Finite Group Databasebeta

← #86785

Subgroups — #86785

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×#550C211yesmaximal, characteristic, complemented
2#5828C212yesmaximal, complemented
3#5824C212yesmaximal, complemented
4#5718C211yesmaximal, characteristic, complemented
5#5716C211yesmaximal, characteristic, complemented
6#5715C212yesmaximal, complemented
7#5713C212yesmaximal, complemented
8#5616C212yesmaximal, complemented
9#5613C214yesmaximal, complemented
10#5606C212yesmaximal, complemented
11#5601C214yesmaximal, complemented
12#5584C211yesmaximal, characteristic, complemented
13#5578C211yesmaximal, characteristic, complemented
14#5559C212yesmaximal, complemented
15#5546C212yesmaximal, complemented
16C4×D3×D4C211yesmaximal, characteristic, complemented
17#5542C211yesmaximal, characteristic, complemented
18#3506—31nomaximal, Sylow
19C2×C3×#41C2212yescomplemented
20C2×#237C2212yes—
21C2×#237C2212yescomplemented
22C2×D3×D4C2211yescharacteristic
23C22×C4×D3C2212yes—
24C3×#127C2212yes—
25C3×#127C2211yescharacteristic, complemented
26C3×#121C2214yescomplemented
27C3×#121C2212yescomplemented
28C3×C4×Q8C2211yescharacteristic, complemented
29C3×C4×D4C2211yescharacteristic
30C3×C4×D4C2212yescomplemented
31#946C2211yescharacteristic
32#946C2212yescomplemented
33#944C2211yescharacteristic, complemented
34#940C2212yes—
35#939C2214yes—
36#939C2214yes—
37#939C2214yescomplemented
38#938C2212yes—
39#938C2212yes—
40#938C2212yescomplemented
41#937C2214yes—
42#937C2214yes—
43#936C2212yes—
44#936C2212yescomplemented
45#936C2214yescomplemented
46#935C2214yescomplemented
47#935C2214yescomplemented
48D4×Dic3C2211yescharacteristic
49D4×Dic3C2212yescomplemented
50D4×Dic3C2214yescomplemented
51D4×Dic3C2212yescomplemented
52D4×Dic3C2212yescomplemented
53#929C2214yescomplemented
54C4×#67C2212yes—
55C4×#67C2214yes—
56C4×#67C2214yescomplemented
57#926C2212yescomplemented
58#924C2212yescomplemented
59C2×Dic3⋊C4C2214yescomplemented
60#898C2214yescomplemented
61#897C2211yescharacteristic
62#896C2212yes—
63#893C2211yescharacteristic
64#892C2212yescomplemented
65D3×C4⋊C4C2211yescharacteristic
66#889C2212yescomplemented
67#888C2211yescharacteristic, complemented
68#887C2212yescomplemented
69#885C2214yescomplemented
70#883C2214yescomplemented
71#67⋊C4C2212yes—
72#67⋊C4C2214yes—
73D3×C22⋊C4C2212yes—
74#879C2214yescomplemented
75#877C2214yescomplemented
76#875C2212yes—
77C4×D12C2211yescharacteristic
78#872C2212yes—
79C42×D3C2211yescharacteristic
80#869C2211yescharacteristic, complemented
81C4×Dic6C2212yescomplemented
82C4×Dic6C2212yescomplemented
83C4×Dic6C2211yescharacteristic, complemented
84#559—32no—
85#559—32no—
86#556—31no—
87#556—31no—
88#555—32no—
89#553—34no—
90#550—31no—
91#550D311yescharacteristic, complemented
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—
