Finite Group Databasebeta

← D3×#541 #86508

Subgroups — D3×#541 #86508

Order 384.

1004 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#541C211yesmaximal, characteristic, complemented
2#5731C211yesmaximal, characteristic, complemented
3D3×C4⋊Q8C211yesmaximal, characteristic, complemented
4D3×#126C214yesmaximal
5#5701C214yesmaximal, complemented
6#5698C211yesmaximal, characteristic, complemented
7#5696C214yesmaximal, complemented
8D3×#125C214yesmaximal, complemented
9D3×#125C211yesmaximal, characteristic, complemented
10D3×C22⋊Q8C212yesmaximal
11#5625C212yesmaximal, complemented
12#5583C212yesmaximal, complemented
13C4×D3×Q8C212yesmaximal, complemented
14D3×#117C211yesmaximal, characteristic
15#5518C211yesmaximal, characteristic, complemented
16C2×#541—31nomaximal, Sylow
17C2×D3×Q8C2212yes—
18C22×C4×D3C2211yescharacteristic
19C3×C4⋊Q8C2211yescharacteristic, complemented
20C3×#126C2214yes—
21C3×#125C2214yescomplemented
22C3×#125C2211yescharacteristic, complemented
23C3×C22⋊Q8C2212yes—
24C3×C4×Q8C2212yescomplemented
25C3×#117C2211yescharacteristic
26#946C2212yes—
27Q8×Dic3C2212yescomplemented
28#944C2212yescomplemented
29#926C2211yescharacteristic
30#924C2212yes—
31#898C2214yes—
32#898C2214yes—
33#898C2214yes—
34#897C2212yes—
35#896C2214yes—
36#892C2212yes—
37D3×C4⋊C4C2214yes—
38D3×C4⋊C4C2212yes—
39D3×C4⋊C4C2214yes—
40D3×C4⋊C4C2214yes—
41D3×C4⋊C4C2214yes—
42D3×C4⋊C4C2212yes—
43#890C2214yescomplemented
44#890C2214yescomplemented
45#890C2212yescomplemented
46#889C2214yescomplemented
47#889C2214yescomplemented
48#889C2214yescomplemented
49#889C2214yescomplemented
50#889C2214yescomplemented
51#888C2214yescomplemented
52#887C2214yescomplemented
53#887C2212yescomplemented
54D3×C22⋊C4C2212yes—
55D3×C22⋊C4C2214yes—
56#879C2214yes—
57#879C2214yes—
58#879C2214yes—
59#878C2214yes—
60#877C2212yes—
61#876C2214yes—
62#872C2211yescharacteristic
63#872C2211yescharacteristic
64C42×D3C2214yes—
65C42×D3C2211yescharacteristic
66C42×D3C2211yescharacteristic
67#870C2214yescomplemented
68#870C2211yescharacteristic, complemented
69#869C2211yescharacteristic, complemented
70C4×Dic6C2214yescomplemented
71C4×Dic6C2212yescomplemented
72#541—31no—
73#541—34no—
74#541—34no—
75#541—31no—
76#541—32no—
77#541—32no—
78#541—31no—
79#541D311yescharacteristic, direct product factor
80C2×C4⋊Q8—31no—
81C2×#126—34no—
82C2×#125—31no—
83C2×#125—34no—
84C2×C22⋊Q8—32no—
85C2×C4×Q8—32no—
86C2×#117—31no—
87C23×D3C2311yescharacteristic
88C2×C3×Q8C2312yes—
89C22×C3×C4C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91D3×Q8—28no—
92C2×C4×D3—24no—
93C2×C4×D3C2312yes—
94C2×C4×D3C2314yes—
95C2×C4×D3C2312yes—
96C2×C4×D3C2314yes—
97C2×C4×D3C2311yescharacteristic
98C2×C4×D3C2311yescharacteristic
99C2×C4×D3C2311yescharacteristic
100C2×C4×D3C2311yescharacteristic
101C2×Dic6C2312yes—
102C2×Dic6C2314yes—
103C3×C4⋊C4C2314yes—
104C3×C4⋊C4C2312yes—
105C3×C4⋊C4C2314yes—
106C3×C4⋊C4C2312yes—
107C3×C4⋊C4C2314yes—
108C3×C4⋊C4C2314yes—
109C3×C22⋊C4C2312yes—
110C3×C22⋊C4C2314yes—
111C3×C42C2311yescharacteristic
112C3×C42C2314yes—
113C3×C42C2311yescharacteristic
114#217C2312yes—
115#217C2314yes—
116(C2×D3)⋊C4C2312yes—
117(C2×D3)⋊C4C2312yes—
118(C2×D3)⋊C4C2314yes—
119(C2×D3)⋊C4C2314yes—
120C4⋊Dic3C2312yes—
121C4⋊Dic3C2314yes—
122C4⋊Dic3C2314yes—
123C4⋊Dic3C2314yes—
124C4⋊Dic3C2314yes—
125C4⋊Dic3C2312yes—
126Dic3⋊C4C2314yes—
127Dic3⋊C4C2314yes—
128Dic3⋊C4C2314yes—
129Dic3⋊C4C2312yes—
130Dic3⋊C4C2314yes—
131Dic3⋊C4C2314yes—
132Dic3⋊C4C2314yes—
133Dic3⋊C4C2312yes—
134Dic3⋊C4C2314yes—
135Dic3⋊C4C2314yes—
136Dic3⋊C4C2314yes—
137C4×Dic3C2312yes—
138C4×Dic3C2314yes—
139C4×Dic3C2311yescharacteristic
140C4×Dic3C2311yescharacteristic
141C4×Dic3C2312yes—
142C4×Dic3C2314yes—
143C4×Dic3C2314yes—
144C22×Q8—32no—
145C23×C4—31no—
146C4⋊Q8—32no—
147C4⋊Q8—34no—
148C4⋊Q8—31no—
149C4⋊Q8C2×D311yescharacteristic, complemented
150#126—34no—
151#126—34no—
152#126—34no—
153#126—34no—
154#126—34no—
155#126—34no—
156#126—34no—
