Finite Group Databasebeta

← C2×C3⋊#6258 #13049289

Subgroups — C2×C3⋊#6258 #13049289

Order 1296.

668 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C2×C3×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C3⋊#6258C212yesmaximal, direct product factor
3C2×C3×C3⋊S4—33nomaximal
4C2×C3×#638—31nomaximal
5C2×#6258—39nomaximal
6C2×C3⋊#272—41nomaximal
7C3×C22⋊H(3,𝔽3)C2211yescharacteristic, complemented
8C23×C33D311yescharacteristic, complemented
9C22×C3×C3⋊D3—31no—
10C2×C32×A4D313yescomplemented
11C3×C3⋊S4—36no—
12C3×#638—32no—
13C3×#638—32no—
14C2×C3×C3⋊Dic3—31no—
15C2×C22⋊H(3,𝔽3)D319yescomplemented
16#6258—318no—
17C2×C3×H(3,𝔽3)—41no—
18C3⋊#272—42no—
19C2×C3⋊S4—33no—
20C2×C3×S4—99no—
21C2×#638—31no—
22C2×C3×#67—91no—
23C2×C3×#67—93no—
24C22×C33—31no—
25C22×C33—31no—
26C22×C33C2×D311yescharacteristic, complemented
27C2×C3×C3⋊D3—32no—
28C2×C3×C3⋊D3—61no—
29C2×C3×C3⋊D3—61no—
30C2×C3×C3⋊D3—123no—
31C32×A4C2×D313yescomplemented
32C3×C3⋊Dic3—32no—
33C2×#272—129no—
34C22⋊H(3,𝔽3)C2×D319yescomplemented
35C3×H(3,𝔽3)—41noSylow
36C23×C32—31no—
37C23×C32—61no—
38C23×C32C3⋊D311yescharacteristic
39C23×C32C3⋊D313yescomplemented
40C22×C3⋊D3—31no—
41C22×C3×D3—91no—
42C22×C3×D3—93no—
43C2×C3×A4—33no—
44C2×C3×A4—63no—
45C2×C3×A4C3⋊D319yescomplemented
46C3⋊S4—36no—
47C3×S4—918no—
48#638—32no—
49#638—32no—
50C2×C3⋊Dic3—31no—
51C3×#67—92no—
52C3×#67—96no—
53C3×#67—92no—
54C3×#67—96no—
55C2×C3×Dic3—91no—
56C2×C3×Dic3—93no—
57C2×C33—31no—
58C2×C33—31no—
59C2×C33—43no—
60C2×C33S411yescharacteristic, complemented
61C3×C3⋊D3—62no—
62C3×C3⋊D3—126no—
63C2×H(3,𝔽3)—49no—
64#272—1218no—
65C2×S4—99no—
66C2×C3×D4—271no—
67C2×#67—93no—
68C2×#67—91no—
69C22×C32—31no—
70C22×C32—31no—
71C22×C32—31no—
72C22×C32—31no—
73C22×C32—33no—
74C22×C32—31no—
75C22×C32—33no—
76C22×C32—61no—
77C22×C32—61no—
78C22×C32—61no—
79C22×C32—63no—
80C22×C32—63no—
81C22×C32C2×C3⋊D311yescharacteristic
82C22×C32C2×C3⋊D313yescomplemented
83C2×C3⋊D3—32no—
84C2×C3⋊D3—61no—
85C2×C3⋊D3—61no—
86C2×C3⋊D3—123no—
87C2×C3×D3—92no—
88C2×C3×D3—96no—
89C2×C3×D3—181no—
90C2×C3×D3—183no—
91C2×C3×D3—181no—
92C2×C3×D3—183no—
93C2×C3×D3—369no—
94C3×A4—33no—
95C3×A4—63no—
96C3×A4C2×C3⋊D319yescomplemented
97C3⋊Dic3—32no—
98C3×Dic3—92no—
99C3×Dic3—96no—
100C33—43no—
101C33C2×S411yescharacteristic, complemented
102H(3,𝔽3)—49no—
103C23×C3—21no—
104C23×C3—33no—
105C23×C3—271no—
106C23×C3#27211yescharacteristic, complemented
107C23×C3C3⋊(C3⋊D3)11yescharacteristic
108C22×D3—93no—
109C22×D3—91no—
110C2×A4—39no—
111S4—918no—
112C3×D4—272no—
113C3×D4—272no—
114C2×C3×C4—271no—
115#67—92no—
116#67—96no—
117#67—96no—
118#67—92no—
119C2×Dic3—93no—
120C2×Dic3—91no—
121C2×C32—31no—
122C2×C32—31no—
123C2×C32—31no—
124C2×C32—31no—
125C2×C32—31no—
126C2×C32—33no—
