Finite Group Databasebeta

← #86249

Subgroups — #86249

Order 384.

1150 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#532C211yesmaximal, characteristic, complemented
2#5833C212yesmaximal, complemented
3#5830C212yesmaximal, complemented
4#5733C212yesmaximal, complemented
5#5715C212yesmaximal, complemented
6#5678C214yesmaximal, complemented
7#5620C214yesmaximal, complemented
8#5612C212yesmaximal, complemented
9#5599C212yesmaximal, complemented
10#5565C212yesmaximal, complemented
11#5555C211yesmaximal, characteristic, complemented
12#5553C211yesmaximal, characteristic, complemented
13#5550C212yesmaximal, complemented
14D4×D12C211yesmaximal, characteristic, complemented
15#5542C211yesmaximal, characteristic, complemented
16#5479C211yesmaximal, characteristic, complemented
17#5477C211yesmaximal, characteristic, complemented
18#3546—31nomaximal, Sylow
19C2×C3×#41C2212yescomplemented
20C2×#237C2212yescomplemented
21C2×D3×D4C2212yes—
22C2×#235C2212yescomplemented
23C2×#235C2212yescomplemented
24C22×D12C2212yes—
25C3×C4⋊Q8C2211yescharacteristic, complemented
26C3×#127C2211yescharacteristic
27C3×#124C2212yescomplemented
28C3×#121C2211yescharacteristic
29C3×#121C2211yescharacteristic, complemented
30C3×#121C2212yescomplemented
31C3×C4×D4C2211yescharacteristic
32C3×C4×D4C2212yescomplemented
33C3×C4×D4C2211yescharacteristic, complemented
34C2×C3×C42C2211yescharacteristic, complemented
35#947C2214yescomplemented
36#946C2214yescomplemented
37#940C2212yescomplemented
38#940C2214yescomplemented
39#939C2214yescomplemented
40#939C2214yescomplemented
41#939C2214yescomplemented
42#938C2214yescomplemented
43#937C2214yescomplemented
44#937C2214yescomplemented
45#937C2214yescomplemented
46#936C2212yes—
47#930C2211yescharacteristic
48#930C2212yes—
49#930C2212yes—
50#930C2212yescomplemented
51#930C2212yescomplemented
52#930C2212yescomplemented
53#929C2212yescomplemented
54C4×#67C2212yescomplemented
55#926C2212yescomplemented
56C2×C4⋊Dic3C2212yescomplemented
57#924C2211yescharacteristic, complemented
58#924C2212yescomplemented
59#898C2214yescomplemented
60#897C2212yescomplemented
61#895C2212yes—
62#894C2214yescomplemented
63#893C2212yescomplemented
64#887C2212yescomplemented
65#886C2214yescomplemented
66#884C2214yescomplemented
67#884C2214yescomplemented
68#883C2214yescomplemented
69#882C2214yes—
70#67⋊C4C2214yescomplemented
71D3×C22⋊C4C2214yescomplemented
72#879C2214yescomplemented
73#876C2212yescomplemented
74#875C2211yescharacteristic
75#875C2212yescomplemented
76#874C2211yescharacteristic
77C4×D12C2211yescharacteristic
78C4×D12C2212yescomplemented
79C4×D12C2212yescomplemented
80#872C2212yescomplemented
81#870C2211yescharacteristic, complemented
82#869C2211yescharacteristic, complemented
83C4×Dic6C2211yescharacteristic, complemented
84#560—34no—
85#556—31no—
86#553—32no—
87#550—32no—
88#550—31no—
89#547—31no—
90#546—32no—
91#546—32no—
92D42—31no—
93#542—32no—
94#538—34no—
95#537—32no—
96#535—32no—
97#535—32no—
98#532—31no—
99#532D311yescharacteristic, complemented
100#529—31no—
101C23×D3C2314yes—
102C3×#41—44no—
103C2×C3×Q8C2312yes—
