Finite Group Databasebeta

← #86334

Subgroups — #86334

Order 384.

1270 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#535C211yesmaximal, characteristic, complemented
2#5830C211yesmaximal, characteristic, complemented
3#5828C211yesmaximal, characteristic, complemented
4D4×#67C212yesmaximal, complemented
5#5720C212yesmaximal, complemented
6#5717C212yesmaximal, complemented
7#5678C214yesmaximal, complemented
8#5619C212yesmaximal, complemented
9#5614C212yesmaximal, complemented
10#5609C212yesmaximal, complemented
11D3×#121C212yesmaximal, complemented
12#5595C212yesmaximal, complemented
13#5590C212yesmaximal, complemented
14#5589C214yesmaximal, complemented
15#5530C212yesmaximal, complemented
16#3537—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic, complemented
18C22×C3×D4C2211yescharacteristic, complemented
19C22×#67C2214yescomplemented
20C2×D3×D4C2214yescomplemented
21C2×D3×D4C2214yescomplemented
22C2×D3×D4C2212yescomplemented
23C22×D12C2211yescharacteristic, complemented
24C22×C4×D3C2211yescharacteristic, complemented
25C3×#127C2212yescomplemented
26C3×#124C2212yescomplemented
27C3×#121C2212yescomplemented
28C3×#121C2212yescomplemented
29C3×#120C2212yescomplemented
30C3×#120C2212yescomplemented
31C3×#117C2211yescharacteristic, complemented
32#953C2214yescomplemented
33#947C2212yescomplemented
34#946C2212yescomplemented
35#940C2212yescomplemented
36#940C2212yescomplemented
37#939C2214yescomplemented
38#939C2214yescomplemented
39#939C2214yescomplemented
40#938C2214yescomplemented
41#938C2214yescomplemented
42#938C2212yescomplemented
43#938C2212yescomplemented
44#937C2214yescomplemented
45#937C2211yescharacteristic, complemented
46#937C2212yescomplemented
47#937C2214yescomplemented
48#937C2211yescharacteristic, complemented
49#937C2212yescomplemented
50#935C2211yescharacteristic, complemented
51#935C2214yescomplemented
52#935C2211yescharacteristic, complemented
53D4×Dic3C2212yescomplemented
54#930C2212yescomplemented
55#930C2212yescomplemented
56C4×#67C2212yescomplemented
57C4×#67C2212yescomplemented
58#926C2211yescharacteristic, complemented
59#926C2211yescharacteristic, complemented
60#895C2212yescomplemented
61#893C2212yescomplemented
62D3×C4⋊C4C2212yescomplemented
63#890C2212yescomplemented
64#886C2214yescomplemented
65#884C2214yescomplemented
66#884C2214yescomplemented
67#884C2214yescomplemented
68#883C2214yescomplemented
69#883C2214yescomplemented
70#882C2214yescomplemented
71#67⋊C4C2214yescomplemented
72D3×C22⋊C4C2214yescomplemented
73#879C2214yescomplemented
74#876C2214yescomplemented
75C4×D12C2214yescomplemented
76#560—32no—
77#560—32no—
78#553—32no—
79#553—32no—
80D42—32no—
81D42—32no—
82D42—32no—
83D42—32no—
84#540—32no—
85#538—34no—
86#535—31no—
87#535D311yescharacteristic, complemented
88#534—34no—
89#525—31no—
90C2×#121—32no—
91C24×C3C2312yes—
92C23×D3C2311yescharacteristic
93C23×D3C2314yes—
94C23×D3C2312yes—
95C3×#41—42no—
96C2×C3×Q8C2311yescharacteristic
97C2×C3×D4—42no—
