Finite Group Databasebeta

← #86174

Subgroups — #86174

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×#529C211yesmaximal, characteristic, complemented
2#5704C214yesmaximal, complemented
3#5695C212yesmaximal, complemented
4#5676C212yesmaximal, complemented
5#5666C214yesmaximal, complemented
6#5629C212yesmaximal, complemented
7#5607C212yesmaximal, complemented
8#5584C211yesmaximal, characteristic, complemented
9#5577C211yesmaximal, characteristic, complemented
10#5553C212yesmaximal, complemented
11#5551C211yesmaximal, characteristic, complemented
12#5550C212yesmaximal, complemented
13D4×D12C211yesmaximal, characteristic, complemented
14#5534C212yesmaximal, complemented
15#5525C212yesmaximal, complemented
16#5477C211yesmaximal, characteristic, complemented
17#5477C211yesmaximal, characteristic, complemented
18#3546—31nomaximal, Sylow
19C2×#239C2212yescomplemented
20C2×D3×D4C2212yes—
21C2×#235C2212yescomplemented
22C2×#235C2212yescomplemented
23C2×#235C2212yescomplemented
24C22×D12C2212yes—
25C3×#126C2212yescomplemented
26C3×#125C2211yescharacteristic, complemented
27C3×#124C2211yescharacteristic
28C3×#123C2212yescomplemented
29C3×C22⋊Q8C2211yescharacteristic, complemented
30C3×#121C2211yescharacteristic
31C3×C4×Q8C2211yescharacteristic, complemented
32C3×C4×D4C2211yescharacteristic
33C3×C4×D4C2212yescomplemented
34C3×#117C2212yescomplemented
35C2×C3×C42C2211yescharacteristic, complemented
36#946C2212yescomplemented
37#940C2214yescomplemented
38#938C2212yes—
39#937C2214yescomplemented
40#930C2211yescharacteristic
41#930C2212yes—
42#930C2212yescomplemented
43#930C2212yescomplemented
44#930C2211yescharacteristic, complemented
45#930C2212yescomplemented
46#924C2212yescomplemented
47#898C2214yescomplemented
48#898C2214yescomplemented
49#896C2214yescomplemented
50#895C2212yes—
51#895C2214yescomplemented
52#895C2214yescomplemented
53#895C2212yescomplemented
54#895C2214yescomplemented
55#894C2214yescomplemented
56#893C2214yescomplemented
57#892C2212yescomplemented
58D3×C4⋊C4C2212yescomplemented
59#885C2214yescomplemented
60#884C2214yescomplemented
61#884C2214yescomplemented
62#884C2214yescomplemented
63#883C2214yescomplemented
64#883C2214yescomplemented
65#882C2214yes—
66#882C2214yescomplemented
67#882C2214yescomplemented
68D3×C22⋊C4C2214yescomplemented
69#875C2212yes—
70#875C2212yescomplemented
71#875C2212yescomplemented
72#874C2211yescharacteristic
73#874C2211yescharacteristic
74#874C2212yescomplemented
75C4×D12C2211yescharacteristic
76C4×D12C2212yescomplemented
77C4×D12C2212yescomplemented
78C4×D12C2212yescomplemented
79C4×D12C2211yescharacteristic, complemented
80C4×D12C2212yescomplemented
81#869C2211yescharacteristic, complemented
82#869C2211yescharacteristic, complemented
83C4×Dic6C2212yescomplemented
84#560—34no—
85#556—31no—
86#553—32no—
87#550—31no—
88#550—32no—
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#532—31no—
100#529D311yescharacteristic, complemented
101C23×D3C2314yes—
102C2×C3×Q8C2311yescharacteristic
103C2×C3×D4C2311yescharacteristic
104C2×C3×D4C2312yes—
105C22×C3×C4C2312yes—
