Finite Group Databasebeta

← #86751

Subgroups — #86751

Order 384.

1124 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×D4×Q8C211yesmaximal, characteristic, complemented
2#5817C212yesmaximal, complemented
3C2×#946C212yesmaximal, complemented
4#5733C212yesmaximal, complemented
5#5730C212yesmaximal, complemented
6#5728C211yesmaximal, characteristic, complemented
7#5726C211yesmaximal, characteristic, complemented
8#5645C214yesmaximal, complemented
9#5642C212yesmaximal, complemented
10#5637C214yesmaximal, complemented
11#5627C212yesmaximal, complemented
12#5584C211yesmaximal, characteristic, complemented
13#5580C211yesmaximal, characteristic, complemented
14#5555C211yesmaximal, characteristic, complemented
15#5554C212yesmaximal, complemented
16#5546C212yesmaximal, complemented
17C4×D3×D4C211yesmaximal, characteristic, complemented
18#3520—31nomaximal, Sylow
19C22×C3×Q8C2212yescomplemented
20C2×#239C2212yescomplemented
21C2×#237C2212yes—
22C2×D3×D4C2211yescharacteristic
23C22×C4×D3C2212yes—
24C3×C4⋊Q8C2211yescharacteristic, complemented
25C3×C4⋊Q8C2212yescomplemented
26C3×C22⋊Q8C2214yescomplemented
27C3×C22⋊Q8C2212yescomplemented
28C3×C4×Q8C2211yescharacteristic, complemented
29C3×C4×D4C2211yescharacteristic
30C3×C4×D4C2212yes—
31#947C2214yescomplemented
32#947C2211yescharacteristic, complemented
33#946C2214yes—
34#946C2212yes—
35#946C2211yescharacteristic
36#946C2212yescomplemented
37#946C2214yescomplemented
38#946C2214yescomplemented
39#946C2212yescomplemented
40Q8×Dic3C2212yescomplemented
41D4×Dic3C2211yescharacteristic
42#930C2214yes—
43#930C2212yescomplemented
44C4×#67C2212yes—
45C4×#67C2214yescomplemented
46C2×(C2×D3)⋊C4C2214yes—
47C2×C4⋊Dic3C2212yescomplemented
48C2×Dic3⋊C4C2214yescomplemented
49#898C2214yescomplemented
50#898C2214yescomplemented
51#898C2212yescomplemented
52#897C2212yes—
53#897C2212yescomplemented
54#895C2212yes—
55#895C2211yescharacteristic
56#894C2212yes—
57#894C2214yescomplemented
58#894C2214yescomplemented
59#893C2214yes—
60#893C2211yescharacteristic
61#892C2212yes—
62#892C2212yescomplemented
63D3×C4⋊C4C2211yescharacteristic
64D3×C4⋊C4C2212yescomplemented
65#890C2211yescharacteristic, complemented
66#890C2214yescomplemented
67#890C2212yescomplemented
68#887C2212yescomplemented
69#886C2214yes—
70#886C2214yescomplemented
71#882C2214yes—
72#67⋊C4C2212yes—
73#67⋊C4C2214yescomplemented
74D3×C22⋊C4C2212yes—
75#877C2214yescomplemented
76#875C2212yes—
77C4×D12C2211yescharacteristic
78C4×D12C2212yes—
79C4×D12C2211yescharacteristic, complemented
80#872C2212yescomplemented
81C42×D3C2211yescharacteristic
82#869C2211yescharacteristic, complemented
83C4×Dic6C2212yescomplemented
84#556—31no—
85#554—32no—
86#552—32no—
87#551—34no—
88D4×Q8D311yescharacteristic, complemented
89#548—32no—
90#547—31no—
91#547—31no—