98C2×C3×D4C2312yes—
99C2×C3×D4C2311yescharacteristic
100C2×C3×D4C2314yes—
101C2×C3×D4C2311yescharacteristic
102C2×C3×D4C2311yescharacteristic
103C2×C3×D4C2312yes—
104C2×C3×D4C2314yes—
105C22×C3×C4C2311yescharacteristic
106C22×C3×C4C2311yescharacteristic
107C22×C3×C4C2311yescharacteristic
108C2×#67—48no—
109C2×#67C2314yes—
110C2×#67C2312yes—
111C2×#67C2312yes—
112C2×#67C2314yes—
113C2×#67C2314yes—
114C2×#67C2314yes—
115C2×#67C2314yes—
116C2×#67C2314yes—
117C22×Dic3C2314yes—
118C22×Dic3C2311yescharacteristic
119D3×D4—44no—
120D3×D4—48no—
121D3×D4—48no—
122C2×D12—42no—
123C2×D12C2314yes—
124C2×D12C2312yes—
125C2×D12C2312yes—
126C2×C4×D3—42no—
127C2×C4×D3C2312yes—
128C2×C4×D3C2314yes—
129C2×C4×D3C2314yes—
130C2×C4×D3C2312yes—
131C3×C4⋊C4C2312yes—
132C3×C22⋊C4C2314yes—
133C3×C22⋊C4C2314yes—
134C3×C22⋊C4C2312yes—
135C3×C42C2312yes—
136#217C2314yes—
137#217C2314yes—
138#217C2312yes—
139#217C2312yes—
140#217C2311yescharacteristic
141#217C2311yescharacteristic
142#217C2312yes—
143(C2×D3)⋊C4C2312yes—
144(C2×D3)⋊C4C2314yes—
145(C2×D3)⋊C4C2314yes—
146(C2×D3)⋊C4C2314yes—
147(C2×D3)⋊C4C2314yes—
148(C2×D3)⋊C4C2312yes—
149C4⋊Dic3C2314yes—
150C4⋊Dic3C2312yes—
151C4⋊Dic3C2312yes—
152Dic3⋊C4C2312yes—
153Dic3⋊C4C2314yes—
154Dic3⋊C4C2314yes—
155Dic3⋊C4C2312yes—
156C4×Dic3C2312yes—
157C4×Dic3C2312yes—
158C2×#41C2×D311yescharacteristic, complemented
159C22×D4—34no—
160C22×D4—32no—
161C22×D4—34no—
162C22×D4—34no—
163C22×D4—31no—
164C22×D4C2×D311yescharacteristic, complemented
165C23×C4—31no—
166#127—32no—
167#127—32no—
168#127C2×D312yescomplemented
169#126—34no—
170#126—34no—
171#125—32no—
172#124—32no—
173#124C2×D312yescomplemented
174#123—31no—
175#123—31no—
176#123—34no—
177#123—34no—
178#123—34no—
179#123—34no—
180C22⋊Q8—32no—
181#121—32no—
182#121—34no—
183#121—32no—
184#121—34no—
185#121—34no—
186#121—34no—
187#121—34no—
188#121—32no—
189#121—32no—
190#121—32no—
191#121—34no—
192#121—34no—
193#121—34no—
194#121C2×D312yescomplemented
195#121C2×D312yescomplemented
196#120—34no—
197#120—31no—
198#120—31no—
199#120—32no—
200#120—34no—
201#120—32no—
202#120—34no—
203#120—34no—
204#120C2×D312yescomplemented
205#120C2×D312yescomplemented
206C4×D4—32no—
207C4×D4—32no—
208C4×D4—32no—
209C4×D4—32no—
210C4×D4—34no—
211C4×D4—34no—
212#117—31no—
213#117—31no—
214#117C2×D311yescharacteristic, complemented
215C2×C4⋊C4—32no—
216C2×C22⋊C4—34no—
217C23×C3—44no—
218C23×C3—42no—
219C23×C3C2411yescharacteristic
220C23×C3C2414yes—
221C23×C3C2411yescharacteristic
222C23×C3C2411yescharacteristic
223C22×D3—24no—
224C22×D3—28no—
225C22×D3—41no—
226C22×D3—42no—
227C22×D3—44no—
228C22×D3—44no—
229C22×D3—44no—
230C22×D3C2414yes—
231C22×D3C2412yes—
232C3×Q8—41no—
233C3×D4—24no—
234C3×D4—44no—
235C3×D4—44no—
236C3×D4—42no—
237C3×D4—41no—
238C3×D4—41no—
239C3×D4—41no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4C2411yescharacteristic
244C2×C3×C4C2414yes—
245C2×C3×C4C2411yescharacteristic
246C2×C3×C4C2411yescharacteristic
247C2×C3×C4C2411yescharacteristic
