Finite Group Databasebeta

← #86748

Subgroups — #86748

Order 384.

1142 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×D4×Q8C211yesmaximal, characteristic, complemented
2#5817C212yesmaximal, complemented
3C2×#947C212yesmaximal, complemented
4#5734C212yesmaximal, complemented
5#5733C212yesmaximal, complemented
6#5732C211yesmaximal, characteristic, complemented
7#5725C211yesmaximal, characteristic, complemented
8#5644C214yesmaximal, complemented
9#5637C214yesmaximal, complemented
10#5630C212yesmaximal, complemented
11#5624C212yesmaximal, complemented
12#5585C211yesmaximal, characteristic, complemented
13#5580C211yesmaximal, characteristic, complemented
14#5554C212yesmaximal, complemented
15D4×D12C211yesmaximal, characteristic, complemented
16#5545C212yesmaximal, complemented
17C4×#237C211yesmaximal, characteristic, complemented
18#3514—31nomaximal, Sylow
19C22×C3×Q8C2212yescomplemented
20C2×#239C2212yescomplemented
21C2×#237C2211yescharacteristic
22C2×D3×D4C2212yes—
23C22×D12C2212yes—
24C3×C4⋊Q8C2211yescharacteristic, complemented
25C3×C4⋊Q8C2212yescomplemented
26C3×C22⋊Q8C2214yescomplemented
27C3×C22⋊Q8C2212yescomplemented
28C3×C4×Q8C2211yescharacteristic, complemented
29C3×C4×D4C2211yescharacteristic
30C3×C4×D4C2212yes—
31#947C2212yes—
32#947C2211yescharacteristic
33#947C2214yescomplemented
34#947C2214yescomplemented
35#947C2214yescomplemented
36#947C2212yescomplemented
37#947C2212yescomplemented
38#946C2214yescomplemented
39#946C2211yescharacteristic, complemented
40Q8×Dic3C2212yescomplemented
41D4×Dic3C2211yescharacteristic
42#930C2212yes—
43#930C2214yescomplemented
44C4×#67C2214yescomplemented
45C4×#67C2212yescomplemented
46C2×(C2×D3)⋊C4C2214yescomplemented
47C2×(C2×D3)⋊C4C2214yescomplemented
48C2×C4×Dic3C2212yescomplemented
49#898C2212yescomplemented
50#898C2214yescomplemented
51#898C2214yescomplemented
52#897C2212yescomplemented
53#895C2212yes—
54#895C2211yescharacteristic
55#895C2212yescomplemented
56#895C2214yescomplemented
57#895C2212yescomplemented
58#894C2214yescomplemented
59#894C2214yescomplemented
60#894C2212yescomplemented
61#893C2212yescomplemented
62#893C2212yescomplemented
63#893C2212yescomplemented
64#892C2214yescomplemented
65#892C2212yescomplemented
66#892C2211yescharacteristic, complemented
67#890C2211yescharacteristic, complemented
68#887C2211yescharacteristic, complemented
69#886C2214yescomplemented
70#882C2214yes—
71#882C2214yescomplemented
72#67⋊C4C2212yescomplemented
73#67⋊C4C2214yescomplemented
74D3×C22⋊C4C2214yescomplemented
75#877C2212yescomplemented
76#875C2212yescomplemented
77#874C2211yescharacteristic
78C4×D12C2211yescharacteristic
79C4×D12C2212yescomplemented
80C4×D12C2212yescomplemented
81#872C2212yescomplemented
82C42×D3C2211yescharacteristic, complemented
83C4×Dic6C2211yescharacteristic, complemented
84#559—31no—
85#556—31no—
86#555—32no—
87#551—34no—
88D4×Q8D311yescharacteristic, complemented
89#548—32no—
90#546—32no—
91#546—32no—
92D42—31no—
93#542—32no—
94#538—34no—
95#532—31no—
