Finite Group Databasebeta

← C2×D3×C22⋊H(3,𝔽3) #13049299

Subgroups — C2×D3×C22⋊H(3,𝔽3) #13049299

Order 1296.

548 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 70 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
2D3×C22⋊H(3,𝔽3)C212yesmaximal, direct product factor
3C23×C32×D3C311yesmaximal, characteristic, complemented
4C2×C3×D3×A4C313yesmaximal, complemented
5C22×C22⋊H(3,𝔽3)—31nomaximal
6C2×D3×H(3,𝔽3)—41nomaximal
7C3×C22⋊H(3,𝔽3)C2211yescharacteristic, complemented
8C23×C33C2×C311yescharacteristic, complemented
9C22×C32×D3—31no—
10C22×C32×D3—31no—
11C22×C32×D3—32no—
12C22×C32×D3C2×C312yescomplemented
13C2×C32×A4C2×C313yescomplemented
14C3×D3×A4C2×C316yescomplemented
15C2×C22⋊H(3,𝔽3)—21no—
16C2×C22⋊H(3,𝔽3)—23no—
17C2×C22⋊H(3,𝔽3)—32no—
18C2×C22⋊H(3,𝔽3)D311yescharacteristic, direct product factor
19C2×C3×H(3,𝔽3)—41no—
20D3×H(3,𝔽3)—42no—
21C24×C32—31no—
22C23×C3×D3—31no—
23C23×C3×D3C3211yescharacteristic
24C22×C3×A4—33no—
25C2×D3×A4—33no—
26C22×C33—31no—
27C22×C33—31no—
28C22×C33C22×C311yescharacteristic, complemented
29C2×C32×D3—32no—
30C2×C32×D3—31no—
31C2×C32×D3—32no—
32C2×C32×D3—32no—
33C2×C32×D3—32no—
34C2×C32×D3—43no—
35C2×C32×D3A411yescharacteristic, complemented
36C32×A4C22×C313yescomplemented
37C22×H(3,𝔽3)—121no—
38C22⋊H(3,𝔽3)—21no—
39C22⋊H(3,𝔽3)—23no—
40C22⋊H(3,𝔽3)C2×D311yescharacteristic, direct product factor
41C3×H(3,𝔽3)—41noSylow
42C23×C32—21no—
43C23×C32—31no—
44C23×C32—32no—
45C23×C32—61no—
46C23×C32—91no—
47C23×C32—92no—
48C23×C32—91no—
49C23×C32C3×D311yescharacteristic, complemented
50C23×C32C2×C3211yescharacteristic
51C22×C3×D3—33no—
52C22×C3×D3—33no—
53C22×C3×D3—36no—
54C22×C3×D3—31no—
55C22×C3×D3—31no—
56C22×C3×D3—32no—
57C22×C3×D3—32no—
58C22×C3×D3C2×C3212yes—
59C2×C3×A4—23no—
60C2×C3×A4—33no—
61C2×C3×A4—36no—
62C2×C3×A4—63no—
63C2×C3×A4C3×D313yescomplemented
64D3×A4—36no—
65C2×C33—31no—
66C2×C33—31no—
67C2×C33—43no—
68C2×C33C2×A411yescharacteristic, complemented
69C32×D3—32no—
70C32×D3—46no—
71C32×D3C2×A412yescomplemented
72C2×H(3,𝔽3)—41no—
73C2×H(3,𝔽3)—83no—
74C2×H(3,𝔽3)—81no—
75C2×H(3,𝔽3)—122no—
76C24×C3—31no—
77C24×C3—91no—
78C23×D3H(3,𝔽3)11yescharacteristic, complemented
79C22×A4—93no—
80C22×C32—21no—
81C22×C32—31no—
82C22×C32—31no—
83C22×C32—31no—
84C22×C32—33no—
85C22×C32—33no—
86C22×C32—31no—
87C22×C32—31no—
88C22×C32—31no—
89C22×C32—61no—
90C22×C32—61no—
91C22×C32—61no—
92C22×C32—63no—
93C22×C32—63no—
94C22×C32—92no—
95C22×C32—92no—
96C22×C32—92no—
97C22×C32—92no—
98C22×C32—91no—
99C22×C32—123no—
100C22×C32C2×C3×D311yescharacteristic, complemented
101C22×C32C22×C3211yescharacteristic
102C2×C3×D3—36no—
103C2×C3×D3—32no—
104C2×C3×D3—33no—
105C2×C3×D3—36no—
106C2×C3×D3—36no—
107C2×C3×D3—36no—
108C2×C3×D3—31no—
109C2×C3×D3—32no—
110C2×C3×D3—32no—
111C2×C3×D3—32no—
112C2×C3×D3—31no—
113C2×C3×D3—123no—
114C2×C3×D3C3×A411yescharacteristic
115C3×A4—23no—
116C3×A4—33no—
117C3×A4—63no—
118C3×A4C2×C3×D313yescomplemented
119C33—43no—
120C33C22×A411yescharacteristic, complemented
121H(3,𝔽3)—41no—
