Finite Group Databasebeta

← #86326

Subgroups — #86326

Order 384.

1258 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#535C211yesmaximal, characteristic, complemented
2#5832C211yesmaximal, characteristic, complemented
3#5830C211yesmaximal, characteristic, complemented
4D4×#67C212yesmaximal, complemented
5#5715C214yesmaximal, complemented
6#5683C212yesmaximal, complemented
7#5670C212yesmaximal, complemented
8#5616C212yesmaximal, complemented
9#5609C212yesmaximal, complemented
10#5607C212yesmaximal, complemented
11#5596C212yesmaximal, complemented
12#5594C212yesmaximal, complemented
13#5592C214yesmaximal, complemented
14#5590C212yesmaximal, complemented
15#5532C212yesmaximal, complemented
16#3549—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic, complemented
18C22×C3×D4C2211yescharacteristic, complemented
19C22×#67C2214yescomplemented
20C2×#237C2214yescomplemented
21C2×D3×D4C2214yescomplemented
22C2×D3×D4C2212yescomplemented
23C2×#235C2211yescharacteristic, complemented
24C22×D12C2211yescharacteristic, complemented
25C3×#127C2212yescomplemented
26C3×#124C2212yescomplemented
27C3×#121C2212yescomplemented
28C3×#121C2212yescomplemented
29C3×#120C2212yescomplemented
30C3×#120C2212yescomplemented
31C3×#117C2211yescharacteristic, complemented
32#953C2214yescomplemented
33#947C2212yescomplemented
34#947C2211yescharacteristic, complemented
35#944C2211yescharacteristic, complemented
36#940C2212yes—
37#940C2211yescharacteristic
38#940C2214yescomplemented
39#940C2212yescomplemented
40#939C2214yescomplemented
41#939C2212yescomplemented
42#939C2214yescomplemented
43#939C2214yescomplemented
44#939C2212yescomplemented
45#938C2212yescomplemented
46#937C2214yescomplemented
47#937C2212yescomplemented
48#937C2214yescomplemented
49#937C2214yescomplemented
50#937C2212yescomplemented
51#936C2214yescomplemented
52#936C2211yescharacteristic, complemented
53D4×Dic3C2212yescomplemented
54#930C2212yescomplemented
55#930C2212yescomplemented
56C4×#67C2212yescomplemented
57C4×#67C2212yescomplemented
58#926C2211yescharacteristic, complemented
59C2×C4×Dic3C2211yescharacteristic, complemented
60#895C2212yescomplemented
61#893C2212yescomplemented
62#892C2212yescomplemented
63#888C2212yescomplemented
64#885C2214yescomplemented
65#885C2214yescomplemented
66#885C2214yescomplemented
67#885C2214yescomplemented
68#884C2214yescomplemented
69#884C2214yescomplemented
70#882C2214yescomplemented
71#882C2214yescomplemented
72#67⋊C4C2214yescomplemented
73#877C2214yescomplemented
74#875C2214yescomplemented
75#872C2214yescomplemented
76H(3,𝔽4)—32no—
77H(3,𝔽4)—32no—
78#555—32no—
79#555—32no—
80#555—32no—
81D42—32no—
82D42—32no—
83D42—32no—
84#535—31no—
85#535—34no—
86#535—32no—
87#535—34no—
88#535D311yescharacteristic, complemented
89#532—31no—
90#532—32no—
91C24×C3C2312yes—
92C23×D3C2314yes—
93C23×D3C2312yes—
94C3×#41—42no—
