Finite Group Databasebeta

← #86787

Subgroups — #86787

Order 384.

1260 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#550C211yesmaximal, characteristic, complemented
2#5830C212yesmaximal, complemented
3#5829C212yesmaximal, complemented
4#5718C211yesmaximal, characteristic, complemented
5#5717C212yesmaximal, complemented
6#5715C212yesmaximal, complemented
7D3×#127C211yesmaximal, characteristic, complemented
8#5620C214yesmaximal, complemented
9#5619C212yesmaximal, complemented
10#5609C212yesmaximal, complemented
11#5605C214yesmaximal, complemented
12#5585C211yesmaximal, characteristic, complemented
13#5580C211yesmaximal, characteristic, complemented
14#5562C212yesmaximal, complemented
15#5557C211yesmaximal, characteristic, complemented
16#5554C212yesmaximal, complemented
17D4×D12C211yesmaximal, characteristic, complemented
18#3543—31nomaximal, Sylow
19C2×C3×#41C2212yescomplemented
20C2×#237C2212yescomplemented
21C2×D3×D4C2212yescomplemented
22C2×D3×D4C2212yescomplemented
23C2×D3×D4C2212yescomplemented
24C2×D3×D4C2212yescomplemented
25C2×D3×D4C2214yescomplemented
26C22×D12C2212yescomplemented
27C3×#127C2211yescharacteristic, complemented
28C3×#127C2212yescomplemented
29C3×#121C2214yescomplemented
30C3×#121C2212yescomplemented
31C3×C4×Q8C2211yescharacteristic, complemented
32C3×C4×D4C2211yescharacteristic, complemented
33C3×C4×D4C2212yescomplemented
34#947C2212yescomplemented
35#947C2211yescharacteristic, complemented
36#946C2211yescharacteristic, complemented
37#940C2212yescomplemented
38#940C2212yescomplemented
39#940C2212yescomplemented
40#940C2212yescomplemented
41#939C2214yescomplemented
42#939C2214yescomplemented
43#939C2214yescomplemented
44#938C2211yescharacteristic
45#938C2212yescomplemented
46#938C2214yescomplemented
47#937C2214yescomplemented
48#937C2214yescomplemented
49#937C2214yescomplemented
50#937C2214yescomplemented
51#936C2211yescharacteristic, complemented
52D4×Dic3C2212yescomplemented
53#930C2212yescomplemented
54#930C2214yescomplemented
55#930C2214yescomplemented
56#929C2214yescomplemented
57C4×#67C2212yescomplemented
58C2×(C2×D3)⋊C4C2214yescomplemented
59#926C2212yescomplemented
60#898C2212yescomplemented
61#897C2211yescharacteristic, complemented
62#895C2211yescharacteristic
63#895C2212yescomplemented
64#894C2212yescomplemented
65#894C2214yescomplemented
66#893C2212yescomplemented
67#890C2211yescharacteristic, complemented
68#890C2211yescharacteristic, complemented
69#886C2214yescomplemented
70#884C2214yescomplemented
71#883C2212yescomplemented
72#883C2214yescomplemented
73#882C2214yescomplemented
74#882C2214yescomplemented
75#879C2212yescomplemented
76#875C2212yescomplemented
77#874C2211yescharacteristic
78C4×D12C2212yescomplemented
79C4×D12C2212yescomplemented
80C4×D12C2211yescharacteristic, complemented
81#872C2212yescomplemented
82C42×D3C2211yescharacteristic, complemented
83C4×Dic6C2211yescharacteristic, complemented
84#560—34no—
85#559—31no—
86#550—31no—
87#550D311yescharacteristic, complemented
88#546—32no—
89#546—32no—
90D42—32no—
91D42—32no—
92D42—31no—
93#540—32no—
94#540—32no—
95#535—32no—
96#535—32no—
97#534—34no—
98C2×#127—31no—
99#529—31no—
100#529—31no—
