Finite Group Databasebeta

← #87009

Subgroups — #87009

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×#556C211yesmaximal, characteristic, complemented
2#5726C211yesmaximal, characteristic, complemented
3#5725C211yesmaximal, characteristic, complemented
4#5711C214yesmaximal, complemented
5#5704C214yesmaximal, complemented
6#5676C212yesmaximal, complemented
7#5674C212yesmaximal, complemented
8#5630C212yesmaximal, complemented
9#5629C212yesmaximal, complemented
10#5582C212yesmaximal, complemented
11#5580C211yesmaximal, characteristic, complemented
12#5577C211yesmaximal, characteristic, complemented
13#5577C212yesmaximal, complemented
14D4×D12C211yesmaximal, characteristic, complemented
15#5548C211yesmaximal, characteristic, complemented
16#5530C212yesmaximal, complemented
17#5525C212yesmaximal, complemented
18#3546—31nomaximal, Sylow
19C2×#239C2212yescomplemented
20C2×#239C2212yescomplemented
21C2×#239C2212yescomplemented
22C2×D3×D4C2212yes—
23C2×#235C2212yescomplemented
24C22×D12C2212yes—
25C3×C4⋊Q8C2211yescharacteristic, complemented
26C3×#126C2214yescomplemented
27C3×#124C2212yes—
28C3×C22⋊Q8C2212yescomplemented
29C3×C4×Q8C2211yescharacteristic, complemented
30C3×C4×Q8C2212yescomplemented
31C3×C4×D4C2211yescharacteristic
32C3×#117C2212yescomplemented
33#947C2211yescharacteristic
34#947C2212yescomplemented
35#946C2212yescomplemented
36#946C2211yescharacteristic, complemented
37#940C2211yescharacteristic
38#938C2211yescharacteristic
39#930C2212yes—
40C4×#67C2212yescomplemented
41#898C2212yescomplemented
42#896C2214yescomplemented
43#895C2212yes—
44#895C2211yescharacteristic
45#895C2212yes—
46#895C2214yescomplemented
47#895C2214yescomplemented
48#895C2212yescomplemented
49#895C2214yescomplemented
50#895C2212yescomplemented
51#895C2212yescomplemented
52#895C2211yescharacteristic, complemented
53#894C2214yescomplemented
54#894C2212yescomplemented
55#894C2214yescomplemented
56#893C2212yescomplemented
57#893C2212yescomplemented
58D3×C4⋊C4C2212yescomplemented
59#890C2211yescharacteristic, complemented
60#888C2211yescharacteristic, complemented
61#885C2212yescomplemented
62#884C2214yescomplemented
63#884C2214yescomplemented
64#884C2214yescomplemented
65#884C2212yescomplemented
66#883C2214yescomplemented
67#882C2214yes—
68#882C2214yescomplemented
69#882C2214yescomplemented
70#882C2214yescomplemented
71D3×C22⋊C4C2214yescomplemented
72#876C2214yescomplemented
73#875C2214yescomplemented
74#874C2211yescharacteristic
75#874C2212yescomplemented
76#874C2214yescomplemented
77C4×D12C2211yescharacteristic
78C4×D12C2212yescomplemented
79C4×D12C2212yescomplemented
80C4×D12C2214yescomplemented
81C4×D12C2211yescharacteristic, complemented
82#872C2212yescomplemented
83C42×D3C2211yescharacteristic, complemented
84#560—34no—
85#556D311yescharacteristic, complemented
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#532—31no—
100#529—31no—
101C23×D3C2314yes—
