Finite Group Databasebeta

← D3×#537 #86378

Subgroups — D3×#537 #86378

Order 384.

1640 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#41C212yesmaximal, complemented
2C3×#537C211yesmaximal, characteristic, complemented
3#5833C212yesmaximal, complemented
4#5629C214yesmaximal, complemented
5D3×C22⋊Q8C214yesmaximal, complemented
6#5608C214yesmaximal, complemented
7D3×#121C214yesmaximal, complemented
8D3×#121C214yesmaximal, complemented
9#5597C214yesmaximal, complemented
10#5510C211yesmaximal, characteristic, complemented
11C2×D3×C4⋊C4C211yesmaximal, characteristic, complemented
12C2×#537—31nomaximal, Sylow
13C2×C3×#41C2212yescomplemented
14D3×#41—216no—
15C2×#239C2212yescomplemented
16C2×D3×Q8C2212yes—
17C2×#237C2214yescomplemented
18C2×#237C2212yescomplemented
19C2×D3×D4C2214yes—
20C2×D3×D4C2212yes—
21C2×D3×D4C2214yes—
22C2×#235C2212yescomplemented
23C2×#235C2214yescomplemented
24C22×C4×D3C2214yes—
25C22×C4×D3C2212yes—
26C22×C4×D3C2211yescharacteristic
27C3×C22⋊Q8C2214yescomplemented
28C3×#121C2214yescomplemented
29C3×#121C2214yescomplemented
30C2×C3×C4⋊C4C2211yescharacteristic, complemented
31#946C2214yescomplemented
32#939C2218yescomplemented
33#939C2218yescomplemented
34#938C2214yescomplemented
35#938C2214yescomplemented
36#930C2214yescomplemented
37#930C2214yescomplemented
38C2×C4⋊Dic3C2211yescharacteristic, complemented
39#924C2214yescomplemented
40C2×Dic3⋊C4C2212yescomplemented
41#897C2214yescomplemented
42#896C2218yescomplemented
43#895C2214yescomplemented
44#895C2214yescomplemented
45D3×C4⋊C4C2214yes—
46D3×C4⋊C4C2214yes—
47D3×C4⋊C4C2214yescomplemented
48#884C2218yescomplemented
49#884C2218yescomplemented
50D3×C22⋊C4C2218yes—
51#878C2218yescomplemented
52C22×#41—32no—
53#537—32no—
54#537—34no—
55#537—34no—
56#537—34no—
57#537—31no—
58#537D311yescharacteristic, direct product factor
59C2×C22⋊Q8—34no—
60C2×#121—34no—
61C2×#121—34no—
62C22×C4⋊C4—31no—
63C23×D3C2314yes—
64C23×D3C2311yescharacteristic
65C3×#41—28no—
66C2×C3×Q8C2312yes—
67C2×C3×D4C2312yes—
68C2×C3×D4C2314yes—
69C2×C3×D4C2314yes—
70C22×C3×C4C2314yes—
71C22×C3×C4C2312yes—
72C22×C3×C4C2311yescharacteristic
73C2×#67C2314yes—
74C2×#67C2314yes—
75C2×#67C2314yes—
76C2×#67C2318yes—
77C22×Dic3C2314yes—
78C22×Dic3C2312yes—
79C22×Dic3C2311yescharacteristic
80#239—28no—
81D3×Q8—28no—
82#237—216no—
83#237—28no—
84D3×D4—216no—
85D3×D4—28no—
86D3×D4—48no—
87#235—28no—
88#235—216no—
89C2×D12C2314yes—
90C2×D12C2312yes—
91C2×D12C2314yes—
92C2×C4×D3—216no—
93C2×C4×D3—44no—
94C2×C4×D3D418yescomplemented
95C2×C4×D3C2314yes—
96C2×C4×D3C2314yes—
97C2×C4×D3C2314yes—
98C2×C4×D3C2314yes—
99C2×C4×D3C2314yes—
100C2×C4×D3C2312yes—
101C2×C4×D3C2312yes—
102C2×Dic6C2314yes—
103C2×Dic6C2312yes—
104C3×C4⋊C4C2314yes—
105C3×C4⋊C4C2314yes—
106C3×C22⋊C4C2318yes—
107#217C2318yes—
108(C2×D3)⋊C4C2318yes—
109(C2×D3)⋊C4C2318yes—
110C4⋊Dic3C2314yes—
111C4⋊Dic3C2314yes—
112Dic3⋊C4C2318yes—
