Finite Group Databasebeta

← C2×C3×#66366 #14518444

Subgroups — C2×C3×#66366 #14518444

Order 1944.

656 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 100 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×#66366C211yesmaximal, characteristic, direct product factor
2C2×C33×A4C311yesmaximal, characteristic, complemented
3C23×C3×3−1+2C311yesmaximal, characteristic, complemented
4C2×C3×C22⋊H(3,𝔽3)C311yesmaximal, characteristic, complemented
5C2×C3×#1094C311yesmaximal, characteristic, complemented
6C2×#66366C319yesmaximal, direct product factor
7C2×C3×AΣL(1,𝔽27)—41nomaximal
8C33×A4C2×C311yescharacteristic, complemented
9C22×C3×3−1+2—31no—
10C22×C3×3−1+2—31no—
11C22×C3×3−1+2C2×C311yescharacteristic, complemented
12C3×C22⋊H(3,𝔽3)C2×C311yescharacteristic, complemented
13C3×#1094C2×C311yescharacteristic, complemented
14#66366C2×C319yesdirect product factor
15C3×AΣL(1,𝔽27)—41noSylow
16C23×C33C3211yescharacteristic
17C2×C32×A4—32no—
18C2×C32×A4—31no—
19C2×C32×A4—39no—
20C2×C32×A4—31no—
21C2×C32×A4C3213yescomplemented
22C23×3−1+2—32no—
23C23×3−1+2C3213yescomplemented
24C23×C3×C9—31no—
25C2×C22⋊H(3,𝔽3)—32no—
26C2×C22⋊H(3,𝔽3)C3213yescomplemented
27C2×#1094—32no—
28C2×#1094C3213yescomplemented
29C2×C3×C22⋊C9—31no—
30C2×C34—41no—
31C2×C3×3−1+2—31no—
32C2×C3×3−1+2—31no—
33C2×C3×3−1+2—41no—
34C2×C3×3−1+2A411yescharacteristic, complemented
35C2×C3×H(3,𝔽3)—41no—
36C2×AΣL(1,𝔽27)—49no—
37C22×C33—31no—
38C22×C33—31no—
39C22×C33C2×C3211yescharacteristic
40C32×A4—32no—
41C32×A4—31no—
42C32×A4—39no—
43C32×A4—31no—
44C32×A4C2×C3213yescomplemented
45C22×3−1+2—36no—
46C22×3−1+2—36no—
47C22×3−1+2—32no—
48C22×3−1+2—33no—
49C22×3−1+2—33no—
50C22×3−1+2C2×C3213yescomplemented
51C22×C3×C9—33no—
52C22×C3×C9—33no—
53C22×C3×C9—31no—
54C22⋊H(3,𝔽3)—32no—
55C22⋊H(3,𝔽3)C2×C3213yescomplemented
56#1094—32no—
57#1094C2×C3213yescomplemented
58C3×C22⋊C9—31no—
59C34—41no—
60C3×3−1+2—41no—
61C3×3−1+2C2×A411yescharacteristic, complemented
62C3×H(3,𝔽3)—41no—
63AΣL(1,𝔽27)—49no—
64C23×C32—33no—
65C23×C32H(3,𝔽3)12yes—
66C23×C32H(3,𝔽3)11yescharacteristic
67C23×C32C3311yescharacteristic
68C2×C3×A4—318no—
69C2×C3×A4—39no—
70C2×C3×A4—33no—
71C2×C3×A4—39no—
72C2×C3×A4—33no—
73C2×C3×A4—93no—
74C23×C9—33no—
75C2×C22⋊C9—33no—
76C2×C33—31no—
77C2×C33—31no—
78C2×C33—43no—
79C2×C33—129no—
80C2×C33—122no—
81C2×C33—121no—
82C2×C33—121no—
83C2×C33C3×A411yescharacteristic
84C2×3−1+2—36no—
85C2×3−1+2—36no—
86C2×3−1+2—33no—
87C2×3−1+2—33no—
88C2×3−1+2—32no—
89C2×3−1+2—43no—
90C2×3−1+2—122no—
91C2×3−1+2C3×A413yescomplemented
92C2×H(3,𝔽3)—43no—
93C2×H(3,𝔽3)—122no—
94C2×C3×C9—33no—
95C2×C3×C9—33no—
96C2×C3×C9—31no—
97C2×C3×C9—121no—
98C22×C32—31no—
99C22×C32—31no—
100C22×C32—33no—
101C22×C32—32no—
102C22×C32—31no—
103C22×C32—32no—
104C22×C32—31no—
105C22×C32—93no—
106C22×C32—93no—
107C22×C32C2×H(3,𝔽3)12yes—
108C22×C32C2×H(3,𝔽3)11yescharacteristic
109C22×C32C2×C3311yescharacteristic
110C3×A4—318no—
111C3×A4—39no—
112C3×A4—33no—
113C3×A4—39no—
114C3×A4—33no—
115C3×A4—93no—
116C22×C9—39no—
117C22×C9—39no—
118C22×C9—33no—
119C22⋊C9—33no—
120C33—43no—
121C33—129no—
122C33—122no—
123C33—121no—
124C33—121no—
125C33C2×C3×A411yescharacteristic
1263−1+2—32no—
1273−1+2—43no—
1283−1+2—122no—
1293−1+2C2×C3×A413yescomplemented
130H(3,𝔽3)—43no—
131H(3,𝔽3)—122no—
132C3×C9—31no—
133C3×C9—121no—
134C23×C3—32no—
135C23×C3—31no—
136C23×C3AΣL(1,𝔽27)13yescomplemented
137C23×C3C3×H(3,𝔽3)11yescharacteristic
138C2×A4—39no—
139C2×A4—93no—
140C2×C32—31no—
141C2×C32—31no—
142C2×C32—33no—
143C2×C32—32no—
144C2×C32—32no—
145C2×C32—31no—
146C2×C32—31no—
147C2×C32—93no—
148C2×C32—93no—
149C2×C32—1218no—
150C2×C32—129no—
151C2×C32—129no—
152C2×C32—123no—
153C2×C32—123no—
154C2×C32—363no—
155C2×C32C22⋊H(3,𝔽3)12yes—
156C2×C32C22⋊H(3,𝔽3)11yescharacteristic
157C2×C32C32×A411yescharacteristic
158C2×C9—39no—
159C2×C9—39no—
160C2×C9—33no—
161C2×C9—123no—
162C22×C3—33no—
163C22×C3—31no—
164C22×C3—33no—
165C22×C3—31no—
166C22×C3—32no—
167C22×C3—31no—
168C22×C3—92no—
169C22×C3—91no—
170C22×C3—92no—
171C22×C3—91no—
172C22×C3C2×AΣL(1,𝔽27)13yescomplemented
173C22×C3C2×C3×H(3,𝔽3)11yescharacteristic
174A4—39no—
175A4—93no—
176C32—33no—
177C32—129no—
178C32—1218no—
179C32—129no—
180C32—123no—
181C32—123no—
182C32—363no—
183C32C2×C22⋊H(3,𝔽3)12yes—
184C32C2×C22⋊H(3,𝔽3)11yescharacteristic
185C32C2×C32×A411yescharacteristic
186C9—33no—
187C9—123no—
188C23C3×AΣL(1,𝔽27)11yescharacteristic, complemented, Sylow
189C2×C3—33no—
190C2×C3—33no—
191C2×C3—31no—
192C2×C3—31no—
193C2×C3—32no—
194C2×C3—31no—
195C2×C3—92no—
196C2×C3—91no—
197C2×C3—92no—
198C2×C3—91no—
199C2×C3—129no—
200C2×C3—363no—
201C2×C3#6636613yesdirect product factor
202C2×C3C3×C22⋊H(3,𝔽3)11yescharacteristic
203C22—31no—
204C22—31no—
205C22C2×C3×AΣL(1,𝔽27)11yescharacteristic, complemented
206C3—32no—
207C3—31no—
208C3—129no—
209C3—363no—
210C3C2×#6636613yesdirect product factor
211C3C2×C3×C22⋊H(3,𝔽3)11yescharacteristic
212C2—31no—
213C2—31no—
214C2C3×#6636611yescharacteristic, direct product factor
subgroupG/H
centerC2×C32C22⋊H(3,𝔽3)
commutator, cosocleC22×C32C2×C33
FrattiniC32C2×C32×A4
FittingC23×C3×3−1+2C3
socleC23×C32H(3,𝔽3)
hypercenterC2×C3×3−1+2A4
hypocenterC22C2×C3×AΣL(1,𝔽27)
LCSG ▷ C22×C32 ▷ C22×C3 ▷ C22
UCSC1 ◁ C2×C32 ◁ C2×C33 ◁ C2×C3×3−1+2

