Finite Group Databasebeta

← #87131

Subgroups — #87131

Order 384.

1060 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#563C211yesmaximal, characteristic, complemented
2#5734C212yesmaximal, complemented
3#5724C212yesmaximal, complemented
4#5705C214yesmaximal, complemented
5#5704C214yesmaximal, complemented
6#5700C212yesmaximal, complemented
7#5691C212yesmaximal, complemented
8#5683C211yesmaximal, characteristic, complemented
9D3×#124C211yesmaximal, characteristic, complemented
10#5659C212yesmaximal, complemented
11#5653C212yesmaximal, complemented
12#5640C214yesmaximal, complemented
13#5630C214yesmaximal, complemented
14#3545—31nomaximal, Sylow
15C2×#239C2212yes—
16C2×D3×Q8C2211yescharacteristic, complemented
17C2×D3×D4C2211yescharacteristic
18C22×D12C2212yescomplemented
19C3×C4⋊Q8C2212yescomplemented
20C3×#126C2214yescomplemented
21C3×#125C2212yescomplemented
22C3×#124C2211yescharacteristic
23C3×#123C2212yes—
24C3×C22⋊Q8C2214yescomplemented
25#947C2214yescomplemented
26#947C2211yescharacteristic, complemented
27#947C2212yescomplemented
28#944C2212yescomplemented
29#937C2212yes—
30#936C2211yescharacteristic
31#929C2212yescomplemented
32C4×#67C2214yescomplemented
33#898C2214yescomplemented
34#897C2212yescomplemented
35#896C2214yescomplemented
36#896C2214yescomplemented
37#896C2214yescomplemented
38#895C2212yes—
39#895C2214yescomplemented
40#895C2214yescomplemented
41#895C2214yescomplemented
42#894C2214yescomplemented
43#894C2214yescomplemented
44#894C2214yescomplemented
45#894C2214yescomplemented
46#893C2212yescomplemented
47#893C2214yescomplemented
48#892C2214yescomplemented
49#892C2214yescomplemented
50#887C2212yescomplemented
51#885C2214yescomplemented
52#885C2214yescomplemented
53#884C2214yes—
54#883C2214yescomplemented
55#882C2214yes—
56#882C2212yescomplemented
57#882C2214yescomplemented
58D3×C22⋊C4C2212yescomplemented
59D3×C22⋊C4C2214yescomplemented
60D3×C22⋊C4C2214yescomplemented
61#879C2214yescomplemented
62#875C2211yescharacteristic, complemented
63#875C2212yescomplemented
64#875C2212yescomplemented
65C4×D12C2214yescomplemented
66#872C2212yescomplemented
67C42×D3C2211yescharacteristic, complemented
68#563D311yescharacteristic, complemented
69#562—32no—
70H(3,𝔽4)—31no—
71#555—32no—
72#551—34no—
73#551—32no—
74#546—34no—
75#542—34no—
76#538—34no—
77#536—32no—
78#534—32no—
79#529—32no—
80C2×#124—31no—
81C23×D3C2312yes—
82C23×D3C2312yes—
83C2×C3×Q8C2312yes—
84C2×C3×Q8C2311yescharacteristic
85C2×C3×D4C2311yescharacteristic
86C22×C3×C4C2312yes—
87C2×#67C2314yes—
88C2×#67C2312yes—
89#239—44no—
90D3×Q8—42no—
91D3×D4—42no—
92C2×D12—44no—
93C2×D12C2314yes—
94C2×D12C2314yes—
95C2×D12C2312yes—
96C2×D12C2312yes—
97C2×D12C2311yescharacteristic
98C2×D12C2314yes—
99C2×C4×D3C2311yescharacteristic
100C2×C4×D3C2312yes—
101C2×C4×D3C2312yes—
102C2×C4×D3C2314yes—
103C2×C4×D3C2314yes—
104C2×C4×D3C2314yes—
105C2×Dic6C2312yes—
106C2×Dic6C2312yes—
107C2×Dic6C2311yescharacteristic
108C3×C4⋊C4C2312yes—
109C3×C4⋊C4C2314yes—
110C3×C4⋊C4C2314yes—
111C3×C4⋊C4C2312yes—
112C3×C4⋊C4C2314yes—
113C3×C22⋊C4C2314yes—
114C3×C22⋊C4C2312yes—
115C3×C22⋊C4C2314yes—
116C3×C42C2311yescharacteristic
117C3×C42C2312yes—
118#217C2312yes—
119#217C2314yes—
120(C2×D3)⋊C4C2314yes—
121(C2×D3)⋊C4C2314yes—
122(C2×D3)⋊C4C2314yes—
123(C2×D3)⋊C4C2314yes—
124(C2×D3)⋊C4C2314yes—
125(C2×D3)⋊C4C2314yes—
126(C2×D3)⋊C4C2314yes—
127(C2×D3)⋊C4C2314yes—
128(C2×D3)⋊C4C2314yes—
129(C2×D3)⋊C4C2314yes—
130C4⋊Dic3C2314yes—
131Dic3⋊C4C2314yes—
132Dic3⋊C4C2314yes—
133Dic3⋊C4C2314yes—
134C4×Dic3C2312yes—
135C4×Dic3C2311yescharacteristic
136C4×Dic3C2312yes—
137C4×Dic3C2314yes—
138C2×#41—32no—
139C22×Q8—31no—
140C22×D4—31no—
141C22×D4—32no—
142C4⋊Q8—32no—
143C4⋊Q8C2×D312yescomplemented
144#126—34no—
145#126—34no—
146#126C2×D314yescomplemented
147#125C2×D312yescomplemented
148#124—31no—
149#124—32no—
150#124—34no—
151#124—31no—
152#124—34no—
153#124—34no—