113Dic3⋊C4C2318yes—
114C2×#41—32no—
115C2×#41—34no—
116C2×#41—32no—
117C2×#41—34no—
118C2×#41—32no—
119C2×#41—616no—
120C2×#41C2×D312yescomplemented
121C22×Q8—32no—
122C22×D4—32no—
123C22×D4—34no—
124C22×D4—34no—
125C23×C4—31no—
126C23×C4—34no—
127C23×C4—32no—
128C22⋊Q8—34no—
129C22⋊Q8—34no—
130C22⋊Q8—38no—
131C22⋊Q8—34no—
132C22⋊Q8—38no—
133C22⋊Q8C2×D314yescomplemented
134#121—34no—
135#121—34no—
136#121—38no—
137#121—38no—
138#121—34no—
139#121—34no—
140#121—34no—
141#121—34no—
142#121—38no—
143#121—38no—
144#121C2×D314yescomplemented
145#121C2×D314yescomplemented
146C2×C4⋊C4—32no—
147C2×C4⋊C4—31no—
148C2×C4⋊C4—34no—
149C2×C4⋊C4—34no—
150C2×C4⋊C4—34no—
151C2×C4⋊C4C2×D311yescharacteristic, complemented
152C2×C22⋊C4—38no—
153C23×C3C2414yes—
154C23×C3C2411yescharacteristic
155C22×D3—22no—
156C22×D3—22no—
157C22×D3—28no—
158C22×D3—28no—
159C22×D3—28no—
160C22×D3C2×D414yes—
161C22×D3C2414yes—
162C22×D3C2411yescharacteristic
163C22×D3C2411yescharacteristic
164C3×Q8—24no—
165C3×D4—28no—
166C3×D4—24no—
167C3×D4—44no—
168C2×C3×C4—28no—
169C2×C3×C4—42no—
170C2×C3×C4C2×D414yescomplemented
171C2×C3×C4C2414yes—
172C2×C3×C4C2414yes—
173C2×C3×C4C2412yes—
174#67—28no—
175#67—28no—
176#67—28no—
177#67—48no—
178C2×Dic3—28no—
179C2×Dic3—42no—
180C2×Dic3C2×D414yescomplemented
181C2×Dic3C2414yes—
182C2×Dic3C2414yes—
183C2×Dic3C2412yes—
184D12—28no—
185D12—24no—
186D12—44no—
187C4×D3—28no—
188C4×D3—28no—
189C4×D3—28no—
190C4×D3—28no—
191C4×D3—44no—
192C4×D3C2×D418yes—
193C4×D3C2×D418yescomplemented
194Dic6—24no—
195Dic6—28no—
196C24—31no—
197C24—34no—
198#41—28no—
199#41—616no—
200#41—616no—
201#41—68no—
202#41—68no—
203#41—68no—
204C2×Q8—34no—
205C2×Q8—32no—
206C2×Q8—68no—
207C2×Q8C22×D312yes—
208C2×D4—34no—
209C2×D4—34no—
210C2×D4—34no—
211C2×D4—38no—
212C2×D4—34no—
213C2×D4—32no—
214C2×D4—34no—
215C2×D4—68no—
216C2×D4—616no—
217C2×D4—128no—
218C2×D4C22×D314yes—
219C2×D4C22×D312yes—
220C2×D4C22×D314yes—
221C22×C4—31no—
222C22×C4—38no—
223C22×C4—32no—
224C22×C4—32no—
225C22×C4—34no—
226C22×C4—32no—
227C22×C4—34no—
228C22×C4—34no—
229C22×C4—34no—
230C22×C4—34no—
231C22×C4—34no—
232C22×C4—616no—
233C22×C4—124no—
234C22×C4C22×D314yes—
235C22×C4C22×D312yes—
236C22×C4C22×D311yescharacteristic
237C4⋊C4—38no—
238C4⋊C4—34no—
239C4⋊C4—38no—
240C4⋊C4—34no—
241C4⋊C4C22×D314yes—
242C4⋊C4C22×D314yes—
243C22⋊C4—38no—
244C22⋊C4—38no—
245C22⋊C4—38no—
246C22⋊C4C22×D318yes—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—24no—
250C22×C3—24no—
251C22×C3—24no—
252C22×C3C22×D412yes—
253C22×C3C2511yescharacteristic
254C2×D3—21no—
255C2×D3—21no—
256C2×D3—28no—
257C2×D3—24no—
258C2×D3—24no—
259C2×D3—24no—
260C2×D3—416no—
261C2×D3C22×D412yes—
262C2×D3C22×D412yes—
263C2×D3H(5,𝔽2)12yes—
264C2×D32−1+412yes—
265C3×C4—24no—
266C3×C4—24no—
267C3×C4C22×D414yes—
268Dic3—24no—
269Dic3—24no—
270Dic3C22×D414yes—
271C23—31no—
272C23—34no—
273C23—31no—
274C23—34no—
