Finite Group Databasebeta

← #13048753

Subgroups — #13048753

Order 1296.

645 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 645 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C2×#66432C212yesmaximal, complemented
2C3×C4⋊#272C211yesmaximal, characteristic, complemented
3C4⋊(C3⋊((C3⋊D3)⋊C3))C211yesmaximal, characteristic, complemented
4C4×C3⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
5#10619904C212yesmaximal, complemented
6#89026—31nomaximal
7C3⋊D3×D12—31nomaximal
8#88654—31nomaximal
9C2×C3×#272C2212yescomplemented
10C2×C3⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×C3⋊((C3⋊D3)⋊C3)C2212yescomplemented
12#66432—24no—
13C3×C4×H(3,𝔽3)C2211yescharacteristic, complemented
14C3⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
15C2×D3×C3⋊D3—32no—
16C2×D3×C3⋊D3—32no—
17C4⋊(C3⋊(C3⋊D3))D311yescharacteristic, complemented
18C4×C3⋊(C3⋊D3)D311yescharacteristic, complemented
19C3×C4⋊(C3⋊D3)—31no—
20C3×C4×C3⋊D3—31no—
21C32×D12—31no—
22C32×D12—31no—
23C32×D12—61no—
24#6291—32no—
25#6290—32no—
26C2×((C3⋊D3)⋊C3)⋊C2—32no—
27C4⋊#272D311yescharacteristic, complemented
28C4⋊((C3⋊D3)⋊C3)—31no—
29C4⋊((C3⋊D3)⋊C3)—61no—
30C4×(C3⋊D3)⋊C3—31no—
31C4×(C3⋊D3)⋊C3—61no—
32#6200—32no—
33C2×C3×H(3,𝔽3)C2311yescharacteristic
34C3×#272—22no—
35C3⋊((C3⋊D3)⋊C3)—22no—
36C3⋊((C3⋊D3)⋊C3)D412yescomplemented
37D4×C3⋊D3—91no—
38#3802—91no—
39#3802—93no—
40D3×D12—91no—
41D3×D12—93no—
42C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
43C2×C3⋊(C3⋊D3)C2×D312yescomplemented
44C2×C3×C3⋊D3—31no—
45C2×C3×C3⋊D3—32no—
46C2×C32×D3—32no—
47C2×C32×D3—32no—
48C2×C32×D3—62no—
49D3×C3⋊D3—64no—
50D3×C3⋊D3—64no—
51C33×C4—21no—
52C33×C4C2×D311yescharacteristic, complemented
53C33×C4C2×D311yescharacteristic, complemented
54C3⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
55C3×C3⋊Dic3—31no—
56C2×#272C2×D312yescomplemented
57C2×(C3⋊D3)⋊C3—31no—
58C2×(C3⋊D3)⋊C3—32no—
59C2×(C3⋊D3)⋊C3—61no—
60C2×(C3⋊D3)⋊C3—62no—
61((C3⋊D3)⋊C3)⋊C2—64no—
62C4×H(3,𝔽3)—21no—
63C4×H(3,𝔽3)—21no—
64C4×H(3,𝔽3)—41no—
65C4×H(3,𝔽3)C2×D311yescharacteristic, complemented
66(C3⋊Dic3)⋊C3—31no—
67(C3⋊Dic3)⋊C3—61no—
68C3×H(3,𝔽3)C2×D411yescharacteristic, complemented, Sylow
69C22×C3⋊D3—92no—
70C2×D32—92no—
71C2×D32—92no—
72C2×D32—96no—
73C2×D32—96no—
74C32×D4—91no—
75#638—92no—
76C4⋊(C3⋊D3)—31no—
77C4⋊(C3⋊D3)—31no—
78C4⋊(C3⋊D3)—61no—
79C4⋊(C3⋊D3)—93no—
80C4×C3⋊D3—31no—
81C4×C3⋊D3—31no—
82C4×C3⋊D3—61no—
83C4×C3⋊D3—93no—
84C3×D12—33no—
85C3×D12—33no—
86C3×D12—31no—
87C3×D12—31no—
88C3×D12—61no—
89C3×D12—63no—
90C3×D12—93no—
91C3×D12—91no—