106C22×C3×C4C2311yescharacteristic
107C22×C3×C4C2311yescharacteristic
108C22×C3×C4C2311yescharacteristic
109C2×#67C2314yes—
110C2×#67C2314yes—
111C2×#67C2312yes—
112C2×#67C2314yes—
113C2×#67C2312yes—
114#239—44no—
115D3×D4—44no—
116#235—44no—
117#235—44no—
118#235—44no—
119C2×D12—44no—
120C2×D12C2311yescharacteristic
121C2×D12C2311yescharacteristic
122C2×D12C2312yes—
123C2×D12C2312yes—
124C2×D12C2312yes—
125C2×D12C2312yes—
126C2×D12C2312yes—
127C2×D12C2314yes—
128C2×D12C2314yes—
129C2×D12C2314yes—
130C2×C4×D3C2312yes—
131C2×C4×D3C2312yes—
132C2×C4×D3C2314yes—
133C2×C4×D3C2314yes—
134C2×C4×D3C2314yes—
135C2×Dic6C2312yes—
136C2×Dic6C2312yes—
137C3×C4⋊C4C2311yescharacteristic
138C3×C4⋊C4C2311yescharacteristic
139C3×C4⋊C4C2312yes—
140C3×C4⋊C4C2312yes—
141C3×C4⋊C4C2312yes—
142C3×C4⋊C4C2312yes—
143C3×C22⋊C4C2312yes—
144C3×C22⋊C4C2312yes—
145C3×C22⋊C4C2312yes—
146C3×C22⋊C4C2312yes—
147C3×C22⋊C4C2312yes—
148C3×C42C2312yes—
149C3×C42C2311yescharacteristic
150C3×C42C2311yescharacteristic
151C3×C42C2311yescharacteristic
152C3×C42C2311yescharacteristic
153#217C2314yes—
154(C2×D3)⋊C4C2312yes—
155(C2×D3)⋊C4C2312yes—
156(C2×D3)⋊C4C2314yes—
157(C2×D3)⋊C4C2314yes—
158(C2×D3)⋊C4C2312yes—
159(C2×D3)⋊C4C2312yes—
160(C2×D3)⋊C4C2314yes—
161(C2×D3)⋊C4C2314yes—
162(C2×D3)⋊C4C2314yes—
163(C2×D3)⋊C4C2314yes—
164C4⋊Dic3C2311yescharacteristic
165C4⋊Dic3C2311yescharacteristic
166C4⋊Dic3C2312yes—
167C4⋊Dic3C2312yes—
168C4⋊Dic3C2312yes—
169Dic3⋊C4C2314yes—
170Dic3⋊C4C2314yes—
171C4×Dic3C2314yes—
172C2×#41—32no—
173C2×#41—32no—
174C2×#41—32no—
175C2×#41—32no—
176C22×D4—32no—
177C22×D4—32no—
178C4⋊Q8—31no—
179C4⋊Q8—31no—
180#127—34no—
181#127—31no—
182#127—31no—
183#127—32no—
184#126—34no—
185#126—34no—
186#126C2×D312yescomplemented
187#125C2×D311yescharacteristic, complemented
188#124—34no—
189#124—32no—
190#124—32no—
191#124—32no—
192#124C2×D311yescharacteristic
193#123—34no—
194#123—34no—
195#123—34no—
196#123C2×D312yescomplemented
197C22⋊Q8—32no—
198C22⋊Q8—32no—
199C22⋊Q8—34no—
200C22⋊Q8C2×D311yescharacteristic, complemented
201#121—32no—
202#121—32no—
203#121—31no—
204#121—31no—
205#121—32no—
206#121—32no—
207#121—32no—
208#121—34no—
209#121—32no—
210#121—34no—
211#121—32no—
212#121—34no—
213#121—34no—
214#121—34no—
215#121—34no—
216#121C2×D311yescharacteristic
217#120—34no—
218#120—34no—
219#120—34no—
220#120—34no—
221C4×Q8—32no—
222C4×Q8C2×D311yescharacteristic, complemented
223C4×D4—34no—
224C4×D4—32no—
225C4×D4—31no—
226C4×D4—32no—
227C4×D4—32no—
228C4×D4—31no—
229C4×D4—32no—
230C4×D4C2×D311yescharacteristic
231C4×D4C2×D312yescomplemented
232#117—32no—
233#117C2×D312yescomplemented
234C2×C4⋊C4—32no—
235C2×C22⋊C4—34no—
236C2×C42C2×D311yescharacteristic, complemented
237C23×C3C2412yes—
238C23×C3C2411yescharacteristic
239C22×D3—44no—
240C22×D3—44no—
241C22×D3—44no—
242C22×D3C2412yes—
243C22×D3C2414yes—
244C22×D3C2412yes—
245C3×Q8—22no—
246C3×D4—22no—
247C3×D4—42no—
248C2×C3×C4—24no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—41no—
252C2×C3×C4C2412yes—
253C2×C3×C4C2412yes—
254C2×C3×C4C2412yes—
255C2×C3×C4C2411yescharacteristic
