Finite Group Databasebeta

← #13048781

Subgroups — #13048781

Order 1296.

699 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 699 conjugacy classes of subgroups — too many to draw. Only its 63 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×#272C211yesmaximal, characteristic, complemented
2C2×#66432C211yesmaximal, characteristic, complemented
3#10619977C212yesmaximal, complemented
4#10619904C212yesmaximal, complemented
5#10619903C211yesmaximal, characteristic, complemented
6D3×#638—32nomaximal
7#88677—31nomaximal
8C22×C3×H(3,𝔽3)C2211yescharacteristic, complemented
9C2×C3×#272—22no—
10C2×C3×#272C2211yescharacteristic
11C2×C3×#272C2211yescharacteristic, complemented
12C2×C3⋊((C3⋊D3)⋊C3)C2212yescomplemented
13#66432—22no—
14C3⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
15C22×C32×D3—32no—
16C22×C32×D3—61no—
17C2×D3×C3⋊D3—32no—
18#6312D312yescomplemented
19C3×#638—32no—
20#6291—32no—
21#6290—32no—
22D3×C3⋊Dic3—32no—
23C22×#272D311yescharacteristic, complemented
24C2×((C3⋊D3)⋊C3)⋊C2—31no—
25#638⋊C3—32no—
26#638⋊C3—62no—
27#6200—32no—
28#6199—31no—
29C2×C3×H(3,𝔽3)—21no—
30C2×C3×H(3,𝔽3)C2311yescharacteristic
31C3×#272—21no—
32C3×#272D412yescomplemented
33C3⋊((C3⋊D3)⋊C3)—22no—
34C2×#638—91no—
35D3×#67—92no—
36D3×#67—96no—
37C22×C33—21no—
38C22×C33C2×D312yescomplemented
39C2×C3⋊(C3⋊D3)C2×D312yescomplemented
40C2×C3×C3⋊D3—32no—
41C2×C32×D3—32no—
42C2×C32×D3—32no—
43C2×C32×D3—64no—
44C2×C32×D3—61no—
45C2×C32×D3—61no—
46C2×C32×D3—64no—
47D3×C3⋊D3—64no—
48C3⋊(C3⋊Dic3)C2×D312yescomplemented
49C3×C3⋊Dic3—32no—
50C22×H(3,𝔽3)—22no—
51C22×H(3,𝔽3)—41no—
52C22×H(3,𝔽3)C2×D311yescharacteristic, complemented
53C2×#272—22no—
54C2×#272C2×D311yescharacteristic
55C2×#272C2×D311yescharacteristic, complemented
56C2×(C3⋊D3)⋊C3—32no—
57C2×(C3⋊D3)⋊C3—62no—
58((C3⋊D3)⋊C3)⋊C2—62no—
59(C3⋊Dic3)⋊C3—32no—
60(C3⋊Dic3)⋊C3—62no—
61C3×H(3,𝔽3)C2×D411yescharacteristic, complemented, Sylow
62C23×C32—91no—
63C22×C3⋊D3—91no—
64C22×C3×D3—36no—
65C22×C3×D3—32no—
66C22×C3×D3—61no—
67C22×C3×D3—63no—
68C2×D32—92no—
69C2×D32—96no—
70#638—32no—
71#638—32no—
72#638—62no—
73#638—96no—
74#638—92no—
75C2×C3⋊Dic3—91no—
76C3×#67—96no—
77C3×#67—92no—
78#626—92no—
79#626—96no—
80#625—92no—
81#625—96no—
82D3×Dic3—92no—
83D3×Dic3—96no—
84C2×C33—22no—
85C2×C33—22no—
86C2×C33—21no—
87C2×C33C22×D312yes—
88C3⋊(C3⋊D3)—22no—
89C3×C3⋊D3—62no—
90C32×D3—34no—
91C32×D3—62no—
92C32×D3—62no—
93C32×D3—62no—
94C2×H(3,𝔽3)—21no—
95C2×H(3,𝔽3)—22no—
96C2×H(3,𝔽3)—41no—
97C2×H(3,𝔽3)—42no—
98C2×H(3,𝔽3)—42no—
99C2×H(3,𝔽3)C22×D311yescharacteristic
100#272—21no—
101#272#6712yescomplemented
102(C3⋊D3)⋊C3—62no—
103(C3⋊D3)⋊C3—122no—
104C2×#67—273no—
105C2×#67—271no—
106D3×D4—272no—
107C22×C32—21no—
108C22×C32—22no—
109C22×C32—36no—
110C22×C32—41no—
111C22×C32—63no—
112C22×C32—91no—
113C22×C32—91no—
114C22×C32—182no—
115C22×C32D3211yescharacteristic
116C22×C32D3212yescomplemented
117C2×C3⋊D3—32no—
118C2×C3⋊D3—32no—
119C2×C3⋊D3—62no—
120C2×C3⋊D3—96no—
121C2×C3⋊D3—182no—
122C2×C3×D3—36no—
123C2×C3×D3—32no—
124C2×C3×D3—36no—
125C2×C3×D3—32no—
126C2×C3×D3—61no—
127C2×C3×D3—61no—
128C2×C3×D3—63no—
129C2×C3×D3—63no—
130C2×C3×D3—64no—
131C2×C3×D3—612no—
132C2×C3×D3—64no—
133C2×C3×D3—612no—
134C2×C3×D3—96no—
135C2×C3×D3—92no—
136D32—184no—
137D32—1812no—
138C3⋊Dic3—32no—
139C3⋊Dic3—32no—
140C3⋊Dic3—62no—
141C3⋊Dic3—96no—
142C3×Dic3—96no—
143C3×Dic3—92no—
144C33—21no—
145C33D3×D412yescomplemented
146H(3,𝔽3)—22no—
147H(3,𝔽3)—41no—
