Finite Group Databasebeta

← D3×#556 #87003

Subgroups — D3×#556 #87003

Order 384.

1112 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#556C211yesmaximal, characteristic, complemented
2#5732C211yesmaximal, characteristic, complemented
3D3×C4⋊Q8C211yesmaximal, characteristic, complemented
4D3×#126C214yesmaximal, complemented
5#5702C214yesmaximal, complemented
6D3×#124C212yesmaximal
7#5671C212yesmaximal, complemented
8#5641C212yesmaximal, complemented
9D3×C22⋊Q8C212yesmaximal, complemented
10#5584C211yesmaximal, characteristic, complemented
11C4×D3×Q8C211yesmaximal, characteristic, complemented
12C4×D3×Q8C212yesmaximal, complemented
13#5568C212yesmaximal, complemented
14C4×D3×D4C211yesmaximal, characteristic
15#5542C211yesmaximal, characteristic, complemented
16#5533C212yesmaximal, complemented
17D3×#117C212yesmaximal, complemented
18C2×#556—31nomaximal, Sylow
19C2×D3×Q8C2212yes—
20C2×D3×Q8C2211yescharacteristic
21C2×D3×D4C2211yescharacteristic
22C22×C4×D3C2212yes—
23C3×C4⋊Q8C2211yescharacteristic, complemented
24C3×#126C2214yescomplemented
25C3×#124C2212yes—
26C3×C22⋊Q8C2212yescomplemented
27C3×C4×Q8C2212yescomplemented
28C3×C4×Q8C2211yescharacteristic, complemented
29C3×C4×D4C2211yescharacteristic
30C3×#117C2212yescomplemented
31#947C2212yes—
32#946C2212yescomplemented
33Q8×Dic3C2211yescharacteristic, complemented
34Q8×Dic3C2212yescomplemented
35#944C2212yescomplemented
36#936C2212yes—
37D4×Dic3C2211yescharacteristic
38C4×#67C2212yes—
39#926C2212yescomplemented
40#924C2212yescomplemented
41#898C2214yescomplemented
42#898C2214yescomplemented
43#898C2214yescomplemented
44#897C2212yescomplemented
45#896C2214yescomplemented
46#893C2211yescharacteristic
47#892C2212yescomplemented
48#892C2212yescomplemented
49D3×C4⋊C4C2212yes—
50D3×C4⋊C4C2212yes—
51D3×C4⋊C4C2214yes—
52D3×C4⋊C4C2212yes—
53D3×C4⋊C4C2211yescharacteristic
54D3×C4⋊C4C2211yescharacteristic
55#888C2212yescomplemented
56#888C2212yescomplemented
57#887C2214yescomplemented
58#887C2212yescomplemented
59#887C2211yescharacteristic, complemented
60#887C2212yescomplemented
61#885C2214yes—
62#885C2214yes—
63#67⋊C4C2212yes—
64D3×C22⋊C4C2214yes—
65D3×C22⋊C4C2214yes—
66D3×C22⋊C4C2212yes—
67#879C2214yescomplemented
68#879C2214yescomplemented
69#879C2214yescomplemented
70#878C2214yescomplemented
71#877C2214yescomplemented
72#876C2214yescomplemented
73#875C2212yes—
74C4×D12C2211yescharacteristic
75#872C2214yescomplemented
76C42×D3C2212yes—
77C42×D3C2214yes—
78C42×D3C2211yescharacteristic
79#869C2211yescharacteristic, complemented
80C4×Dic6C2212yescomplemented
81C4×Dic6C2212yescomplemented
82C4×Dic6C2214yescomplemented
83C4×Dic6C2211yescharacteristic, complemented
84#556—31no—
85#556—34no—
86#556—32no—
87#556—32no—
88#556—32no—
89#556—31no—
90#556—31no—
91#556—32no—
92#556D311yescharacteristic, direct product factor
93C2×C4⋊Q8—31no—
94C2×#126—34no—
95C2×#124—32no—
96C2×C22⋊Q8—32no—
97C2×C4×Q8—32no—
98C2×C4×Q8—31no—
99C2×C4×D4—31no—
100C2×#117—32no—
101C23×D3C2312yes—
102C2×C3×Q8C2312yes—
103C2×C3×Q8C2311yescharacteristic
104C2×C3×D4C2311yescharacteristic
105C22×C3×C4C2312yes—
106C2×#67C2312yes—
107C22×Dic3C2312yes—
108D3×Q8—24no—
109D3×Q8—28no—
110D3×D4—24no—
111C2×D12C2311yescharacteristic
