Finite Group Databasebeta

← C2×C3⋊D3×#41⋊C3 #13181468

Subgroups — C2×C3⋊D3×#41⋊C3 #13181468

Order 1728.

1424 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 1424 conjugacy classes of subgroups — too many to draw. Only its 149 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×C32×#41⋊C3C211yesmaximal, characteristic, complemented
2C3⋊D3×#41⋊C3C214yesmaximal, direct product factor
3C2×C3⋊D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C2×C3⋊(C3⋊(#41⋊C3))C211yesmaximal, characteristic, complemented
5C2×#41×C3⋊D3C311yesmaximal, characteristic, complemented
6C2×D3×#41⋊C3—34nomaximal
7C2×C3×C4×C3⋊D3—41nomaximal
8C32×#41⋊C3C2212yescomplemented
9C2×C32×SL(2,𝔽3)C2211yescharacteristic, complemented
10C3⋊D3×SL(2,𝔽3)C2212yescomplemented
11C3⋊(C3⋊(#41⋊C3))C2212yescomplemented
12C2×C32×#41C2×C311yescharacteristic, complemented
13#41×C3⋊D3—34no—
14#41×C3⋊D3C2×C314yescomplemented
15C2×#3832C2×C311yescharacteristic, complemented
16C2×Q8×C3⋊D3C2×C311yescharacteristic
17C2×#3830—31no—
18C2×D4×C3⋊D3—31no—
19C2×#3828—31no—
20C22×C4×C3⋊D3—31no—
21C2×C3×#41⋊C3—24no—
22C2×C3×#41⋊C3D314yescomplemented
23D3×#41⋊C3—316no—
24C2×D3×SL(2,𝔽3)—34no—
25C2×C3⋊(#41⋊C3)—34no—
26C22×C3×C3⋊D3—41no—
27C2×C33×C4—41no—
28C2×C3×C3⋊Dic3—41no—
29C3×C4×C3⋊D3—44no—
30C32×SL(2,𝔽3)C2311yescharacteristic, complemented
31C2×D3×#41—34no—
32C22×#41⋊C3—91no—
33C23×C3⋊D3—31no—
34C32×#41—32no—
35C32×#41C22×C312yescomplemented
36C2×C32×Q8C22×C311yescharacteristic
37C2×C32×D4—31no—
38C22×C32×C4—31no—
39C2×#638—32no—
40C22×C3⋊Dic3—31no—
41#3832—32no—
42#3832C22×C312yescomplemented
43Q8×C3⋊D3—32no—
44Q8×C3⋊D3C22×C312yes—
45#3830—34no—
46#3830—34no—
47D4×C3⋊D3—34no—
48D4×C3⋊D3—34no—
49#3828—34no—
50#3828—34no—
51C2×C4⋊(C3⋊D3)—31no—
52C2×C4×C3⋊D3—31no—
53C2×C4×C3⋊D3—31no—
54C2×C4×C3⋊D3—31no—
55C2×C4×C3⋊D3—34no—
56C2×C4×C3⋊D3—34no—
57C2×C4×C3⋊D3A411yescharacteristic
58C2×#634—31no—
59C2×C3×C4×D3—124no—
60C3×#41⋊C3—28no—
61C3×#41⋊C3C2×D318yescomplemented
62C2×C3×SL(2,𝔽3)—24no—
63C2×C3×SL(2,𝔽3)C2×D314yescomplemented
64D3×SL(2,𝔽3)—38no—
65C3⋊(#41⋊C3)—38no—
66C22×C33—41no—
67C2×C3×C3⋊D3—44no—
68C2×C3×C3⋊D3—42no—
69C33×C4—42no—
70C3×C3⋊Dic3—42no—
71C2×C3×#41C3×D314yescomplemented
72D3×#41—316no—
73D3×#41—916no—
74C2×#239—34no—
75C2×D3×Q8—34no—
76C2×#237—94no—
77C2×D3×D4—94no—
78C2×#235—94no—
79C22×C4×D3—94no—
80C2×#41⋊C3—24no—
81C2×#41⋊C3—94no—
82C2×#41⋊C3—91no—
83C2×#41⋊C3C3⋊D311yescharacteristic, direct product factor
84C22×SL(2,𝔽3)—91no—
85C23×C32—31no—
86C22×C3⋊D3—31no—
87C22×C3⋊D3—34no—
88C22×C3⋊D3—64no—
89C22×C3⋊D3C2×A411yescharacteristic
90C22×C3×D3—124no—
91C32×Q8—31no—
92C32×Q8C23×C311yescharacteristic
93C32×D4—32no—
94C32×D4—32no—
95C2×C32×C4—31no—
96C2×C32×C4—32no—
97C2×C32×C4—32no—
98C2×C32×C4—44no—
99C2×C32×C4—84no—
100C2×C32×C4C2×A411yescharacteristic
101#638—38no—
102C2×C3⋊Dic3—31no—
103C2×C3⋊Dic3—34no—
104C2×C3⋊Dic3C2×A411yescharacteristic
