Finite Group Databasebeta

← #13048750

Subgroups — #13048750

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#10619903C212yesmaximal, complemented
6#89015—31nomaximal
7#89014—31nomaximal
8#88652—31nomaximal
9C2×C3×#272C2212yes—
10C2×C3⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C3×C4×H(3,𝔽3)C2211yescharacteristic
12C3⋊((C3⋊Dic3)⋊C3)C2212yes—
13C3⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
14C4×C3⋊(C3⋊D3)D311yescharacteristic, complemented
15#6308D311yescharacteristic, complemented
16C3×C4×C3⋊D3—31no—
17C3×#634—31no—
18C32×D12—31no—
19C32×D12—31no—
20C32×D12—61no—
21#6291—32no—
22#6290—32no—
23D3×C3⋊Dic3—32no—
24D3×C3⋊Dic3—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#6199—32no—
32C2×C3×H(3,𝔽3)C2311yescharacteristic
33C3×#272—22no—
34C3⋊((C3⋊D3)⋊C3)—21no—
35#3830—91no—
36#3796—91no—
37#3796—93no—
38#3795—91no—
39#3795—93no—
40C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
41C2×C3×C3⋊D3—31no—
42C2×C32×D3—32no—
43C2×C32×D3—32no—
44C2×C32×D3—62no—
45C33×C4—21no—
46C33×C4C2×D311yescharacteristic
47C33×C4C2×D311yescharacteristic
48C3⋊(C3⋊Dic3)C2×D311yescharacteristic
49C3⋊(C3⋊Dic3)C2×D312yes—
50C3×C3⋊Dic3—31no—
51C3×C3⋊Dic3—32no—
52C2×#272C2×D312yes—
53C2×(C3⋊D3)⋊C3—31no—
54C2×(C3⋊D3)⋊C3—61no—
55C4×H(3,𝔽3)—21no—
56C4×H(3,𝔽3)—21no—
57C4×H(3,𝔽3)—41no—
58C4×H(3,𝔽3)C2×D311yescharacteristic
59(C3⋊Dic3)⋊C3—31no—
60(C3⋊Dic3)⋊C3—32no—
61(C3⋊Dic3)⋊C3—62no—
62(C3⋊Dic3)⋊C3—61no—
63C3×H(3,𝔽3)#4111yescharacteristic, complemented, Sylow
64C32×D4—91no—
65#638—92no—
66C2×C3⋊Dic3—92no—
67C4×C3⋊D3—31no—
68C4×C3⋊D3—31no—
69C4×C3⋊D3—61no—
70C4×C3⋊D3—93no—
71#634—31no—
72#634—31no—
73#634—61no—
74#634—93no—
75C3×D12—33no—
76C3×D12—33no—
77C3×D12—31no—
78C3×D12—31no—
79C3×D12—61no—
80C3×D12—63no—
81C3×C4×D3—93no—
82C3×C4×D3—91no—
83C3×Dic6—93no—
84C3×Dic6—91no—
85#626—92no—
86#626—96no—
87#625—92no—
88#625—96no—
89D3×Dic3—92no—
90D3×Dic3—92no—
91D3×Dic3—96no—
92D3×Dic3—96no—
93C2×C33—21no—
94C2×C33C22×D311yescharacteristic
95C2×C33C22×D311yescharacteristic
96C3⋊(C3⋊D3)—21no—
97C3×C3⋊D3—61no—
98C32×D3—62no—
99C32×D3—62no—
100C32×D3—122no—
101C2×H(3,𝔽3)—21no—
102C2×H(3,𝔽3)—21no—
103C2×H(3,𝔽3)—41no—
104C2×H(3,𝔽3)C22×D311yescharacteristic
105#272—22no—
106(C3⋊D3)⋊C3—62no—
107(C3⋊D3)⋊C3—61no—
108#239—271no—
109#237—273no—
110#237—271no—
111#235—271no—
112C22×C32—92no—
113C2×C3⋊D3—31no—
114C2×C3⋊D3—31no—
115C2×C3⋊D3—61no—
116C2×C3⋊D3—93no—
117C2×C3×D3—36no—
118C2×C3×D3—36no—
119C2×C3×D3—32no—
120C2×C3×D3—32no—
121C2×C3×D3—62no—
122C2×C3×D3—66no—
123C2×C3×D3—93no—
124C2×C3×D3—91no—
125C32×C4—21no—
126C32×C4—21no—
127C32×C4—21no—
128C32×C4—33no—
129C32×C4—33no—
130C32×C4—41no—
131C32×C4—63no—
132C32×C4D3211yescharacteristic
133C32×C4D3211yescharacteristic
134C32×C4D3211yescharacteristic
135C3⋊Dic3—32no—
136C3⋊Dic3—31no—
137C3⋊Dic3—31no—
138C3⋊Dic3—32no—