256C2×C3×C4C2411yescharacteristic
257C2×C3×C4C2411yescharacteristic
258C2×C3×C4C2411yescharacteristic
259C2×C3×C4C2411yescharacteristic
260C2×C3×C4C2411yescharacteristic
261#67—44no—
262#67—44no—
263#67—42no—
264#67—42no—
265#67—44no—
266C2×Dic3C2412yes—
267C2×Dic3C2414yes—
268C2×Dic3C2412yes—
269D12—22no—
270D12—24no—
271D12—22no—
272D12—24no—
273D12—44no—
274D12—42no—
275D12—42no—
276D12—44no—
277D12—42no—
278D12—44no—
279C4×D3—24no—
280C4×D3—24no—
281C4×D3—44no—
282C4×D3—44no—
283C4×D3—44no—
284Dic6—24no—
285Dic6—42no—
286C24—34no—
287#41—124no—
288#41—124no—
289#41—124no—
290#41—124no—
291C2×Q8—32no—
292C2×Q8—32no—
293C2×Q8C22×D311yescharacteristic
294C2×D4—32no—
295C2×D4—34no—
296C2×D4—34no—
297C2×D4—34no—
298C2×D4—32no—
299C2×D4—32no—
300C2×D4—31no—
301C2×D4—32no—
302C2×D4—31no—
303C2×D4—32no—
304C2×D4—32no—
305C2×D4—34no—
306C2×D4—32no—
307C2×D4—34no—
308C2×D4—34no—
309C2×D4—124no—
310C2×D4—124no—
311C2×D4C22×D311yescharacteristic
312C2×D4C22×D312yes—
313C22×C4—32no—
314C22×C4—32no—
315C22×C4—34no—
316C22×C4—34no—
317C22×C4—34no—
318C22×C4C22×D311yescharacteristic
319C22×C4C22×D311yescharacteristic
320C22×C4C22×D312yes—
321C22×C4C22×D311yescharacteristic
322C4⋊C4—32no—
323C4⋊C4—34no—
324C4⋊C4—31no—
325C4⋊C4—31no—
326C4⋊C4—32no—
327C4⋊C4—32no—
328C4⋊C4—34no—
329C4⋊C4C22×D311yescharacteristic
330C4⋊C4C22×D311yescharacteristic
331C4⋊C4C22×D312yes—
332C4⋊C4C22×D312yes—
333C4⋊C4C22×D312yes—
334C4⋊C4C22×D312yes—
335C22⋊C4—34no—
336C22⋊C4—32no—
337C22⋊C4—34no—
338C22⋊C4—34no—
339C22⋊C4—32no—
340C22⋊C4—32no—
341C22⋊C4—34no—
342C22⋊C4—34no—
343C22⋊C4—34no—
344C22⋊C4—34no—
345C22⋊C4—32no—
346C22⋊C4C22×D312yes—
347C22⋊C4C22×D312yes—
348C22⋊C4C22×D312yes—
349C22⋊C4C22×D312yes—
350C22⋊C4C22×D312yes—
351C42—34no—
352C42C22×D311yescharacteristic
353C42C22×D311yescharacteristic
354C42C22×D311yescharacteristic
355C42C22×D311yescharacteristic
356C42C22×D312yes—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—21no—
362C22×C3—22no—
363C22×C3C2511yescharacteristic
364C2×D3—22no—
365C2×D3—22no—
366C2×D3—22no—
367C2×D3—24no—
368C2×D3—24no—
369C2×D3—22no—
370C2×D3—22no—
371C2×D3—24no—
372C2×D3—22no—
373C2×D3—88no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—22no—
377C3×C4—22no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—22no—
381C3×C4H(5,𝔽2)12yes—
382C3×C4H(5,𝔽2)12yes—
383Dic3—24no—
384Dic3—22no—
385Dic3—22no—
386C23—32no—
387C23—34no—
388C23—32no—
389C23—124no—
390C23—124no—
391C23—124no—
392C23C23×D312yes—
393C23C23×D311yescharacteristic
394Q8—22no—
395Q8—64no—
396Q8—122no—
397D4—22no—
398D4—42no—
399D4—62no—
400D4—64no—
401D4—64no—
402D4—62no—
403D4—124no—
404D4—122no—
405D4—124no—
406D4—122no—
407D4—122no—
408D4—124no—
409D4—124no—
410D4—122no—
411D4—122no—
412D4—124no—
413D4—124no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—32no—
418C2×C4—34no—
419C2×C4—32no—
420C2×C4—41no—
421C2×C4—64no—
422C2×C4—64no—
423C2×C4—124no—
424C2×C4—124no—
425C2×C4—124no—
426C2×C4C23×D312yes—
427C2×C4C23×D311yescharacteristic
428C2×C4C23×D311yescharacteristic
429C2×C4C23×D311yescharacteristic