122H(3,𝔽3)—83no—
123H(3,𝔽3)—81no—
124C23×C3—21no—
125C23×C3—32no—
126C23×C3—31no—
127C23×C3—61no—
128C23×C3—92no—
129C23×C3—91no—
130C23×C3—92no—
131C23×C3—91no—
132C23×C3—93no—
133C23×C3—96no—
134C23×C3—93no—
135C23×C3C2×H(3,𝔽3)11yescharacteristic, complemented
136C23×C3C32×D311yescharacteristic
137C22×D3—31no—
138C22×D3—31no—
139C22×D3—32no—
140C22×D3C2×H(3,𝔽3)12yescomplemented
141C2×A4—33no—
142C2×A4—63no—
143C2×A4—96no—
144C2×C32—21no—
145C2×C32—31no—
146C2×C32—31no—
147C2×C32—33no—
148C2×C32—33no—
149C2×C32—31no—
150C2×C32—31no—
151C2×C32—31no—
152C2×C32—32no—
153C2×C32—43no—
154C2×C32—61no—
155C2×C32—61no—
156C2×C32—61no—
157C2×C32—63no—
158C2×C32—63no—
159C2×C32—83no—
160C2×C32—92no—
161C2×C32—126no—
162C2×C32—123no—
163C2×C32—243no—
164C2×C32D3×A411yescharacteristic, complemented
165C2×C32C2×C3×A411yescharacteristic
166C3×D3—36no—
167C3×D3—32no—
168C3×D3—32no—
169C3×D3—126no—
170C3×D3C2×C3×A412yes—
171C24—31noSylow
172C22×C3—21no—
173C22×C3—31no—
174C22×C3—31no—
175C22×C3—33no—
176C22×C3—33no—
177C22×C3—31no—
178C22×C3—31no—
179C22×C3—31no—
180C22×C3—31no—
181C22×C3—61no—
182C22×C3—61no—
183C22×C3—63no—
184C22×C3—63no—
185C22×C3—61no—
186C22×C3—91no—
187C22×C3—92no—
188C22×C3—92no—
189C22×C3—92no—
190C22×C3—96no—
191C22×C3—96no—
192C22×C3—96no—
193C22×C3—96no—
194C22×C3—92no—
195C22×C3—93no—
196C22×C3—91no—
197C22×C3—363no—
198C22×C3C22×H(3,𝔽3)11yescharacteristic, complemented
199C22×C3C2×C32×D311yescharacteristic
200C2×D3—32no—
201C2×D3—31no—
202C2×D3—32no—
203C2×D3—32no—
204C2×D3—32no—
205C2×D3C22⋊H(3,𝔽3)11yescharacteristic, direct product factor
206A4—33no—
207A4—63no—
208C32—21no—
209C32—31no—
210C32—43no—
211C32—61no—
212C32—83no—
213C32—123no—
214C32—243no—
215C32C2×D3×A411yescharacteristic, complemented
216C32C22×C3×A411yescharacteristic
217C23—32no—
218C23—91no—
219C23—92no—
220C23—91no—
221C23D3×H(3,𝔽3)11yescharacteristic, complemented
222C2×C3—21no—
223C2×C3—33no—
224C2×C3—33no—
225C2×C3—31no—
226C2×C3—31no—
227C2×C3—32no—
228C2×C3—31no—
229C2×C3—31no—
230C2×C3—31no—
231C2×C3—61no—
232C2×C3—61no—
233C2×C3—63no—
234C2×C3—63no—
235C2×C3—61no—
236C2×C3—92no—
237C2×C3—96no—
238C2×C3—92no—
239C2×C3—123no—
240C2×C3—243no—
241C2×C3—366no—
242C2×C3C2×C22⋊H(3,𝔽3)11yescharacteristic, complemented
243C2×C3C3×D3×A411yescharacteristic
244D3—32no—
245D3C2×C22⋊H(3,𝔽3)12yesdirect product factor
246C22—31no—
247C22—31no—
248C22—31no—
249C22—92no—
250C22—92no—
251C22—92no—
252C22—92no—
253C22—91no—
254C22C2×D3×H(3,𝔽3)11yescharacteristic, complemented
255C3—21no—
256C3—31no—
257C3—61no—
258C3—123no—
259C3—243no—
260C3C22×C22⋊H(3,𝔽3)11yescharacteristic, complemented
261C3C2×C3×D3×A411yescharacteristic
262C2—31no—
263C2—31no—
264C2—32no—
265C2—92no—
266C2D3×C22⋊H(3,𝔽3)11yescharacteristic, direct product factor
subgroupG/H
centerC2×C3C3×D3×A4
commutator, cosocleC22×C32C22×C32
FrattiniC3C2×C3×D3×A4
FittingC23×C33C2×C3
socleC23×C32C2×C32
hypercenterC2×C32D3×A4
hypocenterC22×C3C22×H(3,𝔽3)
LCSG ▷ C22×C32 ▷ C22×C3
UCSC1 ◁ C2×C3 ◁ C2×C32

