
C2×C32×C22⋊C93
C2×#664403
C2×C3×C9⋊A43
C2×C3×C9×A43
C2×C3×#7374