96#529—32no—
97#529—31no—
98C2×#124—32no—
99#518—32no—
100C4×#41—31no—
101C23×D3C2314yes—
102C2×C3×Q8—44no—
103C2×C3×Q8C2312yes—
104C2×C3×Q8C2311yescharacteristic
105C2×C3×Q8C2314yes—
106C2×C3×D4C2311yescharacteristic
107C22×C3×C4C2312yes—
108C22×C3×C4C2314yes—
109C2×#67C2314yes—
110C2×#67C2312yes—
111C22×Dic3C2312yes—
112C22×Dic3C2312yes—
113#239—44no—
114#237—24no—
115D3×D4—28no—
116C2×D12—44no—
117C2×D12C2311yescharacteristic
118C2×D12C2312yes—
119C2×D12C2312yes—
120C2×D12C2312yes—
121C2×D12C2312yes—
122C2×D12C2314yes—
123C2×D12C2314yes—
124C2×C4×D3C2311yescharacteristic
125C2×C4×D3C2311yescharacteristic
126C2×C4×D3C2312yes—
127C2×C4×D3C2311yescharacteristic
128C2×C4×D3C2314yes—
129C2×C4×D3C2314yes—
130C2×Dic6C2311yescharacteristic
131C3×C4⋊C4C2311yescharacteristic
132C3×C4⋊C4C2311yescharacteristic
133C3×C4⋊C4C2312yes—
134C3×C4⋊C4C2314yes—
135C3×C4⋊C4C2312yes—
136C3×C4⋊C4C2312yes—
137C3×C22⋊C4C2312yes—
138C3×C22⋊C4C2314yes—
139C3×C42C2311yescharacteristic
140C3×C42C2312yes—
141#217C2312yes—
142(C2×D3)⋊C4C2312yes—
143(C2×D3)⋊C4C2312yes—
144(C2×D3)⋊C4C2312yes—
145(C2×D3)⋊C4C2314yes—
146(C2×D3)⋊C4C2312yes—
147(C2×D3)⋊C4C2312yes—
148(C2×D3)⋊C4C2314yes—
149(C2×D3)⋊C4C2314yes—
150(C2×D3)⋊C4C2314yes—
151(C2×D3)⋊C4C2312yes—
152(C2×D3)⋊C4C2314yes—
153(C2×D3)⋊C4C2314yes—
154(C2×D3)⋊C4C2314yes—
155(C2×D3)⋊C4C2314yes—
156C4⋊Dic3C2311yescharacteristic
157C4⋊Dic3C2311yescharacteristic
158C4⋊Dic3C2312yes—
159C4⋊Dic3C2314yes—
160Dic3⋊C4C2312yes—
161Dic3⋊C4C2312yes—
162Dic3⋊C4C2312yes—
163Dic3⋊C4C2312yes—
164C4×Dic3C2312yes—
165C4×Dic3C2311yescharacteristic
166C4×Dic3C2311yescharacteristic
167C4×Dic3C2311yescharacteristic
168C4×Dic3C2312yes—
169C4×Dic3C2312yes—
170C4×Dic3C2314yes—
171C2×#41—32no—
172C2×#41—31no—
173C22×Q8C2×D312yescomplemented
174C22×D4—32no—
175C22×D4—32no—
176C4⋊Q8C2×D311yescharacteristic, complemented
177C4⋊Q8C2×D312yescomplemented
178#127—31no—
179#126—34no—
180#126—32no—
181#126—34no—
182#125—31no—
183#124—31no—
184#124—32no—
185#124—34no—
186#124—32no—
187#124—32no—
188#124—34no—
189#124—34no—
190#124—32no—
191#123—32no—
192#123—34no—
193#123—34no—
194#123—34no—
195C22⋊Q8—31no—
196C22⋊Q8—34no—
197C22⋊Q8—32no—
198C22⋊Q8C2×D312yescomplemented
199C22⋊Q8C2×D314yescomplemented
200#121—32no—
201#121—34no—
202#121—31no—
203#121—32no—
204#121—32no—
205#121—34no—
206#121—32no—
207#120—34no—
208#120—34no—
209C4×Q8—32no—
210C4×Q8—31no—
211C4×Q8—31no—
212C4×Q8C2×D311yescharacteristic, complemented
213C4×D4—31no—
214C4×D4—32no—
215C4×D4—34no—
216C4×D4—32no—
217C4×D4—32no—
218C4×D4—32no—
219C4×D4—32no—
220C4×D4—34no—
221C4×D4—32no—
222C4×D4—31no—
223C4×D4—32no—
224C4×D4C2×D311yescharacteristic
225C4×D4C2×D312yes—
226#117—31no—
227#117—32no—
228#117—34no—
229#117—32no—
230#117—32no—
231C2×C22⋊C4—34no—
232C2×C22⋊C4—34no—
233C2×C22⋊C4—34no—
234C2×C42—32no—
235C2×C42—31no—
236C23×C3C2412yes—
237C22×D3—28no—
238C22×D3—44no—
239C22×D3—44no—
240C22×D3C2412yes—
241C22×D3C2414yes—
242C22×D3C2411yescharacteristic
243C3×Q8—24no—
244C3×Q8—22no—