101C23×D3C2314yes—
102C23×D3C2314yes—
103C3×#41—44no—
104C2×C3×Q8C2311yescharacteristic
105C2×C3×D4C2312yes—
106C2×C3×D4C2311yescharacteristic
107C2×C3×D4C2312yes—
108C2×C3×D4C2314yes—
109C2×C3×D4C2314yes—
110C2×C3×D4C2312yes—
111C22×C3×C4C2312yes—
112C22×C3×C4C2314yes—
113C2×#67C2312yes—
114C2×#67C2314yes—
115C2×#67C2314yes—
116C2×#67C2314yes—
117C2×#67C2314yes—
118C2×#67C2314yes—
119C2×#67C2314yes—
120C22×Dic3C2314yes—
121#237—44no—
122D3×D4—44no—
123D3×D4—44no—
124D3×D4—44no—
125D3×D4—44no—
126D3×D4—48no—
127C2×D12—44no—
128C2×D12C2312yes—
129C2×D12C2312yes—
130C2×D12C2311yescharacteristic
131C2×D12C2312yes—
132C2×D12C2312yes—
133C2×D12C2314yes—
134C2×C4×D3C2311yescharacteristic
135C2×C4×D3C2311yescharacteristic
136C2×C4×D3C2311yescharacteristic
137C2×C4×D3C2312yes—
138C2×C4×D3C2314yes—
139C2×Dic6C2311yescharacteristic
140C3×C4⋊C4C2311yescharacteristic
141C3×C4⋊C4C2311yescharacteristic
142C3×C4⋊C4C2312yes—
143C3×C22⋊C4C2312yes—
144C3×C22⋊C4C2314yes—
145C3×C42C2311yescharacteristic
146C3×C42C2312yes—
147#217C2312yes—
148#217C2312yes—
149#217C2312yes—
150#217C2314yes—
151(C2×D3)⋊C4C2312yes—
152(C2×D3)⋊C4C2312yes—
153(C2×D3)⋊C4C2312yes—
154(C2×D3)⋊C4C2312yes—
155(C2×D3)⋊C4C2314yes—
156(C2×D3)⋊C4C2312yes—
157(C2×D3)⋊C4C2314yes—
158(C2×D3)⋊C4C2314yes—
159(C2×D3)⋊C4C2312yes—
160(C2×D3)⋊C4C2314yes—
161C4⋊Dic3C2311yescharacteristic
162C4⋊Dic3C2312yes—
163C4⋊Dic3C2311yescharacteristic
164C4⋊Dic3C2312yes—
165C4⋊Dic3C2312yes—
166Dic3⋊C4C2312yes—
167Dic3⋊C4C2312yes—
168Dic3⋊C4C2312yes—
169Dic3⋊C4C2312yes—
170C4×Dic3C2312yes—
171C4×Dic3C2311yescharacteristic
172C4×Dic3C2311yescharacteristic
173C4×Dic3C2311yescharacteristic
174C2×#41—32no—
175C2×#41C2×D312yescomplemented
176C22×D4—32no—
177C22×D4—32no—
178C22×D4—32no—
179C22×D4—34no—
180C22×D4—32no—
181C22×D4—32no—
182#127—32no—
183#127—32no—
184#127—32no—
185#127—32no—
186#127—31no—
187#127C2×D311yescharacteristic, complemented
188#127C2×D312yescomplemented
189#126—32no—
190#126—32no—
191#125—31no—
192#125—31no—
193#124—31no—
194#124—32no—
195#124—31no—
196#124—32no—
197#123—34no—
198#123—32no—
199#123—34no—
200#123—32no—
201#123—34no—
202#123—34no—
203C22⋊Q8—31no—
204C22⋊Q8—31no—
205#121—31no—
206#121—34no—
207#121—34no—
208#121—34no—
209#121—32no—
210#121—34no—
211#121—34no—
212#121—34no—
213#121—32no—
214#121—31no—
215#121—32no—
216#121—34no—
217#121C2×D314yescomplemented
218#121C2×D312yescomplemented
219#120—34no—
220#120—34no—
221#120—34no—
222#120—34no—
223#120—34no—
224#120—34no—
225C4×Q8—31no—
226C4×Q8C2×D311yescharacteristic, complemented
227C4×D4—32no—
228C4×D4—32no—
229C4×D4—32no—
230C4×D4—32no—
231C4×D4—31no—
232C4×D4—32no—
233C4×D4C2×D311yescharacteristic, complemented
234C4×D4C2×D312yescomplemented
235#117—32no—
236#117—32no—
237C2×C22⋊C4—34no—
238C2×C42—31no—
239C23×C3C2412yes—
240C23×C3C2414yes—
241C22×D3—28no—
242C22×D3—44no—
243C22×D3—44no—
244C22×D3—44no—
245C22×D3—44no—
246C22×D3—44no—
247C22×D3C2412yes—
248C22×D3C2414yes—
249C22×D3C2411yescharacteristic
250C3×Q8—22no—
251C3×D4—22no—
252C3×D4—24no—
253C3×D4—24no—
254C3×D4—42no—
255C3×D4—44no—
256C3×D4—44no—
257C2×C3×C4—24no—
258C2×C3×C4—44no—
259C2×C3×C4C2411yescharacteristic
