Finite Group Databasebeta

← #13049711

Subgroups — #13049711

Order 1296.

1036 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 111 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×C3⋊#1113C212yesmaximal, complemented
2C3×C4⋊(C3⋊(C3⋊D3))C211yesmaximal, characteristic, complemented
3C3×C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4C32×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
5#10620157C212yesmaximal, complemented
6C4⋊#1113—34nomaximal
7#89026—31nomaximal
8C3⋊D3×D12—31nomaximal
9C2×C3×C3⋊(C3⋊D3)C2211yescharacteristic
10C2×C3×C3⋊(C3⋊D3)C2212yescomplemented
11C2×C32×C3⋊D3C2212yescomplemented
12C3⋊#1113—24no—
13C34×C4C2211yescharacteristic, complemented
14C3×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
15C2×#1113—38no—
16C2×D3×C3⋊D3—32no—
17C2×D3×C3⋊D3—32no—
18C4⋊(C3⋊(C3⋊D3))D311yescharacteristic, complemented
19C4×C3⋊(C3⋊D3)D311yescharacteristic, complemented
20C3×C4⋊(C3⋊D3)—31no—
21C3×C4⋊(C3⋊D3)—34no—
22C3×C4⋊(C3⋊D3)—64no—
23C3×C4⋊(C3⋊D3)D314yescomplemented
24C3×C4×C3⋊D3—31no—
25C3×C4×C3⋊D3—34no—
26C3×C4×C3⋊D3—64no—
27C32×D12—31no—
28C32×D12—31no—
29C32×D12—61no—
30#6295—38no—
31#6291—32no—
32#6290—32no—
33C2×C34C2311yescharacteristic
34C3×C3⋊(C3⋊D3)—22no—
35C3×C3⋊(C3⋊D3)D412yescomplemented
36C32×C3⋊D3—22no—
37D4×C3⋊D3—91no—
38#3802—94no—
39D3×D12—94no—
40D3×D12—91no—
41C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
42C2×C3⋊(C3⋊D3)C2×D312yescomplemented
43C2×C3×C3⋊D3—31no—
44C2×C3×C3⋊D3—34no—
45C2×C3×C3⋊D3—32no—
46C2×C3×C3⋊D3—38no—
47C2×C3×C3⋊D3—64no—
48C2×C3×C3⋊D3—68no—
49C2×C3×C3⋊D3C2×D318yescomplemented
50C2×C32×D3—32no—
51C2×C32×D3—32no—
52C2×C32×D3—62no—
53#1113—616no—
54D3×C3⋊D3—64no—
55D3×C3⋊D3—64no—
56C33×C4—21no—
57C33×C4—24no—
58C33×C4—24no—
59C33×C4—44no—
60C33×C4C2×D311yescharacteristic, complemented
61C33×C4C2×D311yescharacteristic, complemented
62C33×C4C2×D314yescomplemented
63C3⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
64C3×C3⋊Dic3—31no—
65C3×C3⋊Dic3—34no—
66C3×C3⋊Dic3—64no—
67C34C2×D411yescharacteristic, complemented, Sylow
68C22×C3⋊D3—92no—
69C2×D32—98no—
70C2×D32—98no—
71C2×D32—92no—
72C32×D4—91no—
73#638—92no—
74C4⋊(C3⋊D3)—31no—
75C4⋊(C3⋊D3)—34no—
76C4⋊(C3⋊D3)—64no—
77C4⋊(C3⋊D3)C3⋊D311yescharacteristic, complemented
78C4×C3⋊D3—31no—
79C4×C3⋊D3—34no—
80C4×C3⋊D3—64no—
81C3×D12—34no—
82C3×D12—34no—
83C3×D12—64no—
84C3×D12—94no—
85C3×D12—91no—
86C3×D12—184no—
87C3×C4×D3—94no—
88C3×C4×D3—91no—
89C3×C4×D3—184no—
90#626—98no—
91#626—92no—
92#625—98no—
93C2×C33—24no—
94C2×C33—21no—
95C2×C33—24no—
96C2×C33—44no—
