Finite Group Databasebeta

← #86298

Subgroups — #86298

Order 384.

1660 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1004C212yesmaximal, complemented
2C3×#534C211yesmaximal, characteristic, complemented
3C2×#939C212yesmaximal, complemented
4#5655C214yesmaximal, complemented
5#5654C214yesmaximal, complemented
6#5605C212yesmaximal, complemented
7D3×#121C212yesmaximal, complemented
8#5597C212yesmaximal, complemented
9#5596C212yesmaximal, complemented
10#5591C214yesmaximal, complemented
11#5590C214yesmaximal, complemented
12#5493C211yesmaximal, characteristic, complemented
13C2×#67⋊C4C211yesmaximal, characteristic, complemented
14#3496—31nomaximal, Sylow
15C22×C3×D4C2212yescomplemented
16C22×#67C2211yescharacteristic
17C22×#67C2212yescomplemented
18C22×#67C2214yescomplemented
19C22×#67C2212yescomplemented
20C23×Dic3C2211yescharacteristic, complemented
21#1004—216no—
22C2×#237C2214yescomplemented
23C2×#237C2214yescomplemented
24C2×D3×D4C2214yescomplemented
25C2×D3×D4C2214yescomplemented
26C2×D3×D4C2212yescomplemented
27C2×#235C2214yescomplemented
28C22×C4×D3C2211yescharacteristic, complemented
29C3×#123C2214yescomplemented
30C3×#121C2212yescomplemented
31C3×#121C2212yescomplemented
32C3×#120C2214yescomplemented
33C2×C3×C22⋊C4C2211yescharacteristic, complemented
34C2×#217C2212yescomplemented
35#940C2212yescomplemented
36#939C2214yescomplemented
37#939C2214yescomplemented
38#939C2214yescomplemented
39#939C2214yescomplemented
40#939C2214yescomplemented
41#938C2212yescomplemented
42#937C2214yescomplemented
43D4×Dic3C2212yescomplemented
44#930C2212yescomplemented
45#930C2212yescomplemented
46C2×(C2×D3)⋊C4C2212yescomplemented
47C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
48C2×Dic3⋊C4C2212yescomplemented
49C2×Dic3⋊C4C2211yescharacteristic, complemented
50C2×C4×Dic3C2211yescharacteristic, complemented
51#896C2214yescomplemented
52#895C2212yescomplemented
53#894C2214yescomplemented
54#893C2212yescomplemented
55D3×C4⋊C4C2212yescomplemented
56#888C2212yescomplemented
57#886C2214yescomplemented
58#885C2214yescomplemented
59#884C2214yescomplemented
60#884C2214yescomplemented
61#883C2214yescomplemented
62#882C2214yescomplemented
63#882C2214yescomplemented
64#67⋊C4C2214yescomplemented
65#67⋊C4C2214yescomplemented
66#67⋊C4C2214yescomplemented
67D3×C22⋊C4C2214yescomplemented
68#878C2214yescomplemented
69C2×H(5,𝔽2)—32no—
70#547—34no—
71#546—34no—
72#546—34no—
73D42—34no—
74#537—32no—
75#534—32no—
76#534—31no—
77#534D311yescharacteristic, complemented
78#532—32no—
79C2×#121—32no—
80C2×#121—32no—
81C2×C4×D4—31no—
82C24×C3C2312yes—
83C24×C3C2311yescharacteristic
84C23×D3C2314yes—
85C23×D3C2312yes—
86C23×D3C2311yescharacteristic
87C2×C3×D4—28no—
88C2×C3×D4C2314yes—
89C2×C3×D4C2314yes—
90C2×C3×D4C2312yes—
91C2×C3×D4C2312yes—
92C22×C3×C4C2312yes—
93C22×C3×C4C2311yescharacteristic
94C22×C3×C4C2311yescharacteristic
95C2×#67—28no—