102C3×#41—44no—
103C2×C3×Q8C2311yescharacteristic
104C2×C3×D4C2311yescharacteristic
105C2×C3×D4C2312yes—
106C2×C3×D4C2312yes—
107C2×C3×D4C2314yes—
108C2×C3×D4C2312yes—
109C2×C3×D4C2314yes—
110C22×C3×C4C2312yes—
111C22×C3×C4C2314yes—
112C2×#67C2314yes—
113C2×#67C2312yes—
114C2×#67C2314yes—
115C2×#67C2314yes—
116C22×Dic3C2314yes—
117C22×Dic3C2314yes—
118C22×Dic3C2312yes—
119#237—28no—
120#237—44no—
121D3×D4—24no—
122C2×D12C2311yescharacteristic
123C2×C4×D3—44no—
124C2×C4×D3C2311yescharacteristic
125C2×C4×D3C2312yes—
126C2×C4×D3C2311yescharacteristic
127C2×C4×D3C2312yes—
128C2×C4×D3C2312yes—
129C2×C4×D3C2311yescharacteristic
130C2×Dic6C2312yes—
131C2×Dic6C2311yescharacteristic
132C2×Dic6C2312yes—
133C3×C4⋊C4C2311yescharacteristic
134C3×C4⋊C4C2311yescharacteristic
135C3×C4⋊C4C2312yes—
136C3×C22⋊C4C2314yes—
137C3×C22⋊C4C2312yes—
138C3×C42C2312yes—
139C3×C42C2311yescharacteristic
140#217C2314yes—
141#217C2314yes—
142#217C2314yes—
143#217C2312yes—
144#217C2314yes—
145#217C2314yes—
146(C2×D3)⋊C4C2312yes—
147(C2×D3)⋊C4C2312yes—
148(C2×D3)⋊C4C2312yes—
149(C2×D3)⋊C4C2314yes—
150(C2×D3)⋊C4C2314yes—
151(C2×D3)⋊C4C2312yes—
152C4⋊Dic3C2311yescharacteristic
153C4⋊Dic3C2312yes—
154C4⋊Dic3C2311yescharacteristic
155C4⋊Dic3C2312yes—
156C4⋊Dic3C2312yes—
157Dic3⋊C4C2312yes—
158Dic3⋊C4C2312yes—
159Dic3⋊C4C2312yes—
160Dic3⋊C4C2314yes—
161Dic3⋊C4C2312yes—
162Dic3⋊C4C2312yes—
163Dic3⋊C4C2312yes—
164Dic3⋊C4C2314yes—
165C4×Dic3C2311yescharacteristic
166C4×Dic3C2312yes—
167C4×Dic3C2314yes—
168C4×Dic3C2311yescharacteristic
169C4×Dic3C2311yescharacteristic
170C4×Dic3C2314yes—
171C2×#41—32no—
172C2×#41—32no—
173C2×#41C2×D312yescomplemented
174C22×D4—31no—
175C23×C4—32no—
176C4⋊Q8—31no—
177C4⋊Q8—31no—
178C4⋊Q8—31no—
179#127—32no—
180#127C2×D312yes—
181#127C2×D311yescharacteristic, complemented
182#126—34no—
183#126—34no—
184#125—32no—
185#124—32no—
186#124—32no—
187#124—34no—
188#124—34no—
189#124—32no—
190#123—34no—
191#123—34no—
192#123—34no—
193#123—34no—
194C22⋊Q8—31no—
195C22⋊Q8—32no—
196C22⋊Q8—32no—
197C22⋊Q8—31no—
198C22⋊Q8—32no—
199#121—32no—
200#121—32no—
201#121—32no—
202#121—34no—
203#121—34no—
204#121—34no—
205#121C2×D312yescomplemented
206#121C2×D314yescomplemented
207#120—34no—
208#120—34no—
209C4×Q8—32no—
210C4×Q8—32no—
211C4×Q8—31no—
212C4×Q8—32no—
213C4×Q8C2×D311yescharacteristic, complemented
214C4×D4—31no—
215C4×D4—32no—
216C4×D4—34no—
217C4×D4—32no—
218C4×D4—32no—
219C4×D4—32no—
220C4×D4—34no—
221C4×D4—34no—
222C4×D4—31no—
223C4×D4—32no—
224C4×D4—34no—
225C4×D4—31no—
226C4×D4C2×D311yescharacteristic
227C4×D4C2×D312yescomplemented
228#117—32no—
229#117—32no—
230#117—34no—
231#117—32no—
232C2×C4⋊C4—34no—
233C2×C4⋊C4—31no—
234C2×C22⋊C4—32no—
235C2×C42—31no—
236C23×C3C2412yes—
237C23×C3C2414yes—
238C22×D3—24no—
239C22×D3—42no—
240C22×D3—42no—
241C22×D3C2412yes—
