Finite Group Databasebeta

← #86254

Subgroups — #86254

Order 384.

1160 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#532C211yesmaximal, characteristic, complemented
2#5829C214yesmaximal, complemented
3#5733C212yesmaximal, complemented
4#5715C212yesmaximal, complemented
5#5684C212yesmaximal, complemented
6#5683C212yesmaximal, complemented
7#5617C212yesmaximal, complemented
8#5609C212yesmaximal, complemented
9#5605C212yesmaximal, complemented
10#5598C212yesmaximal, complemented
11#5560C214yesmaximal, complemented
12#5559C212yesmaximal, complemented
13#5554C212yesmaximal, complemented
14#5479C211yesmaximal, characteristic, complemented
15C2×#875C211yesmaximal, characteristic, complemented
16#3558—31nomaximal, Sylow
17C2×C3×#41C2212yescomplemented
18C2×#237C2212yescomplemented
19C2×D3×D4C2212yescomplemented
20C2×D3×D4C2214yescomplemented
21C22×D12C2211yescharacteristic
22C22×Dic6C2211yescharacteristic, complemented
23C3×C4⋊Q8C2211yescharacteristic, complemented
24C3×#127C2211yescharacteristic, complemented
25C3×#124C2212yescomplemented
26C3×#121C2212yescomplemented
27C3×#121C2212yescomplemented
28C3×C4×D4C2212yescomplemented
29C3×C4×D4C2212yescomplemented
30C2×C3×C42C2211yescharacteristic, complemented
31#947C2214yescomplemented
32#946C2214yescomplemented
33#940C2212yes—
34#939C2214yescomplemented
35#939C2214yescomplemented
36#939C2214yescomplemented
37#937C2214yescomplemented
38#937C2214yescomplemented
39#937C2214yescomplemented
40#936C2212yescomplemented
41D4×Dic3C2214yescomplemented
42#930C2211yescharacteristic
43#930C2214yescomplemented
44#929C2212yescomplemented
45#929C2214yescomplemented
46C4×#67C2212yescomplemented
47C4×#67C2214yescomplemented
48C2×(C2×D3)⋊C4C2212yescomplemented
49C2×(C2×D3)⋊C4C2212yescomplemented
50C2×(C2×D3)⋊C4C2214yescomplemented
51#924C2211yescharacteristic, complemented
52#898C2214yescomplemented
53#897C2212yescomplemented
54#895C2212yes—
55#894C2214yescomplemented
56#893C2212yescomplemented
57#887C2212yescomplemented
58#886C2214yescomplemented
59#885C2214yescomplemented
60#885C2214yescomplemented
61#884C2214yescomplemented
62#883C2214yescomplemented
63#882C2214yescomplemented
64#882C2214yescomplemented
65#878C2214yescomplemented
66#876C2212yescomplemented
67#875C2211yescharacteristic, complemented
68#875C2212yescomplemented
69#875C2212yescomplemented
70#875C2212yescomplemented
71#875C2212yescomplemented
72C4×D12C2211yescharacteristic, complemented
73#872C2212yescomplemented
74#870C2211yescharacteristic, complemented
75C4×Dic6C2211yescharacteristic, complemented
76#562—32no—
77H(3,𝔽4)—32no—
78#559—32no—
79#548—32no—
80#546—34no—
81#546—32no—
82D42—32no—
83#542—32no—
84#538—32no—
85#538—34no—
86#535—32no—
87#534—32no—
88#532D311yescharacteristic, complemented
89#529—31no—
90C2×#124—31no—
91C23×D3C2312yes—
92C23×D3C2314yes—
93C3×#41—44no—
94C2×C3×Q8C2312yes—
95C2×C3×D4C2314yes—
