Finite Group Databasebeta

← #13048666

Subgroups — #13048666

Order 1296.

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

normal subgroups only

This group has 531 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
2C4×C3⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
3C3×#634⋊C3C211yesmaximal, characteristic, complemented
4#10619887C212yesmaximal, complemented
5#10619885C212yesmaximal, complemented
6#89041—31nomaximal
7#89017—31nomaximal
8#88652—33nomaximal
9C2×C3⋊#272C2212yes—
10C2×C3⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C3×C4×H(3,𝔽3)C2211yescharacteristic
12C3⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
13C3×(C3⋊Dic3)⋊C3C2212yes—
14C4×C3⋊(C3⋊D3)D311yescharacteristic, complemented
15C3×C4⋊(C3⋊D3)—31no—
16C3×C4⋊(C3⋊D3)—31no—
17C3×C4⋊(C3⋊D3)—61no—
18C3×C4×C3⋊D3—31no—
19C3×#634D311yescharacteristic, complemented
20C32×Dic6—31no—
21#6295—32no—
22#6294—32no—
23#6292—32no—
24Dic3×C3⋊D3—32no—
25C4⋊#272—33no—
26C4⋊#272—63no—
27C4×(C3⋊D3)⋊C3—33no—
28#634⋊C3D313yescomplemented
29#6200—36no—
30#6199—36no—
31C2×C3×H(3,𝔽3)C2311yescharacteristic
32C3⋊#272—22no—
33C3⋊((C3⋊D3)⋊C3)—21no—
34#3832—91no—
35#3799—91no—
36#3796—93no—
37#3796—91no—
38#3795—91no—
39C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
40C2×C3×C3⋊D3—32no—
41C2×C3×C3⋊D3—31no—
42C2×C3×C3⋊D3—32no—
43C2×C3×C3⋊D3—62no—
44C33×C4—21no—
45C33×C4C2×D311yescharacteristic
46C33×C4C2×D311yescharacteristic
47C3⋊(C3⋊Dic3)C2×D311yescharacteristic
48C3×C3⋊Dic3—31no—
49C3×C3⋊Dic3C2×D312yes—
50C32×Dic3—32no—
51C2×#272—36no—
52C2×#272—66no—
53C2×(C3⋊D3)⋊C3—33no—
54C4×H(3,𝔽3)—23no—
55C4×H(3,𝔽3)C2×D313yes—
56(C3⋊Dic3)⋊C3—33no—
57(C3⋊Dic3)⋊C3C2×D316yes—
58C3×H(3,𝔽3)#4111yescharacteristic, complemented, Sylow
59C32×Q8—91no—
60C4⋊(C3⋊D3)—31no—
61C4⋊(C3⋊D3)—31no—
62C4⋊(C3⋊D3)—61no—
63C4⋊(C3⋊D3)—92no—
64C4×C3⋊D3—31no—
65C4×C3⋊D3—33no—
66C4×C3⋊D3—91no—
67C4×C3⋊D3—92no—
68C4×C3⋊D3—181no—
69#634C3⋊D311yescharacteristic, complemented
70C3×D12—93no—
71C3×D12—91no—
72C3×D12—91no—
73C3×D12—183no—
74C3×D12—181no—
75C3×C4×D3—91no—
76C3×C4×D3—91no—
77C3×C4×D3—181no—
78C3×Dic6—33no—
79C3×Dic6—31no—
80C3×Dic6—91no—
81#626—96no—
82#626—92no—
83#626—92no—
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—61no—
95C3×C3⋊D3—62no—
96C3×C3⋊D3—122no—
97C2×H(3,𝔽3)—23no—
98C2×H(3,𝔽3)C22×D313yes—
99#272—66no—
100#272—126no—
101(C3⋊D3)⋊C3—63no—
102#239—271no—
103#239—273no—
104#237—271no—
105#235—271no—
106C2×C3⋊D3—32no—
107C2×C3⋊D3—32no—
108C2×C3⋊D3—31no—
109C2×C3⋊D3—33no—
110C2×C3⋊D3—62no—
111C2×C3⋊D3—91no—
112C2×C3⋊D3—181no—
113C2×C3×D3—91no—
114C2×C3×D3—96no—
115C2×C3×D3—92no—
116C2×C3×D3—91no—
117C2×C3×D3—92no—
118C2×C3×D3—181no—
119C2×C3×D3—186no—
120C2×C3×D3—182no—
121C32×C4—21no—
122C32×C4—23no—
123C32×C4—31no—
124C32×C4—31no—
125C32×C4—61no—
126C32×C4—61no—
127C32×C4—61no—
128C32×C4—92no—
129C32×C4—121no—
130C32×C4D3211yescharacteristic
131C32×C4D3213yes—
132C32×C4C2×C3⋊D311yescharacteristic
133C3⋊Dic3—31no—
134C3⋊Dic3—33no—
135C3⋊Dic3—91no—
136C3⋊Dic3—181no—