242C22×D3C2411yescharacteristic
243C3×Q8—22no—
244C3×D4—22no—
245C3×D4—24no—
246C3×D4—24no—
247C3×D4—44no—
248C3×D4—42no—
249C3×D4—44no—
250C2×C3×C4—24no—
251C2×C3×C4—44no—
252C2×C3×C4C2412yes—
253C2×C3×C4C2412yes—
254C2×C3×C4C2411yescharacteristic
255C2×C3×C4C2412yes—
256C2×C3×C4C2411yescharacteristic
257C2×C3×C4C2411yescharacteristic
258#67—28no—
259#67—24no—
260#67—44no—
261#67—44no—
262C2×Dic3—28no—
263C2×Dic3—42no—
264C2×Dic3—44no—
265C2×Dic3C2412yes—
266C2×Dic3C2412yes—
267C2×Dic3C2414yes—
268C2×Dic3C2411yescharacteristic
269C2×Dic3C2411yescharacteristic
270C2×Dic3C2412yes—
271C2×Dic3C2411yescharacteristic
272D12—22no—
273C4×D3—22no—
274C4×D3—24no—
275C4×D3—42no—
276C4×D3—42no—
277C4×D3—41no—
278C4×D3#4114yescomplemented
279Dic6—24no—
280Dic6—22no—
281Dic6—42no—
282C24—32no—
283#41—44no—
284#41—68no—
285#41—124no—
286C2×Q8—32no—
287C2×Q8—31no—
288C2×Q8—32no—
289C2×Q8C22×D311yescharacteristic
290C2×D4—34no—
291C2×D4—32no—
292C2×D4—34no—
293C2×D4—34no—
294C2×D4—31no—
295C2×D4—64no—
296C2×D4C22×D311yescharacteristic
297C2×D4C22×D312yes—
298C2×D4C22×D312yes—
299C2×D4C22×D314yes—
300C2×D4C22×D312yes—
301C2×D4C22×D314yes—
302C22×C4—31no—
303C22×C4—34no—
304C22×C4—34no—
305C22×C4—32no—
306C22×C4—31no—
307C22×C4—32no—
308C22×C4—31no—
309C22×C4—32no—
310C22×C4—32no—
311C22×C4—124no—
312C22×C4C22×D312yes—
313C22×C4C22×D314yes—
314C4⋊C4—32no—
315C4⋊C4—32no—
316C4⋊C4—31no—
317C4⋊C4—34no—
318C4⋊C4—34no—
319C4⋊C4—32no—
320C4⋊C4—32no—
321C4⋊C4—31no—
322C4⋊C4—32no—
323C4⋊C4—32no—
324C4⋊C4—32no—
325C4⋊C4—32no—
326C4⋊C4—32no—
327C4⋊C4C22×D311yescharacteristic
328C4⋊C4C22×D311yescharacteristic
329C4⋊C4C22×D312yes—
330C22⋊C4—32no—
331C22⋊C4—34no—
332C22⋊C4—34no—
333C22⋊C4—34no—
334C22⋊C4—34no—
335C22⋊C4—34no—
336C22⋊C4—32no—
337C22⋊C4—32no—
338C22⋊C4—32no—
339C22⋊C4—32no—
340C22⋊C4—34no—
341C22⋊C4—34no—
342C22⋊C4C22×D312yes—
343C22⋊C4C22×D314yes—
344C42—31no—
345C42—31no—
346C42—31no—
347C42—34no—
348C42—34no—
349C42—32no—
350C42C22×D311yescharacteristic
351C42C22×D312yes—
352C22×C3—24no—
353C22×C3—22no—
354C22×C3—24no—
355C22×C3—22no—
356C22×C3—24no—
357C22×C3—22no—
358C22×C3C2511yescharacteristic
359C2×D3—21no—
360C2×D3—21no—
361C2×D3—22no—
362C2×D3—22no—
363C2×D3—22no—
364C2×D3—48no—
365C2×D3C2×#4112yes—
366C3×C4—22no—
367C3×C4—22no—
368C3×C4—22no—
369C3×C4—21no—
370C3×C4C2×#4112yes—
371C3×C4H(5,𝔽2)12yes—
372Dic3—24no—
373Dic3—22no—
374Dic3—21no—
375Dic3—22no—
376Dic3—21no—
377Dic3—22no—
378Dic3C2×#4112yes—
379C23—31no—
380C23—32no—
381C23—64no—
382C23—122no—
383C23—122no—
384C23C23×D312yes—
385C23C23×D314yes—
386Q8—22no—
387Q8—62no—
388Q8—64no—
389Q8—122no—
390D4—24no—
391D4—22no—
392D4—24no—
393D4—44no—
394D4—42no—
395D4—44no—
396D4—64no—
397D4—68no—
398D4—62no—
399D4—124no—
400D4—124no—
401C2×C4—24no—
402C2×C4—31no—
403C2×C4—34no—
404C2×C4—32no—
405C2×C4—31no—
406C2×C4—34no—