Complements

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

NH ≅ G/Ntype
1. C2×C3×C22⋊H(3,𝔽3)C2semidirect
2. D3×C22⋊H(3,𝔽3)C2direct (N × H)
3. C23×C32×D3C3semidirect
4. C2×C3×D3×A4C3semidirect
7. C3×C22⋊H(3,𝔽3)C22semidirect
8. C23×C33C2×C3semidirect
12. C22×C32×D3C2×C3semidirect
13. C2×C32×A4C2×C3semidirect
14. C3×D3×A4C2×C3semidirect
18. C2×C22⋊H(3,𝔽3)D3direct (N × H)
28. C22×C33C22×C3semidirect
35. C2×C32×D3A4semidirect
36. C32×A4C22×C3semidirect
40. C22⋊H(3,𝔽3)C2×D3direct (N × H)
49. C23×C32C3×D3semidirect
63. C2×C3×A4C3×D3semidirect
68. C2×C33C2×A4semidirect
71. C32×D3C2×A4semidirect
78. C23×D3H(3,𝔽3)semidirect
100. C22×C32C2×C3×D3semidirect
118. C3×A4C2×C3×D3semidirect
120. C33C22×A4semidirect
135. C23×C3C2×H(3,𝔽3)semidirect
140. C22×D3C2×H(3,𝔽3)semidirect
164. C2×C32D3×A4semidirect
198. C22×C3C22×H(3,𝔽3)semidirect
205. C2×D3C22⋊H(3,𝔽3)direct (N × H)
215. C32C2×D3×A4semidirect
221. C23D3×H(3,𝔽3)semidirect
242. C2×C3C2×C22⋊H(3,𝔽3)semidirect
245. D3C2×C22⋊H(3,𝔽3)direct (N × H)
254. C22C2×D3×H(3,𝔽3)semidirect
260. C3C22×C22⋊H(3,𝔽3)semidirect
266. C2D3×C22⋊H(3,𝔽3)direct (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
D3×C22⋊H(3,𝔽3)2
C23×C32×D33
C2×C3×D3×A43
C22×C22⋊H(3,𝔽3)3
C2×D3×H(3,𝔽3)4