Finite Group Databasebeta

← #87005

Subgroups — #87005

Order 384.

1140 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#556C211yesmaximal, characteristic, complemented
2#5732C211yesmaximal, characteristic, complemented
3#5725C211yesmaximal, characteristic, complemented
4#5709C214yesmaximal, complemented
5#5704C214yesmaximal, complemented
6#5681C212yesmaximal, complemented
7#5674C212yesmaximal, complemented
8#5631C212yesmaximal, complemented
9#5627C212yesmaximal, complemented
10#5585C212yesmaximal, complemented
11#5578C211yesmaximal, characteristic, complemented
12#5577C211yesmaximal, characteristic, complemented
13#5574C212yesmaximal, complemented
14#5553C211yesmaximal, characteristic, complemented
15C4×D3×D4C211yesmaximal, characteristic, complemented
16#5532C212yesmaximal, complemented
17#5524C212yesmaximal, complemented
18#3506—31nomaximal, Sylow
19C2×#239C2212yes—
20C2×#239C2212yescomplemented
21C2×D3×D4C2211yescharacteristic
22C2×#235C2212yescomplemented
23C22×C4×D3C2212yes—
24C3×C4⋊Q8C2211yescharacteristic, complemented
25C3×#126C2214yescomplemented
26C3×#124C2212yes—
27C3×C22⋊Q8C2212yescomplemented
28C3×C4×Q8C2212yescomplemented
29C3×C4×Q8C2211yescharacteristic, complemented
30C3×C4×D4C2211yescharacteristic
31C3×#117C2212yescomplemented
32#947C2212yes—
33#947C2212yescomplemented
34#946C2211yescharacteristic
35Q8×Dic3C2212yescomplemented
36#944C2211yescharacteristic, complemented
37#940C2212yes—
38D4×Dic3C2211yescharacteristic
39#930C2212yescomplemented
40C4×#67C2212yes—
41#898C2214yescomplemented
42#897C2211yescharacteristic
43#897C2212yescomplemented
44#896C2212yes—
45#895C2212yes—
46#895C2212yes—
47#895C2214yescomplemented
48#895C2212yescomplemented
49#894C2214yescomplemented
50#894C2214yescomplemented
51#893C2214yes—
52#893C2211yescharacteristic
53#893C2212yescomplemented
54#893C2212yescomplemented
55#893C2212yescomplemented
56#892C2212yescomplemented
57D3×C4⋊C4C2211yescharacteristic
58D3×C4⋊C4C2214yescomplemented
59#889C2212yescomplemented
60#888C2211yescharacteristic, complemented
61#886C2214yescomplemented
62#885C2214yescomplemented
63#884C2214yes—
64#884C2214yes—
65#884C2214yescomplemented
66#883C2214yescomplemented
67#882C2214yes—
68#882C2214yes—
69#67⋊C4C2212yes—
70#67⋊C4C2214yescomplemented
71D3×C22⋊C4C2212yes—
72#875C2214yescomplemented
73#874C2212yes—
74#874C2211yescharacteristic, complemented
75C4×D12C2214yes—
76C4×D12C2211yescharacteristic
77C4×D12C2212yescomplemented
78C4×D12C2214yescomplemented
79#872C2212yes—
80#872C2214yescomplemented
81C42×D3C2211yescharacteristic
82C4×Dic6C2212yescomplemented
83C4×Dic6C2211yescharacteristic, complemented
84#559—32no—
85#559—32no—
86#556—31no—
87#556D311yescharacteristic, complemented
88#555—32no—
89#553—34no—
90#550—31no—
91#550—31no—
92#547—32no—
93#538—34no—
94#535—32no—
95#533—31no—
96#532—31no—
97#525—32no—
98#525—32no—
99#520—32no—
100C2×C4×D4—31no—
101C23×D3C2312yes—
102C2×C3×Q8C2311yescharacteristic
103C2×C3×Q8C2312yes—
104C2×C3×D4C2311yescharacteristic
105C22×C3×C4C2312yes—
106C2×#67C2312yes—
107C2×#67C2314yes—
108C2×#67C2314yes—
109C22×Dic3C2312yes—
110#239—28no—
111#239—44no—
112D3×D4—24no—
113#235—44no—
114C2×D12C2311yescharacteristic
115C2×D12C2314yes—
116C2×D12C2312yes—
117C2×D12C2312yes—
118C2×D12C2314yes—