92C3×C4×D3—93no—
93C3×C4×D3—91no—
94#626—92no—
95#626—96no—
96#625—92no—
97#625—96no—
98C2×C33—21no—
99C2×C33C22×D311yescharacteristic
100C2×C33C22×D311yescharacteristic
101C3⋊(C3⋊D3)—22no—
102C3⋊(C3⋊D3)D1212yescomplemented
103C3×C3⋊D3—32no—
104C3×C3⋊D3—62no—
105C32×D3—62no—
106C32×D3—62no—
107C32×D3—122no—
108C2×H(3,𝔽3)—21no—
109C2×H(3,𝔽3)—21no—
110C2×H(3,𝔽3)—41no—
111C2×H(3,𝔽3)C22×D311yescharacteristic
112#272—22no—
113(C3⋊D3)⋊C3—32no—
114(C3⋊D3)⋊C3—62no—
115(C3⋊D3)⋊C3—62no—
116(C3⋊D3)⋊C3—122no—
117D3×D4—273no—
118D3×D4—271no—
119D3×D4—271no—
120C2×D12—271no—
121C22×C32—92no—
122C2×C3⋊D3—31no—
123C2×C3⋊D3—32no—
124C2×C3⋊D3—31no—
125C2×C3⋊D3—32no—
126C2×C3⋊D3—61no—
127C2×C3⋊D3—62no—
128C2×C3⋊D3—93no—
129C2×C3⋊D3—96no—
130C2×C3⋊D3—184no—
131C2×C3×D3—36no—
132C2×C3×D3—36no—
133C2×C3×D3—32no—
134C2×C3×D3—32no—
135C2×C3×D3—62no—
136C2×C3×D3—66no—
137C2×C3×D3—93no—
138C2×C3×D3—91no—
139C2×C3×D3—96no—
140C2×C3×D3—92no—
141D32—184no—
142D32—1812no—
143D32—184no—
144D32—1812no—
145C32×C4—21no—
146C32×C4—21no—
147C32×C4—21no—
148C32×C4—33no—
149C32×C4—33no—
150C32×C4—41no—
151C32×C4—63no—
152C32×C4D3211yescharacteristic
153C32×C4D3211yescharacteristic, complemented
154C32×C4D3211yescharacteristic, complemented
155C3⋊Dic3—31no—
156C3⋊Dic3—31no—
157C3⋊Dic3—61no—
158C3⋊Dic3—93no—
159C3×Dic3—93no—
160C3×Dic3—91no—
161C33—21no—
162C33C2×D1211yescharacteristic, complemented
163C33D3×D411yescharacteristic, complemented
164H(3,𝔽3)—21no—
165H(3,𝔽3)—21no—
166H(3,𝔽3)—41no—
167H(3,𝔽3)D3×D411yescharacteristic, complemented
168C22×D3—272no—
169C22×D3—272no—
170C22×D3—276no—
171C22×D3—272no—
172C3×D4—93no—
173C3×D4—91no—
174C3×D4—271no—
175C2×C3×C4—271no—
176#67—276no—
177#67—272no—
178#67—272no—
179D12—31no—
180D12—31no—
181D12—61no—
182D12—93no—
183D12—91no—
184D12—271no—
185D12—272no—
186D12—541no—
187C4×D3—93no—
188C4×D3—91no—
189C4×D3—271no—
190C4×D3—541no—
191C2×C32—21no—
192C2×C32—21no—
193C2×C32—21no—
194C2×C32—33no—
195C2×C32—33no—
196C2×C32—41no—
197C2×C32—63no—
198C2×C32—182no—
199C2×C32C2×D3211yescharacteristic
200C2×C32C2×D3211yescharacteristic
201C2×C32C2×D3211yescharacteristic
202C3⋊D3—32no—
203C3⋊D3—32no—
204C3⋊D3—62no—
205C3⋊D3—62no—
206C3⋊D3—62no—
207C3⋊D3—96no—
208C3⋊D3—122no—
209C3⋊D3—186no—
210C3×D3—66no—
211C3×D3—66no—
212C3×D3—62no—
213C3×D3—62no—
214C3×D3—96no—
215C3×D3—92no—
216C3×D3—122no—
217C3×D3—126no—
218C3×D3—186no—
219C3×D3—182no—
220C2×D4—811noSylow
221C22×C3—96no—
222C22×C3—92no—
223C22×C3—271no—
224C22×C3—272no—
225C2×D3—32no—
226C2×D3—32no—
227C2×D3—62no—
228C2×D3—93no—
229C2×D3—91no—
