Finite Group Databasebeta

← C2×C3×#66324 #14517455

Subgroups — C2×C3×#66324 #14517455

Order 1944.

334 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 98 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×#731C211yesmaximal, characteristic
2C3×#66324C212yesmaximal, direct product factor
3C2×C3×C3⋊Dic9C311yesmaximal, characteristic, complemented
4C2×C3×(C3⋊Dic3)⋊C3—31nomaximal
5C2×#66324C313yesmaximal, direct product factor
6C2×C3×#731C412yescomplemented
7C2×C3×#731C2211yescharacteristic
8C22×C3×3−1+2—21no—
9C22×C3×H(3,𝔽3)D311yescharacteristic
10C22×C32×C9C2×C311yescharacteristic
11C3×C3⋊Dic9C2×C312yescomplemented
12C3×(C3⋊Dic3)⋊C3—32no—
13C22×#731—23no—
14C22×#731C2×C313yes—
15#66324C2×C316yesdirect product factor
16C3×#731C2×C411yescharacteristic, complemented, Sylow
17C2×C3×C3⋊Dic3—31no—
18C2×C32×Dic3—91no—
19C2×C3⋊Dic9C3211yescharacteristic, complemented
20C2×(C3⋊Dic3)⋊C3—33no—
21C2×C3×Dic9—91no—
22C2×C3×3−1+2—22no—
23C2×C3×3−1+2—21no—
24C2×C3×H(3,𝔽3)Dic312yes—
25C2×C3×H(3,𝔽3)C2×D311yescharacteristic
26C2×C32×C9C3×C412yescomplemented
27C2×C32×C9C22×C311yescharacteristic
28C2×#731—26no—
29C2×#731—23no—
30C2×#731C3×C416yescomplemented
31C2×#731C22×C313yes—
32C22×C33—31no—
33C22×C33C3×D311yescharacteristic
34C3×C3⋊Dic3—32no—
35C32×Dic3—92no—
36C22×3−1+2—23no—
37C22×3−1+2—62no—
38C22×H(3,𝔽3)—61no—
39C22×H(3,𝔽3)C3×D313yes—
40C22×C3×C9—21no—
41C22×C3×C9—31no—
42C22×C3×C9—61no—
43C22×C3×C9—61no—
44C22×C3×C9C2×C3211yescharacteristic
45C3⋊Dic9C2×C3212yescomplemented
46(C3⋊Dic3)⋊C3—36no—
47C3×Dic9—92no—
48C3×3−1+2—21no—
49C3×H(3,𝔽3)C2×Dic311yescharacteristic
50C32×C9C2×C3×C411yescharacteristic, complemented
51#731—23no—
52#731C2×C3×C413yescomplemented
53C2×C32×C4—271no—
54C2×C3⋊Dic3—31no—
55C2×C3×Dic3—93no—
56C2×C3×Dic3—91no—
57C2×C3×Dic3—271no—
58C2×Dic9—91no—
59C2×C33—32no—
60C2×C33—31no—
61C2×C33C3×Dic312yes—
62C2×C33C2×C3×D311yescharacteristic
63C2×3−1+2—26no—
64C2×3−1+2—23no—
65C2×3−1+2—64no—
66C2×3−1+2—62no—
67C2×H(3,𝔽3)—62no—
68C2×H(3,𝔽3)—61no—
69C2×H(3,𝔽3)C3×Dic316yes—
70C2×H(3,𝔽3)C2×C3×D313yes—
71C2×C3×C9—22no—
72C2×C3×C9—21no—
73C2×C3×C9—32no—
74C2×C3×C9—31no—
75C2×C3×C9—62no—
76C2×C3×C9—61no—
77C2×C3×C9—62no—
78C2×C3×C9—61no—
79C2×C3×C9C32×C412yescomplemented
80C2×C3×C9C22×C3211yescharacteristic
81C22×C32—21no—
82C22×C32—31no—
83C22×C32—33no—
84C22×C32—61no—
85C22×C32—91no—
86C22×C32—181no—
87C22×C32(C3⋊D3)⋊C311yescharacteristic
88C22×C32C32×D311yescharacteristic
89C32×C4—272no—
90C3⋊Dic3—32no—
91C3×Dic3—96no—
92C3×Dic3—92no—
93C3×Dic3—272no—
94C22×C9—31no—
95C22×C9—61no—
96C22×C9—63no—
97Dic9—92no—