119C2×D12C2314yes—
120C2×D12C2312yes—
121C2×C4×D3—44no—
122C2×C4×D3C2311yescharacteristic
123C2×C4×D3C2314yes—
124C2×C4×D3C2312yes—
125C2×C4×D3C2311yescharacteristic
126C2×C4×D3C2312yes—
127C2×C4×D3C2314yes—
128C2×C4×D3C2312yes—
129C2×C4×D3C2314yes—
130C2×C4×D3C2311yescharacteristic
131C2×Dic6C2311yescharacteristic
132C2×Dic6C2312yes—
133C3×C4⋊C4C2311yescharacteristic
134C3×C4⋊C4C2312yes—
135C3×C4⋊C4C2311yescharacteristic
136C3×C4⋊C4C2314yes—
137C3×C4⋊C4C2312yes—
138C3×C4⋊C4C2312yes—
139C3×C22⋊C4C2312yes—
140C3×C22⋊C4C2314yes—
141C3×C22⋊C4C2314yes—
142C3×C42C2311yescharacteristic
143C3×C42C2312yes—
144C3×C42C2314yes—
145#217C2312yes—
146(C2×D3)⋊C4C2312yes—
147(C2×D3)⋊C4C2312yes—
148(C2×D3)⋊C4C2314yes—
149(C2×D3)⋊C4C2314yes—
150(C2×D3)⋊C4C2312yes—
151(C2×D3)⋊C4C2314yes—
152(C2×D3)⋊C4C2314yes—
153(C2×D3)⋊C4C2312yes—
154(C2×D3)⋊C4C2314yes—
155(C2×D3)⋊C4C2314yes—
156C4⋊Dic3C2312yes—
157C4⋊Dic3C2311yescharacteristic
158C4⋊Dic3C2311yescharacteristic
159C4⋊Dic3C2314yes—
160Dic3⋊C4C2312yes—
161Dic3⋊C4C2312yes—
162Dic3⋊C4C2312yes—
163Dic3⋊C4C2312yes—
164Dic3⋊C4C2312yes—
165Dic3⋊C4C2312yes—
166C4×Dic3C2311yescharacteristic
167C4×Dic3C2312yes—
168C4×Dic3C2311yescharacteristic
169C4×Dic3C2311yescharacteristic
170C4×Dic3C2314yes—
171C2×#41—32no—
172C2×#41—32no—
173C2×#41—32no—
174C22×D4—31no—
175C23×C4—32no—
176C4⋊Q8—31no—
177C4⋊Q8—31no—
178C4⋊Q8C2×D311yescharacteristic, complemented
179#127—32no—
180#127—31no—
181#127—32no—
182#126—34no—
183#126C2×D314yescomplemented
184#125—32no—
185#124—32no—
186#124—32no—
187#124—34no—
188#124—34no—
189#124C2×D312yes—
190#123—34no—
191#123—34no—
192#123—34no—
193#123—34no—
194C22⋊Q8—31no—
195C22⋊Q8—31no—
196C22⋊Q8—32no—
197C22⋊Q8—32no—
198C22⋊Q8C2×D312yescomplemented
199#121—32no—
200#121—32no—
201#121—32no—
202#121—32no—
203#121—34no—
204#121—34no—
205#121—34no—
206#121—34no—
207#120—34no—
208#120—34no—
209C4×Q8—32no—
210C4×Q8—31no—
211C4×Q8—32no—
212C4×Q8C2×D312yescomplemented
213C4×Q8C2×D311yescharacteristic, complemented
214C4×D4—31no—
215C4×D4—32no—
216C4×D4—31no—
217C4×D4—34no—
218C4×D4—32no—
219C4×D4—32no—
220C4×D4—32no—
221C4×D4—32no—
222C4×D4—34no—
223C4×D4—31no—
224C4×D4—34no—
225C4×D4—32no—
226C4×D4—34no—
227C4×D4C2×D311yescharacteristic
228#117—32no—
229#117—32no—
230#117—34no—
231#117C2×D312yescomplemented
232C2×C4⋊C4—31no—
233C2×C4⋊C4—34no—
234C2×C22⋊C4—32no—
235C2×C42—31no—
236C23×C3C2412yes—
237C22×D3—24no—
238C22×D3—42no—
239C22×D3—42no—
240C22×D3C2412yes—
241C22×D3C2414yes—
242C22×D3C2411yescharacteristic
243C3×Q8—24no—
244C3×Q8—22no—
245C3×D4—22no—
246C2×C3×C4—24no—
247C2×C3×C4C2411yescharacteristic
248C2×C3×C4C2414yes—
249C2×C3×C4C2412yes—
250C2×C3×C4C2412yes—
251C2×C3×C4C2412yes—
252C2×C3×C4C2411yescharacteristic
253C2×C3×C4C2411yescharacteristic
254#67—24no—
255#67—44no—
256#67—44no—
257C2×Dic3—42no—
258C2×Dic3C2412yes—
259C2×Dic3C2411yescharacteristic
260C2×Dic3C2411yescharacteristic
261C2×Dic3C2412yes—
262C2×Dic3C2412yes—
263C2×Dic3C2411yescharacteristic
264D12—28no—
265D12—22no—
266D12—24no—
267D12—24no—
268D12—44no—
269D12—42no—
270D12—44no—
271C4×D3—22no—
272C4×D3—24no—
273C4×D3—28no—
274C4×D3—42no—
275C4×D3—42no—
276C4×D3—41no—
277C4×D3—44no—
278C4×D3—44no—
279C4×D3#4114yescomplemented
280Dic6—22no—
281Dic6—42no—
282C24—32no—
