Finite Group Databasebeta

← #86986

Subgroups — #86986

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×#556C211yesmaximal, characteristic, complemented
2#5727C211yesmaximal, characteristic, complemented
3#5725C211yesmaximal, characteristic, complemented
4#5705C214yesmaximal, complemented
5#5704C214yesmaximal, complemented
6#5676C212yesmaximal, complemented
7D3×#124C212yesmaximal, complemented
8#5643C212yesmaximal, complemented
9#5630C212yesmaximal, complemented
10#5585C211yesmaximal, characteristic, complemented
11#5580C212yesmaximal, complemented
12#5576C212yesmaximal, complemented
13C4×#239C211yesmaximal, characteristic, complemented
14#5557C211yesmaximal, characteristic, complemented
15D4×D12C211yesmaximal, characteristic, complemented
16#5532C212yesmaximal, complemented
17#5521C212yesmaximal, complemented
18#3514—31nomaximal, Sylow
19C2×#239C2211yescharacteristic
20C2×#239C2212yescomplemented
21C2×D3×Q8C2212yescomplemented
22C2×D3×D4C2212yes—
23C22×D12C2212yes—
24C3×C4⋊Q8C2211yescharacteristic, complemented
25C3×#126C2214yescomplemented
26C3×#124C2212yes—
27C3×C22⋊Q8C2212yescomplemented
28C3×C4×Q8C2211yescharacteristic, complemented
29C3×C4×Q8C2212yescomplemented
30C3×C4×D4C2211yescharacteristic
31C3×#117C2212yescomplemented
32#947C2212yescomplemented
33#947C2212yescomplemented
34#946C2212yescomplemented
35Q8×Dic3C2211yescharacteristic, complemented
36#938C2211yescharacteristic
37#936C2211yescharacteristic
38#930C2212yes—
39C4×#67C2212yescomplemented
40#926C2212yescomplemented
41#898C2214yescomplemented
42#897C2214yescomplemented
43#897C2211yescharacteristic, complemented
44#896C2214yescomplemented
45#895C2212yes—
46#895C2212yescomplemented
47#895C2214yescomplemented
48#895C2212yescomplemented
49#894C2214yescomplemented
50#894C2214yescomplemented
51#893C2212yes—
52#893C2211yescharacteristic
53#893C2212yescomplemented
54#893C2212yescomplemented
55#893C2212yescomplemented
56#893C2212yescomplemented
57#892C2212yescomplemented
58#892C2211yescharacteristic, complemented
59#890C2211yescharacteristic, complemented
60#888C2212yescomplemented
61#885C2214yescomplemented
62#885C2214yescomplemented
63#884C2214yescomplemented
64#883C2214yescomplemented
65#883C2212yescomplemented
66#882C2214yes—
67#882C2214yescomplemented
68D3×C22⋊C4C2214yescomplemented
69D3×C22⋊C4C2214yescomplemented
70D3×C22⋊C4C2214yescomplemented
71#879C2212yescomplemented
72#875C2212yescomplemented
73#875C2214yescomplemented
74#874C2211yescharacteristic
75C4×D12C2211yescharacteristic
76C4×D12C2212yescomplemented
77C4×D12C2212yescomplemented
78C4×D12C2214yescomplemented
79C4×D12C2214yescomplemented
80#872C2214yescomplemented
81C42×D3C2212yescomplemented
82C42×D3C2211yescharacteristic, complemented
83C4×Dic6C2211yescharacteristic, complemented
84#559—31no—
85#556D311yescharacteristic, complemented
86#555—32no—
87#551—34no—
88D4×Q8—31no—