96C2×C3×D4C2312yes—
97C2×C3×D4C2312yes—
98C2×C3×D4C2312yes—
99C22×C3×C4C2311yescharacteristic
100C22×C3×C4C2311yescharacteristic
101C22×C3×C4C2311yescharacteristic
102C22×C3×C4C2314yes—
103C2×#67C2312yes—
104C2×#67C2314yes—
105C2×#67C2314yes—
106C2×#67C2314yes—
107C2×#67C2314yes—
108C22×Dic3C2314yes—
109C22×Dic3C2312yes—
110#237—44no—
111D3×D4—44no—
112D3×D4—48no—
113C2×D12—42no—
114C2×D12C2312yes—
115C2×D12C2311yescharacteristic
116C2×D12C2312yes—
117C2×D12C2314yes—
118C2×C4×D3C2312yes—
119C2×C4×D3C2314yes—
120C2×Dic6—42no—
121C2×Dic6C2312yes—
122C2×Dic6C2311yescharacteristic
123C2×Dic6C2312yes—
124C3×C4⋊C4C2312yes—
125C3×C4⋊C4C2312yes—
126C3×C22⋊C4C2314yes—
127C3×C22⋊C4C2314yes—
128C3×C42C2311yescharacteristic
129C3×C42C2311yescharacteristic
130C3×C42C2311yescharacteristic
131C3×C42C2311yescharacteristic
132#217C2314yes—
133#217C2314yes—
134#217C2312yes—
135(C2×D3)⋊C4C2312yes—
136(C2×D3)⋊C4C2312yes—
137(C2×D3)⋊C4C2312yes—
138(C2×D3)⋊C4C2314yes—
139(C2×D3)⋊C4C2314yes—
140(C2×D3)⋊C4C2314yes—
141(C2×D3)⋊C4C2314yes—
142(C2×D3)⋊C4C2312yes—
143(C2×D3)⋊C4C2314yes—
144(C2×D3)⋊C4C2314yes—
145(C2×D3)⋊C4C2314yes—
146C4⋊Dic3C2311yescharacteristic
147C4⋊Dic3C2311yescharacteristic
148C4⋊Dic3C2314yes—
149Dic3⋊C4C2312yes—
150Dic3⋊C4C2312yes—
151Dic3⋊C4C2312yes—
152Dic3⋊C4C2312yes—
153Dic3⋊C4C2314yes—
154C4×Dic3C2312yes—
155C4×Dic3C2314yes—
156C2×#41—32no—
157C2×#41C2×D312yescomplemented
158C22×Q8—31no—
159C22×D4—34no—
160C22×D4—32no—
161C22×D4—31no—
162C4⋊Q8C2×D311yescharacteristic, complemented
163#127—32no—
164#127C2×D311yescharacteristic, complemented
165#126—34no—
166#126—32no—
167#125—31no—
168#124—34no—
169#124—32no—
170#124—34no—
171#124—34no—
172#124—32no—
173#124—32no—
174#124—32no—
175#124—32no—
176#124—31no—
177#124C2×D312yescomplemented
178#123—32no—
179#123—34no—
180#123—34no—
181#123—34no—
182#123—34no—
183C22⋊Q8—34no—
184C22⋊Q8—31no—
185C22⋊Q8—32no—
186C22⋊Q8—34no—
187#121—34no—
188#121—34no—
189#121—34no—
190#121—31no—
191#121—34no—
192#121—32no—
193#121—34no—
194#121C2×D312yescomplemented
195#121C2×D312yescomplemented
196#120—34no—
197#120—34no—
198#120—34no—
199#120—34no—
200#120—34no—
201C4×Q8—32no—
202C4×Q8—31no—
203C4×D4—34no—
204C4×D4—32no—
205C4×D4—34no—
206C4×D4—32no—
207C4×D4—31no—
208C4×D4C2×D312yescomplemented
209C4×D4C2×D312yescomplemented
210#117—32no—
211C2×C22⋊C4—32no—
212C2×C22⋊C4—32no—
213C2×C22⋊C4—34no—
214C2×C42C2×D311yescharacteristic, complemented
215C23×C3C2414yes—
216C23×C3C2411yescharacteristic
217C22×D3—24no—
218C22×D3—42no—
219C22×D3—44no—
220C22×D3—44no—
221C22×D3—44no—
222C22×D3—42no—
223C22×D3C2412yes—
224C22×D3C2414yes—
225C3×Q8—42no—
226C3×D4—28no—
227C3×D4—24no—
228C3×D4—42no—
229C3×D4—42no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—41no—
233C2×C3×C4—44no—
234C2×C3×C4C2414yes—
235C2×C3×C4C2411yescharacteristic
236C2×C3×C4C2411yescharacteristic
237C2×C3×C4C2411yescharacteristic