148H(3,𝔽3)C2×#6711yescharacteristic, complemented
149C23×C3—93no—
150C23×C3—91no—
151C22×D3—32no—
152C22×D3—61no—
153C22×D3—272no—
154C22×D3—273no—
155C22×D3—271no—
156C3×D4—272no—
157#67—96no—
158#67—92no—
159#67—272no—
160#67—276no—
161#67—272no—
162#67—272no—
163#67—542no—
164C2×Dic3—273no—
165C2×Dic3—271no—
166D12—272no—
167C4×D3—272no—
168C2×C32—21no—
169C2×C32—22no—
170C2×C32—22no—
171C2×C32—22no—
172C2×C32—21no—
173C2×C32—36no—
174C2×C32—42no—
175C2×C32—42no—
176C2×C32—41no—
177C2×C32—66no—
178C2×C32—66no—
179C2×C32—63no—
180C2×C32—92no—
181C2×C32—181no—
182C2×C32C2×D3212yes—
183C2×C32C2×D3211yescharacteristic
184C3⋊D3—62no—
185C3⋊D3—62no—
186C3⋊D3—122no—
187C3⋊D3—186no—
188C3×D3—312no—
189C3×D3—34no—
190C3×D3—62no—
191C3×D3—66no—
192C3×D3—62no—
193C3×D3—66no—
194C3×D3—66no—
195C3×D3—62no—
196C3×D3—186no—
197C3×D3—182no—
198C2×D4—811noSylow
199C22×C3—32no—
200C22×C3—61no—
201C22×C3—62no—
202C22×C3—93no—
203C22×C3—91no—
204C22×C3—93no—
205C22×C3—91no—
206C22×C3—121no—
207C22×C3—186no—
208C22×C3—182no—
209C22×C3—272no—
210C22×C3((C3⋊D3)⋊C3)⋊C213yescomplemented
211C22×C3#111311yescharacteristic
212C2×D3—32no—
213C2×D3—32no—
214C2×D3—61no—
215C2×D3—61no—
216C2×D3—64no—
217C2×D3—64no—
218C2×D3—96no—
219C2×D3—92no—
220C2×D3—272no—
221C2×D3—542no—
222C2×D3—544no—
223C2×D3—546no—
224C2×D3—542no—
225C3×C4—272no—
226Dic3—96no—
227Dic3—92no—
228Dic3—272no—
229Dic3—542no—
230C32—21no—
231C32—22no—
232C32—36no—
233C32—41no—
234C32—63no—
235C32D3×#6712yescomplemented
236C32#381111yescharacteristic
237C23—91no—
238C23—811no—
239D4—272no—
240D4—812no—
241C2×C4—811no—
242C2×C3—23no—
243C2×C3—21no—
244C2×C3—32no—
245C2×C3—62no—
246C2×C3—62no—
247C2×C3—62no—
248C2×C3—61no—
249C2×C3—96no—
250C2×C3—92no—
251C2×C3—122no—
252C2×C3—122no—
253C2×C3—121no—
254C2×C3—183no—
255C2×C3—181no—
256C2×C3—542no—
257C2×C3C2×((C3⋊D3)⋊C3)⋊C213yes—
258C2×C3C2×#111311yescharacteristic
259D3—34no—
260D3—62no—
261D3—62no—
262D3—62no—
263D3—186no—
264D3—182no—
265D3—542no—
266D3—1082no—
267C22—91no—
268C22—91no—
269C22—182no—
270C22—272no—
271C22—1622no—
272C22#6643211yescharacteristic, complemented
273C4—272no—
274C3—32no—
275C3—61no—
276C3—62no—
277C3—121no—
278C3#8867713yescomplemented
279C3#8904911yescharacteristic
280C2—21no—
281C2—92no—
282C2—181no—
283C2—542no—
284C2C2×#6643211yescharacteristic
subgroupG/H
centerC2C2×#66432
commutator, cosocleC2×C3×H(3,𝔽3)C23
FrattiniC2×C3C2×#1113
FittingC22×C3×H(3,𝔽3)C22
socleC2×C32C2×D32
hypercenterC22#66432
hypocenterC3×H(3,𝔽3)C2×D4
LCSG ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3)
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3×#272C2semidirect
2. C2×#66432C2semidirect
3. #10619977C2semidirect
4. #10619904C2semidirect
5. #10619903C2semidirect
8. C22×C3×H(3,𝔽3)C22semidirect
11. C2×C3×#272C22semidirect
12. C2×C3⋊((C3⋊D3)⋊C3)C22semidirect
14. C3⋊((C3⋊Dic3)⋊C3)C22semidirect
18. #6312D3semidirect
23. C22×#272D3semidirect
32. C3×#272D4semidirect
38. C22×C33C2×D3semidirect
39. C2×C3⋊(C3⋊D3)C2×D3semidirect
48. C3⋊(C3⋊Dic3)C2×D3semidirect
52. C22×H(3,𝔽3)C2×D3semidirect
55. C2×#272C2×D3semidirect
61. C3×H(3,𝔽3)C2×D4semidirect
101. #272#67semidirect
116. C22×C32D32semidirect
145. C33D3×D4semidirect
148. H(3,𝔽3)C2×#67semidirect
210. C22×C3((C3⋊D3)⋊C3)⋊C2semidirect
235. C32D3×#67semidirect
272. C22#66432semidirect
278. C3#88677semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#2722
C2×#664322
#106199772
#106199042
#106199032
D3×#6383
#886773