92#546—32no—
93#542—32no—
94#539—34no—
95#533—31no—
96#529—31no—
97#525—32no—
98C2×C22⋊Q8—32no—
99#520—32no—
100C2×C4×D4—31no—
101C23×D3C2312yes—
102C2×C3×Q8—44no—
103C2×C3×Q8C2312yes—
104C2×C3×Q8C2311yescharacteristic
105C2×C3×Q8C2314yes—
106C2×C3×D4C2311yescharacteristic
107C22×C3×C4C2312yes—
108C22×C3×C4C2314yes—
109C2×#67C2312yes—
110C2×#67C2314yes—
111C22×Dic3C2314yes—
112C22×Dic3C2312yes—
113#239—44no—
114#237—28no—
115D3×D4—24no—
116C2×D12C2311yescharacteristic
117C2×D12C2312yes—
118C2×D12C2311yescharacteristic
119C2×D12C2314yes—
120C2×C4×D3—44no—
121C2×C4×D3C2311yescharacteristic
122C2×C4×D3C2311yescharacteristic
123C2×C4×D3C2311yescharacteristic
124C2×C4×D3C2312yes—
125C2×C4×D3C2312yes—
126C2×C4×D3C2312yes—
127C2×C4×D3C2312yes—
128C2×C4×D3C2314yes—
129C2×Dic6C2312yes—
130C3×C4⋊C4C2312yes—
131C3×C4⋊C4C2312yes—
132C3×C4⋊C4C2311yescharacteristic
133C3×C4⋊C4C2314yes—
134C3×C4⋊C4C2311yescharacteristic
135C3×C4⋊C4C2312yes—
136C3×C22⋊C4C2314yes—
137C3×C22⋊C4C2312yes—
138C3×C42C2312yes—
139C3×C42C2311yescharacteristic
140#217C2312yes—
141(C2×D3)⋊C4C2312yes—
142(C2×D3)⋊C4C2312yes—
143(C2×D3)⋊C4C2312yes—
144(C2×D3)⋊C4C2312yes—
145(C2×D3)⋊C4C2312yes—
146(C2×D3)⋊C4C2312yes—
147(C2×D3)⋊C4C2314yes—
148(C2×D3)⋊C4C2314yes—
149(C2×D3)⋊C4C2312yes—
150(C2×D3)⋊C4C2314yes—
151(C2×D3)⋊C4C2314yes—
152(C2×D3)⋊C4C2314yes—
153C4⋊Dic3C2311yescharacteristic
154C4⋊Dic3C2312yes—
155C4⋊Dic3C2312yes—
156C4⋊Dic3C2312yes—
157C4⋊Dic3C2311yescharacteristic
158C4⋊Dic3C2314yes—
159C4⋊Dic3C2314yes—
160Dic3⋊C4C2314yes—
161Dic3⋊C4C2312yes—
162Dic3⋊C4C2312yes—
163Dic3⋊C4C2314yes—
164Dic3⋊C4C2312yes—
165Dic3⋊C4C2314yes—
166C4×Dic3C2314yes—
167C4×Dic3C2311yescharacteristic
168C4×Dic3C2311yescharacteristic
169C4×Dic3C2311yescharacteristic
170C4×Dic3C2314yes—
171C2×#41—32no—
172C2×#41—32no—
173C22×Q8C2×D312yescomplemented
174C22×D4—31no—
175C23×C4—32no—
176C4⋊Q8—31no—
177C4⋊Q8C2×D311yescharacteristic, complemented
178C4⋊Q8C2×D312yescomplemented
179#126—32no—
180#126—34no—
181#126—34no—
182#125—31no—
183#125—32no—
184#125—34no—
185#124—31no—
186#124—34no—
187#124—32no—
188#123—34no—
189#123—32no—
190#123—34no—
191#123—34no—
192#123—34no—
193C22⋊Q8—31no—
194C22⋊Q8—32no—
195C22⋊Q8—34no—
196C22⋊Q8—32no—
197C22⋊Q8—32no—
198C22⋊Q8—34no—
199C22⋊Q8—34no—
200C22⋊Q8—32no—
201C22⋊Q8—32no—
202C22⋊Q8C2×D312yescomplemented
203C22⋊Q8C2×D314yescomplemented
204#121—32no—
205#121—34no—
206#121—31no—
207#121—32no—
208#120—34no—
209C4×Q8—32no—
210C4×Q8—32no—
211C4×Q8—32no—
212C4×Q8C2×D311yescharacteristic, complemented
213C4×D4—31no—
214C4×D4—32no—
215C4×D4—34no—
216C4×D4—31no—
217C4×D4—34no—
218C4×D4—32no—
219C4×D4—34no—
220C4×D4—31no—
221C4×D4—32no—
222C4×D4—31no—
223C4×D4C2×D311yescharacteristic
224C4×D4C2×D312yes—
225#117—32no—
226#117—32no—
227#117—34no—
228#117—32no—
229C2×C4⋊C4—34no—
