Finite Group Databasebeta

← #13048665

Subgroups — #13048665

Order 1296.

525 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 525 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C4⋊(C3⋊#272)C211yesmaximal, characteristic, complemented
2#10619956C211yesmaximal, characteristic, complemented
3C3×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
4#10619886C212yesmaximal, complemented
5#10619885C212yesmaximal, complemented
6#89041—31nomaximal
7#89021—31nomaximal
8#88652—33nomaximal
9C2×C3⋊#272C2212yes—
10C2×C3×(C3⋊D3)⋊C3C2211yescharacteristic
11C3×C4×H(3,𝔽3)C2211yescharacteristic
12C3⋊((C3⋊Dic3)⋊C3)C2212yes—
13C3×(C3⋊Dic3)⋊C3C2211yescharacteristic
14#6308D311yescharacteristic, complemented
15C3×C4⋊(C3⋊D3)—31no—
16C3×C4⋊(C3⋊D3)—31no—
17C3×C4⋊(C3⋊D3)—61no—
18C3×C4×C3⋊D3D311yescharacteristic, complemented
19C3×#634—31no—
20C32×C4×D3—31no—
21#6295—32no—
22#6294—32no—
23#6290—32no—
24Dic3×C3⋊D3—32no—
25C4⋊#272—33no—
26C4⋊#272—63no—
27C4×(C3⋊D3)⋊C3D313yescomplemented
28#634⋊C3—33no—
29#6200—36no—
30#6199—36no—
31C2×C3×H(3,𝔽3)C2311yescharacteristic
32C3⋊#272—22no—
33C3×(C3⋊D3)⋊C3—21no—
34#3828—91no—
35#3799—91no—
36#3796—91no—
37#3795—93no—
38#3795—91no—
39C2×C3×C3⋊D3—32no—
40C2×C3×C3⋊D3—32no—
41C2×C3×C3⋊D3—62no—
42C2×C3×C3⋊D3C2×D311yescharacteristic
43C2×C32×D3—31no—
44C33×C4—21no—
45C33×C4C2×D311yescharacteristic
46C33×C4C2×D311yescharacteristic
47C3⋊(C3⋊Dic3)C2×D312yes—
48C3×C3⋊Dic3—32no—
49C3×C3⋊Dic3C2×D311yescharacteristic
50C32×Dic3—31no—
51C2×#272—36no—
52C2×#272—66no—
53C2×(C3⋊D3)⋊C3C2×D313yes—
54C4×H(3,𝔽3)—23no—
55C4×H(3,𝔽3)C2×D313yes—
56(C3⋊Dic3)⋊C3—36no—
57(C3⋊Dic3)⋊C3C2×D313yes—
58C3×H(3,𝔽3)#4111yescharacteristic, complemented, Sylow
59C2×C32×C4—91no—
60#638—92no—
61C4⋊(C3⋊D3)—31no—
62C4⋊(C3⋊D3)—31no—
63C4⋊(C3⋊D3)—61no—
64C4×C3⋊D3—92no—
65C4×C3⋊D3C3⋊D311yescharacteristic, complemented
66#634—31no—
67#634—33no—
68#634—91no—
69#634—181no—
70C3×D12—93no—
71C3×D12—91no—
72C3×D12—91no—
73C3×D12—183no—
74C3×D12—181no—
75C3×C4×D3—33no—
76C3×C4×D3—31no—
77C3×C4×D3—91no—
78C3×Dic6—91no—
79C3×Dic6—91no—
80C3×Dic6—181no—
81#626—92no—
82#626—92no—
83#625—96no—
84#625—92no—
85#624—92no—
86D3×Dic3—96no—
87D3×Dic3—92no—
88D3×Dic3—92no—
89C2×C33—21no—
90C2×C33C22×D311yescharacteristic
91C2×C33C22×D311yescharacteristic
92C3×C3⋊D3—21no—
93C3×C3⋊D3—62no—
94C3×C3⋊D3—62no—
95C3×C3⋊D3—122no—
96C32×D3—61no—
97C2×H(3,𝔽3)—23no—
98C2×H(3,𝔽3)C22×D313yes—
99#272—66no—
100#272—126no—
101(C3⋊D3)⋊C3—23no—
102#239—271no—
103#237—271no—
104#235—271no—
105#235—273no—
106C22×C32—91no—
107C2×C3⋊D3—32no—
108C2×C3⋊D3—32no—
109C2×C3⋊D3—62no—
110C2×C3⋊D3C2×C3⋊D311yescharacteristic
111C2×C3×D3—33no—
112C2×C3×D3—31no—
113C2×C3×D3—91no—
114C2×C3×D3—96no—
115C2×C3×D3—92no—
116C2×C3×D3—92no—
117C2×C3×D3—186no—
118C2×C3×D3—182no—
119C32×C4—21no—
120C32×C4—23no—
121C32×C4—31no—
122C32×C4—31no—
123C32×C4—61no—
124C32×C4—61no—
125C32×C4—61no—
126C32×C4—91no—
127C32×C4—121no—
128C32×C4D3211yescharacteristic
129C32×C4D3213yes—
130C32×C4C2×C3⋊D311yescharacteristic
131C3⋊Dic3—32no—
132C3⋊Dic3—36no—
133C3⋊Dic3—92no—
134C3⋊Dic3—182no—
135C3⋊Dic3C2×C3⋊D311yescharacteristic
136C3×Dic3—33no—
137C3×Dic3—31no—
138C3×Dic3—91no—