154#124—31no—
155#124—32no—
156#124—32no—
157#124C2×D311yescharacteristic
158#123—32no—
159#123—34no—
160#123—34no—
161#123—34no—
162#123—34no—
163#123—34no—
164#123C2×D312yes—
165C22⋊Q8—32no—
166C22⋊Q8—34no—
167C22⋊Q8—34no—
168C22⋊Q8—34no—
169C22⋊Q8C2×D314yescomplemented
170#121—32no—
171#121—34no—
172#121—34no—
173#121—34no—
174#121—34no—
175#120—32no—
176#120—32no—
177#120—34no—
178#120—34no—
179C4×Q8—32no—
180C4×D4—34no—
181C4×D4—32no—
182C4×D4—34no—
183C4×D4—34no—
184#117—34no—
185#117—34no—
186#117—32no—
187C2×C22⋊C4—34no—
188C2×C22⋊C4—32no—
189C2×C22⋊C4—34no—
190C2×C42—31no—
191C23×C3C2412yes—
192C22×D3—24no—
193C22×D3—42no—
194C22×D3—44no—
195C22×D3—44no—
196C22×D3C2414yes—
197C22×D3C2412yes—
198C22×D3C2411yescharacteristic
199C3×Q8—41no—
200C3×Q8—42no—
201C3×D4—41no—
202C2×C3×C4—42no—
203C2×C3×C4C2414yes—
204C2×C3×C4C2412yes—
205C2×C3×C4C2412yes—
206C2×C3×C4C2411yescharacteristic
207C2×C3×C4C2414yes—
208#67—42no—
209#67—44no—
210C2×Dic3C2412yes—
211C2×Dic3C2411yescharacteristic
212C2×Dic3C2414yes—
213C2×Dic3C2412yes—
214D12—28no—
215D12—44no—
216D12—42no—
217D12—44no—
218D12—41no—
219D12—42no—
220C4×D3—22no—
221C4×D3—24no—
222C4×D3—44no—
223C4×D3—42no—
224C4×D3—44no—
225C4×D3—44no—
226Dic6—42no—
227Dic6—41no—
228Dic6—42no—
229C24—32no—
230C24—32no—
231#41—124no—
232C2×Q8—32no—
233C2×Q8—31no—
234C2×Q8—32no—
235C2×Q8—122no—
236C2×Q8C22×D312yes—
237C2×Q8C22×D311yescharacteristic
238C2×D4—34no—
239C2×D4—32no—
240C2×D4—34no—
241C2×D4—34no—
242C2×D4—32no—
243C2×D4—32no—
244C2×D4—34no—
245C2×D4—31no—
246C2×D4—124no—
247C2×D4—122no—
248C2×D4C22×D311yescharacteristic
249C22×C4—31no—
250C22×C4—32no—
251C22×C4—32no—
252C22×C4—34no—
253C22×C4—34no—
254C22×C4—34no—
255C22×C4C22×D312yes—
256C4⋊C4—34no—
257C4⋊C4—34no—
258C4⋊C4—34no—
259C4⋊C4—34no—
260C4⋊C4C22×D314yes—
261C4⋊C4C22×D314yes—
262C4⋊C4C22×D314yes—
263C4⋊C4C22×D312yes—
264C4⋊C4C22×D312yes—
265C22⋊C4—34no—
266C22⋊C4—32no—
267C22⋊C4—34no—
268C22⋊C4—34no—
269C22⋊C4—34no—
270C22⋊C4—34no—
271C22⋊C4—34no—
272C22⋊C4—34no—
273C22⋊C4—34no—
274C22⋊C4—34no—
275C22⋊C4—34no—
276C22⋊C4—34no—
277C22⋊C4C22×D314yes—
278C22⋊C4C22×D314yes—
279C22⋊C4C22×D312yes—
280C42—31no—
281C42—32no—
282C42—32no—
283C42—34no—
284C42C22×D311yescharacteristic
285C42C22×D312yes—
286C22×C3—22no—
287C22×C3—24no—
288C22×C3C2511yescharacteristic
289C2×D3—22no—
290C2×D3—24no—
291C2×D3—22no—
292C2×D3—24no—
293C2×D3—24no—
294C2×D3—24no—
295C2×D3—84no—
296C2×D3—84no—
297C2×D3H(5,𝔽2)12yes—
298C3×C4—21no—
299C3×C4—22no—
300C3×C4—24no—
301C3×C4—22no—
302C3×C4—24no—
303Dic3—22no—
304Dic3—21no—
305Dic3—24no—
306Dic3—22no—
307C23—31no—
308C23—34no—
309C23—32no—
310C23—64no—
311C23—122no—
312C23—124no—
313C23—124no—
314C23C23×D312yes—
315Q8—41no—
316Q8—42no—
317Q8—121no—
318Q8—122no—
319Q8—122no—
320D4—41no—
321D4—68no—
322D4—124no—
323D4—121no—
324D4—122no—
325D4—124no—
326D4—122no—
327D4—122no—
328D4—124no—
329C2×C4—32no—
330C2×C4—34no—
331C2×C4—31no—
332C2×C4—32no—
333C2×C4—42no—
334C2×C4—62no—
335C2×C4—64no—
336C2×C4—124no—
337C2×C4—124no—
338C2×C4—124no—
339C2×C4—122no—
340C2×C4C23×D312yes—
341C2×C4C23×D312yes—
342C2×C4C23×D311yescharacteristic
343C2×C4C23×D314yes—
344C2×C4C23×D314yes—
345C2×C3—42no—
346C2×C3C2×H(5,𝔽2)11yescharacteristic
347C2×C3#58512yes—
348D3—22no—
349D3—44no—
350D3—42no—
351C22—22no—
352C22—24no—
353C22—32no—
354C22—62no—
355C22—64no—
356C22—62no—
357C22—64no—
358C22—64no—
359C22—64no—
360C22—244no—
361C22—244no—
362C22C24×D311yescharacteristic
363C4—24no—
364C4—21no—
365C4—22no—
366C4—24no—
367C4—22no—
368C4—62no—
369C4—61no—
370C4—62no—
371C4—64no—
372C3#354511yescharacteristic, complemented, Sylow
373C2—42no—
374C2—62no—
375C2—124no—
376C2—122no—
377C2D3×H(5,𝔽2)11yescharacteristic
378C2#596812yes—
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#563C2
hypercenter#563D3
hypocenterC3#3545
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #563