430C2×C4C23×D311yescharacteristic
431C2×C4C23×D312yes—
432C2×C4C23×D312yes—
433C2×C4C23×D311yescharacteristic
434C2×C4C23×D311yescharacteristic
435C2×C3—42no—
436C2×C3—41no—
437C2×C3C2×H(5,𝔽2)11yescharacteristic
438C2×C3C2×H(5,𝔽2)11yescharacteristic
439C2×C3#58511yescharacteristic
440D3—42no—
441D3—44no—
442D3—42no—
443C22—21no—
444C22—21no—
445C22—22no—
446C22—22no—
447C22—21no—
448C22—22no—
449C22—62no—
450C22—62no—
451C22—62no—
452C22—64no—
453C22—64no—
454C22—62no—
455C22—62no—
456C22—64no—
457C22—62no—
458C22—248no—
459C22C24×D311yescharacteristic
460C4—21no—
461C4—22no—
462C4—21no—
463C4—22no—
464C4—21no—
465C4—21no—
466C4—22no—
467C4—62no—
468C4—62no—
469C4—64no—
470C4#100912yes—
471C4#100912yes—
472C3#354611yescharacteristic, complemented, Sylow
473C2—42no—
474C2—41no—
475C2—122no—
476C2—124no—
477C2—122no—
478C2C2×#100911yescharacteristic
479C2C2×#100911yescharacteristic
480C2#596411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#529C2
hypercenter#529D3
hypocenterC3#3546
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #529

Complements

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

NH ≅ G/Ntype
1. C3×#529C2semidirect
2. #5704C2semidirect
3. #5695C2semidirect
4. #5676C2semidirect
5. #5666C2semidirect
6. #5629C2semidirect
7. #5607C2semidirect
8. #5584C2semidirect
9. #5577C2semidirect
10. #5553C2semidirect
11. #5551C2semidirect
12. #5550C2semidirect
13. D4×D12C2semidirect
14. #5534C2semidirect
15. #5525C2semidirect
16. #5477C2semidirect
17. #5477C2semidirect
19. C2×#239C22semidirect
21. C2×#235C22semidirect
22. C2×#235C22semidirect
23. C2×#235C22semidirect
25. C3×#126C22semidirect
26. C3×#125C22semidirect
28. C3×#123C22semidirect
29. C3×C22⋊Q8C22semidirect
31. C3×C4×Q8C22semidirect
33. C3×C4×D4C22semidirect
34. C3×#117C22semidirect
35. C2×C3×C42C22semidirect
36. #946C22semidirect
37. #940C22semidirect
39. #937C22semidirect
42. #930C22semidirect
43. #930C22semidirect
44. #930C22semidirect
45. #930C22semidirect
46. #924C22semidirect
47. #898C22semidirect
48. #898C22semidirect
49. #896C22semidirect
51. #895C22semidirect
52. #895C22semidirect
53. #895C22semidirect
54. #895C22semidirect
55. #894C22semidirect
56. #893C22semidirect
57. #892C22semidirect
58. D3×C4⋊C4C22semidirect
59. #885C22semidirect
60. #884C22semidirect
61. #884C22semidirect
62. #884C22semidirect
63. #883C22semidirect
64. #883C22semidirect
66. #882C22semidirect
67. #882C22semidirect
68. D3×C22⋊C4C22semidirect
70. #875C22semidirect
71. #875C22semidirect
74. #874C22semidirect
76. C4×D12C22semidirect
77. C4×D12C22semidirect
78. C4×D12C22semidirect
79. C4×D12C22semidirect
80. C4×D12C22semidirect
81. #869C22semidirect
82. #869C22semidirect
83. C4×Dic6C22semidirect
100. #529D3semidirect
186. #126C2×D3semidirect
187. #125C2×D3semidirect
196. #123C2×D3semidirect
200. C22⋊Q8C2×D3semidirect
222. C4×Q8C2×D3semidirect
231. C4×D4C2×D3semidirect
233. #117C2×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×#5292
#57042
#56952
#56762
#56662
#56292
#56072
#55842
#55772
#55532
#55512
#55502
D4×D122
#55342
#55252
#54772
#35463