245C3×Q8—44no—
246C3×D4#4114yescomplemented
247C2×C3×C4—24no—
248C2×C3×C4—44no—
249C2×C3×C4C2412yes—
250C2×C3×C4C2411yescharacteristic
251C2×C3×C4C2414yes—
252C2×C3×C4C2412yes—
253C2×C3×C4C2412yes—
254C2×C3×C4C2411yescharacteristic
255C2×C3×C4C2411yescharacteristic
256#67—28no—
257#67—24no—
258C2×Dic3—24no—
259C2×Dic3—42no—
260C2×Dic3C2411yescharacteristic
261C2×Dic3C2412yes—
262C2×Dic3C2411yescharacteristic
263C2×Dic3C2411yescharacteristic
264C2×Dic3C2412yes—
265C2×Dic3C2412yes—
266D12—24no—
267D12—22no—
268D12—24no—
269D12—44no—
270D12—42no—
271D12—42no—
272D12—44no—
273C4×D3—24no—
274C4×D3—22no—
275C4×D3—22no—
276C4×D3—44no—
277C4×D3—41no—
278C4×D3—44no—
279Dic6—22no—
280C24—34no—
281#41—64no—
282#41—124no—
283C2×Q8—31no—
284C2×Q8—44no—
285C2×Q8C22×D312yes—
286C2×Q8C22×D311yescharacteristic
287C2×Q8C22×D314yes—
288C2×D4—32no—
289C2×D4—34no—
290C2×D4—32no—
291C2×D4—31no—
292C2×D4—32no—
293C2×D4—32no—
294C2×D4—32no—
295C2×D4—34no—
296C2×D4—34no—
297C2×D4—68no—
298C2×D4—124no—
299C2×D4C22×D311yescharacteristic
300C22×C4—32no—
301C22×C4—32no—
302C22×C4—31no—
303C22×C4—31no—
304C22×C4—32no—
305C22×C4—31no—
306C22×C4—34no—
307C22×C4—34no—
308C22×C4C22×D312yes—
309C22×C4C22×D314yes—
310C4⋊C4—31no—
311C4⋊C4—32no—
312C4⋊C4—32no—
313C4⋊C4—32no—
314C4⋊C4—32no—
315C4⋊C4—32no—
316C4⋊C4—31no—
317C4⋊C4—34no—
318C4⋊C4C22×D312yes—
319C4⋊C4C22×D311yescharacteristic
320C4⋊C4C22×D314yes—
321C4⋊C4C22×D311yescharacteristic
322C4⋊C4C22×D312yes—
323C4⋊C4C22×D312yes—
324C22⋊C4—32no—
325C22⋊C4—32no—
326C22⋊C4—34no—
327C22⋊C4—32no—
328C22⋊C4—32no—
329C22⋊C4—34no—
330C22⋊C4—34no—
331C22⋊C4—34no—
332C22⋊C4—34no—
333C22⋊C4—34no—
334C22⋊C4—32no—
335C22⋊C4—34no—
336C22⋊C4—32no—
337C22⋊C4—34no—
338C22⋊C4—32no—
339C22⋊C4C22×D314yes—
340C22⋊C4C22×D312yes—
341C42—31no—
342C42—32no—
343C42—31no—
344C42—31no—
345C42—32no—
346C42—32no—
347C42—34no—
348C42C22×D311yescharacteristic
349C42C22×D312yes—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3C2×#4114yes—
353C22×C3C2511yescharacteristic
354C2×D3—21no—
355C2×D3—22no—
356C2×D3—24no—
357C2×D3—21no—
358C2×D3—24no—
359C2×D3—22no—
360C2×D3—21no—
361C2×D3—24no—
362C2×D3—22no—
363C2×D3—88no—
364C3×C4—24no—
365C3×C4—22no—
366C3×C4—22no—
367C3×C4—22no—
368C3×C4—21no—
369C3×C4C2×#4112yes—
370C3×C4H(5,𝔽2)12yes—
371Dic3—22no—
372Dic3—21no—
373Dic3—22no—
374Dic3—21no—
375Dic3—22no—
376Dic3—21no—
377C23—32no—
378C23—34no—
379C23—31no—
380C23—68no—
381C23—124no—
382C23—124no—
383C23C23×D312yes—
384Q8—22no—
385Q8—24no—
386Q8—44no—
387Q8—62no—
388D4—68no—
389D4—64no—
390D4—62no—
391D4—64no—
392D4—64no—
393D4—124no—
394D4—124no—
395D4—122no—
396D4—122no—
397D4#23914yescomplemented
398C2×C4—24no—
399C2×C4—31no—
400C2×C4—31no—
401C2×C4—32no—
402C2×C4—32no—
403C2×C4—32no—
404C2×C4—31no—
405C2×C4—44no—
406C2×C4—64no—
407C2×C4—62no—
408C2×C4—64no—
409C2×C4—62no—
410C2×C4—124no—
411C2×C4—124no—
412C2×C4—121no—
413C2×C4—122no—
414C2×C4C23×D312yes—
415C2×C4C23×D311yescharacteristic
416C2×C4C23×D314yes—
417C2×C4C23×D312yes—
418C2×C4C23×D312yes—