283#41—68no—
284#41—124no—
285#41—124no—
286C2×Q8—31no—
287C2×Q8—32no—
288C2×Q8C22×D312yes—
289C2×Q8C22×D311yescharacteristic
290C2×D4—32no—
291C2×D4—34no—
292C2×D4—34no—
293C2×D4—34no—
294C2×D4—31no—
295C2×D4—32no—
296C2×D4—32no—
297C2×D4—34no—
298C2×D4—32no—
299C2×D4—34no—
300C2×D4—64no—
301C2×D4C22×D311yescharacteristic
302C22×C4—31no—
303C22×C4—32no—
304C22×C4—31no—
305C22×C4—34no—
306C22×C4—32no—
307C22×C4—31no—
308C22×C4—32no—
309C22×C4—34no—
310C22×C4—34no—
311C22×C4—32no—
312C22×C4—124no—
313C22×C4C22×D312yes—
314C4⋊C4—32no—
315C4⋊C4—32no—
316C4⋊C4—31no—
317C4⋊C4—34no—
318C4⋊C4—32no—
319C4⋊C4—31no—
320C4⋊C4—32no—
321C4⋊C4—32no—
322C4⋊C4—32no—
323C4⋊C4—32no—
324C4⋊C4C22×D311yescharacteristic
325C4⋊C4C22×D314yes—
326C4⋊C4C22×D312yes—
327C4⋊C4C22×D311yescharacteristic
328C4⋊C4C22×D312yes—
329C4⋊C4C22×D312yes—
330C22⋊C4—32no—
331C22⋊C4—32no—
332C22⋊C4—32no—
333C22⋊C4—32no—
334C22⋊C4—32no—
335C22⋊C4—34no—
336C22⋊C4—34no—
337C22⋊C4—34no—
338C22⋊C4—34no—
339C22⋊C4—34no—
340C22⋊C4—34no—
341C22⋊C4C22×D312yes—
342C22⋊C4C22×D314yes—
343C22⋊C4C22×D314yes—
344C42—31no—
345C42—31no—
346C42—31no—
347C42—34no—
348C42—32no—
349C42C22×D311yescharacteristic
350C42C22×D314yes—
351C42C22×D312yes—
352C22×C3—22no—
353C22×C3—22no—
354C22×C3—22no—
355C22×C3C2511yescharacteristic
356C2×D3—21no—
357C2×D3—21no—
358C2×D3—24no—
359C2×D3—22no—
360C2×D3—22no—
361C2×D3—24no—
362C2×D3—24no—
363C2×D3—22no—
364C2×D3—48no—
365C2×D3C2×#4112yes—
366C3×C4—24no—
367C3×C4—22no—
368C3×C4—22no—
369C3×C4—22no—
370C3×C4—21no—
371C3×C4C2×#4112yes—
372C3×C4H(5,𝔽2)12yes—
373Dic3—22no—
374Dic3—22no—
375Dic3—21no—
376Dic3—21no—
377Dic3—22no—
378Dic3C2×#4112yes—
379C23—31no—
380C23—32no—
381C23—34no—
382C23—64no—
383C23—122no—
384C23—122no—
385C23C23×D312yes—
386Q8—24no—
387Q8—22no—
388Q8—62no—
389Q8—122no—
390D4—22no—
391D4—68no—
392D4—64no—
393D4—62no—
394D4—64no—
395D4—64no—
396D4—124no—
397D4—122no—
398D4—124no—
399D4—124no—
400D4—124no—
401C2×C4—24no—
402C2×C4—31no—
403C2×C4—34no—
404C2×C4—32no—
405C2×C4—31no—
406C2×C4—31no—
407C2×C4—32no—
408C2×C4—32no—
409C2×C4—68no—
410C2×C4—62no—
411C2×C4—64no—
412C2×C4—122no—
413C2×C4—122no—
414C2×C4—124no—
415C2×C4—124no—
416C2×C4—121no—
417C2×C4—122no—
418C2×C4C23×D314yes—
419C2×C4C23×D311yescharacteristic
420C2×C4C23×D312yes—
421C2×C4C23×D312yes—
422C2×C4C23×D312yes—
423C2×C4C23×D311yescharacteristic
424C2×C4C23×D311yescharacteristic
425C2×C3—42no—
426C2×C3C22×#4111yescharacteristic
427C2×C3C2×H(5,𝔽2)11yescharacteristic
428C2×C3#58511yescharacteristic
429D3—22no—
430D3—44no—
431D3—42no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—32no—
436C22—61no—
437C22—61no—
438C22—64no—
439C22—62no—
440C22—62no—
441C22—64no—
442C22—64no—
443C22—62no—
444C22—128no—
445C22C24×D311yescharacteristic
446C4—24no—
447C4—22no—
448C4—22no—
449C4—21no—
450C4—22no—
451C4—32no—
452C4—61no—
453C4—62no—
454C4—62no—
455C4—61no—
456C4—62no—
457C4D3×#4112yes—
458C4#100912yes—
459C3#350611yescharacteristic, complemented, Sylow
460C2—42no—
461C2—62no—
462C2—122no—
463C2—124no—
464C2C2×D3×#4111yescharacteristic
465C2C2×#100911yescharacteristic
466C2#596811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#556C2
hypercenter#556D3
hypocenterC3#3506
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #556