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×Q8C2311yescharacteristic
103C2×C3×Q8C2312yes—
104C2×C3×D4C2311yescharacteristic
105C22×C3×C4C2312yes—
106C2×#67C2314yes—
107C2×#67C2312yes—
108#239—24no—
109#239—44no—
110D3×Q8—44no—
111D3×D4—44no—
112C2×D12—28no—
113C2×D12C2314yes—
114C2×D12C2312yes—
115C2×D12C2312yes—
116C2×D12C2312yes—
117C2×D12C2312yes—
118C2×D12C2314yes—
119C2×D12C2311yescharacteristic
120C2×C4×D3C2312yes—
121C2×C4×D3C2311yescharacteristic
122C2×C4×D3C2312yes—
123C2×C4×D3C2311yescharacteristic
124C2×C4×D3C2312yes—
125C2×C4×D3C2311yescharacteristic
126C2×C4×D3C2314yes—
127C2×C4×D3C2314yes—
128C2×C4×D3C2314yes—
129C2×Dic6C2311yescharacteristic
130C2×Dic6C2314yes—
131C3×C4⋊C4C2312yes—
132C3×C4⋊C4C2311yescharacteristic
133C3×C4⋊C4C2311yescharacteristic
134C3×C4⋊C4C2312yes—
135C3×C4⋊C4C2312yes—
136C3×C4⋊C4C2314yes—
137C3×C22⋊C4C2312yes—
138C3×C22⋊C4C2314yes—
139C3×C22⋊C4C2314yes—
140C3×C42C2311yescharacteristic
141C3×C42C2312yes—
142C3×C42C2314yes—
143#217C2312yes—
144#217C2312yes—
145#217C2312yes—
146(C2×D3)⋊C4C2314yes—
147(C2×D3)⋊C4C2314yes—
148(C2×D3)⋊C4C2312yes—
149(C2×D3)⋊C4C2312yes—
150(C2×D3)⋊C4C2314yes—
151(C2×D3)⋊C4C2312yes—
152(C2×D3)⋊C4C2314yes—
153(C2×D3)⋊C4C2314yes—
154(C2×D3)⋊C4C2314yes—
155(C2×D3)⋊C4C2312yes—
156(C2×D3)⋊C4C2314yes—
157C4⋊Dic3C2311yescharacteristic
158C4⋊Dic3C2311yescharacteristic
159C4⋊Dic3C2312yes—
160C4⋊Dic3C2312yes—
161C4⋊Dic3C2312yes—
162Dic3⋊C4C2312yes—
163Dic3⋊C4C2312yes—
164Dic3⋊C4C2314yes—
165C4×Dic3C2312yes—
166C4×Dic3C2311yescharacteristic
167C4×Dic3C2312yes—
168C4×Dic3C2311yescharacteristic
169C4×Dic3C2312yes—
170C4×Dic3C2311yescharacteristic
171C2×#41—31no—
172C2×#41—32no—
173C22×Q8—32no—
174C22×D4—32no—
175C22×D4—32no—
176C4⋊Q8—32no—
177C4⋊Q8C2×D311yescharacteristic, complemented
178#127—31no—
179#126—34no—
180#126—32no—
181#126C2×D314yescomplemented
182#125—31no—
183#124—32no—
184#124—32no—
185#124—31no—
186#124—34no—
187#124—34no—
188#124—32no—
189#124—34no—
190#124C2×D312yes—
191#123—34no—
192#123—34no—
193#123—32no—
194#123—34no—
195C22⋊Q8—32no—
196C22⋊Q8—31no—
197C22⋊Q8—34no—
198C22⋊Q8—34no—
199C22⋊Q8C2×D312yescomplemented
200#121—31no—
201#121—32no—
202#121—32no—
203#121—32no—
204#121—32no—
205#121—34no—
206#121—34no—
207#120—34no—
208#120—34no—
209C4×Q8—31no—
210C4×Q8—31no—
211C4×Q8C2×D311yescharacteristic, complemented
212C4×Q8C2×D312yescomplemented
213C4×D4—32no—
214C4×D4—31no—
215C4×D4—32no—
216C4×D4—32no—
217C4×D4—32no—
218C4×D4—32no—
219C4×D4—32no—
220C4×D4—31no—
221C4×D4—32no—
222C4×D4—34no—
223C4×D4—34no—
224C4×D4—32no—
225C4×D4C2×D311yescharacteristic
226#117—32no—
227#117—32no—
228#117—31no—
229#117—34no—
230#117C2×D312yescomplemented
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×D3C2414yes—
241C22×D3C2411yescharacteristic
242C22×D3C2412yes—
243C3×Q8—22no—
244C3×Q8—24no—
245C3×D4—22no—
246C2×C3×C4—24no—
247C2×C3×C4C2411yescharacteristic
248C2×C3×C4C2412yes—
249C2×C3×C4C2412yes—