97C2×C33C22×D311yescharacteristic
98C2×C33C22×D311yescharacteristic
99C2×C33C22×D314yes—
100C3⋊(C3⋊D3)—22no—
101C3⋊(C3⋊D3)D1212yescomplemented
102C3×C3⋊D3—28no—
103C3×C3⋊D3—32no—
104C3×C3⋊D3—38no—
105C3×C3⋊D3—68no—
106C3×C3⋊D3—62no—
107C3×C3⋊D3—68no—
108C3×C3⋊D3—128no—
109C32×D3—62no—
110C32×D3—62no—
111C32×D3—122no—
112D3×D4—271no—
113D3×D4—274no—
114C2×D12—271no—
115C22×C32—92no—
116C2×C3⋊D3—31no—
117C2×C3⋊D3—32no—
118C2×C3⋊D3—34no—
119C2×C3⋊D3—38no—
120C2×C3⋊D3—64no—
121C2×C3⋊D3—68no—
122C2×C3⋊D3—184no—
123C2×C3⋊D3C2×C3⋊D312yescomplemented
124C2×C3×D3—38no—
125C2×C3×D3—38no—
126C2×C3×D3—68no—
127C2×C3×D3—94no—
128C2×C3×D3—91no—
129C2×C3×D3—98no—
130C2×C3×D3—92no—
131C2×C3×D3—184no—
132C2×C3×D3—188no—
133D32—1816no—
134D32—1816no—
135D32—184no—
136C32×C4—24no—
137C32×C4—24no—
138C32×C4—24no—
139C32×C4—24no—
140C32×C4—24no—
141C32×C4—44no—
142C32×C4—412no—
143C32×C4—44no—
144C32×C4D3214yescomplemented
145C32×C4D3214yescomplemented
146C32×C4D3211yescharacteristic, complemented
147C32×C4C2×C3⋊D311yescharacteristic, complemented
148C3⋊Dic3—31no—
149C3⋊Dic3—34no—
150C3⋊Dic3—64no—
151C3×Dic3—94no—
152C3×Dic3—91no—
153C3×Dic3—184no—
154C33—21no—
155C33—24no—
156C33—24no—
157C33—44no—
158C33C2×D1211yescharacteristic, complemented
159C33D3×D411yescharacteristic, complemented
160C33D3×D414yescomplemented
161C22×D3—272no—
162C22×D3—272no—
163C22×D3—278no—
164C3×D4—94no—
165C3×D4—271no—
166C2×C3×C4—271no—
167#67—272no—
168#67—278no—
169D12—31no—
170D12—31no—
171D12—61no—
172D12—91no—
173D12—94no—
174D12—184no—
175D12—272no—
176C4×D3—91no—
177C4×D3—94no—
178C4×D3—184no—
179C2×C32—24no—
180C2×C32—24no—
181C2×C32—24no—
182C2×C32—24no—
183C2×C32—24no—
184C2×C32—44no—
185C2×C32—412no—
186C2×C32—44no—
187C2×C32—182no—
188C2×C32C2×D3214yes—
189C2×C32C2×D3214yes—
190C2×C32C2×D3211yescharacteristic
191C2×C32C22×C3⋊D311yescharacteristic
192C3⋊D3—22no—
193C3⋊D3—32no—
194C3⋊D3—38no—
195C3⋊D3—68no—
196C3⋊D3—62no—
197C3⋊D3—68no—
198C3⋊D3—128no—
199C3×D3—68no—
200C3×D3—68no—
201C3×D3—98no—
202C3×D3—92no—
203C3×D3—128no—
204C3×D3—188no—
205C3×D3—188no—
206C3×D3—182no—
207C3×D3—368no—
208C2×D4—811noSylow
209C22×C3—98no—
210C22×C3—271no—
211C22×C3—272no—
212C2×D3—32no—
213C2×D3—32no—
214C2×D3—62no—
215C2×D3—91no—
216C2×D3—94no—
217C2×D3—92no—
218C2×D3—98no—
219C2×D3—184no—
220C2×D3—188no—
221C2×D3—544no—
222C2×D3—544no—
223C2×D3—5416no—
224C3×C4—24no—
225C3×C4—21no—
226C3×C4—24no—
227C3×C4—44no—
228C3×C4—271no—
229C3×C4D3×C3⋊D311yescharacteristic, complemented
230C3×C4D3×C3⋊D311yescharacteristic, complemented
231C3×C4#111314yescomplemented