Complements

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

NH ≅ G/Ntype
1. C3×#563C2semidirect
2. #5734C2semidirect
3. #5724C2semidirect
4. #5705C2semidirect
5. #5704C2semidirect
6. #5700C2semidirect
7. #5691C2semidirect
8. #5683C2semidirect
9. D3×#124C2semidirect
10. #5659C2semidirect
11. #5653C2semidirect
12. #5640C2semidirect
13. #5630C2semidirect
16. C2×D3×Q8C22semidirect
18. C22×D12C22semidirect
19. C3×C4⋊Q8C22semidirect
20. C3×#126C22semidirect
21. C3×#125C22semidirect
24. C3×C22⋊Q8C22semidirect
25. #947C22semidirect
26. #947C22semidirect
27. #947C22semidirect
28. #944C22semidirect
31. #929C22semidirect
32. C4×#67C22semidirect
33. #898C22semidirect
34. #897C22semidirect
35. #896C22semidirect
36. #896C22semidirect
37. #896C22semidirect
39. #895C22semidirect
40. #895C22semidirect
41. #895C22semidirect
42. #894C22semidirect
43. #894C22semidirect
44. #894C22semidirect
45. #894C22semidirect
46. #893C22semidirect
47. #893C22semidirect
48. #892C22semidirect
49. #892C22semidirect
50. #887C22semidirect
51. #885C22semidirect
52. #885C22semidirect
54. #883C22semidirect
56. #882C22semidirect
57. #882C22semidirect
58. D3×C22⋊C4C22semidirect
59. D3×C22⋊C4C22semidirect
60. D3×C22⋊C4C22semidirect
61. #879C22semidirect
62. #875C22semidirect
63. #875C22semidirect
64. #875C22semidirect
65. C4×D12C22semidirect
66. #872C22semidirect
67. C42×D3C22semidirect
68. #563D3semidirect
143. C4⋊Q8C2×D3semidirect
146. #126C2×D3semidirect
147. #125C2×D3semidirect
169. C22⋊Q8C2×D3semidirect
372. C3#3545semidirect

Maximal subgroups

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

subgroupindex
C3×#5632
#57342
#57242
#57052
#57042
#57002
#56912
#56832
D3×#1242
#56592
#56532
#56402
#56302
#35453