98C33—31no—
99C33C2×C3×Dic311yescharacteristic
1003−1+2—23no—
1013−1+2—62no—
102H(3,𝔽3)—61no—
103H(3,𝔽3)C2×C3×Dic313yes—
104C3×C9—21no—
105C3×C9—31no—
106C3×C9—61no—
107C3×C9—61no—
108C3×C9C2×C32×C411yescharacteristic, complemented
109C2×C3×C4—273no—
110C2×C3×C4—271no—
111C2×Dic3—91no—
112C2×Dic3—271no—
113C2×C32—22no—
114C2×C32—21no—
115C2×C32—32no—
116C2×C32—31no—
117C2×C32—36no—
118C2×C32—33no—
119C2×C32—62no—
120C2×C32—61no—
121C2×C32—92no—
122C2×C32—91no—
123C2×C32—182no—
124C2×C32—181no—
125C2×C32(C3⋊Dic3)⋊C312yes—
126C2×C32C2×(C3⋊D3)⋊C311yescharacteristic
127C2×C32C32×Dic312yes—
128C2×C32C2×C32×D311yescharacteristic
129C2×C9—32no—
130C2×C9—31no—
131C2×C9—62no—
132C2×C9—66no—
133C2×C9—61no—
134C2×C9—63no—
135C22×C3—21no—
136C22×C3—31no—
137C22×C3—61no—
138C22×C3—93no—
139C22×C3#417711yescharacteristic
140C22×C3C3×(C3⋊D3)⋊C311yescharacteristic
141C3×C4—276no—
142C3×C4—272no—
143Dic3—92no—
144Dic3—272no—
145C32—21no—
146C32—31no—
147C32—33no—
148C32—61no—
149C32—91no—
150C32—181no—
151C32C2×(C3⋊Dic3)⋊C311yescharacteristic
152C32C2×C32×Dic311yescharacteristic
153C9—31no—
154C9—61no—
155C9—63no—
156C2×C4—271noSylow
157C2×C3—22no—
158C2×C3—21no—
159C2×C3—32no—
160C2×C3—31no—
161C2×C3—62no—
162C2×C3—61no—
163C2×C3—96no—
164C2×C3—93no—
165C2×C3#6632412yesdirect product factor
166C2×C3C2×#417711yescharacteristic
167C2×C3C3×(C3⋊Dic3)⋊C312yes—
168C2×C3C2×C3×(C3⋊D3)⋊C311yescharacteristic
169C22C3×#417711yescharacteristic
170C4—272no—
171C3—21no—
172C3—31no—
173C3—61no—
174C3—93no—
175C3C2×#6632411yescharacteristic, direct product factor
176C3C2×C3×(C3⋊Dic3)⋊C311yescharacteristic
177C2C3×#6632412yesdirect product factor
178C2C2×C3×#417711yescharacteristic
subgroupG/H
center, hypercenterC22×C3#4177
commutator, hypocenterC3×C9C2×C32×C4
FrattiniC2×C32C2×C32×D3
FittingC22×C3×#731C2
socleC22×C32(C3⋊D3)⋊C3
prime radicalC22×C3×H(3,𝔽3)D3
cosocleC2×C3×C9C22×C32
LCSG ▷ C3×C9
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
2. C3×#66324C2direct (N × H)
3. C2×C3×C3⋊Dic9C3semidirect
5. C2×#66324C3direct (N × H)
6. C2×C3×#731C4semidirect
11. C3×C3⋊Dic9C2×C3semidirect
15. #66324C2×C3direct (N × H)
16. C3×#731C2×C4semidirect
19. C2×C3⋊Dic9C32semidirect
26. C2×C32×C9C3×C4semidirect
30. C2×#731C3×C4semidirect
45. C3⋊Dic9C2×C32semidirect
50. C32×C9C2×C3×C4semidirect
52. #731C2×C3×C4semidirect
79. C2×C3×C9C32×C4semidirect
108. C3×C9C2×C32×C4semidirect
165. C2×C3#66324direct (N × H)
175. C3C2×#66324direct (N × H)
177. C2C3×#66324direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#7312
C3×#663242
C2×C3×C3⋊Dic93
C2×C3×(C3⋊Dic3)⋊C33
C2×#663243