95C2×C3×Q8C2311yescharacteristic
96C2×C3×D4—42no—
97C2×C3×D4C2312yes—
98C2×C3×D4C2311yescharacteristic
99C2×C3×D4C2314yes—
100C2×C3×D4C2312yes—
101C2×C3×D4C2311yescharacteristic
102C2×C3×D4C2314yes—
103C2×C3×D4C2311yescharacteristic
104C22×C3×C4C2311yescharacteristic
105C22×C3×C4C2311yescharacteristic
106C22×C3×C4C2311yescharacteristic
107C2×#67—48no—
108C2×#67C2314yes—
109C2×#67C2312yes—
110C2×#67C2312yes—
111C2×#67C2314yes—
112C2×#67C2314yes—
113C2×#67C2314yes—
114C2×#67C2314yes—
115C2×#67C2314yes—
116C2×#67C2312yes—
117C22×Dic3C2314yes—
118C22×Dic3C2312yes—
119#237—48no—
120D3×D4—44no—
121D3×D4—48no—
122#235—42no—
123C2×D12—42no—
124C2×D12C2312yes—
125C2×D12C2314yes—
126C2×D12C2311yescharacteristic
127C2×D12C2312yes—
128C2×C4×D3C2314yes—
129C2×C4×D3C2312yes—
130C2×Dic6C2314yes—
131C2×Dic6C2311yescharacteristic
132C3×C4⋊C4C2312yes—
133C3×C22⋊C4C2314yes—
134C3×C22⋊C4C2314yes—
135C3×C22⋊C4C2312yes—
136C3×C42C2312yes—
137#217C2314yes—
138#217C2312yes—
139#217C2314yes—
140#217C2312yes—
141#217C2312yes—
142(C2×D3)⋊C4C2314yes—
143(C2×D3)⋊C4C2314yes—
144(C2×D3)⋊C4C2312yes—
145(C2×D3)⋊C4C2314yes—
146(C2×D3)⋊C4C2314yes—
147(C2×D3)⋊C4C2314yes—
148(C2×D3)⋊C4C2312yes—
149C4⋊Dic3C2312yes—
150Dic3⋊C4C2312yes—
151Dic3⋊C4C2314yes—
152Dic3⋊C4C2312yes—
153C4×Dic3C2314yes—
154C4×Dic3C2312yes—
155C4×Dic3C2311yescharacteristic
156C4×Dic3C2311yescharacteristic
157C4×Dic3C2312yes—
158C2×#41—34no—
159C2×#41—31no—
160C2×#41C2×D311yescharacteristic, complemented
161C22×D4—34no—
162C22×D4—34no—
163C22×D4—32no—
164C22×D4—31no—
165C22×D4C2×D311yescharacteristic, complemented
166C4⋊Q8—31no—
167C4⋊Q8—32no—
168#127—31no—
169#127—32no—
170#127—34no—
171#127—32no—
172#127C2×D312yescomplemented
173#124—31no—
174#124—32no—
175#124—31no—
176#124—34no—
177#124—34no—
178#124—34no—
179#124—34no—
180#124—34no—
181#124—34no—
182#124C2×D312yescomplemented
183#121—32no—
184#121—34no—
185#121—32no—
186#121—32no—
187#121—34no—
188#121—34no—
189#121—32no—
190#121—32no—
191#121—32no—
192#121—34no—
193#121—34no—
194#121C2×D312yescomplemented
195#121C2×D312yescomplemented
196#120—34no—
197#120—32no—
198#120—34no—
199#120—32no—
200#120—34no—
201#120—34no—
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—
211#117—31no—
212#117—32no—
213#117—34no—
214#117—34no—
215#117C2×D311yescharacteristic, complemented
216C2×C42—31no—
217C23×C3—44no—
218C23×C3—42no—
219C23×C3C2411yescharacteristic
220C23×C3C2414yes—
221C23×C3C2411yescharacteristic
222C23×C3C2411yescharacteristic
223C22×D3—42no—
224C22×D3—44no—
225C22×D3—44no—
226C22×D3—44no—
227C22×D3—44no—
228C22×D3C2412yes—
229C22×D3C2414yes—
230C3×Q8—41no—
231C3×D4—24no—
232C3×D4—44no—
233C3×D4—44no—
234C3×D4—42no—
235C3×D4—41no—
236C3×D4—41no—
237C3×D4—41no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4C2411yescharacteristic
242C2×C3×C4C2411yescharacteristic
243C2×C3×C4C2414yes—
244C2×C3×C4C2411yescharacteristic