102C2×C3×Q8C2311yescharacteristic
103C2×C3×Q8C2312yes—
104C2×C3×D4C2311yescharacteristic
105C22×C3×C4C2312yes—
106C2×#67C2314yes—
107C2×#67C2312yes—
108C2×#67C2312yes—
109C2×#67C2312yes—
110#239—44no—
111#239—44no—
112#239—44no—
113D3×D4—44no—
114#235—44no—
115C2×D12—44no—
116C2×D12C2311yescharacteristic
117C2×D12C2312yes—
118C2×D12C2311yescharacteristic
119C2×D12C2312yes—
120C2×D12C2312yes—
121C2×D12C2314yes—
122C2×D12C2312yes—
123C2×D12C2312yes—
124C2×D12C2314yes—
125C2×D12C2314yes—
126C2×D12C2314yes—
127C2×D12C2314yes—
128C2×C4×D3C2311yescharacteristic
129C2×C4×D3C2311yescharacteristic
130C2×C4×D3C2312yes—
131C2×C4×D3C2312yes—
132C2×C4×D3C2311yescharacteristic
133C2×C4×D3C2314yes—
134C2×C4×D3C2314yes—
135C2×C4×D3C2314yes—
136C2×Dic6C2312yes—
137C3×C4⋊C4C2311yescharacteristic
138C3×C4⋊C4C2311yescharacteristic
139C3×C4⋊C4C2312yes—
140C3×C4⋊C4C2312yes—
141C3×C4⋊C4C2314yes—
142C3×C4⋊C4C2312yes—
143C3×C22⋊C4C2314yes—
144C3×C22⋊C4C2312yes—
145C3×C22⋊C4C2314yes—
146C3×C42C2311yescharacteristic
147C3×C42C2312yes—
148C3×C42C2314yes—
149#217C2312yes—
150(C2×D3)⋊C4C2312yes—
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)⋊C4C2312yes—
158(C2×D3)⋊C4C2312yes—
159(C2×D3)⋊C4C2314yes—
160(C2×D3)⋊C4C2314yes—
161(C2×D3)⋊C4C2314yes—
162C4⋊Dic3C2311yescharacteristic
163C4⋊Dic3C2312yes—
164C4⋊Dic3C2311yescharacteristic
165Dic3⋊C4C2312yes—
166Dic3⋊C4C2312yes—
167Dic3⋊C4C2312yes—
168Dic3⋊C4C2314yes—
169C4×Dic3C2311yescharacteristic
170C4×Dic3C2311yescharacteristic
171C4×Dic3C2311yescharacteristic
172C2×#41—32no—
173C2×#41—32no—
174C2×#41—32no—
175C2×#41—32no—
176C22×D4—32no—
177C22×D4—32no—
178C4⋊Q8—31no—
179C4⋊Q8C2×D311yescharacteristic, complemented
180#127—31no—
181#127—31no—
182#127—32no—
183#127—34no—
184#126—32no—
185#126—34no—
186#126C2×D314yescomplemented
187#125—31no—
188#124—31no—
189#124—32no—
190#124—32no—
191#124—34no—
192#124C2×D312yes—
193#123—32no—
194#123—34no—
195#123—34no—
196#123—34no—
197C22⋊Q8—31no—
198C22⋊Q8—32no—
199C22⋊Q8—34no—
200C22⋊Q8C2×D312yescomplemented
201#121—31no—
202#121—32no—
203#121—31no—
204#121—32no—
205#121—31no—
206#121—32no—
207#121—32no—
208#121—32no—
209#121—34no—
210#121—34no—
211#121—34no—
212#121—32no—
213#121—32no—
214#121—34no—
215#121—34no—
216#121—34no—
217#120—34no—
218#120—34no—
219#120—34no—
220#120—34no—
221C4×Q8C2×D311yescharacteristic, complemented
222C4×Q8C2×D312yescomplemented
223C4×D4—32no—
224C4×D4—32no—
225C4×D4—32no—
226C4×D4—31no—
227C4×D4—32no—
228C4×D4—31no—
229C4×D4—34no—
230C4×D4—32no—
231C4×D4C2×D311yescharacteristic
232#117—32no—
233#117C2×D312yescomplemented
234C2×C4⋊C4—32no—
235C2×C22⋊C4—34no—
236C2×C42—31no—
237C23×C3C2412yes—
238C22×D3—44no—
239C22×D3—44no—
240C22×D3—44no—
241C22×D3C2412yes—
242C22×D3C2414yes—
243C22×D3C2411yescharacteristic
244C22×D3C2412yes—
245C3×Q8—22no—
246C3×Q8—24no—
247C3×D4—22no—
248C2×C3×C4—24no—
249C2×C3×C4C2411yescharacteristic
250C2×C3×C4C2412yes—
251C2×C3×C4C2412yes—
252C2×C3×C4C2412yes—