96C2×#67—216no—
97C2×#67—44no—
98C2×#67D418yescomplemented
99C2×#67C2314yes—
100C2×#67C2314yes—
101C2×#67C2314yes—
102C2×#67C2314yes—
103C2×#67C2314yes—
104C2×#67C2314yes—
105C2×#67C2314yes—
106C22×Dic3—42no—
107C22×Dic3C2314yes—
108C22×Dic3C2312yes—
109C22×Dic3C2312yes—
110C22×Dic3C2311yescharacteristic
111C22×Dic3C2312yes—
112C22×Dic3C2311yescharacteristic
113C22×Dic3C2311yescharacteristic
114#237—216no—
115#237—216no—
116D3×D4—216no—
117D3×D4—216no—
118D3×D4—44no—
119#235—216no—
120C2×D12C2314yes—
121C2×D12C2312yes—
122C2×D12C2312yes—
123C2×C4×D3—42no—
124C2×C4×D3C2314yes—
125C2×C4×D3C2312yes—
126C2×C4×D3C2314yes—
127C2×C4×D3C2312yes—
128C2×Dic6C2314yes—
129C3×C4⋊C4C2312yes—
130C3×C4⋊C4C2312yes—
131C3×C22⋊C4C2314yes—
132C3×C22⋊C4C2314yes—
133C3×C22⋊C4C2314yes—
134#217C2314yes—
135#217C2314yes—
136(C2×D3)⋊C4C2314yes—
137(C2×D3)⋊C4C2314yes—
138(C2×D3)⋊C4C2314yes—
139(C2×D3)⋊C4C2314yes—
140(C2×D3)⋊C4C2314yes—
141C4⋊Dic3C2312yes—
142C4⋊Dic3C2312yes—
143Dic3⋊C4C2314yes—
144Dic3⋊C4C2314yes—
145Dic3⋊C4C2314yes—
146C4×Dic3C2312yes—
147C4×Dic3C2312yes—
148H(5,𝔽2)—616no—
149C2×#41—34no—
150C2×#41—34no—
151C2×#41—34no—
152C22×D4—31no—
153C22×D4—32no—
154C22×D4—34no—
155C22×D4—32no—
156C22×D4—34no—
157C22×D4—34no—
158C22×D4—32no—
159C22×D4C2×D312yescomplemented
160C23×C4—31no—
161C23×C4—31no—
162C4⋊Q8—32no—
163#127—32no—
164#124—34no—
165#123—34no—
166#123—34no—
167#123—34no—
168#123C2×D314yescomplemented
169C22⋊Q8—34no—
170C22⋊Q8—34no—
171#121—34no—
172#121—34no—
173#121—34no—
174#121—32no—
175#121—34no—
176#121—34no—
177#121—32no—
178#121—32no—
179#121—32no—
180#121—34no—
181#121—34no—
182#121C2×D312yescomplemented
183#121C2×D312yescomplemented
184#120—34no—
185#120—34no—
186#120—34no—
187#120C2×D314yescomplemented
188C4×D4—32no—
189C4×D4—32no—
190C4×D4—34no—
191C4×D4—34no—
192C4×D4—34no—
193C2×C4⋊C4—31no—
194C2×C4⋊C4—32no—
195C2×C4⋊C4—32no—
196C2×C22⋊C4—32no—
197C2×C22⋊C4—31no—
198C2×C22⋊C4—32no—
199C2×C22⋊C4—34no—
200C2×C22⋊C4C2×D311yescharacteristic, complemented
201C2×C42—31no—
202C23×C3—24no—
203C23×C3—41no—
204C23×C3—41no—
205C23×C3—42no—
206C23×C3—42no—
207C23×C3C2×D414yescomplemented
208C23×C3C2414yes—
209C23×C3C2412yes—
210C23×C3C2411yescharacteristic
211C22×D3—24no—
212C22×D3—28no—
213C22×D3—41no—
214C22×D3—41no—
215C22×D3—44no—
216C22×D3—42no—
217C22×D3—44no—
218C22×D3—42no—
219C22×D3C2×D414yescomplemented
220C22×D3C2414yes—
221C22×D3C2412yes—
222C3×D4—28no—
223C3×D4—28no—
224C3×D4—42no—
225C3×D4—42no—
226C2×C3×C4—24no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4C2412yes—
230C2×C3×C4C2412yes—
231C2×C3×C4C2414yes—
232#67—28no—
233#67—28no—
234#67—28no—
235#67—28no—
236#67—28no—
237#67—44no—
238#67C2×D4116yescomplemented
239C2×Dic3—24no—
240C2×Dic3—28no—
241C2×Dic3—22no—
242C2×Dic3—24no—
243C2×Dic3—42no—
244C2×Dic3—41no—
245C2×Dic3C2×D414yescomplemented
246C2×Dic3C2411yescharacteristic
247C2×Dic3C2412yes—