238C2×C3×C4C2411yescharacteristic
239C2×C3×C4C2411yescharacteristic
240C2×C3×C4C2411yescharacteristic
241#67—24no—
242#67—44no—
243#67—44no—
244#67—44no—
245#67—44no—
246C2×Dic3—42no—
247C2×Dic3—44no—
248C2×Dic3C2412yes—
249C2×Dic3C2412yes—
250C2×Dic3C2414yes—
251C2×Dic3C2412yes—
252D12—24no—
253D12—42no—
254D12—41no—
255D12—44no—
256C4×D3—24no—
257C4×D3—44no—
258Dic6—24no—
259Dic6—42no—
260Dic6—41no—
261C24—32no—
262C24—34no—
263#41—44no—
264#41—124no—
265C2×Q8—32no—
266C2×Q8—31no—
267C2×Q8—32no—
268C2×Q8—122no—
269C2×Q8C22×D312yes—
270C2×D4—32no—
271C2×D4—34no—
272C2×D4—34no—
273C2×D4—34no—
274C2×D4—34no—
275C2×D4—32no—
276C2×D4—31no—
277C2×D4—34no—
278C2×D4—32no—
279C2×D4—122no—
280C2×D4—124no—
281C2×D4—128no—
282C2×D4C22×D314yes—
283C2×D4C22×D312yes—
284C2×D4C22×D312yes—
285C2×D4C22×D312yes—
286C22×C4—32no—
287C22×C4—34no—
288C22×C4—32no—
289C22×C4—34no—
290C22×C4C22×D311yescharacteristic
291C22×C4C22×D311yescharacteristic
292C22×C4C22×D311yescharacteristic
293C22×C4C22×D314yes—
294C4⋊C4—32no—
295C4⋊C4—32no—
296C4⋊C4—32no—
297C4⋊C4—32no—
298C4⋊C4—34no—
299C4⋊C4—31no—
300C4⋊C4—34no—
301C4⋊C4—31no—
302C4⋊C4C22×D312yes—
303C4⋊C4C22×D312yes—
304C22⋊C4—32no—
305C22⋊C4—34no—
306C22⋊C4—34no—
307C22⋊C4—32no—
308C22⋊C4—34no—
309C22⋊C4—34no—
310C22⋊C4—34no—
311C22⋊C4—34no—
312C22⋊C4—34no—
313C22⋊C4—34no—
314C22⋊C4—32no—
315C22⋊C4—32no—
316C22⋊C4—34no—
317C22⋊C4—32no—
318C22⋊C4C22×D314yes—
319C22⋊C4C22×D314yes—
320C42—32no—
321C42—34no—
322C42C22×D311yescharacteristic
323C42C22×D311yescharacteristic
324C42C22×D311yescharacteristic
325C42C22×D311yescharacteristic
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—24no—
329C22×C3—24no—
330C22×C3—24no—
331C22×C3H(5,𝔽2)12yes—
332C22×C3C2511yescharacteristic
333C2×D3—22no—
334C2×D3—24no—
335C2×D3—24no—
336C2×D3—22no—
337C2×D3—22no—
338C2×D3—24no—
339C2×D3—84no—
340C2×D3—88no—
341C3×C4—21no—
342C3×C4—24no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4H(5,𝔽2)12yes—
348Dic3—24no—
349Dic3—22no—
350Dic3—22no—
351Dic3—22no—
352C23—34no—
353C23—32no—
354C23—64no—
355C23—122no—
356C23—124no—
357C23—124no—
358C23—122no—
359C23—124no—
360C23C23×D314yes—
361C23C23×D311yescharacteristic
362Q8—42no—
363Q8—64no—
364Q8—122no—
365Q8—121no—
366D4—28no—
367D4—24no—
368D4—42no—
369D4—42no—
370D4—64no—
371D4—64no—
372D4—124no—
373D4—122no—
374D4—121no—
375D4—124no—
376D4—124no—
377D4—124no—
378D4—124no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—32no—
382C2×C4—34no—
383C2×C4—32no—
384C2×C4—32no—
385C2×C4—44no—
386C2×C4—41no—
387C2×C4—64no—
388C2×C4—124no—
389C2×C4—122no—
390C2×C4—124no—
391C2×C4C23×D314yes—
392C2×C4C23×D311yescharacteristic
393C2×C4C23×D311yescharacteristic
394C2×C4C23×D311yescharacteristic
395C2×C4C23×D311yescharacteristic
396C2×C4C23×D311yescharacteristic
397C2×C4C23×D311yescharacteristic
398C2×C3—22no—
399C2×C3—44no—
400C2×C3C2×H(5,𝔽2)11yescharacteristic