139C3⋊Dic3—62no—
140C3⋊Dic3—61no—
141C3⋊Dic3—93no—
142C3⋊Dic3—96no—
143C3×Dic3—93no—
144C3×Dic3—96no—
145C3×Dic3—91no—
146C3×Dic3—92no—
147C33—21no—
148C33#23511yescharacteristic, complemented
149C33#23911yescharacteristic, complemented
150H(3,𝔽3)—21no—
151H(3,𝔽3)—21no—
152H(3,𝔽3)—41no—
153H(3,𝔽3)#23711yescharacteristic, complemented
154C3×Q8—271no—
155C3×D4—93no—
156C3×D4—91no—
157C2×C3×C4—271no—
158#67—276no—
159#67—272no—
160#67—272no—
161C2×Dic3—276no—
162C2×Dic3—272no—
163D12—31no—
164D12—31no—
165D12—61no—
166D12—272no—
167C4×D3—93no—
168C4×D3—91no—
169C4×D3—271no—
170C4×D3—272no—
171C4×D3—272no—
172C4×D3—541no—
173Dic6—93no—
174Dic6—91no—
175Dic6—271no—
176Dic6—541no—
177C2×C32—21no—
178C2×C32—21no—
179C2×C32—21no—
180C2×C32—33no—
181C2×C32—33no—
182C2×C32—41no—
183C2×C32—63no—
184C2×C32—182no—
185C2×C32C2×D3211yescharacteristic
186C2×C32C2×D3211yescharacteristic
187C2×C32C2×D3211yescharacteristic
188C3⋊D3—61no—
189C3⋊D3—62no—
190C3⋊D3—61no—
191C3⋊D3—183no—
192C3×D3—66no—
193C3×D3—66no—
194C3×D3—62no—
195C3×D3—62no—
196C3×D3—122no—
197C3×D3—126no—
198C3×D3—183no—
199C3×D3—181no—
200#41—811noSylow
201C22×C3—96no—
202C22×C3—92no—
203C22×C3—271no—
204C2×D3—32no—
205C2×D3—32no—
206C2×D3—62no—
207C2×D3—93no—
208C2×D3—91no—
209C2×D3—271no—
210C2×D3—541no—
211C3×C4—31no—
212C3×C4—31no—
213C3×C4—61no—
214C3×C4—61no—
215C3×C4—61no—
216C3×C4—121no—
217C3×C4—271no—
218C3×C4—272no—
219C3×C4((C3⋊D3)⋊C3)⋊C213yes—
220C3×C4#111311yescharacteristic
221Dic3—96no—
222Dic3—93no—
223Dic3—92no—
224Dic3—91no—
225Dic3—272no—
226Dic3—271no—
227Dic3—542no—
228Dic3—541no—
229C32—21no—
230C32—21no—
231C32—21no—
232C32—33no—
233C32—33no—
234C32—41no—
235C32—63no—
236C32#379511yescharacteristic, complemented
237C32#379611yescharacteristic, complemented
238C32#379911yescharacteristic
239Q8—271no—
240D4—91no—
241D4—812no—
242C2×C4—271no—
243C2×C4—812no—
244C2×C3—31no—
245C2×C3—31no—
246C2×C3—61no—
247C2×C3—61no—
248C2×C3—61no—
249C2×C3—121no—
250C2×C3—186no—
251C2×C3—182no—
252C2×C3—541no—
253C2×C3C2×((C3⋊D3)⋊C3)⋊C213yes—
254C2×C3C2×#111311yescharacteristic
255D3—62no—
256D3—62no—
257D3—122no—
258D3—183no—
259D3—181no—
260D3—541no—
261D3—542no—
262C22—92no—
263C22—271no—
264C4—271no—
265C4—272no—
266C4#6643211yescharacteristic
267C3—31no—
268C3—31no—
269C3—61no—
270C3—61no—
271C3—61no—
272C3—121no—
273C3#8865213yescomplemented
274C3#8904111yescharacteristic
275C2—182no—
276C2—541no—
277C2C2×#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. #10619903C2semidirect
14. C4×C3⋊(C3⋊D3)D3semidirect
15. #6308D3semidirect
25. C4⋊#272D3semidirect
63. C3×H(3,𝔽3)#41semidirect
148. C33#235semidirect
149. C33#239semidirect
153. H(3,𝔽3)#237semidirect
236. C32#3795semidirect
237. C32#3796semidirect
273. C3#88652semidirect

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
#106199032
#890153
#890143
#886523