245C2×C3×C4C2411yescharacteristic
246#67—24no—
247#67—28no—
248#67—24no—
249#67—42no—
250#67—44no—
251#67—44no—
252#67—44no—
253#67—44no—
254#67—44no—
255C2×Dic3—24no—
256C2×Dic3—28no—
257C2×Dic3C2412yes—
258C2×Dic3C2414yes—
259C2×Dic3C2412yes—
260C2×Dic3C2412yes—
261D12—24no—
262D12—42no—
263D12—41no—
264D12—44no—
265C4×D3—24no—
266C4×D3—28no—
267Dic6—41no—
268Dic6—44no—
269C24—32no—
270C24—34no—
271C24C22×D312yes—
272#41—42no—
273#41—122no—
274#41—128no—
275C2×Q8—34no—
276C2×Q8—31no—
277C2×Q8C22×D311yescharacteristic
278C2×D4—32no—
279C2×D4—34no—
280C2×D4—32no—
281C2×D4—34no—
282C2×D4—34no—
283C2×D4—34no—
284C2×D4—34no—
285C2×D4—32no—
286C2×D4—34no—
287C2×D4—32no—
288C2×D4—31no—
289C2×D4—32no—
290C2×D4—34no—
291C2×D4—42no—
292C2×D4—122no—
293C2×D4—124no—
294C2×D4—128no—
295C2×D4—128no—
296C2×D4C22×D312yes—
297C2×D4C22×D311yescharacteristic
298C2×D4C22×D312yes—
299C2×D4C22×D311yescharacteristic
300C2×D4C22×D314yes—
301C2×D4C22×D314yes—
302C2×D4C22×D311yescharacteristic
303C22×C4—32no—
304C22×C4—34no—
305C22×C4—34no—
306C22×C4—32no—
307C22×C4C22×D311yescharacteristic
308C22×C4C22×D311yescharacteristic
309C22×C4C22×D311yescharacteristic
310C4⋊C4—32no—
311C4⋊C4—34no—
312C4⋊C4—32no—
313C4⋊C4—32no—
314C4⋊C4C22×D312yes—
315C22⋊C4—32no—
316C22⋊C4—34no—
317C22⋊C4—32no—
318C22⋊C4—34no—
319C22⋊C4—32no—
320C22⋊C4—34no—
321C22⋊C4—34no—
322C22⋊C4—34no—
323C22⋊C4—32no—
324C22⋊C4—34no—
325C22⋊C4—32no—
326C22⋊C4—34no—
327C22⋊C4C22×D314yes—
328C22⋊C4C22×D312yes—
329C22⋊C4C22×D314yes—
330C42—31no—
331C42—32no—
332C42—31no—
333C42—32no—
334C42—34no—
335C42C22×D312yes—
336C22×C3—22no—
337C22×C3—24no—
338C22×C3—22no—
339C22×C3—24no—
340C22×C3—22no—
341C22×C3—21no—
342C22×C3—24no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—84no—
346C22×C3C2511yescharacteristic
347C2×D3—22no—
348C2×D3—24no—
349C2×D3—24no—
350C2×D3—22no—
351C2×D3—24no—
352C2×D3—22no—
353C2×D3—84no—
354C2×D3—88no—
355C3×C4—24no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4H(5,𝔽2)12yes—
360Dic3—22no—
361Dic3—22no—
362Dic3—24no—
363Dic3H(5,𝔽2)14yes—
364C23—32no—
365C23—34no—
366C23—42no—
367C23—44no—
368C23—122no—
369C23—124no—
370C23—124no—
371C23—124no—
372C23—124no—
373C23C23×D311yescharacteristic
374C23C23×D314yes—
375C23C23×D311yescharacteristic
376C23C23×D311yescharacteristic
377Q8—41no—
378Q8—121no—
379Q8—124no—
380D4—24no—
381D4—42no—
382D4—44no—
383D4—41no—
384D4—41no—
385D4—41no—
386D4—44no—
387D4—64no—
388D4—68no—
389D4—64no—
390D4—64no—
391D4—122no—
392D4—124no—
393D4—121no—
394D4—124no—
395D4—124no—
396D4—124no—
397D4—122no—
398D4—124no—
399D4—124no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—32no—
404C2×C4—32no—
405C2×C4—32no—
406C2×C4—34no—
407C2×C4—68no—
408C2×C4—68no—
409C2×C4—64no—
410C2×C4—64no—
411C2×C4C23×D311yescharacteristic
412C2×C4C23×D311yescharacteristic
413C2×C4C23×D314yes—
