Finite Group Databasebeta

← #13049215

Subgroups — #13049215

Order 1296.

392 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 32 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×(C3⋊D3)⋊A4C211yesmaximal, characteristic, complemented
2C3⋊#6258C211yesmaximal, characteristic, complemented
3C3⋊((C3⋊S4)⋊C3)C211yesmaximal, characteristic, complemented
4D3×C3⋊S4—31nomaximal
5#89049—31nomaximal
6((C3⋊S4)⋊C3)⋊C2—33nomaximal
7C3×C22⋊H(3,𝔽3)C2211yescharacteristic, complemented
8((C3⋊D3)⋊C3)⋊D3—41nomaximal
9C22×C3×C3⋊D3D311yescharacteristic, complemented
10C2×#1113—31no—
11C3×D3×A4—31no—
12C3⋊(C3⋊S4)D311yescharacteristic, complemented
13C3×C3⋊S4—31no—
14C3×C3⋊S4—61no—
15C3×#638—31no—
16C3×#638—31no—
17#6295—31no—
18#6295—31no—
19#6294—31no—
20(C3⋊D3)⋊A4D313yescomplemented
21#6258—33no—
22#6258—63no—
23(C3⋊S4)⋊C3—33no—
24C3⋊#272—41no—
25C3⋊((C3⋊D3)⋊C3)—41no—
26C3×(C3⋊D3)⋊C3—41no—
27C2×C3⋊S4—91no—
28D3×S4—93no—
29#3811—91no—
30D3×#67—91no—
31D3×#67—91no—
32C22×C33C2×D311yescharacteristic, complemented
33C2×C3×C3⋊D3—31no—
34C2×C3×C3⋊D3—31no—
35C2×C3×C3⋊D3—31no—
36C2×C3×C3⋊D3—31no—
37C32×A4—21no—
38C32×A4C2×D311yescharacteristic, complemented
39#1113—61no—
40#1113—61no—
41D3×C3⋊D3—121no—
42C3×C3⋊Dic3—31no—
43C3×C3⋊Dic3—31no—
44C22⋊H(3,𝔽3)—23no—
45C22⋊H(3,𝔽3)C2×D313yescomplemented
46((C3⋊D3)⋊C3)⋊C2—123no—
47C3×H(3,𝔽3)—41noSylow
48C22×C3⋊D3C3⋊D311yescharacteristic, complemented
49C22×C3×D3—31no—
50C22×C3×D3—91no—
51C2×C3×A4—91no—
52C2×D32—91no—
53C2×D32—91no—
54C2×D32—91no—
55D3×A4—33no—
56C3⋊S4—31no—
57C3⋊S4—33no—
58C3⋊S4—61no—
59C3⋊S4—91no—
60C3⋊S4—181no—
61C3×S4—93no—
62C3×S4—183no—
63#638—31no—
64#638—31no—
65C3×#67—91no—
66C3×#67—91no—
67C3×#67—91no—
68C3×#67—91no—
69C3×#67—181no—
70C3×#67—181no—
71#626—91no—
72#626—91no—
73#626—91no—
74#626—91no—
75#625—91no—
76#625—91no—
77#624—91no—
78D3×Dic3—91no—
79D3×Dic3—91no—
80C2×C33—31no—
81C3⋊(C3⋊D3)—41no—
82C3×C3⋊D3—31no—
83C3×C3⋊D3—61no—
84C3×C3⋊D3—61no—
85C3×C3⋊D3—121no—
86C3×C3⋊D3—241no—
87C3×C3⋊D3S411yescharacteristic, complemented
88C32×D3—121no—
89#272—123no—
90#272—243no—
91(C3⋊D3)⋊C3—43no—
92(C3⋊D3)⋊C3—123no—
93C2×S4—273no—
94C2×#67—271no—
95D3×D4—271no—
96D3×D4—271no—
97C22×C32—21no—
98C22×C32—31no—
99C22×C32—61no—
100C22×C32D3211yescharacteristic
101C22×C32C2×C3⋊D311yescharacteristic, complemented
102C2×C3⋊D3—31no—
103C2×C3⋊D3—31no—
104C2×C3⋊D3—31no—
105C2×C3⋊D3—31no—
106C2×C3⋊D3—361no—
107C2×C3×D3—91no—
108C2×C3×D3—91no—
109C2×C3×D3—91no—
110C2×C3×D3—91no—
111C2×C3×D3—91no—
112C2×C3×D3—91no—
113C2×C3×D3—91no—
114C2×C3×D3—91no—
115C2×C3×D3—181no—
116C2×C3×D3—181no—
117C2×C3×D3—181no—
118C2×C3×D3—181no—
119C3×A4—23no—
120C3×A4—31no—
121C3×A4—61no—
122C3×A4—61no—
123C3×A4—121no—
124C3×A4D3213yescomplemented
125D32—181no—
126D32—181no—
127D32—181no—
128D32—181no—
129D32—181no—
130D32—181no—
131D32—363no—
132C3⋊Dic3—31no—
133C3⋊Dic3—31no—
134C3×Dic3—91no—
135C3×Dic3—91no—
136C3×Dic3—91no—
137C3×Dic3—91no—
138C3×Dic3—181no—
139C3×Dic3—181no—
140C33—41no—
141C33—81no—
142C33C2×S411yescharacteristic, complemented
143H(3,𝔽3)—43no—
144H(3,𝔽3)—83no—
145C23×C3—91no—
146C22×D3—31no—
147C22×D3—91no—
148C22×D3—271no—
149C22×D3—271no—
150C22×D3—271no—
151C2×A4—93no—
152S4—93no—
153S4—183no—
154S4—273no—
155C3×D4—271no—
156C3×D4—271no—
157#67—91no—
158#67—91no—
159#67—91no—
160#67—91no—
161#67—181no—
162#67—181no—
163#67—271no—
164#67—271no—
165#67—271no—
166#67—271no—
167C2×Dic3—271no—
168D12—271no—
169D12—271no—
170C4×D3—271no—
171C4×D3—271no—
172C2×C32—31no—
173C2×C32—31no—
174C2×C32—31no—
175C2×C32—61no—
176C2×C32—61no—
177C2×C32—61no—
178C2×C32—121no—
179C2×C32—361no—
180C3⋊D3—31no—
181C3⋊D3—61no—
182C3⋊D3—61no—
183C3⋊D3—121no—
184C3⋊D3—123no—
185C3⋊D3—241no—
186C3⋊D3—361no—
187C3⋊D3—721no—
188C3⋊D3C3⋊S411yescharacteristic, complemented
189C3×D3—31no—
190C3×D3—91no—
191C3×D3—91no—
192C3×D3—91no—
193C3×D3—123no—
194C3×D3—181no—
195C3×D3—181no—
196C3×D3—181no—
197C3×D3—181no—
198C3×D3—181no—
199C3×D3—361no—
200C3×D3—363no—
201C3×D3—361no—
202C3×D3—723no—
203C2×D4—811noSylow
204C22×C3—21no—
205C22×C3—31no—
206C22×C3—61no—
207C22×C3—271no—
208C22×C3—271no—
209C22×C3—271no—
210C22×C3—271no—
211C22×C3((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
212C22×C3D3×C3⋊D311yescharacteristic
213C2×D3—91no—
214C2×D3—91no—
215C2×D3—91no—
216C2×D3—91no—
217C2×D3—91no—
218C2×D3—91no—
219C2×D3—91no—
220C2×D3—91no—
221C2×D3—181no—
222C2×D3—181no—
223C2×D3—181no—
224C2×D3—181no—
225C2×D3—541no—
226C2×D3—541no—
227C2×D3—541no—
228C2×D3—541no—
229C2×D3—541no—
230C2×D3—541no—
231C2×D3—1083no—
232A4—33no—
233A4—63no—
234C3×C4—271no—
235C3×C4—271no—
236Dic3—91no—
237Dic3—91no—
238Dic3—91no—
239Dic3—91no—
240Dic3—181no—
241Dic3—181no—
242C32—21no—
243C32—31no—
244C32—43no—
245C32—61no—
246C32—83no—
247C32—121no—
248C32—241no—
249C32—241no—
250C32—481no—
251C32D3×S411yescharacteristic
252C32C2×C3⋊S411yescharacteristic, complemented
253C23—91no—
254C23—811no—
255D4—271no—
256D4—271no—
257D4—811no—
258D4—811no—
259C2×C4—811no—
260C2×C3—31no—
261C2×C3—31no—
262C2×C3—31no—
263C2×C3—61no—
264C2×C3—61no—
265C2×C3—61no—
266C2×C3—91no—
267C2×C3—121no—
268C2×C3—271no—
269C2×C3—363no—
270C2×C3—541no—
271C2×C3—541no—
272D3—31no—
273D3—91no—
274D3—91no—
275D3—91no—
276D3—181no—
277D3—181no—
278D3—181no—
279D3—181no—
280D3—181no—
281D3—361no—
282D3—361no—
283D3—363no—
284D3—723no—
285D3—1083no—
286C22—271no—
287C22—271no—
288C22—271no—
289C22—271no—
290C22—1621no—
291C22—1621no—
292C22((C3⋊D3)⋊C3)⋊D311yescharacteristic, complemented
293C4—271no—
294C4—271no—
295C3—21no—
296C3—31no—
297C3—61no—
298C3—123no—
299C3—243no—
300C3((C3⋊S4)⋊C3)⋊C211yescharacteristic, complemented
301C3D3×C3⋊S411yescharacteristic
302C2—31no—
303C2—91no—
304C2—271no—
305C2—541no—
306C2—541no—
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC3×C22⋊H(3,𝔽3)C22
FrattiniC3D3×C3⋊S4
FittingC22×C33C2×D3
socleC22×C32D32
LCSG ▷ C3×C22⋊H(3,𝔽3)
UCSC1