248#67—24no—
249#67—28no—
250#67—24no—
251#67—44no—
252#67—44no—
253#67—44no—
254#67—44no—
255#67—44no—
256C2×Dic3—41no—
257C2×Dic3—44no—
258C2×Dic3C2412yes—
259C2×Dic3C2414yes—
260C2×Dic3C2412yes—
261C2×Dic3C2412yes—
262D12—28no—
263D12—24no—
264D12—42no—
265C4×D3—24no—
266C4×D3—42no—
267C4×D3—44no—
268C4×D3—44no—
269C24—31no—
270C24—34no—
271C24—32no—
272C24C22×D312yes—
273#41—42no—
274C2×Q8C22×D311yescharacteristic
275C2×D4—32no—
276C2×D4—34no—
277C2×D4—32no—
278C2×D4—34no—
279C2×D4—34no—
280C2×D4—34no—
281C2×D4—34no—
282C2×D4—34no—
283C2×D4—34no—
284C2×D4—32no—
285C2×D4—32no—
286C2×D4—42no—
287C2×D4—122no—
288C2×D4—124no—
289C2×D4—128no—
290C2×D4—128no—
291C2×D4—128no—
292C2×D4C22×D312yes—
293C2×D4C22×D311yescharacteristic
294C2×D4C22×D311yescharacteristic
295C2×D4C22×D312yes—
296C2×D4C22×D311yescharacteristic
297C2×D4C22×D314yes—
298C2×D4C22×D314yes—
299C22×C4—34no—
300C22×C4—31no—
301C22×C4—32no—
302C22×C4—32no—
303C22×C4—34no—
304C22×C4—34no—
305C22×C4—122no—
306C22×C4C22×D311yescharacteristic
307C22×C4C22×D311yescharacteristic
308C22×C4C22×D311yescharacteristic
309C4⋊C4—32no—
310C4⋊C4—34no—
311C4⋊C4—32no—
312C4⋊C4—32no—
313C4⋊C4—32no—
314C4⋊C4—34no—
315C4⋊C4—34no—
316C4⋊C4C22×D312yes—
317C22⋊C4—34no—
318C22⋊C4—32no—
319C22⋊C4—31no—
320C22⋊C4—31no—
321C22⋊C4—34no—
322C22⋊C4—32no—
323C22⋊C4—32no—
324C22⋊C4—34no—
325C22⋊C4—34no—
326C22⋊C4—32no—
327C22⋊C4—32no—
328C22⋊C4—34no—
329C22⋊C4—34no—
330C22⋊C4C22×D314yes—
331C22⋊C4C22×D312yes—
332C22⋊C4C22×D314yes—
333C42—32no—
334C42—32no—
335C42C22×D312yes—
336C22×C3—24no—
337C22×C3—21no—
338C22×C3—22no—
339C22×C3—24no—
340C22×C3—22no—
341C22×C3—22no—
342C22×C3—24no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—84no—
346C22×C3C2511yescharacteristic
347C2×D3—22no—
348C2×D3—22no—
349C2×D3—24no—
350C2×D3—24no—
351C2×D3—24no—
352C2×D3—44no—
353C2×D3—88no—
354C2×D3—84no—
355C2×D3H(5,𝔽2)14yes—
356C3×C4—24no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4H(5,𝔽2)12yes—
361Dic3—22no—
362Dic3—22no—
363Dic3—24no—
364Dic3—22no—
365C23—32no—
366C23—34no—
367C23—42no—
368C23—44no—
369C23—64no—
370C23—68no—
371C23—121no—
372C23—122no—
373C23—124no—
374C23—124no—
375C23—124no—
376C23C23×D311yescharacteristic
377C23C23×D314yes—
378C23C23×D311yescharacteristic
379C23C23×D311yescharacteristic
380Q8—41no—
381D4—24no—
382D4—42no—
383D4—44no—
384D4—44no—
385D4—41no—
386D4—41no—
387D4—41no—
388D4—68no—
389D4—64no—
390D4—68no—
391D4—64no—
392D4—64no—
393D4—122no—
394D4—124no—
395D4—124no—
396D4—124no—
397D4—124no—
398D4—124no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—32no—
403C2×C4—32no—
404C2×C4—34no—
405C2×C4—32no—
406C2×C4—64no—
407C2×C4—122no—
408C2×C4—124no—
409C2×C4—124no—
410C2×C4—121no—
411C2×C4—124no—
412C2×C4C23×D311yescharacteristic
413C2×C4C23×D311yescharacteristic
414C2×C4C23×D314yes—
415C2×C4C23×D311yescharacteristic
416C2×C4C23×D311yescharacteristic
417C2×C3—41no—
418C2×C3—41no—
419C2×C3—44no—
420C2×C3—41no—
421C2×C3C2×H(5,𝔽2)11yescharacteristic