105C4⋊(C3⋊D3)—32no—
106C4⋊(C3⋊D3)—32no—
107C4×C3⋊D3—32no—
108C4×C3⋊D3—32no—
109C4×C3⋊D3—34no—
110C4×C3⋊D3—32no—
111C4×C3⋊D3—32no—
112C4×C3⋊D3C2×A414yes—
113#634—32no—
114#634—32no—
115C2×C3×Dic3—124no—
116C3×C4×D3—1216no—
117C3×SL(2,𝔽3)—24no—
118C3×SL(2,𝔽3)C22×D314yescomplemented
119C22×#41—91noSylow
120C2×C33—42no—
121C2×C33—41no—
122C3×C3⋊D3—44no—
123C23×D3—94no—
124C3×#41—38no—
125C3×#41C2×C3×D318yescomplemented
126C2×C3×Q8C2×C3×D314yes—
127C2×C3×D4—34no—
128C22×C3×C4—34no—
129C22×C3×C4—361no—
130C2×#67—98no—
131C22×Dic3—94no—
132#239—38no—
133#239—98no—
134D3×Q8—38no—
135D3×Q8—98no—
136#237—916no—
137#237—916no—
138D3×D4—916no—
139D3×D4—916no—
140#235—916no—
141#235—916no—
142C2×D12—94no—
143C2×C4×D3—34no—
144C2×C4×D3—916no—
145C2×C4×D3—916no—
146C2×C4×D3—94no—
147C2×C4×D3—94no—
148C2×C4×D3—94no—
149C2×Dic6—94no—
150#41⋊C3—28no—
151#41⋊C3—92no—
152#41⋊C3C2×C3⋊D312yesdirect product factor
153C2×SL(2,𝔽3)—24no—
154C2×SL(2,𝔽3)—92no—
155C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic, complemented
156C22×C32—32no—
157C22×C32—44no—
158C22×C32—62no—
159C22×C32—84no—
160C22×C32C22×A411yescharacteristic
161C2×C3⋊D3—32no—
162C2×C3⋊D3—62no—
163C2×C3⋊D3—68no—
164C2×C3⋊D3#41⋊C314yesdirect product factor
165C2×C3⋊D3C22×A412yes—
166C2×C3×D3—128no—
167C2×C3×D3—1216no—
168C32×C4—31no—
169C32×C4—31no—
170C32×C4—48no—
171C32×C4—88no—
172C32×C4C22×A412yes—
173C3⋊Dic3—32no—
174C3⋊Dic3C22×A412yes—
175C3×Dic3—128no—
176C2×#41—94no—
177C2×#41—91no—
178C2×#41—271no—
179C2×#41—271no—
180C2×#41—274no—
181C2×#41C3×C3⋊D311yescharacteristic, complemented
182C22×Q8—91no—
183C22×D4—271no—
184C23×C4—271no—
185C33—41noSylow
186C23×C3—34no—
187C23×C3—361no—
188C22×D3—34no—
189C22×D3—916no—
190C22×D3—94no—
191C22×D3—1816no—
192C3×Q8—34no—
193C3×Q8C22×C3×D314yes—
194C3×D4—38no—
195C3×D4—38no—
196C2×C3×C4—34no—
197C2×C3×C4—38no—
198C2×C3×C4—38no—
199C2×C3×C4—41no—
200C2×C3×C4—84no—
201C2×C3×C4—364no—
202C2×C3×C4—361no—
203C2×C3×C4D3×A414yes—
204#67—932no—
205C2×Dic3—34no—
206C2×Dic3—916no—
207C2×Dic3—94no—
208D12—98no—
209D12—98no—
210C4×D3—316no—
211C4×D3—98no—
212C4×D3—98no—
213C4×D3—98no—
214C4×D3—98no—
215C4×D3—916no—
216Dic6—98no—
217Dic6—98no—
218SL(2,𝔽3)—24no—
219SL(2,𝔽3)C22×C3⋊D311yescharacteristic, complemented
220C2×C32—48no—
221C2×C32—44no—
222C2×C32—62no—
223C2×C32—88no—
224C2×C32—84no—
225C2×C32C2×#41⋊C312yescomplemented
226C2×C32C23×A411yescharacteristic
227C3⋊D3—62no—
228C3⋊D3C2×#41⋊C314yesdirect product factor
229C3×D3—1216no—
230C24—271no—
231#41—32no—
232#41—92no—
233#41—274no—
234#41—274no—
235#41—274no—
236#41—274no—
237#41—272no—
238#41C2×C3×C3⋊D312yescomplemented
239C2×Q8—92no—
240C2×Q8—271no—
241C2×Q8—272no—
242C2×Q8C2×C3×C3⋊D311yescharacteristic
243C2×D4—31no—
244C2×D4—271no—
245C2×D4—274no—
246C2×D4—274no—
247C2×D4—272no—
248C22×C4—31no—
249C22×C4—91no—
250C22×C4—271no—
251C22×C4—274no—
252C22×C4—274no—