104C2×C3×D4C2311yescharacteristic
105C2×C3×D4C2311yescharacteristic
106C2×C3×D4C2312yes—
107C2×C3×D4C2312yes—
108C2×C3×D4C2312yes—
109C2×C3×D4C2312yes—
110C22×C3×C4C2312yes—
111C22×C3×C4C2311yescharacteristic
112C22×C3×C4C2312yes—
113C22×C3×C4C2311yescharacteristic
114C22×C3×C4C2311yescharacteristic
115C2×#67C2314yes—
116C2×#67C2314yes—
117C2×#67C2314yes—
118C2×#67C2314yes—
119C2×#67C2312yes—
120C2×#67C2314yes—
121C22×Dic3C2314yes—
122#237—44no—
123D3×D4—44no—
124#235—44no—
125#235—44no—
126C2×D12—44no—
127C2×D12C2311yescharacteristic
128C2×D12C2312yes—
129C2×D12C2312yes—
130C2×D12C2312yes—
131C2×D12C2314yes—
132C2×C4×D3C2312yes—
133C2×C4×D3C2314yes—
134C2×C4×D3C2314yes—
135C2×Dic6C2311yescharacteristic
136C2×Dic6C2312yes—
137C3×C4⋊C4C2311yescharacteristic
138C3×C4⋊C4C2311yescharacteristic
139C3×C4⋊C4C2312yes—
140C3×C22⋊C4C2312yes—
141C3×C22⋊C4C2312yes—
142C3×C22⋊C4C2312yes—
143C3×C22⋊C4C2312yes—
144C3×C42C2311yescharacteristic
145C3×C42C2311yescharacteristic
146C3×C42C2311yescharacteristic
147C3×C42C2311yescharacteristic
148#217C2314yes—
149#217C2314yes—
150#217C2312yes—
151(C2×D3)⋊C4C2312yes—
152(C2×D3)⋊C4C2312yes—
153(C2×D3)⋊C4C2312yes—
154(C2×D3)⋊C4C2314yes—
155(C2×D3)⋊C4C2314yes—
156(C2×D3)⋊C4C2312yes—
157(C2×D3)⋊C4C2314yes—
158(C2×D3)⋊C4C2314yes—
159(C2×D3)⋊C4C2314yes—
160C4⋊Dic3C2311yescharacteristic
161C4⋊Dic3C2312yes—
162C4⋊Dic3C2312yes—
163C4⋊Dic3C2311yescharacteristic
164C4⋊Dic3C2314yes—
165Dic3⋊C4C2312yes—
166Dic3⋊C4C2312yes—
167Dic3⋊C4C2312yes—
168Dic3⋊C4C2312yes—
169Dic3⋊C4C2314yes—
170C4×Dic3C2312yes—
171C4×Dic3C2314yes—
172C2×#41—32no—
173C2×#41—32no—
174C2×#41—32no—
175C2×#41C2×D312yescomplemented
176C22×D4—32no—
177C22×D4—32no—
178C4⋊Q8—31no—
179C4⋊Q8C2×D311yescharacteristic, complemented
180#127—32no—
181#127—34no—
182#127—31no—
183#127C2×D311yescharacteristic
184#126—34no—
185#126—34no—
186#126—32no—
187#125—31no—
188#124—34no—
189#124—32no—
190#124—31no—
191#124—32no—
192#124C2×D312yescomplemented
193#123—32no—
194#123—34no—
195#123—34no—
196#123—34no—
197C22⋊Q8—34no—
198C22⋊Q8—31no—
199C22⋊Q8—32no—
200C22⋊Q8—32no—
201#121—34no—
202#121—34no—
203#121—34no—
204#121—34no—
205#121—31no—
206#121—32no—
207#121—32no—
208#121—32no—
209#121—32no—
210#121—32no—
211#121—32no—
212#121—34no—
213#121—34no—
214#121C2×D311yescharacteristic
215#121C2×D311yescharacteristic, complemented
216#121C2×D312yescomplemented
217#120—34no—
218#120—34no—
219#120—34no—
220#120—34no—
221C4×Q8—32no—
222C4×Q8—31no—
223C4×D4—32no—
224C4×D4—32no—
225C4×D4—34no—
226C4×D4—31no—
227C4×D4—32no—
228C4×D4—32no—
229C4×D4C2×D311yescharacteristic
230C4×D4C2×D312yescomplemented
231C4×D4C2×D311yescharacteristic, complemented
232#117—32no—
233#117—32no—
234C2×C4⋊C4—32no—
235C2×C22⋊C4—34no—
236C2×C42C2×D311yescharacteristic, complemented
237C23×C3C2412yes—
238C23×C3C2411yescharacteristic
239C23×C3C2412yes—
240C22×D3—44no—
241C22×D3—44no—
242C22×D3—44no—
243C22×D3C2412yes—
244C22×D3C2414yes—
245C3×Q8—42no—
246C3×D4—22no—
247C3×D4—22no—
248C3×D4—42no—
249C3×D4—42no—
250C3×D4—42no—
251C3×D4—42no—
252C2×C3×C4—24no—
253C2×C3×C4—22no—
254C2×C3×C4—24no—
255C2×C3×C4—22no—