112C2×C4×D3—28no—
113C2×C4×D3C2312yes—
114C2×C4×D3C2312yes—
115C2×C4×D3C2311yescharacteristic
116C2×C4×D3C2312yes—
117C2×C4×D3C2314yes—
118C2×C4×D3C2312yes—
119C2×C4×D3C2312yes—
120C2×C4×D3C2311yescharacteristic
121C2×C4×D3C2311yescharacteristic
122C2×Dic6C2312yes—
123C2×Dic6C2312yes—
124C2×Dic6C2314yes—
125C2×Dic6C2311yescharacteristic
126C3×C4⋊C4C2311yescharacteristic
127C3×C4⋊C4C2311yescharacteristic
128C3×C4⋊C4C2314yes—
129C3×C4⋊C4C2312yes—
130C3×C4⋊C4C2312yes—
131C3×C4⋊C4C2312yes—
132C3×C22⋊C4C2314yes—
133C3×C22⋊C4C2314yes—
134C3×C22⋊C4C2312yes—
135C3×C42C2312yes—
136C3×C42C2311yescharacteristic
137C3×C42C2314yes—
138#217C2314yes—
139#217C2314yes—
140#217C2312yes—
141(C2×D3)⋊C4C2314yes—
142(C2×D3)⋊C4C2314yes—
143(C2×D3)⋊C4C2312yes—
144(C2×D3)⋊C4C2314yes—
145(C2×D3)⋊C4C2312yes—
146(C2×D3)⋊C4C2314yes—
147C4⋊Dic3C2311yescharacteristic
148C4⋊Dic3C2314yes—
149C4⋊Dic3C2312yes—
150C4⋊Dic3C2311yescharacteristic
151C4⋊Dic3C2312yes—
152C4⋊Dic3C2312yes—
153Dic3⋊C4C2312yes—
154Dic3⋊C4C2312yes—
155Dic3⋊C4C2312yes—
156Dic3⋊C4C2314yes—
157Dic3⋊C4C2314yes—
158Dic3⋊C4C2312yes—
159Dic3⋊C4C2314yes—
160Dic3⋊C4C2314yes—
161C4×Dic3C2314yes—
162C4×Dic3C2312yes—
163C4×Dic3C2312yes—
164C4×Dic3C2311yescharacteristic
165C4×Dic3C2314yes—
166C4×Dic3C2314yes—
167C4×Dic3C2311yescharacteristic
168C4×Dic3C2312yes—
169C4×Dic3C2311yescharacteristic
170C22×Q8—31no—
171C22×Q8—32no—
172C22×D4—31no—
173C23×C4—32no—
174C4⋊Q8—32no—
175C4⋊Q8—32no—
176C4⋊Q8—32no—
177C4⋊Q8—31no—
178C4⋊Q8C2×D311yescharacteristic, complemented
179#126—34no—
180#126—34no—
181#126—34no—
182#126—34no—
183#126—34no—
184#126—34no—
185#126—34no—
186#126C2×D314yescomplemented
187#124—32no—
188#124—32no—
189#124—34no—
190#124—34no—
191#124—32no—
192#124C2×D312yes—
193C22⋊Q8—32no—
194C22⋊Q8—32no—
195C22⋊Q8—34no—
196C22⋊Q8—32no—
197C22⋊Q8—34no—
198C22⋊Q8C2×D312yescomplemented
199C4×Q8—32no—
200C4×Q8—31no—
201C4×Q8—34no—
202C4×Q8—32no—
203C4×Q8—32no—
204C4×Q8—31no—
205C4×Q8—32no—
206C4×Q8—34no—
207C4×Q8—32no—
208C4×Q8—31no—
209C4×Q8C2×D312yescomplemented
210C4×Q8C2×D311yescharacteristic, complemented
211C4×D4—31no—
212C4×D4—32no—
213C4×D4—31no—
214C4×D4—32no—
215C4×D4—31no—
216C4×D4C2×D311yescharacteristic
217#117—32no—
218#117—32no—
219#117—32no—
220#117—34no—
221#117—34no—
222#117C2×D312yescomplemented
223C2×C4⋊C4—31no—
224C2×C4⋊C4—32no—
225C2×C4⋊C4—31no—
226C2×C4⋊C4—34no—
227C2×C4⋊C4—32no—
228C2×C4⋊C4—32no—
229C2×C22⋊C4—34no—
230C2×C22⋊C4—32no—
231C2×C22⋊C4—34no—
232C2×C42—34no—
233C2×C42—31no—
234C2×C42—32no—
235C23×C3C2412yes—
236C22×D3—24no—
237C22×D3—24no—
238C22×D3—24no—
239C22×D3C2412yes—
240C22×D3C2411yescharacteristic
241C3×Q8—24no—
242C3×Q8—22no—
243C3×D4—22no—
244C2×C3×C4—24no—
245C2×C3×C4C2412yes—
246C2×C3×C4C2412yes—
247C2×C3×C4C2411yescharacteristic
248C2×C3×C4C2414yes—
249C2×C3×C4C2412yes—
250C2×C3×C4C2411yescharacteristic
251C2×C3×C4C2411yescharacteristic
252#67—24no—
253C2×Dic3—24no—
254C2×Dic3C2412yes—
255C2×Dic3C2412yes—
256C2×Dic3C2411yescharacteristic
257C2×Dic3C2414yes—
258C2×Dic3C2412yes—
259C2×Dic3C2411yescharacteristic
260C2×Dic3C2411yescharacteristic
261D12—22no—
262C4×D3—24no—
263C4×D3—24no—
264C4×D3—28no—
265C4×D3—24no—
266C4×D3—22no—
267C4×D3—24no—
268C4×D3—24no—
269C4×D3#4114yes—