Complements

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

NH ≅ G/Ntype
1. C3×#556C2semidirect
2. #5732C2semidirect
3. #5725C2semidirect
4. #5709C2semidirect
5. #5704C2semidirect
6. #5681C2semidirect
7. #5674C2semidirect
8. #5631C2semidirect
9. #5627C2semidirect
10. #5585C2semidirect
11. #5578C2semidirect
12. #5577C2semidirect
13. #5574C2semidirect
14. #5553C2semidirect
15. C4×D3×D4C2semidirect
16. #5532C2semidirect
17. #5524C2semidirect
20. C2×#239C22semidirect
22. C2×#235C22semidirect
24. C3×C4⋊Q8C22semidirect
25. C3×#126C22semidirect
27. C3×C22⋊Q8C22semidirect
28. C3×C4×Q8C22semidirect
29. C3×C4×Q8C22semidirect
31. C3×#117C22semidirect
33. #947C22semidirect
35. Q8×Dic3C22semidirect
36. #944C22semidirect
39. #930C22semidirect
41. #898C22semidirect
43. #897C22semidirect
47. #895C22semidirect
48. #895C22semidirect
49. #894C22semidirect
50. #894C22semidirect
53. #893C22semidirect
54. #893C22semidirect
55. #893C22semidirect
56. #892C22semidirect
58. D3×C4⋊C4C22semidirect
59. #889C22semidirect
60. #888C22semidirect
61. #886C22semidirect
62. #885C22semidirect
65. #884C22semidirect
66. #883C22semidirect
70. #67⋊C4C22semidirect
72. #875C22semidirect
74. #874C22semidirect
77. C4×D12C22semidirect
78. C4×D12C22semidirect
80. #872C22semidirect
82. C4×Dic6C22semidirect
83. C4×Dic6C22semidirect
87. #556D3semidirect
178. C4⋊Q8C2×D3semidirect
183. #126C2×D3semidirect
198. C22⋊Q8C2×D3semidirect
212. C4×Q8C2×D3semidirect
213. C4×Q8C2×D3semidirect
231. #117C2×D3semidirect
279. C4×D3#41semidirect
459. C3#3506semidirect

Maximal subgroups

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

subgroupindex
C3×#5562
#57322
#57252
#57092
#57042
#56812
#56742
#56312
#56272
#55852
#55782
#55772
#55742
#55532
C4×D3×D42
#55322
#55242
#35063