256C2×C3×C4—41no—
257C2×C3×C4C2412yes—
258C2×C3×C4C2411yescharacteristic
259C2×C3×C4C2411yescharacteristic
260C2×C3×C4C2411yescharacteristic
261C2×C3×C4C2411yescharacteristic
262C2×C3×C4C2412yes—
263C2×C3×C4C2411yescharacteristic
264C2×C3×C4C2411yescharacteristic
265#67—44no—
266#67—44no—
267#67—44no—
268#67—44no—
269#67—44no—
270#67—42no—
271C2×Dic3—44no—
272C2×Dic3C2412yes—
273C2×Dic3C2412yes—
274C2×Dic3C2414yes—
275C2×Dic3C2412yes—
276D12—22no—
277D12—24no—
278D12—24no—
279D12—44no—
280D12—42no—
281C4×D3—24no—
282C4×D3—44no—
283C4×D3—44no—
284Dic6—22no—
285Dic6—24no—
286C24—34no—
287#41—44no—
288#41—124no—
289#41—124no—
290#41—124no—
291C2×Q8—31no—
292C2×Q8—32no—
293C2×Q8C22×D312yes—
294C2×D4—32no—
295C2×D4—34no—
296C2×D4—34no—
297C2×D4—34no—
298C2×D4—34no—
299C2×D4—34no—
300C2×D4—31no—
301C2×D4—32no—
302C2×D4—32no—
303C2×D4—32no—
304C2×D4—34no—
305C2×D4—124no—
306C2×D4—124no—
307C2×D4C22×D311yescharacteristic
308C2×D4C22×D311yescharacteristic
309C2×D4C22×D312yes—
310C2×D4C22×D312yes—
311C2×D4C22×D312yes—
312C2×D4C22×D312yes—
313C22×C4—34no—
314C22×C4—32no—
315C22×C4—34no—
316C22×C4—34no—
317C22×C4C22×D311yescharacteristic
318C22×C4C22×D311yescharacteristic
319C22×C4C22×D312yes—
320C22×C4C22×D312yes—
321C22×C4C22×D311yescharacteristic
322C4⋊C4—31no—
323C4⋊C4—32no—
324C4⋊C4—34no—
325C4⋊C4—31no—
326C4⋊C4—32no—
327C4⋊C4—32no—
328C4⋊C4—32no—
329C4⋊C4—32no—
330C4⋊C4—34no—
331C4⋊C4—32no—
332C4⋊C4C22×D311yescharacteristic
333C4⋊C4C22×D311yescharacteristic
334C4⋊C4C22×D312yes—
335C22⋊C4—34no—
336C22⋊C4—34no—
337C22⋊C4—32no—
338C22⋊C4—32no—
339C22⋊C4—34no—
340C22⋊C4—32no—
341C22⋊C4—32no—
342C22⋊C4—32no—
343C22⋊C4—34no—
344C22⋊C4—34no—
345C22⋊C4—34no—
346C22⋊C4—34no—
347C22⋊C4C22×D312yes—
348C22⋊C4C22×D312yes—
349C22⋊C4C22×D312yes—
350C22⋊C4C22×D312yes—
351C42—32no—
352C42—34no—
353C42C22×D311yescharacteristic
354C42C22×D311yescharacteristic
355C42C22×D311yescharacteristic
356C42C22×D311yescharacteristic
357C22×C3—22no—
358C22×C3—21no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—21no—
362C22×C3—22no—
363C22×C3—21no—
364C22×C3—22no—
365C22×C3—22no—
366C22×C3C2511yescharacteristic
367C2×D3—22no—
368C2×D3—24no—
369C2×D3—24no—
370C2×D3—22no—
371C2×D3—24no—
372C2×D3—22no—
373C2×D3—88no—
374C3×C4—21no—
375C3×C4—22no—
376C3×C4—22no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4H(5,𝔽2)12yes—
381C3×C4H(5,𝔽2)12yes—
382Dic3—22no—
383Dic3—24no—
384Dic3—22no—
385Dic3—22no—
386C23—32no—
387C23—34no—
388C23—124no—
389C23—124no—
390C23—124no—
391C23C23×D312yes—
392C23C23×D311yescharacteristic
393C23C23×D312yes—
394Q8—42no—
395Q8—62no—
396Q8—64no—
397D4—22no—
398D4—22no—
399D4—42no—
400D4—42no—
401D4—42no—
402D4—42no—
403D4—62no—
404D4—64no—
405D4—64no—
406D4—124no—
407D4—122no—
408D4—124no—
409D4—124no—
410D4—124no—
411D4—124no—
412D4—122no—
413D4—124no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—32no—
419C2×C4—32no—
420C2×C4—34no—
421C2×C4—32no—
422C2×C4—41no—
423C2×C4—64no—
424C2×C4—124no—
425C2×C4—124no—
426C2×C4—124no—
427C2×C4C23×D312yes—
428C2×C4C23×D311yescharacteristic
429C2×C4C23×D311yescharacteristic