232Dic3—91no—
233Dic3—94no—
234Dic3—184no—
235C32—24no—
236C32—24no—
237C32—24no—
238C32—24no—
239C32—24no—
240C32—44no—
241C32—412no—
242C32—44no—
243C32D3×D1214yescomplemented
244C32D3×D1211yescharacteristic, complemented
245C32#380214yescomplemented
246C32D4×C3⋊D311yescharacteristic, complemented
247C23—812no—
248D4—91no—
249D4—271no—
250D4—812no—
251C2×C4—271no—
252C2×C3—24no—
253C2×C3—21no—
254C2×C3—24no—
255C2×C3—44no—
256C2×C3—188no—
257C2×C3—272no—
258C2×C3—542no—
259C2×C3C2×D3×C3⋊D311yescharacteristic
260C2×C3C2×D3×C3⋊D311yescharacteristic
261C2×C3C2×#111314yes—
262D3—62no—
263D3—62no—
264D3—92no—
265D3—98no—
266D3—122no—
267D3—188no—
268D3—182no—
269D3—188no—
270D3—368no—
271C22—92no—
272C22—271no—
273C22—272no—
274C22—1624no—
275C4—271no—
276C4C3⋊#111311yescharacteristic, complemented
277C3—21no—
278C3—24no—
279C3—24no—
280C3—44no—
281C3C3⋊D3×D1211yescharacteristic, complemented
282C3#8902611yescharacteristic, complemented
283C3C4⋊#111314yescomplemented
284C2—182no—
285C2—272no—
286C2—542no—
287C2C2×C3⋊#111311yescharacteristic
subgroupG/H
center, FrattiniC2C2×C3⋊#1113
commutator, socle, cosocleC2×C34C23
FittingC34×C4C22
hypercenterC4C3⋊#1113
hypocenterC34C2×D4
LCSG ▷ C2×C34 ▷ C34
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊#1113C2semidirect
2. C3×C4⋊(C3⋊(C3⋊D3))C2semidirect
3. C3×C4×C3⋊(C3⋊D3)C2semidirect
4. C32×C4⋊(C3⋊D3)C2semidirect
5. #10620157C2semidirect
10. C2×C3×C3⋊(C3⋊D3)C22semidirect
11. C2×C32×C3⋊D3C22semidirect
13. C34×C4C22semidirect
14. C3×C3⋊(C3⋊Dic3)C22semidirect
18. C4⋊(C3⋊(C3⋊D3))D3semidirect
19. C4×C3⋊(C3⋊D3)D3semidirect
23. C3×C4⋊(C3⋊D3)D3semidirect
35. C3×C3⋊(C3⋊D3)D4semidirect
42. C2×C3⋊(C3⋊D3)C2×D3semidirect
49. C2×C3×C3⋊D3C2×D3semidirect
60. C33×C4C2×D3semidirect
61. C33×C4C2×D3semidirect
62. C33×C4C2×D3semidirect
63. C3⋊(C3⋊Dic3)C2×D3semidirect
67. C34C2×D4semidirect
77. C4⋊(C3⋊D3)C3⋊D3semidirect
101. C3⋊(C3⋊D3)D12semidirect
123. C2×C3⋊D3C2×C3⋊D3semidirect
144. C32×C4D32semidirect
145. C32×C4D32semidirect
146. C32×C4D32semidirect
147. C32×C4C2×C3⋊D3semidirect
158. C33C2×D12semidirect
159. C33D3×D4semidirect
160. C33D3×D4semidirect
229. C3×C4D3×C3⋊D3semidirect
230. C3×C4D3×C3⋊D3semidirect
231. C3×C4#1113semidirect
243. C32D3×D12semidirect
244. C32D3×D12semidirect
245. C32#3802semidirect
246. C32D4×C3⋊D3semidirect
276. C4C3⋊#1113semidirect
281. C3C3⋊D3×D12semidirect
282. C3#89026semidirect
283. C3C4⋊#1113semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊#11132
C3×C4⋊(C3⋊(C3⋊D3))2
C3×C4×C3⋊(C3⋊D3)2
C32×C4⋊(C3⋊D3)2
#106201572
C4⋊#11133
#890263
C3⋊D3×D123