270C4×D3#4114yes—
271Dic6—24no—
272Dic6—22no—
273Dic6—28no—
274Dic6—24no—
275C24—32no—
276C2×Q8—32no—
277C2×Q8—31no—
278C2×Q8—34no—
279C2×Q8—32no—
280C2×Q8—64no—
281C2×Q8—68no—
282C2×Q8C22×D311yescharacteristic
283C2×Q8C22×D312yes—
284C2×D4—32no—
285C2×D4—31no—
286C2×D4—64no—
287C2×D4C22×D311yescharacteristic
288C22×C4—31no—
289C22×C4—31no—
290C22×C4—32no—
291C22×C4—32no—
292C22×C4—32no—
293C22×C4—31no—
294C22×C4—34no—
295C22×C4—32no—
296C22×C4—32no—
297C22×C4—32no—
298C22×C4—68no—
299C22×C4C22×D312yes—
300C4⋊C4—32no—
301C4⋊C4—31no—
302C4⋊C4—34no—
303C4⋊C4—32no—
304C4⋊C4—32no—
305C4⋊C4—31no—
306C4⋊C4—34no—
307C4⋊C4—34no—
308C4⋊C4—34no—
309C4⋊C4—32no—
310C4⋊C4—32no—
311C4⋊C4—32no—
312C4⋊C4—34no—
313C4⋊C4—32no—
314C4⋊C4C22×D311yescharacteristic
315C4⋊C4C22×D312yes—
316C4⋊C4C22×D311yescharacteristic
317C4⋊C4C22×D314yes—
318C4⋊C4C22×D312yes—
319C4⋊C4C22×D312yes—
320C22⋊C4—34no—
321C22⋊C4—32no—
322C22⋊C4—34no—
323C22⋊C4—34no—
324C22⋊C4—34no—
325C22⋊C4—32no—
326C22⋊C4—32no—
327C22⋊C4—34no—
328C22⋊C4—34no—
329C22⋊C4C22×D314yes—
330C22⋊C4C22×D312yes—
331C22⋊C4C22×D314yes—
332C42—34no—
333C42—34no—
334C42—34no—
335C42—31no—
336C42—31no—
337C42—31no—
338C42—32no—
339C42—32no—
340C42—32no—
341C42C22×D314yes—
342C42C22×D311yescharacteristic
343C42C22×D312yes—
344C22×C3—22no—
345C22×C3—22no—
346C22×C3—22no—
347C22×C3C2511yescharacteristic
348C2×D3—22no—
349C2×D3—22no—
350C2×D3—22no—
351C2×D3—48no—
352C2×D3C2×#4112yes—
353C2×D3C2×#4112yes—
354C2×D32−1+412yes—
355C3×C4—24no—
356C3×C4—22no—
357C3×C4—22no—
358C3×C4—22no—
359C3×C4—21no—
360C3×C4C2×#4112yes—
361C3×C4C2×#4112yes—
362Dic3—24no—
363Dic3—22no—
364Dic3—22no—
365Dic3—22no—
366Dic3—21no—
367Dic3C2×#4112yes—
368Dic3C2×#4112yes—
369C23—31no—
370C23—32no—
371C23—64no—
372C23—64no—
373C23—64no—
374C23C23×D312yes—
375Q8—22no—
376Q8—24no—
377Q8—64no—
378Q8—68no—
379Q8—62no—
380Q8—64no—
381D4—22no—
382D4—64no—
383D4—62no—
384C2×C4—24no—
385C2×C4—31no—
386C2×C4—31no—
387C2×C4—34no—
388C2×C4—34no—
389C2×C4—32no—
390C2×C4—32no—
391C2×C4—31no—
392C2×C4—34no—
393C2×C4—32no—
394C2×C4—64no—
395C2×C4—64no—
396C2×C4—64no—
397C2×C4—68no—
398C2×C4—64no—
399C2×C4—62no—
400C2×C4—64no—
401C2×C4—64no—
402C2×C4C23×D312yes—
403C2×C4C23×D312yes—
404C2×C4C23×D311yescharacteristic
405C2×C4C23×D314yes—
406C2×C4C23×D312yes—
407C2×C4C23×D311yescharacteristic
408C2×C4C23×D311yescharacteristic
409C2×C3—42no—
410C2×C3C22×#4111yescharacteristic
411C2×C3C22×#4111yescharacteristic
412C2×C3C2×2−1+411yescharacteristic
413D3—42no—
414D3#55614yesdirect product factor
415C22—22no—
416C22—22no—
417C22—22no—
418C22—32no—
419C22—32no—
420C22—32no—
421C22—62no—
422C22—62no—
423C22—62no—
424C22—128no—
425C22C24×D311yescharacteristic
426C4—22no—
427C4—22no—
428C4—24no—
429C4—22no—
430C4—21no—
431C4—32no—
432C4—32no—
433C4—62no—
434C4—62no—
435C4—64no—
436C4—62no—
437C4—61no—
438C4D3×#4112yes—
439C4D3×#4112yes—
440C3C2×#55611yescharacteristic, complemented, Sylow
441C2—34no—
442C2—42no—
443C2—122no—
444C2C2×D3×#4111yescharacteristic
445C2C2×D3×#4111yescharacteristic
446C2D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#556C2
prime radicalC2×D3×D4C22
hypercenter#556D3
hypocenterC3C2×#556
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #556