430C2×C4C23×D311yescharacteristic
431C2×C4C23×D311yescharacteristic
432C2×C4C23×D312yes—
433C2×C4C23×D311yescharacteristic
434C2×C4C23×D311yescharacteristic
435C2×C3—42no—
436C2×C3—42no—
437C2×C3—41no—
438C2×C3C2×H(5,𝔽2)11yescharacteristic
439C2×C3C2×H(5,𝔽2)11yescharacteristic
440C2×C3#58511yescharacteristic
441D3—44no—
442D3—42no—
443C22—21no—
444C22—22no—
445C22—22no—
446C22—21no—
447C22—22no—
448C22—22no—
449C22—21no—
450C22—22no—
451C22—22no—
452C22—64no—
453C22—62no—
454C22—62no—
455C22—64no—
456C22—64no—
457C22—62no—
458C22—248no—
459C22C24×D311yescharacteristic
460C4—22no—
461C4—21no—
462C4—22no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—62no—
467C4—62no—
468C4—64no—
469C4—62no—
470C4#100412yes—
471C4#100912yes—
472C3#354611yescharacteristic, complemented, Sylow
473C2—42no—
474C2—41no—
475C2—42no—
476C2—122no—
477C2—124no—
478C2C2×#100411yescharacteristic
479C2C2×#100911yescharacteristic
480C2#596411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#532C2
hypercenter#532D3
hypocenterC3#3546
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C3×#532C2semidirect
2. #5833C2semidirect
3. #5830C2semidirect
4. #5733C2semidirect
5. #5715C2semidirect
6. #5678C2semidirect
7. #5620C2semidirect
8. #5612C2semidirect
9. #5599C2semidirect
10. #5565C2semidirect
11. #5555C2semidirect
12. #5553C2semidirect
13. #5550C2semidirect
14. D4×D12C2semidirect
15. #5542C2semidirect
16. #5479C2semidirect
17. #5477C2semidirect
19. C2×C3×#41C22semidirect
20. C2×#237C22semidirect
22. C2×#235C22semidirect
23. C2×#235C22semidirect
25. C3×C4⋊Q8C22semidirect
27. C3×#124C22semidirect
29. C3×#121C22semidirect
30. C3×#121C22semidirect
32. C3×C4×D4C22semidirect
33. C3×C4×D4C22semidirect
34. C2×C3×C42C22semidirect
35. #947C22semidirect
36. #946C22semidirect
37. #940C22semidirect
38. #940C22semidirect
39. #939C22semidirect
40. #939C22semidirect
41. #939C22semidirect
42. #938C22semidirect
43. #937C22semidirect
44. #937C22semidirect
45. #937C22semidirect
50. #930C22semidirect
51. #930C22semidirect
52. #930C22semidirect
53. #929C22semidirect
54. C4×#67C22semidirect
55. #926C22semidirect
56. C2×C4⋊Dic3C22semidirect
57. #924C22semidirect
58. #924C22semidirect
59. #898C22semidirect
60. #897C22semidirect
62. #894C22semidirect
63. #893C22semidirect
64. #887C22semidirect
65. #886C22semidirect
66. #884C22semidirect
67. #884C22semidirect
68. #883C22semidirect
70. #67⋊C4C22semidirect
71. D3×C22⋊C4C22semidirect
72. #879C22semidirect
73. #876C22semidirect
75. #875C22semidirect
78. C4×D12C22semidirect
79. C4×D12C22semidirect
80. #872C22semidirect
81. #870C22semidirect
82. #869C22semidirect
83. C4×Dic6C22semidirect
99. #532D3semidirect
175. C2×#41C2×D3semidirect
179. C4⋊Q8C2×D3semidirect
192. #124C2×D3semidirect
215. #121C2×D3semidirect
216. #121C2×D3semidirect
230. C4×D4C2×D3semidirect
231. C4×D4C2×D3semidirect
236. C2×C42C2×D3semidirect
472. C3#3546semidirect

Maximal subgroups

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

subgroupindex
C3×#5322
#58332
#58302
#57332
#57152
#56782
#56202
#56122
#55992
#55652
#55552
#55532
#55502
D4×D122
#55422
#54792
#54772
#35463