230C2×D3—96no—
231C2×D3—92no—
232C2×D3—271no—
233C2×D3—272no—
234C2×D3—541no—
235C2×D3—542no—
236C2×D3—544no—
237C2×D3—544no—
238C2×D3—5412no—
239C2×D3—544no—
240C3×C4—31no—
241C3×C4—31no—
242C3×C4—61no—
243C3×C4—61no—
244C3×C4—61no—
245C3×C4—121no—
246C3×C4—271no—
247C3×C4((C3⋊D3)⋊C3)⋊C213yescomplemented
248C3×C4#111311yescharacteristic
249Dic3—93no—
250Dic3—91no—
251Dic3—271no—
252Dic3—541no—
253C32—21no—
254C32—21no—
255C32—21no—
256C32—33no—
257C32—33no—
258C32—41no—
259C32—63no—
260C32D3×D1211yescharacteristic
261C32D3×D1211yescharacteristic, complemented
262C32#380211yescharacteristic, complemented
263C23—812no—
264D4—91no—
265D4—271no—
266D4—812no—
267C2×C4—271no—
268C2×C3—31no—
269C2×C3—31no—
270C2×C3—61no—
271C2×C3—61no—
272C2×C3—61no—
273C2×C3—121no—
274C2×C3—186no—
275C2×C3—182no—
276C2×C3—272no—
277C2×C3—542no—
278C2×C3C2×((C3⋊D3)⋊C3)⋊C213yes—
279C2×C3C2×#111311yescharacteristic
280D3—62no—
281D3—62no—
282D3—96no—
283D3—92no—
284D3—122no—
285D3—186no—
286D3—182no—
287D3—272no—
288D3—542no—
289D3—542no—
290D3—1082no—
291C22—92no—
292C22—271no—
293C22—272no—
294C22—1624no—
295C4—271no—
296C4#6643211yescharacteristic, complemented
297C3—31no—
298C3—31no—
299C3—61no—
300C3—61no—
301C3—61no—
302C3—121no—
303C3#8865413yescomplemented
304C3C4⋊#111311yescharacteristic
305C2—182no—
306C2—272no—
307C2—542no—
308C2C2×#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)C2×D4
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. C2×#66432C2semidirect
2. C3×C4⋊#272C2semidirect
3. C4⋊(C3⋊((C3⋊D3)⋊C3))C2semidirect
4. C4×C3⋊((C3⋊D3)⋊C3)C2semidirect
5. #10619904C2semidirect
9. C2×C3×#272C22semidirect
11. C2×C3⋊((C3⋊D3)⋊C3)C22semidirect
13. C3×C4×H(3,𝔽3)C22semidirect
14. C3⋊((C3⋊Dic3)⋊C3)C22semidirect
17. C4⋊(C3⋊(C3⋊D3))D3semidirect
18. C4×C3⋊(C3⋊D3)D3semidirect
27. C4⋊#272D3semidirect
36. C3⋊((C3⋊D3)⋊C3)D4semidirect
43. C2×C3⋊(C3⋊D3)C2×D3semidirect
52. C33×C4C2×D3semidirect
53. C33×C4C2×D3semidirect
54. C3⋊(C3⋊Dic3)C2×D3semidirect
56. C2×#272C2×D3semidirect
65. C4×H(3,𝔽3)C2×D3semidirect
68. C3×H(3,𝔽3)C2×D4semidirect
102. C3⋊(C3⋊D3)D12semidirect
153. C32×C4D32semidirect
154. C32×C4D32semidirect
162. C33C2×D12semidirect
163. C33D3×D4semidirect
167. H(3,𝔽3)D3×D4semidirect
247. C3×C4((C3⋊D3)⋊C3)⋊C2semidirect
261. C32D3×D12semidirect
262. C32#3802semidirect
296. C4#66432semidirect
303. C3#88654semidirect

Maximal subgroups

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

subgroupindex
C2×#664322
C3×C4⋊#2722
C4⋊(C3⋊((C3⋊D3)⋊C3))2
C4×C3⋊((C3⋊D3)⋊C3)2
#106199042
#890263
C3⋊D3×D123
#886543