414C2×C4C23×D311yescharacteristic
415C2×C4C23×D311yescharacteristic
416C2×C3—41no—
417C2×C3—44no—
418C2×C3—41no—
419C2×C3—41no—
420C2×C3C2×H(5,𝔽2)11yescharacteristic
421C2×C3C2×H(5,𝔽2)12yes—
422D3—42no—
423D3—44no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—24no—
428C22—22no—
429C22—24no—
430C22—22no—
431C22—24no—
432C22—22no—
433C22—62no—
434C22—64no—
435C22—64no—
436C22—62no—
437C22—64no—
438C22—62no—
439C22—84no—
440C22—248no—
441C22—244no—
442C22C24×D311yescharacteristic
443C4—21no—
444C4—21no—
445C4—21no—
446C4—24no—
447C4—34no—
448C4—62no—
449C4—64no—
450C4—62no—
451C4#100412yes—
452C3#354911yescharacteristic, complemented, Sylow
453C2—41no—
454C2—44no—
455C2—41no—
456C2—41no—
457C2—122no—
458C2—124no—
459C2C2×#100411yescharacteristic
460C2D3×H(5,𝔽2)12yes—
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#535C2
hypercenter#535D3
hypocenterC3#3549
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. #5832C2semidirect
3. #5830C2semidirect
4. D4×#67C2semidirect
5. #5715C2semidirect
6. #5683C2semidirect
7. #5670C2semidirect
8. #5616C2semidirect
9. #5609C2semidirect
10. #5607C2semidirect
11. #5596C2semidirect
12. #5594C2semidirect
13. #5592C2semidirect
14. #5590C2semidirect
15. #5532C2semidirect
17. C2×C3×#41C22semidirect
18. C22×C3×D4C22semidirect
19. C22×#67C22semidirect
20. C2×#237C22semidirect
21. C2×D3×D4C22semidirect
22. C2×D3×D4C22semidirect
23. C2×#235C22semidirect
24. C22×D12C22semidirect
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. #947C22semidirect
35. #944C22semidirect
38. #940C22semidirect
39. #940C22semidirect
40. #939C22semidirect
41. #939C22semidirect
42. #939C22semidirect
43. #939C22semidirect
44. #939C22semidirect
45. #938C22semidirect
46. #937C22semidirect
47. #937C22semidirect
48. #937C22semidirect
49. #937C22semidirect
50. #937C22semidirect
51. #936C22semidirect
52. #936C22semidirect
53. D4×Dic3C22semidirect
54. #930C22semidirect
55. #930C22semidirect
56. C4×#67C22semidirect
57. C4×#67C22semidirect
58. #926C22semidirect
59. C2×C4×Dic3C22semidirect
60. #895C22semidirect
61. #893C22semidirect
62. #892C22semidirect
63. #888C22semidirect
64. #885C22semidirect
65. #885C22semidirect
66. #885C22semidirect
67. #885C22semidirect
68. #884C22semidirect
69. #884C22semidirect
70. #882C22semidirect
71. #882C22semidirect
72. #67⋊C4C22semidirect
73. #877C22semidirect
74. #875C22semidirect
75. #872C22semidirect
88. #535D3semidirect
160. C2×#41C2×D3semidirect
165. C22×D4C2×D3semidirect
172. #127C2×D3semidirect
182. #124C2×D3semidirect
194. #121C2×D3semidirect
195. #121C2×D3semidirect
204. #120C2×D3semidirect
205. #120C2×D3semidirect
215. #117C2×D3semidirect
452. C3#3549semidirect

Maximal subgroups

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

subgroupindex
C3×#5352
#58322
#58302
D4×#672
#57152
#56832
#56702
#56162
#56092
#56072
#55962
#55942
#55922
#55902
#55322
#35493