250C2×C3×C4C2412yes—
251C2×C3×C4C2414yes—
252C2×C3×C4C2411yescharacteristic
253C2×C3×C4C2411yescharacteristic
254#67—44no—
255#67—42no—
256C2×Dic3C2411yescharacteristic
257C2×Dic3C2411yescharacteristic
258C2×Dic3C2412yes—
259C2×Dic3C2411yescharacteristic
260C2×Dic3C2412yes—
261C2×Dic3C2412yes—
262D12—28no—
263D12—24no—
264D12—24no—
265D12—24no—
266D12—42no—
267D12—44no—
268D12#4114yescomplemented
269C4×D3—24no—
270C4×D3—22no—
271C4×D3—24no—
272C4×D3—22no—
273C4×D3—42no—
274C4×D3—41no—
275C4×D3—44no—
276C4×D3—44no—
277C4×D3—44no—
278Dic6—22no—
279Dic6—44no—
280C24—34no—
281#41—64no—
282#41—124no—
283C2×Q8—31no—
284C2×Q8—34no—
285C2×Q8—124no—
286C2×Q8C22×D311yescharacteristic
287C2×Q8C22×D312yes—
288C2×D4—32no—
289C2×D4—34no—
290C2×D4—31no—
291C2×D4—32no—
292C2×D4—34no—
293C2×D4—32no—
294C2×D4—32no—
295C2×D4—32no—
296C2×D4—34no—
297C2×D4—68no—
298C2×D4—124no—
299C2×D4C22×D311yescharacteristic
300C22×C4—32no—
301C22×C4—31no—
302C22×C4—32no—
303C22×C4—31no—
304C22×C4—32no—
305C22×C4—31no—
306C22×C4—34no—
307C22×C4—34no—
308C22×C4—34no—
309C22×C4C22×D312yes—
310C4⋊C4—32no—
311C4⋊C4—32no—
312C4⋊C4—32no—
313C4⋊C4—31no—
314C4⋊C4—32no—
315C4⋊C4—34no—
316C4⋊C4—31no—
317C4⋊C4—32no—
318C4⋊C4C22×D311yescharacteristic
319C4⋊C4C22×D312yes—
320C4⋊C4C22×D312yes—
321C4⋊C4C22×D312yes—
322C4⋊C4C22×D311yescharacteristic
323C4⋊C4C22×D314yes—
324C22⋊C4—32no—
325C22⋊C4—32no—
326C22⋊C4—32no—
327C22⋊C4—32no—
328C22⋊C4—34no—
329C22⋊C4—34no—
330C22⋊C4—34no—
331C22⋊C4—34no—
332C22⋊C4—34no—
333C22⋊C4—32no—
334C22⋊C4—32no—
335C22⋊C4—34no—
336C22⋊C4—34no—
337C22⋊C4—32no—
338C22⋊C4C22×D314yes—
339C22⋊C4C22×D314yes—
340C22⋊C4C22×D312yes—
341C42—31no—
342C42—32no—
343C42—31no—
344C42—31no—
345C42—32no—
346C42—32no—
347C42C22×D312yes—
348C42C22×D311yescharacteristic
349C42C22×D314yes—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3C2511yescharacteristic
354C2×D3—22no—
355C2×D3—22no—
356C2×D3—21no—
357C2×D3—24no—
358C2×D3—21no—
359C2×D3—24no—
360C2×D3—24no—
361C2×D3—21no—
362C2×D3—88no—
363C2×D3C2×#4114yes—
364C3×C4—22no—
365C3×C4—22no—
366C3×C4—24no—
367C3×C4—22no—
368C3×C4—21no—
369C3×C4C2×#4112yes—
370C3×C4H(5,𝔽2)12yes—
371Dic3—21no—
372Dic3—22no—
373Dic3—22no—
374Dic3—21no—
375Dic3—21no—
376Dic3—22no—
377C23—32no—
378C23—34no—
379C23—31no—
380C23—68no—
381C23—124no—
382C23—124no—
383C23C23×D312yes—
384Q8—22no—
385Q8—24no—
386Q8—62no—
387Q8—124no—
388D4—22no—
389D4—34no—
390D4—68no—
391D4—64no—
392D4—64no—
393D4—64no—
394D4—124no—
395D4—122no—
396D4—124no—
397D4—122no—
398C2×C4—24no—
399C2×C4—31no—
400C2×C4—31no—
401C2×C4—31no—
402C2×C4—32no—
403C2×C4—32no—
404C2×C4—32no—
405C2×C4—64no—
406C2×C4—62no—
407C2×C4—64no—
408C2×C4—62no—
409C2×C4—124no—
410C2×C4—124no—
411C2×C4—122no—
412C2×C4—124no—
413C2×C4—121no—
414C2×C4C23×D312yes—
415C2×C4C23×D312yes—
416C2×C4C23×D311yescharacteristic
417C2×C4C23×D312yes—
418C2×C4C23×D314yes—
419C2×C4C23×D311yescharacteristic