253C22×C4—271no—
254C22×C4—271no—
255C22×C4—271no—
256C22×C3—38no—
257C22×C3—41no—
258C22×C3—68no—
259C22×C3—84no—
260C22×C3—362no—
261C22×C3—364no—
262C22×C3C2×D3×A414yes—
263C2×D3—316no—
264C2×D3—38no—
265C2×D3—98no—
266C2×D3—1832no—
267C2×D3—188no—
268C3×C4—34no—
269C3×C4—34no—
270C3×C4—42no—
271C3×C4—88no—
272C3×C4—362no—
273C3×C4C2×D3×A418yes—
274Dic3—38no—
275Dic3—98no—
276C32—44no—
277C32—84no—
278C32C22×#41⋊C311yescharacteristic, complemented
279C23—31no—
280C23—91no—
281C23—274no—
282C23—271no—
283C23—544no—
284Q8—31no—
285Q8—272no—
286Q8—272no—
287Q8C22×C3×C3⋊D311yescharacteristic
288D4—32no—
289D4—32no—
290D4—272no—
291D4—272no—
292D4—278no—
293C2×C4—31no—
294C2×C4—32no—
295C2×C4—32no—
296C2×C4—91no—
297C2×C4—94no—
298C2×C4—272no—
299C2×C4—272no—
300C2×C4—272no—
301C2×C4—272no—
302C2×C4—274no—
303C2×C4—271no—
304C2×C4—274no—
305C2×C4A4×C3⋊D311yescharacteristic
306C2×C3—41no—
307C2×C3—42no—
308C2×C3—68no—
309C2×C3—84no—
310C2×C3—88no—
311C2×C3—364no—
312C2×C3D3×#41⋊C318yescomplemented
313C2×C3C22×D3×A414yes—
314D3—316no—
315D3—188no—
316C22—32no—
317C22—62no—
318C22—92no—
319C22—94no—
320C22—272no—
321C22—548no—
322C22—542no—
323C22C2×A4×C3⋊D311yescharacteristic
324C4—31no—
325C4—31no—
326C4—92no—
327C4—272no—
328C4C2×A4×C3⋊D312yes—
329C3—41no—
330C3—84no—
331C3C2×D3×#41⋊C314yescomplemented
332C2—62no—
333C2—94no—
334C2—542no—
335C2C3⋊D3×#41⋊C312yesdirect product factor
336C2C22×A4×C3⋊D311yescharacteristic
subgroupG/H
center, hypercenterC2×C4A4×C3⋊D3
commutator, cosocle, hypocenterC32×Q8C23×C3
FrattiniC2C22×A4×C3⋊D3
FittingC2×C32×#41C2×C3
socleC22×C32C22×A4
LCSG ▷ C32×Q8
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#41⋊C3C2semidirect
2. C3⋊D3×#41⋊C3C2direct (N × H)
3. C2×C3⋊D3×SL(2,𝔽3)C2semidirect
4. C2×C3⋊(C3⋊(#41⋊C3))C2semidirect
5. C2×#41×C3⋊D3C3semidirect
8. C32×#41⋊C3C22semidirect
9. C2×C32×SL(2,𝔽3)C22semidirect
10. C3⋊D3×SL(2,𝔽3)C22semidirect
11. C3⋊(C3⋊(#41⋊C3))C22semidirect
12. C2×C32×#41C2×C3semidirect
14. #41×C3⋊D3C2×C3semidirect
15. C2×#3832C2×C3semidirect
22. C2×C3×#41⋊C3D3semidirect
30. C32×SL(2,𝔽3)C23semidirect
35. C32×#41C22×C3semidirect
42. #3832C22×C3semidirect
61. C3×#41⋊C3C2×D3semidirect
63. C2×C3×SL(2,𝔽3)C2×D3semidirect
71. C2×C3×#41C3×D3semidirect
83. C2×#41⋊C3C3⋊D3direct (N × H)
118. C3×SL(2,𝔽3)C22×D3semidirect
125. C3×#41C2×C3×D3semidirect
152. #41⋊C3C2×C3⋊D3direct (N × H)
155. C2×SL(2,𝔽3)C2×C3⋊D3semidirect
164. C2×C3⋊D3#41⋊C3direct (N × H)
181. C2×#41C3×C3⋊D3semidirect
219. SL(2,𝔽3)C22×C3⋊D3semidirect
225. C2×C32C2×#41⋊C3semidirect
228. C3⋊D3C2×#41⋊C3direct (N × H)
238. #41C2×C3×C3⋊D3semidirect
278. C32C22×#41⋊C3semidirect
312. C2×C3D3×#41⋊C3semidirect
331. C3C2×D3×#41⋊C3semidirect
335. C2C3⋊D3×#41⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#41⋊C32
C3⋊D3×#41⋊C32
C2×C3⋊D3×SL(2,𝔽3)2
C2×C3⋊(C3⋊(#41⋊C3))2
C2×#41×C3⋊D33
C2×D3×#41⋊C33
C2×C3×C4×C3⋊D34