157#126C2×D314yes—
158#125—34no—
159#125—34no—
160#125—34no—
161#125—34no—
162#125—34no—
163#125—32no—
164#125—34no—
165#125—34no—
166#125—31no—
167#125—34no—
168#125C2×D311yescharacteristic, complemented
169#125C2×D314yescomplemented
170C22⋊Q8—32no—
171C22⋊Q8—32no—
172C22⋊Q8—34no—
173C22⋊Q8—32no—
174C22⋊Q8—34no—
175C22⋊Q8C2×D312yes—
176C4×Q8—32no—
177C4×Q8—34no—
178C4×Q8—32no—
179C4×Q8—34no—
180C4×Q8—32no—
181C4×Q8C2×D312yescomplemented
182#117—31no—
183#117—32no—
184#117—32no—
185#117—31no—
186#117—31no—
187#117C2×D311yescharacteristic
188C2×C4⋊C4—32no—
189C2×C4⋊C4—32no—
190C2×C4⋊C4—34no—
191C2×C4⋊C4—34no—
192C2×C4⋊C4—34no—
193C2×C4⋊C4—34no—
194C2×C22⋊C4—32no—
195C2×C22⋊C4—34no—
196C2×C42—31no—
197C2×C42—31no—
198C2×C42—34no—
199C23×C3C2411yescharacteristic
200C22×D3—22no—
201C22×D3—24no—
202C22×D3C2411yescharacteristic
203C22×D3C2411yescharacteristic
204C3×Q8—24no—
205C2×C3×C4—22no—
206C2×C3×C4C2412yes—
207C2×C3×C4C2412yes—
208C2×C3×C4C2414yes—
209C2×C3×C4C2414yes—
210C2×C3×C4C2411yescharacteristic
211C2×C3×C4C2411yescharacteristic
212C2×Dic3—22no—
213C2×Dic3C2412yes—
214C2×Dic3C2414yes—
215C2×Dic3C2412yes—
216C2×Dic3C2414yes—
217C2×Dic3C2411yescharacteristic
218C2×Dic3C2411yescharacteristic
219C4×D3—22no—
220C4×D3—28no—
221C4×D3—24no—
222C4×D3—28no—
223C4×D3—24no—
224C4×D3—22no—
225C4×D3—22no—
226C4×D3#4114yes—
227Dic6—24no—
228Dic6—28no—
229C24—31no—
230C2×Q8—34no—
231C2×Q8—32no—
232C2×Q8—68no—
233C2×Q8C22×D312yes—
234C22×C4—31no—
235C22×C4—31no—
236C22×C4—32no—
237C22×C4—32no—
238C22×C4—34no—
239C22×C4—34no—
240C22×C4—31no—
241C22×C4—31no—
242C22×C4—31no—
243C22×C4—64no—
244C22×C4C22×D311yescharacteristic
245C4⋊C4—32no—
246C4⋊C4—32no—
247C4⋊C4—32no—
248C4⋊C4—34no—
249C4⋊C4—32no—
250C4⋊C4—34no—
251C4⋊C4—34no—
252C4⋊C4—34no—
253C4⋊C4—34no—
254C4⋊C4—34no—
255C4⋊C4—34no—
256C4⋊C4—34no—
257C4⋊C4—34no—
258C4⋊C4—34no—
259C4⋊C4—34no—
260C4⋊C4—34no—
261C4⋊C4—34no—
262C4⋊C4C22×D312yes—
263C4⋊C4C22×D312yes—
264C4⋊C4C22×D314yes—
265C4⋊C4C22×D314yes—
266C4⋊C4C22×D314yes—
267C4⋊C4C22×D314yes—
268C22⋊C4—32no—
269C22⋊C4—34no—
270C22⋊C4—32no—
271C22⋊C4—32no—
272C22⋊C4—34no—
273C22⋊C4—34no—
274C22⋊C4C22×D312yes—
275C22⋊C4C22×D314yes—
276C42—32no—
277C42—31no—
278C42—32no—
279C42—31no—
280C42—34no—
281C42—34no—
282C42—34no—
283C42C22×D311yescharacteristic
284C42C22×D311yescharacteristic
285C42C22×D314yes—
286C22×C3—21no—
287C22×C3—22no—
288C22×C3C2511yescharacteristic
289C2×D3—21no—
290C2×D3—22no—
291C2×D3—44no—
292C2×D3C2×#4112yes—
293C2×D32−1+414yes—
294C3×C4—24no—
295C3×C4—21no—
296C3×C4—22no—
297C3×C4—22no—
298C3×C4—24no—
299C3×C4C2×#4112yes—
300Dic3—24no—
301Dic3—21no—
302Dic3—22no—
303Dic3—24no—
304Dic3—22no—
305Dic3C2×#4112yes—
306C23—31no—
307C23—31no—
308C23—62no—
309C23—64no—
310C23C23×D311yescharacteristic
311Q8—24no—
312Q8—68no—
313Q8—64no—
314C2×C4—22no—
315C2×C4—31no—
316C2×C4—34no—
317C2×C4—32no—
318C2×C4—32no—
319C2×C4—34no—
320C2×C4—34no—
321C2×C4—31no—
322C2×C4—62no—
323C2×C4—62no—
324C2×C4—68no—
325C2×C4—64no—
326C2×C4—64no—
327C2×C4—68no—
328C2×C4—62no—
329C2×C4—62no—
330C2×C4C23×D312yes—
331C2×C4C23×D312yes—
332C2×C4C23×D314yes—
333C2×C4C23×D314yes—
334C2×C4C23×D311yescharacteristic
335C2×C4C23×D311yescharacteristic
336C2×C3—41no—
337C2×C3C22×#4111yescharacteristic
338C2×C3C2×2−1+412yes—
339D3—41no—
340D3#54114yesdirect product factor
341C22—21no—
342C22—22no—
343C22—32no—
344C22—34no—
345C22—61no—
346C22—62no—
347C22—124no—
348C22C24×D311yescharacteristic
349C4—21no—
350C4—24no—
351C4—22no—
352C4—22no—
353C4—24no—
354C4—32no—
355C4—61no—
356C4—64no—
357C4—62no—
358C4—62no—
359C4—64no—
360C4D3×#4112yes—
361C3C2×#54111yescharacteristic, complemented, Sylow
362C2—34no—
363C2—41no—
364C2—121no—
365C2C2×D3×#4111yescharacteristic
366C2D3×2−1+412yes—
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#541C2
prime radicalC23×D3C23
hypercenter#541D3
hypocenterC3C2×#541
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #541