Complements

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

NH ≅ G/Ntype
1. C3×#556C2semidirect
2. #5732C2semidirect
3. D3×C4⋊Q8C2semidirect
4. D3×#126C2semidirect
5. #5702C2semidirect
7. #5671C2semidirect
8. #5641C2semidirect
9. D3×C22⋊Q8C2semidirect
10. #5584C2semidirect
11. C4×D3×Q8C2semidirect
12. C4×D3×Q8C2semidirect
13. #5568C2semidirect
15. #5542C2semidirect
16. #5533C2semidirect
17. D3×#117C2semidirect
23. C3×C4⋊Q8C22semidirect
24. C3×#126C22semidirect
26. C3×C22⋊Q8C22semidirect
27. C3×C4×Q8C22semidirect
28. C3×C4×Q8C22semidirect
30. C3×#117C22semidirect
32. #946C22semidirect
33. Q8×Dic3C22semidirect
34. Q8×Dic3C22semidirect
35. #944C22semidirect
39. #926C22semidirect
40. #924C22semidirect
41. #898C22semidirect
42. #898C22semidirect
43. #898C22semidirect
44. #897C22semidirect
45. #896C22semidirect
47. #892C22semidirect
48. #892C22semidirect
55. #888C22semidirect
56. #888C22semidirect
57. #887C22semidirect
58. #887C22semidirect
59. #887C22semidirect
60. #887C22semidirect
67. #879C22semidirect
68. #879C22semidirect
69. #879C22semidirect
70. #878C22semidirect
71. #877C22semidirect
72. #876C22semidirect
75. #872C22semidirect
79. #869C22semidirect
80. C4×Dic6C22semidirect
81. C4×Dic6C22semidirect
82. C4×Dic6C22semidirect
83. C4×Dic6C22semidirect
92. #556D3direct (N × H)
178. C4⋊Q8C2×D3semidirect
186. #126C2×D3semidirect
198. C22⋊Q8C2×D3semidirect
209. C4×Q8C2×D3semidirect
210. C4×Q8C2×D3semidirect
222. #117C2×D3semidirect
414. D3#556direct (N × H)
440. C3C2×#556semidirect

Maximal subgroups

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

subgroupindex
C3×#5562
#57322
D3×C4⋊Q82
D3×#1262
#57022
D3×#1242
#56712
#56412
D3×C22⋊Q82
#55842
C4×D3×Q82
#55682
C4×D3×D42
#55422
#55332
D3×#1172
C2×#5563