230C2×C4⋊C4—32no—
231C2×C4⋊C4—31no—
232C2×C4⋊C4—32no—
233C2×C22⋊C4—34no—
234C2×C22⋊C4—32no—
235C2×C42—31no—
236C23×C3C2412yes—
237C22×D3—24no—
238C22×D3—24no—
239C22×D3—42no—
240C22×D3C2412yes—
241C22×D3C2412yes—
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×C4C2414yes—
251C2×C3×C4C2411yescharacteristic
252C2×C3×C4C2412yes—
253C2×C3×C4C2412yes—
254C2×C3×C4C2411yescharacteristic
255C2×C3×C4C2411yescharacteristic
256#67—24no—
257#67—28no—
258C2×Dic3—28no—
259C2×Dic3—42no—
260C2×Dic3C2411yescharacteristic
261C2×Dic3C2411yescharacteristic
262C2×Dic3C2412yes—
263C2×Dic3C2411yescharacteristic
264C2×Dic3C2414yes—
265C2×Dic3C2412yes—
266D12—24no—
267D12—22no—
268D12—22no—
269D12—44no—
270C4×D3—22no—
271C4×D3—22no—
272C4×D3—24no—
273C4×D3—22no—
274C4×D3—24no—
275C4×D3—42no—
276C4×D3—42no—
277C4×D3—44no—
278Dic6—24no—
279C24—32no—
280#41—68no—
281#41—124no—
282C2×Q8—32no—
283C2×Q8—44no—
284C2×Q8C22×D312yes—
285C2×Q8C22×D311yescharacteristic
286C2×Q8C22×D314yes—
287C2×D4—34no—
288C2×D4—32no—
289C2×D4—32no—
290C2×D4—31no—
291C2×D4—31no—
292C2×D4—34no—
293C2×D4—64no—
294C2×D4C22×D311yescharacteristic
295C22×C4—34no—
296C22×C4—32no—
297C22×C4—31no—
298C22×C4—31no—
299C22×C4—31no—
300C22×C4—32no—
301C22×C4—32no—
302C22×C4—32no—
303C22×C4—34no—
304C22×C4—32no—
305C22×C4—124no—
306C22×C4C22×D312yes—
307C22×C4C22×D314yes—
308C4⋊C4—31no—
309C4⋊C4—32no—
310C4⋊C4—34no—
311C4⋊C4—31no—
312C4⋊C4—32no—
313C4⋊C4—32no—
314C4⋊C4—34no—
315C4⋊C4—34no—
316C4⋊C4—34no—
317C4⋊C4—32no—
318C4⋊C4—32no—
319C4⋊C4—32no—
320C4⋊C4—34no—
321C4⋊C4C22×D311yescharacteristic
322C4⋊C4C22×D312yes—
323C4⋊C4C22×D311yescharacteristic
324C4⋊C4C22×D312yes—
325C4⋊C4C22×D314yes—
326C4⋊C4C22×D312yes—
327C22⋊C4—32no—
328C22⋊C4—32no—
329C22⋊C4—34no—
330C22⋊C4—32no—
331C22⋊C4—32no—
332C22⋊C4—34no—
333C22⋊C4—34no—
334C22⋊C4—32no—
335C22⋊C4—32no—
336C22⋊C4—34no—
337C22⋊C4—32no—
338C22⋊C4—34no—
339C22⋊C4—32no—
340C22⋊C4C22×D312yes—
341C22⋊C4C22×D314yes—
342C42—31no—
343C42—31no—
344C42—31no—
345C42—34no—
346C42—34no—
347C42C22×D311yescharacteristic
348C42C22×D312yes—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3C2×#4114yes—
352C22×C3C2511yescharacteristic
353C2×D3—21no—
354C2×D3—21no—
355C2×D3—22no—
356C2×D3—22no—
357C2×D3—22no—
358C2×D3—22no—
359C2×D3—22no—
360C2×D3—22no—
361C2×D3—48no—
362C2×D32−1+412yes—
363C3×C4—22no—
364C3×C4—24no—
365C3×C4—22no—
366C3×C4—21no—
367C3×C4—22no—
368C3×C4C2×#4112yes—
369C3×C4H(5,𝔽2)12yes—
370Dic3—24no—
371Dic3—22no—
372Dic3—21no—
373Dic3—22no—
374Dic3—21no—
375Dic3—21no—
376C23—31no—
377C23—32no—
378C23—32no—
379C23—64no—
380C23—64no—
381C23—122no—
382C23C23×D312yes—
383Q8—22no—
384Q8—24no—
385Q8—44no—
386Q8—64no—
387D4—64no—
388D4—64no—
389D4—68no—
390D4—62no—
391D4—62no—
392D4—124no—
393D4#23914yescomplemented
394C2×C4—24no—
395C2×C4—31no—