253C2×C3×C4C2414yes—
254C2×C3×C4C2411yescharacteristic
255C2×C3×C4C2411yescharacteristic
256#67—42no—
257#67—42no—
258#67—44no—
259#67—42no—
260C2×Dic3C2411yescharacteristic
261C2×Dic3C2411yescharacteristic
262C2×Dic3C2411yescharacteristic
263C2×Dic3C2412yes—
264C2×Dic3C2412yes—
265D12—22no—
266D12—24no—
267D12—24no—
268D12—22no—
269D12—44no—
270D12—42no—
271D12—42no—
272D12—44no—
273D12—42no—
274D12—44no—
275D12—44no—
276D12—44no—
277C4×D3—22no—
278C4×D3—24no—
279C4×D3—24no—
280C4×D3—22no—
281C4×D3—41no—
282C4×D3—44no—
283C4×D3—44no—
284C4×D3—44no—
285Dic6—42no—
286C24—34no—
287#41—124no—
288#41—124no—
289#41—124no—
290#41—124no—
291C2×Q8—32no—
292C2×Q8C22×D311yescharacteristic
293C2×Q8C22×D312yes—
294C2×D4—32no—
295C2×D4—32no—
296C2×D4—32no—
297C2×D4—34no—
298C2×D4—31no—
299C2×D4—32no—
300C2×D4—32no—
301C2×D4—31no—
302C2×D4—32no—
303C2×D4—32no—
304C2×D4—34no—
305C2×D4—34no—
306C2×D4—34no—
307C2×D4—34no—
308C2×D4—34no—
309C2×D4—32no—
310C2×D4—124no—
311C2×D4—124no—
312C2×D4C22×D311yescharacteristic
313C22×C4—31no—
314C22×C4—31no—
315C22×C4—32no—
316C22×C4—32no—
317C22×C4—31no—
318C22×C4—34no—
319C22×C4—34no—
320C22×C4—34no—
321C22×C4C22×D312yes—
322C4⋊C4—31no—
323C4⋊C4—34no—
324C4⋊C4—32no—
325C4⋊C4—32no—
326C4⋊C4—31no—
327C4⋊C4—32no—
328C4⋊C4—32no—
329C4⋊C4C22×D312yes—
330C4⋊C4C22×D311yescharacteristic
331C4⋊C4C22×D312yes—
332C4⋊C4C22×D311yescharacteristic
333C4⋊C4C22×D314yes—
334C4⋊C4C22×D312yes—
335C22⋊C4—32no—
336C22⋊C4—32no—
337C22⋊C4—34no—
338C22⋊C4—32no—
339C22⋊C4—32no—
340C22⋊C4—32no—
341C22⋊C4—32no—
342C22⋊C4—34no—
343C22⋊C4—34no—
344C22⋊C4—34no—
345C22⋊C4—34no—
346C22⋊C4—32no—
347C22⋊C4—32no—
348C22⋊C4C22×D314yes—
349C22⋊C4C22×D314yes—
350C22⋊C4C22×D312yes—
351C42—31no—
352C42—31no—
353C42—31no—
354C42C22×D311yescharacteristic
355C42C22×D312yes—
356C42C22×D314yes—
357C22×C3—22no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3C2511yescharacteristic
361C2×D3—21no—
362C2×D3—22no—
363C2×D3—21no—
364C2×D3—22no—
365C2×D3—22no—
366C2×D3—24no—
367C2×D3—24no—
368C2×D3—22no—
369C2×D3—21no—
370C2×D3—22no—
371C2×D3—24no—
372C2×D3—22no—
373C2×D3—88no—
374C3×C4—22no—
375C3×C4—22no—
376C3×C4—24no—
377C3×C4—21no—
378C3×C4—22no—
379C3×C4H(5,𝔽2)12yes—
380C3×C4H(5,𝔽2)12yes—
381Dic3—21no—
382Dic3—22no—
383Dic3—21no—
384Dic3—22no—
385Dic3—21no—
386C23—32no—
387C23—34no—
388C23—31no—
389C23—32no—
390C23—124no—
391C23—124no—
392C23—124no—
393C23C23×D312yes—
394Q8—22no—
395Q8—24no—
396Q8—122no—
397D4—22no—
398D4—62no—
399D4—64no—
400D4—64no—
401D4—62no—
402D4—124no—
403D4—122no—
404D4—124no—
405D4—124no—
406D4—124no—
407D4—122no—
408D4—122no—
409D4—124no—
410D4—124no—
411D4—122no—
412D4—122no—
413D4—122no—
414C2×C4—24no—
415C2×C4—31no—
416C2×C4—31no—
417C2×C4—31no—
418C2×C4—32no—
419C2×C4—32no—
420C2×C4—64no—
421C2×C4—62no—
422C2×C4—64no—
423C2×C4—62no—
424C2×C4—124no—
425C2×C4—124no—
426C2×C4—124no—
427C2×C4—121no—
428C2×C4C23×D311yescharacteristic
429C2×C4C23×D312yes—
