Finite Group Databasebeta

← #13048747

Subgroups — #13048747

Order 1296.

499 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 59 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×C4×#272C211yesmaximal, characteristic, complemented
2C4⋊(C3⋊((C3⋊D3)⋊C3))C211yesmaximal, characteristic, complemented
3#10619956C211yesmaximal, characteristic, complemented
4#10619904C212yesmaximal, complemented
5#10619901C212yesmaximal, complemented
6#89022—31nomaximal
7#89021—31nomaximal
8#88649—31nomaximal
9C2×C3×#272C2211yescharacteristic
10C2×C3⋊((C3⋊D3)⋊C3)C2212yes—
11C3×C4×H(3,𝔽3)C2211yescharacteristic
12C3×#1084C2211yescharacteristic
13C3⋊((C3⋊Dic3)⋊C3)C2212yes—
14C4⋊(C3⋊(C3⋊D3))D311yescharacteristic, complemented
15#6308D311yescharacteristic, complemented
16C3×C4⋊(C3⋊D3)—31no—
17C3×#634—31no—
18C32×C4×D3—31no—
19C32×C4×D3—31no—
20C32×C4×D3—61no—
21#6291—32no—
22#6290—32no—
23#6289—32no—
24Dic3×C3⋊D3—32no—
25C4×#272D311yescharacteristic, complemented
26C4⋊((C3⋊D3)⋊C3)—31no—
27C4⋊((C3⋊D3)⋊C3)—61no—
28#634⋊C3—31no—
29#634⋊C3—61no—
30#6200—32no—
31((C3⋊D3)⋊C3)⋊C4—32no—
32C2×C3×H(3,𝔽3)C2311yescharacteristic
33C3×#272—21no—
34C3⋊((C3⋊D3)⋊C3)—22no—
35#3828—91no—
36#3799—91no—
37#3799—93no—
38#3795—91no—
39#3795—93no—
40C2×C3⋊(C3⋊D3)C2×D312yes—
41C2×C3×C3⋊D3—32no—
42C2×C32×D3—31no—
43C2×C32×D3—31no—
44C2×C32×D3—61no—
45C33×C4—21no—
46C33×C4C2×D311yescharacteristic
47C33×C4C2×D311yescharacteristic
48C3⋊(C3⋊Dic3)C2×D312yes—
49C3×C3⋊Dic3—32no—
50C32×Dic3—31no—
51C32×Dic3—31no—
52C32×Dic3—61no—
53C2×#272C2×D311yescharacteristic
54C2×(C3⋊D3)⋊C3—32no—
55C2×(C3⋊D3)⋊C3—62no—
56C4×H(3,𝔽3)—21no—
57C4×H(3,𝔽3)—21no—
58C4×H(3,𝔽3)—41no—
59C4×H(3,𝔽3)C2×D311yescharacteristic
60#1084C2×D311yescharacteristic
61(C3⋊Dic3)⋊C3—32no—
62(C3⋊Dic3)⋊C3—62no—
63C3×H(3,𝔽3)#4111yescharacteristic, complemented, Sylow
64C2×C32×C4—91no—
65#638—92no—
66C4⋊(C3⋊D3)—31no—
67C4⋊(C3⋊D3)—31no—
68C4⋊(C3⋊D3)—61no—
69C4⋊(C3⋊D3)—93no—
70C4×C3⋊D3—92no—
71#634—31no—
72#634—31no—
73#634—61no—
74#634—93no—
75C3×D12—93no—
76C3×D12—91no—
77C3×C4×D3—33no—
78C3×C4×D3—33no—
79C3×C4×D3—31no—
80C3×C4×D3—31no—
81C3×C4×D3—61no—
82C3×C4×D3—63no—
83C3×Dic6—93no—
84C3×Dic6—91no—
85#626—92no—
86#626—96no—
87#625—92no—
88#625—96no—
89#624—92no—
90#624—96no—
91D3×Dic3—92no—
92D3×Dic3—96no—
93C2×C33—21no—
94C2×C33C22×D311yescharacteristic
95C2×C33C22×D311yescharacteristic
96C3⋊(C3⋊D3)—22no—
97C3×C3⋊D3—62no—
98C32×D3—62no—
99C32×D3—61no—
100C32×D3—61no—
101C2×H(3,𝔽3)—21no—
102C2×H(3,𝔽3)—21no—
103C2×H(3,𝔽3)—41no—
104C2×H(3,𝔽3)C22×D311yescharacteristic
105#272—21no—
106(C3⋊D3)⋊C3—62no—
107(C3⋊D3)⋊C3—122no—
108#239—271no—
109#237—271no—
110#235—273no—
111#235—271no—
112C22×C32—91no—
113C2×C3⋊D3—32no—
114C2×C3⋊D3—32no—
115C2×C3⋊D3—62no—
116C2×C3⋊D3—96no—
117C2×C3×D3—33no—
118C2×C3×D3—33no—
119C2×C3×D3—31no—
120C2×C3×D3—31no—
121C2×C3×D3—61no—
122C2×C3×D3—63no—
123C2×C3×D3—96no—
124C2×C3×D3—92no—
125C32×C4—21no—
126C32×C4—21no—
127C32×C4—21no—
128C32×C4—33no—
129C32×C4—33no—
130C32×C4—41no—
131C32×C4—63no—
132C32×C4—91no—
133C32×C4D3211yescharacteristic
134C32×C4D3211yescharacteristic
135C32×C4D3211yescharacteristic
136C3⋊Dic3—32no—
137C3⋊Dic3—32no—
138C3⋊Dic3—62no—
139C3⋊Dic3—96no—
140C3×Dic3—33no—
141C3×Dic3—33no—