275C23—62no—
276C23—62no—
277C23—68no—
278C23—68no—
279C23—68no—
280C23C23×D314yes—
281C23C23×D311yescharacteristic
282Q8—24no—
283Q8—68no—
284Q8—64no—
285D4—28no—
286D4—24no—
287D4—44no—
288D4—68no—
289D4—68no—
290D4—68no—
291D4—68no—
292D4—64no—
293D4—124no—
294D4—128no—
295C2×C4—28no—
296C2×C4—34no—
297C2×C4—32no—
298C2×C4—38no—
299C2×C4—38no—
300C2×C4—34no—
301C2×C4—34no—
302C2×C4—42no—
303C2×C4—68no—
304C2×C4—68no—
305C2×C4—68no—
306C2×C4—68no—
307C2×C4—68no—
308C2×C4—124no—
309C2×C4—122no—
310C2×C4D3×D414yescomplemented
311C2×C4C23×D314yes—
312C2×C4C23×D314yes—
313C2×C4C23×D312yes—
314C2×C3—22no—
315C2×C3—44no—
316C2×C3C23×D411yescharacteristic
317C2×C3C2×H(5,𝔽2)11yescharacteristic
318C2×C3C2×2−1+411yescharacteristic
319D3—22no—
320D3—44no—
321D3#53714yesdirect product factor
322C22—21no—
323C22—21no—
324C22—24no—
325C22—24no—
326C22—24no—
327C22—32no—
328C22—32no—
329C22—32no—
330C22—32no—
331C22—68no—
332C22—61no—
333C22—61no—
334C22—64no—
335C22—64no—
336C22—64no—
337C22—1216no—
338C22C2×D3×D412yes—
339C22C24×D311yescharacteristic
340C4—24no—
341C4—24no—
342C4—34no—
343C4—64no—
344C4—64no—
345C4C2×D3×D414yes—
346C3C2×#53711yescharacteristic, complemented, Sylow
347C2—22no—
348C2—34no—
349C2—44no—
350C2—62no—
351C2—124no—
352C2C22×D3×D411yescharacteristic
353C2D3×H(5,𝔽2)11yescharacteristic
354C2D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#537C2
hypercenter#537D3
hypocenterC3C2×#537
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×D3×#41C2semidirect
2. C3×#537C2semidirect
3. #5833C2semidirect
4. #5629C2semidirect
5. D3×C22⋊Q8C2semidirect
6. #5608C2semidirect
7. D3×#121C2semidirect
8. D3×#121C2semidirect
9. #5597C2semidirect
10. #5510C2semidirect
11. C2×D3×C4⋊C4C2semidirect
13. C2×C3×#41C22semidirect
15. C2×#239C22semidirect
17. C2×#237C22semidirect
18. C2×#237C22semidirect
22. C2×#235C22semidirect
23. C2×#235C22semidirect
27. C3×C22⋊Q8C22semidirect
28. C3×#121C22semidirect
29. C3×#121C22semidirect
30. C2×C3×C4⋊C4C22semidirect
31. #946C22semidirect
32. #939C22semidirect
33. #939C22semidirect
34. #938C22semidirect
35. #938C22semidirect
36. #930C22semidirect
37. #930C22semidirect
38. C2×C4⋊Dic3C22semidirect
39. #924C22semidirect
40. C2×Dic3⋊C4C22semidirect
41. #897C22semidirect
42. #896C22semidirect
43. #895C22semidirect
44. #895C22semidirect
47. D3×C4⋊C4C22semidirect
48. #884C22semidirect
49. #884C22semidirect
51. #878C22semidirect
58. #537D3direct (N × H)
94. C2×C4×D3D4semidirect
120. C2×#41C2×D3semidirect
133. C22⋊Q8C2×D3semidirect
144. #121C2×D3semidirect
145. #121C2×D3semidirect
151. C2×C4⋊C4C2×D3semidirect
170. C2×C3×C4C2×D4semidirect
180. C2×Dic3C2×D4semidirect
193. C4×D3C2×D4semidirect
310. C2×C4D3×D4semidirect
321. D3#537direct (N × H)
346. C3C2×#537semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#412
C3×#5372
#58332
#56292
D3×C22⋊Q82
#56082
D3×#1212
#55972
#55102
C2×D3×C4⋊C42
C2×#5373