248C2×Dic3C2414yes—
249C2×Dic3C2412yes—
250C2×Dic3C2411yescharacteristic
251D12—28no—
252D12—42no—
253D12—42no—
254C4×D3—28no—
255C4×D3—28no—
256C4×D3—24no—
257C4×D3—42no—
258Dic6—28no—
259C24—31no—
260C24—34no—
261C24—32no—
262C24C22×D312yes—
263C24C22×D311yescharacteristic
264#41—616no—
265#41—616no—
266#41—616no—
267C2×Q8—34no—
268C2×D4—28no—
269C2×D4—38no—
270C2×D4—34no—
271C2×D4—34no—
272C2×D4—34no—
273C2×D4—34no—
274C2×D4—34no—
275C2×D4—34no—
276C2×D4—34no—
277C2×D4—34no—
278C2×D4—32no—
279C2×D4—32no—
280C2×D4—616no—
281C2×D4—616no—
282C2×D4—616no—
283C2×D4—68no—
284C2×D4—124no—
285C2×D4—124no—
286C2×D4C22×D314yes—
287C2×D4C22×D314yes—
288C2×D4C22×D312yes—
289C2×D4C22×D312yes—
290C22×C4—31no—
291C22×C4—31no—
292C22×C4—32no—
293C22×C4—32no—
294C22×C4—34no—
295C22×C4—32no—
296C22×C4—31no—
297C22×C4—34no—
298C22×C4—32no—
299C22×C4—34no—
300C22×C4—32no—
301C22×C4—122no—
302C22×C4—122no—
303C22×C4C22×D312yes—
304C22×C4C22×D311yescharacteristic
305C22×C4C22×D311yescharacteristic
306C4⋊C4—34no—
307C4⋊C4—34no—
308C4⋊C4—34no—
309C4⋊C4—32no—
310C4⋊C4—32no—
311C4⋊C4C22×D312yes—
312C4⋊C4C22×D312yes—
313C22⋊C4—34no—
314C22⋊C4—34no—
315C22⋊C4—34no—
316C22⋊C4—34no—
317C22⋊C4—34no—
318C22⋊C4—34no—
319C22⋊C4—34no—
320C22⋊C4C22×D314yes—
321C22⋊C4C22×D314yes—
322C22⋊C4C22×D314yes—
323C42—32no—
324C42—32no—
325C22×C3—21no—
326C22×C3—22no—
327C22×C3—21no—
328C22×C3—22no—
329C22×C3—24no—
330C22×C3—24no—
331C22×C3—24no—
332C22×C3—44no—
333C22×C3—84no—
334C22×C3C22×D414yes—
335C22×C3C22×D412yes—
336C22×C3C2511yescharacteristic
337C2×D3—22no—
338C2×D3—22no—
339C2×D3—24no—
340C2×D3—24no—
341C2×D3—24no—
342C2×D3—44no—
343C2×D3—88no—
344C2×D3—84no—
345C2×D3C22×D414yes—
346C3×C4—24no—
347C3×C4—22no—
348C3×C4—22no—
349Dic3—24no—
350Dic3—21no—
351Dic3—22no—
352Dic3C22×D414yes—
353Dic3H(5,𝔽2)12yescomplemented
354C23—24no—
355C23—32no—
356C23—34no—
357C23—34no—
358C23—41no—
359C23—41no—
360C23—42no—
361C23—42no—
362C23—68no—
363C23—64no—
364C23—121no—
365C23—121no—
366C23—122no—
367C23—124no—
368C23—122no—
369C23—124no—
370C23D3×D414yescomplemented
371C23C23×D314yes—
372C23C23×D312yes—
373C23C23×D311yescharacteristic
374Q8—68no—
375D4—28no—
376D4—28no—
377D4—316no—
378D4—42no—
379D4—42no—
380D4—68no—
381D4—68no—
382D4—68no—
383D4—68no—
384D4—68no—
385D4—68no—
386D4—122no—
387D4—122no—
388D4—124no—
389C2×C4—24no—
390C2×C4—34no—
391C2×C4—31no—
392C2×C4—32no—
393C2×C4—31no—
394C2×C4—32no—
395C2×C4—34no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—64no—
399C2×C4—68no—
400C2×C4—68no—
401C2×C4—62no—
402C2×C4—64no—
403C2×C4—68no—
404C2×C4—64no—
405C2×C4—122no—
406C2×C4—122no—
407C2×C4—121no—
408C2×C4C23×D312yes—
409C2×C4C23×D312yes—
410C2×C4C23×D314yes—
411C2×C3—22no—
412C2×C3—24no—
413C2×C3—44no—
414C2×C3C23×D411yescharacteristic
415C2×C3C2×H(5,𝔽2)11yescharacteristic
416C2×C3#58511yescharacteristic
417D3—24no—
418D3—44no—
419C22—21no—