420C2×C4C23×D311yescharacteristic
421C2×C3—42no—
422C2×C3C22×#4111yescharacteristic
423C2×C3C2×H(5,𝔽2)11yescharacteristic
424C2×C3#58511yescharacteristic
425D3—24no—
426D3—41no—
427D3—44no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—34no—
432C22—62no—
433C22—62no—
434C22—61no—
435C22—64no—
436C22—61no—
437C22—64no—
438C22—61no—
439C22—64no—
440C22—248no—
441C22C24×D311yescharacteristic
442C4—22no—
443C4—22no—
444C4—24no—
445C4—22no—
446C4—21no—
447C4—61no—
448C4—61no—
449C4—62no—
450C4—62no—
451C4—61no—
452C4—62no—
453C4D3×#4112yes—
454C4#100912yes—
455C3#351411yescharacteristic, complemented, Sylow
456C2—42no—
457C2—64no—
458C2—124no—
459C2—121no—
460C2C2×D3×#4111yescharacteristic
461C2C2×#100911yescharacteristic
462C2#596811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#556C2
hypercenter#556D3
hypocenterC3#3514
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. #5727C2semidirect
3. #5725C2semidirect
4. #5705C2semidirect
5. #5704C2semidirect
6. #5676C2semidirect
7. D3×#124C2semidirect
8. #5643C2semidirect
9. #5630C2semidirect
10. #5585C2semidirect
11. #5580C2semidirect
12. #5576C2semidirect
13. C4×#239C2semidirect
14. #5557C2semidirect
15. D4×D12C2semidirect
16. #5532C2semidirect
17. #5521C2semidirect
20. C2×#239C22semidirect
21. C2×D3×Q8C22semidirect
24. C3×C4⋊Q8C22semidirect
25. C3×#126C22semidirect
27. C3×C22⋊Q8C22semidirect
28. C3×C4×Q8C22semidirect
29. C3×C4×Q8C22semidirect
31. C3×#117C22semidirect
32. #947C22semidirect
33. #947C22semidirect
34. #946C22semidirect
35. Q8×Dic3C22semidirect
39. C4×#67C22semidirect
40. #926C22semidirect
41. #898C22semidirect
42. #897C22semidirect
43. #897C22semidirect
44. #896C22semidirect
46. #895C22semidirect
47. #895C22semidirect
48. #895C22semidirect
49. #894C22semidirect
50. #894C22semidirect
53. #893C22semidirect
54. #893C22semidirect
55. #893C22semidirect
56. #893C22semidirect
57. #892C22semidirect
58. #892C22semidirect
59. #890C22semidirect
60. #888C22semidirect
61. #885C22semidirect
62. #885C22semidirect
63. #884C22semidirect
64. #883C22semidirect
65. #883C22semidirect
67. #882C22semidirect
68. D3×C22⋊C4C22semidirect
69. D3×C22⋊C4C22semidirect
70. D3×C22⋊C4C22semidirect
71. #879C22semidirect
72. #875C22semidirect
73. #875C22semidirect
76. C4×D12C22semidirect
77. C4×D12C22semidirect
78. C4×D12C22semidirect
79. C4×D12C22semidirect
80. #872C22semidirect
81. C42×D3C22semidirect
82. C42×D3C22semidirect
83. C4×Dic6C22semidirect
85. #556D3semidirect
177. C4⋊Q8C2×D3semidirect
181. #126C2×D3semidirect
199. C22⋊Q8C2×D3semidirect
211. C4×Q8C2×D3semidirect
212. C4×Q8C2×D3semidirect
230. #117C2×D3semidirect
268. D12#41semidirect
455. C3#3514semidirect

Maximal subgroups

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

subgroupindex
C3×#5562
#57272
#57252
#57052
#57042
#56762
D3×#1242
#56432
#56302
#55852
#55802
#55762
C4×#2392
#55572
D4×D122
#55322
#55212
#35143