Complements

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

NH ≅ G/Ntype
1. C3×(C3⋊D3)⋊A4C2semidirect
2. C3⋊#6258C2semidirect
3. C3⋊((C3⋊S4)⋊C3)C2semidirect
7. C3×C22⋊H(3,𝔽3)C22semidirect
9. C22×C3×C3⋊D3D3semidirect
12. C3⋊(C3⋊S4)D3semidirect
20. (C3⋊D3)⋊A4D3semidirect
32. C22×C33C2×D3semidirect
38. C32×A4C2×D3semidirect
45. C22⋊H(3,𝔽3)C2×D3semidirect
48. C22×C3⋊D3C3⋊D3semidirect
87. C3×C3⋊D3S4semidirect
101. C22×C32C2×C3⋊D3semidirect
124. C3×A4D32semidirect
142. C33C2×S4semidirect
188. C3⋊D3C3⋊S4semidirect
211. C22×C3((C3⋊D3)⋊C3)⋊C2semidirect
252. C32C2×C3⋊S4semidirect
292. C22((C3⋊D3)⋊C3)⋊D3semidirect
300. C3((C3⋊S4)⋊C3)⋊C2semidirect

Maximal subgroups

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

subgroupindex
C3×(C3⋊D3)⋊A42
C3⋊#62582
C3⋊((C3⋊S4)⋊C3)2
D3×C3⋊S43
#890493
((C3⋊S4)⋊C3)⋊C23
((C3⋊D3)⋊C3)⋊D34