420C22—22no—
421C22—21no—
422C22—22no—
423C22—24no—
424C22—24no—
425C22—24no—
426C22—34no—
427C22—44no—
428C22—62no—
429C22—62no—
430C22—64no—
431C22—64no—
432C22—64no—
433C22—84no—
434C22—124no—
435C22—244no—
436C22—248no—
437C22C2×D3×D414yes—
438C22C2×D3×D412yes—
439C22C24×D311yescharacteristic
440C4—24no—
441C4—22no—
442C4—22no—
443C4—34no—
444C4—32no—
445C4—64no—
446C4—61no—
447C4—62no—
448C3#349611yescharacteristic, complemented, Sylow
449C2—24no—
450C2—22no—
451C2—44no—
452C2—64no—
453C2—124no—
454C2C22×D3×D411yescharacteristic
455C2D3×H(5,𝔽2)11yescharacteristic
456C2#596611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#534C2
hypercenter#534D3
hypocenterC3#3496
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C2×#1004C2semidirect
2. C3×#534C2semidirect
3. C2×#939C2semidirect
4. #5655C2semidirect
5. #5654C2semidirect
6. #5605C2semidirect
7. D3×#121C2semidirect
8. #5597C2semidirect
9. #5596C2semidirect
10. #5591C2semidirect
11. #5590C2semidirect
12. #5493C2semidirect
13. C2×#67⋊C4C2semidirect
15. C22×C3×D4C22semidirect
17. C22×#67C22semidirect
18. C22×#67C22semidirect
19. C22×#67C22semidirect
20. C23×Dic3C22semidirect
22. C2×#237C22semidirect
23. C2×#237C22semidirect
24. C2×D3×D4C22semidirect
25. C2×D3×D4C22semidirect
26. C2×D3×D4C22semidirect
27. C2×#235C22semidirect
28. C22×C4×D3C22semidirect
29. C3×#123C22semidirect
30. C3×#121C22semidirect
31. C3×#121C22semidirect
32. C3×#120C22semidirect
33. C2×C3×C22⋊C4C22semidirect
34. C2×#217C22semidirect
35. #940C22semidirect
36. #939C22semidirect
37. #939C22semidirect
38. #939C22semidirect
39. #939C22semidirect
40. #939C22semidirect
41. #938C22semidirect
42. #937C22semidirect
43. D4×Dic3C22semidirect
44. #930C22semidirect
45. #930C22semidirect
46. C2×(C2×D3)⋊C4C22semidirect
47. C2×(C2×D3)⋊C4C22semidirect
48. C2×Dic3⋊C4C22semidirect
49. C2×Dic3⋊C4C22semidirect
50. C2×C4×Dic3C22semidirect
51. #896C22semidirect
52. #895C22semidirect
53. #894C22semidirect
54. #893C22semidirect
55. D3×C4⋊C4C22semidirect
56. #888C22semidirect
57. #886C22semidirect
58. #885C22semidirect
59. #884C22semidirect
60. #884C22semidirect
61. #883C22semidirect
62. #882C22semidirect
63. #882C22semidirect
64. #67⋊C4C22semidirect
65. #67⋊C4C22semidirect
66. #67⋊C4C22semidirect
67. D3×C22⋊C4C22semidirect
68. #878C22semidirect
77. #534D3semidirect
98. C2×#67D4semidirect
159. C22×D4C2×D3semidirect
168. #123C2×D3semidirect
182. #121C2×D3semidirect
183. #121C2×D3semidirect
187. #120C2×D3semidirect
200. C2×C22⋊C4C2×D3semidirect
207. C23×C3C2×D4semidirect
219. C22×D3C2×D4semidirect
238. #67C2×D4semidirect
245. C2×Dic3C2×D4semidirect
353. Dic3H(5,𝔽2)semidirect
370. C23D3×D4semidirect
448. C3#3496semidirect

Maximal subgroups

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

subgroupindex
C2×#10042
C3×#5342
C2×#9392
#56552
#56542
#56052
D3×#1212
#55972
#55962
#55912
#55902
#54932
C2×#67⋊C42
#34963