430C2×C4C23×D312yes—
431C2×C4C23×D312yes—
432C2×C4C23×D314yes—
433C2×C4C23×D311yescharacteristic
434C2×C4C23×D311yescharacteristic
435C2×C3—42no—
436C2×C3C2×H(5,𝔽2)11yescharacteristic
437C2×C3C2×H(5,𝔽2)11yescharacteristic
438C2×C3#58511yescharacteristic
439D3—41no—
440D3—42no—
441D3—44no—
442D3—42no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—61no—
447C22—62no—
448C22—61no—
449C22—62no—
450C22—62no—
451C22—64no—
452C22—64no—
453C22—62no—
454C22—61no—
455C22—62no—
456C22—64no—
457C22—62no—
458C22—248no—
459C22C24×D311yescharacteristic
460C4—22no—
461C4—22no—
462C4—24no—
463C4—21no—
464C4—22no—
465C4—61no—
466C4—62no—
467C4—62no—
468C4—61no—
469C4—61no—
470C4#100912yes—
471C4#100912yes—
472C3#354611yescharacteristic, complemented, Sylow
473C2—42no—
474C2—122no—
475C2—124no—
476C2—121no—
477C2—122no—
478C2C2×#100911yescharacteristic
479C2C2×#100911yescharacteristic
480C2#596811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#556C2
hypercenter#556D3
hypocenterC3#3546
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. #5726C2semidirect
3. #5725C2semidirect
4. #5711C2semidirect
5. #5704C2semidirect
6. #5676C2semidirect
7. #5674C2semidirect
8. #5630C2semidirect
9. #5629C2semidirect
10. #5582C2semidirect
11. #5580C2semidirect
12. #5577C2semidirect
13. #5577C2semidirect
14. D4×D12C2semidirect
15. #5548C2semidirect
16. #5530C2semidirect
17. #5525C2semidirect
19. C2×#239C22semidirect
20. C2×#239C22semidirect
21. C2×#239C22semidirect
23. C2×#235C22semidirect
25. C3×C4⋊Q8C22semidirect
26. C3×#126C22semidirect
28. C3×C22⋊Q8C22semidirect
29. C3×C4×Q8C22semidirect
30. C3×C4×Q8C22semidirect
32. C3×#117C22semidirect
34. #947C22semidirect
35. #946C22semidirect
36. #946C22semidirect
40. C4×#67C22semidirect
41. #898C22semidirect
42. #896C22semidirect
46. #895C22semidirect
47. #895C22semidirect
48. #895C22semidirect
49. #895C22semidirect
50. #895C22semidirect
51. #895C22semidirect
52. #895C22semidirect
53. #894C22semidirect
54. #894C22semidirect
55. #894C22semidirect
56. #893C22semidirect
57. #893C22semidirect
58. D3×C4⋊C4C22semidirect
59. #890C22semidirect
60. #888C22semidirect
61. #885C22semidirect
62. #884C22semidirect
63. #884C22semidirect
64. #884C22semidirect
65. #884C22semidirect
66. #883C22semidirect
68. #882C22semidirect
69. #882C22semidirect
70. #882C22semidirect
71. D3×C22⋊C4C22semidirect
72. #876C22semidirect
73. #875C22semidirect
75. #874C22semidirect
76. #874C22semidirect
78. C4×D12C22semidirect
79. C4×D12C22semidirect
80. C4×D12C22semidirect
81. C4×D12C22semidirect
82. #872C22semidirect
83. C42×D3C22semidirect
85. #556D3semidirect
179. C4⋊Q8C2×D3semidirect
186. #126C2×D3semidirect
200. C22⋊Q8C2×D3semidirect
221. C4×Q8C2×D3semidirect
222. C4×Q8C2×D3semidirect
233. #117C2×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×#5562
#57262
#57252
#57112
#57042
#56762
#56742
#56302
#56292
#55822
#55802
#55772
D4×D122
#55482
#55302
#55252
#35463