Complements

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

NH ≅ G/Ntype
1. C3×#66366C2direct (N × H)
2. C2×C33×A4C3semidirect
3. C23×C3×3−1+2C3semidirect
4. C2×C3×C22⋊H(3,𝔽3)C3semidirect
5. C2×C3×#1094C3semidirect
6. C2×#66366C3direct (N × H)
8. C33×A4C2×C3semidirect
11. C22×C3×3−1+2C2×C3semidirect
12. C3×C22⋊H(3,𝔽3)C2×C3semidirect
13. C3×#1094C2×C3semidirect
14. #66366C2×C3direct (N × H)
21. C2×C32×A4C32semidirect
23. C23×3−1+2C32semidirect
26. C2×C22⋊H(3,𝔽3)C32semidirect
28. C2×#1094C32semidirect
34. C2×C3×3−1+2A4semidirect
44. C32×A4C2×C32semidirect
50. C22×3−1+2C2×C32semidirect
55. C22⋊H(3,𝔽3)C2×C32semidirect
57. #1094C2×C32semidirect
61. C3×3−1+2C2×A4semidirect
91. C2×3−1+2C3×A4semidirect
129. 3−1+2C2×C3×A4semidirect
136. C23×C3AΣL(1,𝔽27)semidirect
172. C22×C3C2×AΣL(1,𝔽27)semidirect
188. C23C3×AΣL(1,𝔽27)semidirect
201. C2×C3#66366direct (N × H)
205. C22C2×C3×AΣL(1,𝔽27)semidirect
210. C3C2×#66366direct (N × H)
214. C2C3×#66366direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#663662
C2×C33×A43
C23×C3×3−1+23
C2×C3×C22⋊H(3,𝔽3)3
C2×C3×#10943
C2×#663663
C2×C3×AΣL(1,𝔽27)4