139C3×Dic3—92no—
140C3×Dic3—92no—
141C3×Dic3—182no—
142C33—21no—
143C33#23511yescharacteristic, complemented
144C33#23911yescharacteristic, complemented
145H(3,𝔽3)—23no—
146H(3,𝔽3)#23513yescomplemented
147C3×Q8—271no—
148C3×D4—271no—
149C2×C3×C4—93no—
150C2×C3×C4—91no—
151#67—272no—
152#67—272no—
153#67—276no—
154C2×Dic3—272no—
155D12—93no—
156D12—91no—
157D12—91no—
158D12—181no—
159D12—183no—
160D12—272no—
161C4×D3—31no—
162C4×D3—91no—
163C4×D3—272no—
164C4×D3—272no—
165C4×D3—276no—
166Dic6—91no—
167Dic6—91no—
168Dic6—181no—
169Dic6—273no—
170C2×C32—21no—
171C2×C32—23no—
172C2×C32—31no—
173C2×C32—31no—
174C2×C32—61no—
175C2×C32—61no—
176C2×C32—61no—
177C2×C32—121no—
178C2×C32—181no—
179C2×C32C2×D3211yescharacteristic
180C2×C32C2×D3213yes—
181C2×C32C22×C3⋊D311yescharacteristic
182C3⋊D3—21no—
183C3⋊D3—62no—
184C3⋊D3—62no—
185C3⋊D3—122no—
186C3×D3—61no—
187C3×D3—63no—
188C3×D3—181no—
189C3×D3—182no—
190C3×D3—186no—
191C3×D3—182no—
192C3×D3—366no—
193C3×D3—362no—
194#41—811noSylow
195C22×C3—93no—
196C22×C3—91no—
197C22×C3—272no—
198C2×D3—31no—
199C2×D3—91no—
200C2×D3—92no—
201C2×D3—96no—
202C2×D3—92no—
203C2×D3—182no—
204C2×D3—186no—
205C3×C4—21no—
206C3×C4—33no—
207C3×C4—31no—
208C3×C4—61no—
209C3×C4—63no—
210C3×C4—93no—
211C3×C4—91no—
212C3×C4—272no—
213C3×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
214C3×C4D3×C3⋊D311yescharacteristic
215Dic3—31no—
216Dic3—91no—
217Dic3—92no—
218Dic3—92no—
219Dic3—182no—
220Dic3—276no—
221C32—21no—
222C32—23no—
223C32—31no—
224C32—31no—
225C32—61no—
226C32—61no—
227C32—61no—
228C32—121no—
229C32#379911yescharacteristic
230C32#379913yescomplemented
231C32#382811yescharacteristic, complemented
232Q8—271no—
233D4—271no—
234D4—812no—
235C2×C4—91no—
236C2×C4—812no—
237C2×C3—21no—
238C2×C3—31no—
239C2×C3—33no—
240C2×C3—61no—
241C2×C3—63no—
242C2×C3—183no—
243C2×C3—181no—
244C2×C3—542no—
245C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
246C2×C3C2×D3×C3⋊D311yescharacteristic
247D3—61no—
248D3—181no—
249D3—182no—
250D3—186no—
251D3—182no—
252D3—362no—
253D3—366no—
254C22—91no—
255C22—272no—
256C4—91no—
257C4—272no—
258C4((C3⋊D3)⋊C3)⋊D311yescharacteristic
259C3—21no—
260C3—31no—
261C3—33no—
262C3—61no—
263C3—63no—
264C3#8865211yescharacteristic, complemented
265C3#8902211yescharacteristic
266C2—181no—
267C2—542no—
268C2C2×((C3⋊D3)⋊C3)⋊D311yescharacteristic
subgroupG/H
centerC2C2×((C3⋊D3)⋊C3)⋊D3
commutator, cosocleC2×C3×H(3,𝔽3)C23
FrattiniC2×C3C2×D3×C3⋊D3
FittingC3×C4×H(3,𝔽3)C22
socleC2×C32C2×D32
hypercenterC4((C3⋊D3)⋊C3)⋊D3
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. C4⋊(C3⋊#272)C2semidirect
2. #10619956C2semidirect
3. C3×C4×(C3⋊D3)⋊C3C2semidirect
4. #10619886C2semidirect
5. #10619885C2semidirect
14. #6308D3semidirect
18. C3×C4×C3⋊D3D3semidirect
27. C4×(C3⋊D3)⋊C3D3semidirect
58. C3×H(3,𝔽3)#41semidirect
65. C4×C3⋊D3C3⋊D3semidirect
143. C33#235semidirect
144. C33#239semidirect
146. H(3,𝔽3)#235semidirect
230. C32#3799semidirect
231. C32#3828semidirect
264. C3#88652semidirect

Maximal subgroups

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

subgroupindex
C4⋊(C3⋊#272)2
#106199562
C3×C4×(C3⋊D3)⋊C32
#106198862
#106198852
#890413
#890213
#886523