260C2×C3×C4C2412yes—
261C2×C3×C4C2412yes—
262C2×C3×C4C2412yes—
263C2×C3×C4C2411yescharacteristic
264C2×C3×C4C2411yescharacteristic
265#67—24no—
266#67—44no—
267#67—44no—
268#67—44no—
269#67—44no—
270#67—44no—
271#67—44no—
272C2×Dic3—44no—
273C2×Dic3C2411yescharacteristic
274C2×Dic3C2411yescharacteristic
275C2×Dic3C2411yescharacteristic
276C2×Dic3C2412yes—
277C2×Dic3C2412yes—
278C2×Dic3C2412yes—
279D12—24no—
280D12—22no—
281D12—24no—
282D12—24no—
283D12—44no—
284D12—42no—
285C4×D3—22no—
286C4×D3—22no—
287C4×D3—22no—
288C4×D3—24no—
289C4×D3—44no—
290Dic6—22no—
291C24—34no—
292C24—34no—
293#41—44no—
294#41—124no—
295C2×Q8—31no—
296C2×Q8C22×D311yescharacteristic
297C2×D4—32no—
298C2×D4—34no—
299C2×D4—34no—
300C2×D4—34no—
301C2×D4—34no—
302C2×D4—34no—
303C2×D4—34no—
304C2×D4—32no—
305C2×D4—32no—
306C2×D4—31no—
307C2×D4—32no—
308C2×D4—34no—
309C2×D4—32no—
310C2×D4—124no—
311C2×D4—124no—
312C2×D4—124no—
313C2×D4—124no—
314C2×D4—124no—
315C2×D4—128no—
316C2×D4C22×D312yes—
317C2×D4C22×D311yescharacteristic
318C2×D4C22×D312yes—
319C2×D4C22×D314yes—
320C2×D4C22×D314yes—
321C2×D4C22×D312yes—
322C22×C4—34no—
323C22×C4—31no—
324C22×C4—31no—
325C22×C4—31no—
326C22×C4—32no—
327C22×C4—34no—
328C22×C4C22×D312yes—
329C22×C4C22×D314yes—
330C4⋊C4—32no—
331C4⋊C4—32no—
332C4⋊C4—32no—
333C4⋊C4—32no—
334C4⋊C4—31no—
335C4⋊C4—31no—
336C4⋊C4—32no—
337C4⋊C4—32no—
338C4⋊C4—32no—
339C4⋊C4C22×D311yescharacteristic
340C4⋊C4C22×D311yescharacteristic
341C4⋊C4C22×D312yes—
342C22⋊C4—32no—
343C22⋊C4—32no—
344C22⋊C4—32no—
345C22⋊C4—34no—
346C22⋊C4—32no—
347C22⋊C4—32no—
348C22⋊C4—34no—
349C22⋊C4—34no—
350C22⋊C4—32no—
351C22⋊C4—32no—
352C22⋊C4—32no—
353C22⋊C4—32no—
354C22⋊C4—34no—
355C22⋊C4—34no—
356C22⋊C4C22×D312yes—
357C22⋊C4C22×D314yes—
358C42—31no—
359C42—31no—
360C42—31no—
361C42—32no—
362C42C22×D311yescharacteristic
363C42C22×D312yes—
364C22×C3—24no—
365C22×C3—22no—
366C22×C3—22no—
367C22×C3—24no—
368C22×C3—24no—
369C22×C3—22no—
370C22×C3C2511yescharacteristic
371C2×D3—21no—
372C2×D3—21no—
373C2×D3—22no—
374C2×D3—24no—
375C2×D3—22no—
376C2×D3—24no—
377C2×D3—24no—
378C2×D3—22no—
379C2×D3—88no—
380C2×D3—88no—
381C2×D3H(5,𝔽2)12yes—
382C3×C4—21no—
383C3×C4—22no—
384C3×C4—22no—
385C3×C4—22no—
386C3×C4H(5,𝔽2)12yes—
387C3×C4H(5,𝔽2)12yes—
388Dic3—22no—
389Dic3—22no—
390Dic3—21no—
391Dic3—21no—
392Dic3—21no—
393Dic3—22no—
394C23—31no—
395C23—32no—
396C23—34no—
397C23—68no—
398C23—124no—
399C23—124no—
400C23—124no—
401C23—124no—
402C23—124no—
403C23C23×D312yes—
404C23C23×D314yes—
405Q8—22no—
406Q8—62no—
407D4—24no—
408D4—22no—
409D4—24no—
410D4—42no—
411D4—44no—
412D4—44no—
413D4—64no—
414D4—64no—
415D4—62no—
416D4—64no—
417D4—64no—
418D4—124no—
419D4—122no—
420D4—124no—
421D4—124no—
422D4—124no—
423D4—124no—
424D4—124no—
425D4—124no—
426C2×C4—24no—
427C2×C4—31no—
428C2×C4—31no—
429C2×C4—31no—
430C2×C4—32no—
431C2×C4—32no—
432C2×C4—32no—
433C2×C4—44no—
434C2×C4—62no—
435C2×C4—62no—
436C2×C4—62no—
437C2×C4—64no—
438C2×C4—124no—
439C2×C4—124no—
440C2×C4C23×D311yescharacteristic
441C2×C4C23×D312yes—
442C2×C4C23×D312yes—
443C2×C4C23×D312yes—
444C2×C4C23×D311yescharacteristic
445C2×C4C23×D311yescharacteristic
446C2×C3—42no—
447C2×C3—44no—