401C2×C3C2×H(5,𝔽2)11yescharacteristic
402C2×C3#58511yescharacteristic
403D3—42no—
404D3—44no—
405C22—21no—
406C22—21no—
407C22—24no—
408C22—24no—
409C22—24no—
410C22—64no—
411C22—62no—
412C22—62no—
413C22—64no—
414C22—62no—
415C22—64no—
416C22—244no—
417C22—248no—
418C22#100412yes—
419C22C24×D311yescharacteristic
420C4—21no—
421C4—21no—
422C4—24no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—62no—
427C4—62no—
428C4—64no—
429C4—62no—
430C4#100912yes—
431C3#355811yescharacteristic, complemented, Sylow
432C2—22no—
433C2—44no—
434C2—124no—
435C2—122no—
436C2C2×#100411yescharacteristic
437C2C2×#100911yescharacteristic
438C2#596411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#532C2
hypercenter#532D3
hypocenterC3#3558
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C3×#532C2semidirect
2. #5829C2semidirect
3. #5733C2semidirect
4. #5715C2semidirect
5. #5684C2semidirect
6. #5683C2semidirect
7. #5617C2semidirect
8. #5609C2semidirect
9. #5605C2semidirect
10. #5598C2semidirect
11. #5560C2semidirect
12. #5559C2semidirect
13. #5554C2semidirect
14. #5479C2semidirect
15. C2×#875C2semidirect
17. C2×C3×#41C22semidirect
18. C2×#237C22semidirect
19. C2×D3×D4C22semidirect
20. C2×D3×D4C22semidirect
22. C22×Dic6C22semidirect
23. C3×C4⋊Q8C22semidirect
24. C3×#127C22semidirect
25. C3×#124C22semidirect
26. C3×#121C22semidirect
27. C3×#121C22semidirect
28. C3×C4×D4C22semidirect
29. C3×C4×D4C22semidirect
30. C2×C3×C42C22semidirect
31. #947C22semidirect
32. #946C22semidirect
34. #939C22semidirect
35. #939C22semidirect
36. #939C22semidirect
37. #937C22semidirect
38. #937C22semidirect
39. #937C22semidirect
40. #936C22semidirect
41. D4×Dic3C22semidirect
43. #930C22semidirect
44. #929C22semidirect
45. #929C22semidirect
46. C4×#67C22semidirect
47. C4×#67C22semidirect
48. C2×(C2×D3)⋊C4C22semidirect
49. C2×(C2×D3)⋊C4C22semidirect
50. C2×(C2×D3)⋊C4C22semidirect
51. #924C22semidirect
52. #898C22semidirect
53. #897C22semidirect
55. #894C22semidirect
56. #893C22semidirect
57. #887C22semidirect
58. #886C22semidirect
59. #885C22semidirect
60. #885C22semidirect
61. #884C22semidirect
62. #883C22semidirect
63. #882C22semidirect
64. #882C22semidirect
65. #878C22semidirect
66. #876C22semidirect
67. #875C22semidirect
68. #875C22semidirect
69. #875C22semidirect
70. #875C22semidirect
71. #875C22semidirect
72. C4×D12C22semidirect
73. #872C22semidirect
74. #870C22semidirect
75. C4×Dic6C22semidirect
88. #532D3semidirect
157. C2×#41C2×D3semidirect
162. C4⋊Q8C2×D3semidirect
164. #127C2×D3semidirect
177. #124C2×D3semidirect
194. #121C2×D3semidirect
195. #121C2×D3semidirect
208. C4×D4C2×D3semidirect
209. C4×D4C2×D3semidirect
214. C2×C42C2×D3semidirect
431. C3#3558semidirect

Maximal subgroups

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

subgroupindex
C3×#5322
#58292
#57332
#57152
#56842
#56832
#56172
#56092
#56052
#55982
#55602
#55592
#55542
#54792
C2×#8752
#35583