127C2×C32—33no—
128C2×C32—49no—
129C2×C32—61no—
130C2×C32—61no—
131C2×C32—61no—
132C2×C32—63no—
133C2×C32—63no—
134C2×C32—123no—
135C2×C32—243no—
136C2×C32C3⋊S411yescharacteristic
137C2×C32C3⋊S413yescomplemented
138C3⋊D3—62no—
139C3⋊D3—126no—
140C3×D3—182no—
141C3×D3—186no—
142C3×D3—3618no—
143C2×D4—271noSylow
144C22×C3—21no—
145C22×C3—31no—
146C22×C3—31no—
147C22×C3—33no—
148C22×C3—31no—
149C22×C3—31no—
150C22×C3—33no—
151C22×C3—33no—
152C22×C3—61no—
153C22×C3—63no—
154C22×C3—63no—
155C22×C3—61no—
156C22×C3—272no—
157C22×C3—541no—
158C22×C3—541no—
159C22×C3C2×#27211yescharacteristic, complemented
160C22×C3C2×C3⋊(C3⋊D3)11yescharacteristic
161C2×D3—96no—
162C2×D3—92no—
163C2×D3—183no—
164C2×D3—181no—
165C2×D3—183no—
166C2×D3—181no—
167C2×D3—369no—
168A4—39no—
169C3×C4—272no—
170Dic3—96no—
171Dic3—92no—
172C32—31no—
173C32—49no—
174C32—61no—
175C32—123no—
176C32—243no—
177C32C2×C3⋊S411yescharacteristic
178C32C2×C3⋊S413yescomplemented
179C23—271no—
180C23C3⋊#27211yescharacteristic, complemented
181D4—272no—
182D4—272no—
183C2×C4—271no—
184C2×C3—21no—
185C2×C3—33no—
186C2×C3—33no—
187C2×C3—31no—
188C2×C3—31no—
189C2×C3—31no—
190C2×C3—31no—
191C2×C3—33no—
192C2×C3—61no—
193C2×C3—61no—
194C2×C3—63no—
195C2×C3—63no—
196C2×C3—129no—
197C2×C3—542no—
198C2×C3#625811yescharacteristic, complemented
199C2×C3C3⋊(C3⋊S4)11yescharacteristic
200D3—186no—
201D3—182no—
202D3—3618no—
203C22—31no—
204C22—31no—
205C22—272no—
206C22—541no—
207C22—541no—
208C22C2×C3⋊#27211yescharacteristic, complemented
209C4—272no—
210C3—21no—
211C3—33no—
212C3—129no—
213C3C2×#625811yescharacteristic, complemented
214C3C2×C3⋊(C3⋊S4)11yescharacteristic
215C2—31no—
216C2—31no—
217C2—542no—
218C2C3⋊#625811yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3C3⋊(C3⋊S4)
commutator, cosocle, hypocenterC3×C22⋊H(3,𝔽3)C22
FrattiniC3C2×C3⋊(C3⋊S4)
FittingC23×C33D3
socleC23×C32C3⋊D3
LCSG ▷ C3×C22⋊H(3,𝔽3)
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×C22⋊H(3,𝔽3)C2semidirect
2. C3⋊#6258C2direct (N × H)
7. C3×C22⋊H(3,𝔽3)C22semidirect
8. C23×C33D3semidirect
10. C2×C32×A4D3semidirect
15. C2×C22⋊H(3,𝔽3)D3semidirect
26. C22×C33C2×D3semidirect
31. C32×A4C2×D3semidirect
34. C22⋊H(3,𝔽3)C2×D3semidirect
39. C23×C32C3⋊D3semidirect
45. C2×C3×A4C3⋊D3semidirect
60. C2×C33S4semidirect
82. C22×C32C2×C3⋊D3semidirect
96. C3×A4C2×C3⋊D3semidirect
101. C33C2×S4semidirect
106. C23×C3#272semidirect
137. C2×C32C3⋊S4semidirect
159. C22×C3C2×#272semidirect
178. C32C2×C3⋊S4semidirect
180. C23C3⋊#272semidirect
198. C2×C3#6258semidirect
208. C22C2×C3⋊#272semidirect
213. C3C2×#6258semidirect
218. C2C3⋊#6258direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊H(3,𝔽3)2
C3⋊#62582
C2×C3×C3⋊S43
C2×C3×#6383
C2×#62583
C2×C3⋊#2724