396C2×C4—34no—
397C2×C4—31no—
398C2×C4—31no—
399C2×C4—32no—
400C2×C4—32no—
401C2×C4—44no—
402C2×C4—62no—
403C2×C4—68no—
404C2×C4—62no—
405C2×C4—64no—
406C2×C4—62no—
407C2×C4—64no—
408C2×C4—122no—
409C2×C4—122no—
410C2×C4—124no—
411C2×C4—122no—
412C2×C4C23×D312yes—
413C2×C4C23×D311yescharacteristic
414C2×C4C23×D314yes—
415C2×C4C23×D312yes—
416C2×C4C23×D312yes—
417C2×C4C23×D311yescharacteristic
418C2×C4C23×D311yescharacteristic
419C2×C3—24no—
420C2×C3C22×#4111yescharacteristic
421C2×C3C2×H(5,𝔽2)11yescharacteristic
422C2×C3C2×2−1+411yescharacteristic
423D3—22no—
424D3—42no—
425D3—42no—
426C22—22no—
427C22—22no—
428C22—32no—
429C22—61no—
430C22—61no—
431C22—62no—
432C22—62no—
433C22—62no—
434C22—62no—
435C22—62no—
436C22—62no—
437C22—128no—
438C22C2×#23914yes—
439C22C24×D311yescharacteristic
440C4—22no—
441C4—22no—
442C4—24no—
443C4—21no—
444C4—22no—
445C4—61no—
446C4—62no—
447C4—62no—
448C4—61no—
449C4—64no—
450C4—61no—
451C4#100412yes—
452C4C2×#23912yes—
453C3#352011yescharacteristic, complemented, Sylow
454C2—24no—
455C2—62no—
456C2—122no—
457C2—122no—
458C2C2×#100411yescharacteristic
459C2C22×#23911yescharacteristic
460C2D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×D4×Q8C2
hypercenterD4×Q8D3
hypocenterC3#3520
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×#946C2semidirect
4. #5733C2semidirect
5. #5730C2semidirect
6. #5728C2semidirect
7. #5726C2semidirect
8. #5645C2semidirect
9. #5642C2semidirect
10. #5637C2semidirect
11. #5627C2semidirect
12. #5584C2semidirect
13. #5580C2semidirect
14. #5555C2semidirect
15. #5554C2semidirect
16. #5546C2semidirect
17. C4×D3×D4C2semidirect
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
31. #947C22semidirect
32. #947C22semidirect
36. #946C22semidirect
37. #946C22semidirect
38. #946C22semidirect
39. #946C22semidirect
40. Q8×Dic3C22semidirect
43. #930C22semidirect
45. C4×#67C22semidirect
47. C2×C4⋊Dic3C22semidirect
48. C2×Dic3⋊C4C22semidirect
49. #898C22semidirect
50. #898C22semidirect
51. #898C22semidirect
53. #897C22semidirect
57. #894C22semidirect
58. #894C22semidirect
62. #892C22semidirect
64. D3×C4⋊C4C22semidirect
65. #890C22semidirect
66. #890C22semidirect
67. #890C22semidirect
68. #887C22semidirect
70. #886C22semidirect
73. #67⋊C4C22semidirect
75. #877C22semidirect
79. C4×D12C22semidirect
80. #872C22semidirect
82. #869C22semidirect
83. C4×Dic6C22semidirect
88. D4×Q8D3semidirect
173. C22×Q8C2×D3semidirect
177. C4⋊Q8C2×D3semidirect
178. C4⋊Q8C2×D3semidirect
202. C22⋊Q8C2×D3semidirect
203. C22⋊Q8C2×D3semidirect
212. C4×Q8C2×D3semidirect
246. C3×D4#41semidirect
393. D4#239semidirect
453. C3#3520semidirect

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×#9462
#57332
#57302
#57282
#57262
#56452
#56422
#56372
#56272
#55842
#55802
#55552
#55542
#55462
C4×D3×D42
#35203