Complements

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

NH ≅ G/Ntype
1. C3×#541C2semidirect
2. #5731C2semidirect
3. D3×C4⋊Q8C2semidirect
5. #5701C2semidirect
6. #5698C2semidirect
7. #5696C2semidirect
8. D3×#125C2semidirect
9. D3×#125C2semidirect
11. #5625C2semidirect
12. #5583C2semidirect
13. C4×D3×Q8C2semidirect
15. #5518C2semidirect
19. C3×C4⋊Q8C22semidirect
21. C3×#125C22semidirect
22. C3×#125C22semidirect
24. C3×C4×Q8C22semidirect
27. Q8×Dic3C22semidirect
28. #944C22semidirect
43. #890C22semidirect
44. #890C22semidirect
45. #890C22semidirect
46. #889C22semidirect
47. #889C22semidirect
48. #889C22semidirect
49. #889C22semidirect
50. #889C22semidirect
51. #888C22semidirect
52. #887C22semidirect
53. #887C22semidirect
67. #870C22semidirect
68. #870C22semidirect
69. #869C22semidirect
70. C4×Dic6C22semidirect
71. C4×Dic6C22semidirect
79. #541D3direct (N × H)
149. C4⋊Q8C2×D3semidirect
168. #125C2×D3semidirect
169. #125C2×D3semidirect
181. C4×Q8C2×D3semidirect
340. D3#541direct (N × H)
361. C3C2×#541semidirect

Maximal subgroups

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

subgroupindex
C3×#5412
#57312
D3×C4⋊Q82
D3×#1262
#57012
#56982
#56962
D3×#1252
D3×C22⋊Q82
#56252
#55832
C4×D3×Q82
D3×#1172
#55182
C2×#5413