422C2×C3C2×H(5,𝔽2)12yes—
423D3—24no—
424D3—44no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—24no—
429C22—24no—
430C22—22no—
431C22—22no—
432C22—24no—
433C22—22no—
434C22—34no—
435C22—62no—
436C22—62no—
437C22—64no—
438C22—64no—
439C22—64no—
440C22—84no—
441C22—124no—
442C22—248no—
443C22—244no—
444C22C24×D311yescharacteristic
445C4—21no—
446C4—21no—
447C4—21no—
448C4—24no—
449C4—62no—
450C4—62no—
451C4—64no—
452C4—62no—
453C4#100412yes—
454C3#353711yescharacteristic, complemented, Sylow
455C2—41no—
456C2—44no—
457C2—41no—
458C2—41no—
459C2—64no—
460C2—124no—
461C2C2×#100411yescharacteristic
462C2D3×H(5,𝔽2)12yes—
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#535C2
hypercenter#535D3
hypocenterC3#3537
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C3×#535C2semidirect
2. #5830C2semidirect
3. #5828C2semidirect
4. D4×#67C2semidirect
5. #5720C2semidirect
6. #5717C2semidirect
7. #5678C2semidirect
8. #5619C2semidirect
9. #5614C2semidirect
10. #5609C2semidirect
11. D3×#121C2semidirect
12. #5595C2semidirect
13. #5590C2semidirect
14. #5589C2semidirect
15. #5530C2semidirect
17. C2×C3×#41C22semidirect
18. C22×C3×D4C22semidirect
19. C22×#67C22semidirect
20. C2×D3×D4C22semidirect
21. C2×D3×D4C22semidirect
22. C2×D3×D4C22semidirect
23. C22×D12C22semidirect
24. C22×C4×D3C22semidirect
25. C3×#127C22semidirect
26. C3×#124C22semidirect
27. C3×#121C22semidirect
28. C3×#121C22semidirect
29. C3×#120C22semidirect
30. C3×#120C22semidirect
31. C3×#117C22semidirect
32. #953C22semidirect
33. #947C22semidirect
34. #946C22semidirect
35. #940C22semidirect
36. #940C22semidirect
37. #939C22semidirect
38. #939C22semidirect
39. #939C22semidirect
40. #938C22semidirect
41. #938C22semidirect
42. #938C22semidirect
43. #938C22semidirect
44. #937C22semidirect
45. #937C22semidirect
46. #937C22semidirect
47. #937C22semidirect
48. #937C22semidirect
49. #937C22semidirect
50. #935C22semidirect
51. #935C22semidirect
52. #935C22semidirect
53. D4×Dic3C22semidirect
54. #930C22semidirect
55. #930C22semidirect
56. C4×#67C22semidirect
57. C4×#67C22semidirect
58. #926C22semidirect
59. #926C22semidirect
60. #895C22semidirect
61. #893C22semidirect
62. D3×C4⋊C4C22semidirect
63. #890C22semidirect
64. #886C22semidirect
65. #884C22semidirect
66. #884C22semidirect
67. #884C22semidirect
68. #883C22semidirect
69. #883C22semidirect
70. #882C22semidirect
71. #67⋊C4C22semidirect
72. D3×C22⋊C4C22semidirect
73. #879C22semidirect
74. #876C22semidirect
75. C4×D12C22semidirect
87. #535D3semidirect
158. C2×#41C2×D3semidirect
164. C22×D4C2×D3semidirect
168. #127C2×D3semidirect
173. #124C2×D3semidirect
194. #121C2×D3semidirect
195. #121C2×D3semidirect
204. #120C2×D3semidirect
205. #120C2×D3semidirect
214. #117C2×D3semidirect
454. C3#3537semidirect

Maximal subgroups

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

subgroupindex
C3×#5352
#58302
#58282
D4×#672
#57202
#57172
#56782
#56192
#56142
#56092
D3×#1212
#55952
#55902
#55892
#55302
#35373