142C3×Dic3—31no—
143C3×Dic3—31no—
144C3×Dic3—61no—
145C3×Dic3—63no—
146C3×Dic3—96no—
147C3×Dic3—92no—
148C33—21no—
149C33#23711yescharacteristic, complemented
150C33#23911yescharacteristic, complemented
151H(3,𝔽3)—21no—
152H(3,𝔽3)—21no—
153H(3,𝔽3)—41no—
154H(3,𝔽3)#23511yescharacteristic, complemented
155C3×Q8—271no—
156C3×D4—271no—
157C2×C3×C4—93no—
158C2×C3×C4—91no—
159#67—276no—
160#67—272no—
161#67—272no—
162C2×Dic3—272no—
163D12—93no—
164D12—91no—
165D12—271no—
166D12—272no—
167D12—541no—
168C4×D3—31no—
169C4×D3—31no—
170C4×D3—61no—
171C4×D3—272no—
172C4×D3—276no—
173C4×D3—272no—
174Dic6—93no—
175Dic6—91no—
176Dic6—271no—
177Dic6—541no—
178C2×C32—21no—
179C2×C32—21no—
180C2×C32—21no—
181C2×C32—33no—
182C2×C32—33no—
183C2×C32—41no—
184C2×C32—63no—
185C2×C32—181no—
186C2×C32C2×D3211yescharacteristic
187C2×C32C2×D3211yescharacteristic
188C2×C32C2×D3211yescharacteristic
189C3⋊D3—62no—
190C3⋊D3—62no—
191C3⋊D3—122no—
192C3⋊D3—186no—
193C3×D3—62no—
194C3×D3—63no—
195C3×D3—63no—
196C3×D3—66no—
197C3×D3—61no—
198C3×D3—61no—
199C3×D3—186no—
200C3×D3—182no—
201#41—811noSylow
202C22×C3—93no—
203C22×C3—91no—
204C22×C3—272no—
205C2×D3—31no—
206C2×D3—31no—
207C2×D3—61no—
208C2×D3—96no—
209C2×D3—92no—
210C2×D3—272no—
211C2×D3—542no—
212C3×C4—31no—
213C3×C4—31no—
214C3×C4—61no—
215C3×C4—61no—
216C3×C4—61no—
217C3×C4—93no—
218C3×C4—91no—
219C3×C4—121no—
220C3×C4—272no—
221C3×C4((C3⋊D3)⋊C3)⋊C213yes—
222C3×C4#111311yescharacteristic
223Dic3—31no—
224Dic3—31no—
225Dic3—61no—
226Dic3—96no—
227Dic3—92no—
228Dic3—272no—
229Dic3—542no—
230C32—21no—
231C32—21no—
232C32—21no—
233C32—33no—
234C32—33no—
235C32—41no—
236C32—63no—
237C32#379511yescharacteristic, complemented
238C32#379611yescharacteristic
239C32#379911yescharacteristic, complemented
240Q8—271no—
241D4—271no—
242D4—812no—
243C2×C4—91no—
244C2×C4—812no—
245C2×C3—31no—
246C2×C3—31no—
247C2×C3—61no—
248C2×C3—61no—
249C2×C3—61no—
250C2×C3—121no—
251C2×C3—183no—
252C2×C3—181no—
253C2×C3—542no—
254C2×C3C2×((C3⋊D3)⋊C3)⋊C213yes—
255C2×C3C2×#111311yescharacteristic
256D3—61no—
257D3—61no—
258D3—62no—
259D3—186no—
260D3—182no—
261D3—542no—
262D3—1082no—
263C22—91no—
264C22—272no—
265C4—91no—
266C4—272no—
267C4#6643211yescharacteristic
268C3—31no—
269C3—31no—
270C3—61no—
271C3—61no—
272C3—61no—
273C3—121no—
274C3#8864913yescomplemented
275C3#8904111yescharacteristic
276C2—181no—
277C2—542no—
278C2C2×#6643211yescharacteristic
subgroupG/H
centerC2C2×#66432
commutator, cosocleC2×C3×H(3,𝔽3)C23
FrattiniC2×C3C2×#1113
FittingC3×C4×H(3,𝔽3)C22
socleC2×C32C2×D32
hypercenterC4#66432
hypocenterC3×H(3,𝔽3)#41
LCSG ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3)
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C3×C4×#272C2semidirect
2. C4⋊(C3⋊((C3⋊D3)⋊C3))C2semidirect
3. #10619956C2semidirect
4. #10619904C2semidirect
5. #10619901C2semidirect
14. C4⋊(C3⋊(C3⋊D3))D3semidirect
15. #6308D3semidirect
25. C4×#272D3semidirect
63. C3×H(3,𝔽3)#41semidirect
149. C33#237semidirect
150. C33#239semidirect
154. H(3,𝔽3)#235semidirect
237. C32#3795semidirect
239. C32#3799semidirect
274. C3#88649semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#2722
C4⋊(C3⋊((C3⋊D3)⋊C3))2
#106199562
#106199042
#106199012
#890223
#890213
#886493