448C2×C3C2×H(5,𝔽2)11yescharacteristic
449C2×C3C2×H(5,𝔽2)11yescharacteristic
450C2×C3C2×H(5,𝔽2)11yescharacteristic
451D3—22no—
452D3—42no—
453D3—44no—
454C22—22no—
455C22—24no—
456C22—24no—
457C22—22no—
458C22—22no—
459C22—24no—
460C22—32no—
461C22—61no—
462C22—61no—
463C22—62no—
464C22—64no—
465C22—62no—
466C22—64no—
467C22—64no—
468C22—62no—
469C22—248no—
470C22—248no—
471C22C24×D311yescharacteristic
472C4—22no—
473C4—22no—
474C4—21no—
475C4—22no—
476C4—61no—
477C4—62no—
478C4—61no—
479C4—62no—
480C4—61no—
481C4—62no—
482C4#100412yes—
483C4#100912yes—
484C3#354311yescharacteristic, complemented, Sylow
485C2—42no—
486C2—44no—
487C2—62no—
488C2—122no—
489C2—124no—
490C2C2×#100411yescharacteristic
491C2C2×#100911yescharacteristic
492C2D3×H(5,𝔽2)11yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#550C2
hypercenter#550D3
hypocenterC3#3543
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. #5830C2semidirect
3. #5829C2semidirect
4. #5718C2semidirect
5. #5717C2semidirect
6. #5715C2semidirect
7. D3×#127C2semidirect
8. #5620C2semidirect
9. #5619C2semidirect
10. #5609C2semidirect
11. #5605C2semidirect
12. #5585C2semidirect
13. #5580C2semidirect
14. #5562C2semidirect
15. #5557C2semidirect
16. #5554C2semidirect
17. D4×D12C2semidirect
19. C2×C3×#41C22semidirect
20. C2×#237C22semidirect
21. C2×D3×D4C22semidirect
22. C2×D3×D4C22semidirect
23. C2×D3×D4C22semidirect
24. C2×D3×D4C22semidirect
25. C2×D3×D4C22semidirect
26. C22×D12C22semidirect
27. C3×#127C22semidirect
28. C3×#127C22semidirect
29. C3×#121C22semidirect
30. C3×#121C22semidirect
31. C3×C4×Q8C22semidirect
32. C3×C4×D4C22semidirect
33. C3×C4×D4C22semidirect
34. #947C22semidirect
35. #947C22semidirect
36. #946C22semidirect
37. #940C22semidirect
38. #940C22semidirect
39. #940C22semidirect
40. #940C22semidirect
41. #939C22semidirect
42. #939C22semidirect
43. #939C22semidirect
45. #938C22semidirect
46. #938C22semidirect
47. #937C22semidirect
48. #937C22semidirect
49. #937C22semidirect
50. #937C22semidirect
51. #936C22semidirect
52. D4×Dic3C22semidirect
53. #930C22semidirect
54. #930C22semidirect
55. #930C22semidirect
56. #929C22semidirect
57. C4×#67C22semidirect
58. C2×(C2×D3)⋊C4C22semidirect
59. #926C22semidirect
60. #898C22semidirect
61. #897C22semidirect
63. #895C22semidirect
64. #894C22semidirect
65. #894C22semidirect
66. #893C22semidirect
67. #890C22semidirect
68. #890C22semidirect
69. #886C22semidirect
70. #884C22semidirect
71. #883C22semidirect
72. #883C22semidirect
73. #882C22semidirect
74. #882C22semidirect
75. #879C22semidirect
76. #875C22semidirect
78. C4×D12C22semidirect
79. C4×D12C22semidirect
80. C4×D12C22semidirect
81. #872C22semidirect
82. C42×D3C22semidirect
83. C4×Dic6C22semidirect
87. #550D3semidirect
175. C2×#41C2×D3semidirect
187. #127C2×D3semidirect
188. #127C2×D3semidirect
217. #121C2×D3semidirect
218. #121C2×D3semidirect
226. C4×Q8C2×D3semidirect
233. C4×D4C2×D3semidirect
234. C4×D4C2×D3semidirect
484. C3#3543semidirect

Maximal subgroups

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

subgroupindex
C3×#5502
#58302
#58292
#57182
#57172
#57152
D3×#1272
#56202
#56192
#56092
#56052
#55852
#55802
#55622
#55572
#55542
D4×D122
#35433