419C2×C4C23×D311yescharacteristic
420C2×C4C23×D311yescharacteristic
421C2×C3—24no—
422C2×C3C22×#4111yescharacteristic
423C2×C3C2×H(5,𝔽2)11yescharacteristic
424C2×C3#58511yescharacteristic
425D3—41no—
426D3—44no—
427D3—42no—
428C22—22no—
429C22—22no—
430C22—61no—
431C22—64no—
432C22—62no—
433C22—61no—
434C22—62no—
435C22—64no—
436C22—61no—
437C22—64no—
438C22—62no—
439C22—248no—
440C22C2×#23914yes—
441C22C24×D311yescharacteristic
442C4—22no—
443C4—22no—
444C4—24no—
445C4—22no—
446C4—21no—
447C4—61no—
448C4—61no—
449C4—62no—
450C4—62no—
451C4—62no—
452C4—61no—
453C4#100412yes—
454C4C2×#23912yes—
455C3#351411yescharacteristic, complemented, Sylow
456C2—24no—
457C2—122no—
458C2—124no—
459C2—121no—
460C2C2×#100411yescharacteristic
461C2C22×#23911yescharacteristic
462C2#596811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×D4×Q8C2
hypercenterD4×Q8D3
hypocenterC3#3514
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C3×D4×Q8C2semidirect
2. #5817C2semidirect
3. C2×#947C2semidirect
4. #5734C2semidirect
5. #5733C2semidirect
6. #5732C2semidirect
7. #5725C2semidirect
8. #5644C2semidirect
9. #5637C2semidirect
10. #5630C2semidirect
11. #5624C2semidirect
12. #5585C2semidirect
13. #5580C2semidirect
14. #5554C2semidirect
15. D4×D12C2semidirect
16. #5545C2semidirect
17. C4×#237C2semidirect
19. C22×C3×Q8C22semidirect
20. C2×#239C22semidirect
24. C3×C4⋊Q8C22semidirect
25. C3×C4⋊Q8C22semidirect
26. C3×C22⋊Q8C22semidirect
27. C3×C22⋊Q8C22semidirect
28. C3×C4×Q8C22semidirect
33. #947C22semidirect
34. #947C22semidirect
35. #947C22semidirect
36. #947C22semidirect
37. #947C22semidirect
38. #946C22semidirect
39. #946C22semidirect
40. Q8×Dic3C22semidirect
43. #930C22semidirect
44. C4×#67C22semidirect
45. C4×#67C22semidirect
46. C2×(C2×D3)⋊C4C22semidirect
47. C2×(C2×D3)⋊C4C22semidirect
48. C2×C4×Dic3C22semidirect
49. #898C22semidirect
50. #898C22semidirect
51. #898C22semidirect
52. #897C22semidirect
55. #895C22semidirect
56. #895C22semidirect
57. #895C22semidirect
58. #894C22semidirect
59. #894C22semidirect
60. #894C22semidirect
61. #893C22semidirect
62. #893C22semidirect
63. #893C22semidirect
64. #892C22semidirect
65. #892C22semidirect
66. #892C22semidirect
67. #890C22semidirect
68. #887C22semidirect
69. #886C22semidirect
71. #882C22semidirect
72. #67⋊C4C22semidirect
73. #67⋊C4C22semidirect
74. D3×C22⋊C4C22semidirect
75. #877C22semidirect
76. #875C22semidirect
79. C4×D12C22semidirect
80. C4×D12C22semidirect
81. #872C22semidirect
82. C42×D3C22semidirect
83. C4×Dic6C22semidirect
88. D4×Q8D3semidirect
173. C22×Q8C2×D3semidirect
176. C4⋊Q8C2×D3semidirect
177. C4⋊Q8C2×D3semidirect
198. C22⋊Q8C2×D3semidirect
199. C22⋊Q8C2×D3semidirect
212. C4×Q8C2×D3semidirect
246. C3×D4#41semidirect
397. D4#239semidirect
455. C3#3514semidirect

Maximal subgroups

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

subgroupindex
C3×D4×Q82
#58172
C2×#9472
#57342
#57332
#57322
#57252
#56442
#56372
#56302
#56242
#55852
#55802
#55542
D4×D122
#55452
C4×#2372
#35143