407C2×C4—31no—
408C2×C4—32no—
409C2×C4—32no—
410C2×C4—44no—
411C2×C4—68no—
412C2×C4—62no—
413C2×C4—64no—
414C2×C4—122no—
415C2×C4—122no—
416C2×C4—121no—
417C2×C4—122no—
418C2×C4—124no—
419C2×C4C23×D312yes—
420C2×C4C23×D312yes—
421C2×C4C23×D311yescharacteristic
422C2×C4C23×D312yes—
423C2×C4C23×D311yescharacteristic
424C2×C4C23×D311yescharacteristic
425C2×C3—42no—
426C2×C3—44no—
427C2×C3C22×#4111yescharacteristic
428C2×C3C2×H(5,𝔽2)11yescharacteristic
429C2×C3#58511yescharacteristic
430D3—22no—
431D3—42no—
432C22—22no—
433C22—24no—
434C22—24no—
435C22—22no—
436C22—24no—
437C22—22no—
438C22—32no—
439C22—61no—
440C22—61no—
441C22—62no—
442C22—62no—
443C22—62no—
444C22—128no—
445C22C24×D311yescharacteristic
446C4—22no—
447C4—22no—
448C4—21no—
449C4—22no—
450C4—32no—
451C4—61no—
452C4—62no—
453C4—62no—
454C4—64no—
455C4—61no—
456C4—62no—
457C4#100412yes—
458C4D3×#4112yes—
459C3#350611yescharacteristic, complemented, Sylow
460C2—42no—
461C2—44no—
462C2—62no—
463C2—122no—
464C2C2×#100411yescharacteristic
465C2C2×D3×#4111yescharacteristic
466C2#596611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#550C2
hypercenter#550D3
hypocenterC3#3506
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C3×#550C2semidirect
2. #5828C2semidirect
3. #5824C2semidirect
4. #5718C2semidirect
5. #5716C2semidirect
6. #5715C2semidirect
7. #5713C2semidirect
8. #5616C2semidirect
9. #5613C2semidirect
10. #5606C2semidirect
11. #5601C2semidirect
12. #5584C2semidirect
13. #5578C2semidirect
14. #5559C2semidirect
15. #5546C2semidirect
16. C4×D3×D4C2semidirect
17. #5542C2semidirect
19. C2×C3×#41C22semidirect
21. C2×#237C22semidirect
25. C3×#127C22semidirect
26. C3×#121C22semidirect
27. C3×#121C22semidirect
28. C3×C4×Q8C22semidirect
30. C3×C4×D4C22semidirect
32. #946C22semidirect
33. #944C22semidirect
37. #939C22semidirect
40. #938C22semidirect
44. #936C22semidirect
45. #936C22semidirect
46. #935C22semidirect
47. #935C22semidirect
49. D4×Dic3C22semidirect
50. D4×Dic3C22semidirect
51. D4×Dic3C22semidirect
52. D4×Dic3C22semidirect
53. #929C22semidirect
56. C4×#67C22semidirect
57. #926C22semidirect
58. #924C22semidirect
59. C2×Dic3⋊C4C22semidirect
60. #898C22semidirect
64. #892C22semidirect
66. #889C22semidirect
67. #888C22semidirect
68. #887C22semidirect
69. #885C22semidirect
70. #883C22semidirect
74. #879C22semidirect
75. #877C22semidirect
80. #869C22semidirect
81. C4×Dic6C22semidirect
82. C4×Dic6C22semidirect
83. C4×Dic6C22semidirect
91. #550D3semidirect
173. C2×#41C2×D3semidirect
181. #127C2×D3semidirect
205. #121C2×D3semidirect
206. #121C2×D3semidirect
213. C4×Q8C2×D3semidirect
227. C4×D4C2×D3semidirect
278. 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×#5502
#58282
#58242
#57182
#57162
#57152
#57132
#56162
#56132
#56062
#56012
#55842
#55782
#55592
#55462
C4×D3×D42
#55422
#35063