137C3⋊Dic3C2×C3⋊D312yes—
138C3×Dic3—36no—
139C3×Dic3—32no—
140C3×Dic3—92no—
141C3×Dic3—91no—
142C3×Dic3—91no—
143C3×Dic3—181no—
144C33—21no—
145C33#23511yescharacteristic, complemented
146C33#23911yescharacteristic, complemented
147H(3,𝔽3)—23no—
148H(3,𝔽3)#23913yescomplemented
149C3×Q8—93no—
150C3×Q8—91no—
151C3×D4—271no—
152C2×C3×C4—271no—
153#67—272no—
154#67—272no—
155C2×Dic3—272no—
156D12—91no—
157D12—93no—
158D12—91no—
159D12—181no—
160D12—183no—
161D12—272no—
162D12—276no—
163C4×D3—91no—
164C4×D3—91no—
165C4×D3—181no—
166C4×D3—273no—
167C4×D3—272no—
168C4×D3—272no—
169C4×D3—276no—
170Dic6—31no—
171Dic6—91no—
172C2×C32—21no—
173C2×C32—23no—
174C2×C32—31no—
175C2×C32—31no—
176C2×C32—61no—
177C2×C32—61no—
178C2×C32—61no—
179C2×C32—121no—
180C2×C32C2×D3211yescharacteristic
181C2×C32C2×D3213yes—
182C2×C32C22×C3⋊D311yescharacteristic
183C3⋊D3—62no—
184C3⋊D3—61no—
185C3⋊D3—62no—
186C3⋊D3—63no—
187C3⋊D3—122no—
188C3⋊D3—181no—
189C3⋊D3—182no—
190C3×D3—181no—
191C3×D3—182no—
192C3×D3—182no—
193C3×D3—186no—
194C3×D3—181no—
195C3×D3—182no—
196C3×D3—366no—
197C3×D3—362no—
198#41—811noSylow
199C22×C3—271no—
200C22×C3—272no—
201C2×D3—96no—
202C2×D3—92no—
203C2×D3—91no—
204C2×D3—92no—
205C2×D3—91no—
206C2×D3—181no—
207C2×D3—182no—
208C2×D3—186no—
209C2×D3—273no—
210C3×C4—21no—
211C3×C4—31no—
212C3×C4—33no—
213C3×C4—61no—
214C3×C4—63no—
215C3×C4—96no—
216C3×C4—92no—
217C3×C4—271no—
218C3×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
219C3×C4D3×C3⋊D311yescharacteristic
220Dic3—32no—
221Dic3—92no—
222Dic3—91no—
223Dic3—91no—
224Dic3—181no—
225Dic3—273no—
226C32—21no—
227C32—23no—
228C32—31no—
229C32—31no—
230C32—61no—
231C32—61no—
232C32—61no—
233C32—121no—
234C32#379911yescharacteristic
235C32#379913yescomplemented
236C32#383211yescharacteristic, complemented
237Q8—91no—
238D4—271no—
239D4—812no—
240C2×C4—271no—
241C2×C4—812no—
242C2×C3—21no—
243C2×C3—31no—
244C2×C3—33no—
245C2×C3—61no—
246C2×C3—63no—
247C2×C3—541no—
248C2×C3—542no—
249C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
250C2×C3C2×D3×C3⋊D311yescharacteristic
251D3—182no—
252D3—186no—
253D3—182no—
254D3—182no—
255D3—181no—
256D3—181no—
257D3—362no—
258D3—366no—
259D3—543no—
260C22—271no—
261C22—272no—
262C4—92no—
263C4—271no—
264C4((C3⋊D3)⋊C3)⋊D311yescharacteristic
265C3—21no—
266C3—33no—
267C3—31no—
268C3—61no—
269C3—63no—
270C3#8865211yescharacteristic, complemented
271C3#8901811yescharacteristic
272C2—542no—
273C2—541no—
274C2C2×((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. C4×C3⋊((C3⋊D3)⋊C3)C2semidirect
3. C3×#634⋊C3C2semidirect
4. #10619887C2semidirect
5. #10619885C2semidirect
14. C4×C3⋊(C3⋊D3)D3semidirect
19. C3×#634D3semidirect
28. #634⋊C3D3semidirect
58. C3×H(3,𝔽3)#41semidirect
69. #634C3⋊D3semidirect
145. C33#235semidirect
146. C33#239semidirect
148. H(3,𝔽3)#239semidirect
235. C32#3799semidirect
236. C32#3832semidirect
270. 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
C4×C3⋊((C3⋊D3)⋊C3)2
C3×#634⋊C32
#106198872
#106198852
#890413
#890173
#886523