Finite Group Databasebeta

← #86707

Subgroups — #86707

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×#547C211yesmaximal, characteristic, complemented
2#5834C212yesmaximal, complemented
3#5830C212yesmaximal, complemented
4#5732C211yesmaximal, characteristic, complemented
5#5725C211yesmaximal, characteristic, complemented
6#5666C214yesmaximal, complemented
7#5657C214yesmaximal, complemented
8#5643C212yesmaximal, complemented
9#5630C212yesmaximal, complemented
10#5612C212yesmaximal, complemented
11#5607C212yesmaximal, complemented
12#5557C211yesmaximal, characteristic, complemented
13#5553C211yesmaximal, characteristic, complemented
14D4×D12C211yesmaximal, characteristic, complemented
15#5548C211yesmaximal, characteristic, complemented
16#5514C212yesmaximal, complemented
17#5510C212yesmaximal, complemented
18#3546—31nomaximal, Sylow
19C2×C3×#41C2212yescomplemented
20C2×#239C2212yescomplemented
21C2×D3×D4C2212yes—
22C2×#235C2212yescomplemented
23C2×#235C2212yescomplemented
24C22×D12C2212yes—
25C3×C4⋊Q8C2211yescharacteristic, complemented
26C3×#123C2214yescomplemented
27C3×C22⋊Q8C2212yescomplemented
28C3×#121C2212yes—
29C3×C4×D4C2211yescharacteristic
30C3×C4×D4C2211yescharacteristic, complemented
31C2×C3×C4⋊C4C2212yescomplemented
32#947C2212yes—
33#947C2212yescomplemented
34Q8×Dic3C2212yescomplemented
35#940C2211yescharacteristic
36#940C2214yescomplemented
37#940C2212yescomplemented
38#938C2211yescharacteristic
39#938C2214yescomplemented
40#938C2211yescharacteristic, complemented
41#937C2214yescomplemented
42#937C2214yescomplemented
43#936C2211yescharacteristic
44#930C2212yes—
45#930C2214yescomplemented
46#930C2214yescomplemented
47#930C2214yescomplemented
48#930C2212yescomplemented
49C4×#67C2214yescomplemented
50C4×#67C2212yescomplemented
51C4×#67C2212yescomplemented
52#926C2212yescomplemented
53#898C2214yescomplemented
54#898C2214yescomplemented
55#897C2214yescomplemented
56#897C2212yescomplemented
57#897C2211yescharacteristic, complemented
58#895C2212yes—
59#895C2211yescharacteristic
60#895C2212yescomplemented
61#895C2214yescomplemented
62#895C2212yescomplemented
63#894C2214yescomplemented
64#893C2212yescomplemented
65#893C2212yescomplemented
66#893C2212yescomplemented
67#892C2212yescomplemented
68#890C2211yescharacteristic, complemented
69#888C2211yescharacteristic, complemented
70#885C2214yescomplemented
71#885C2212yescomplemented
72#884C2214yescomplemented
73#884C2212yescomplemented
74#883C2214yescomplemented
75#883C2212yescomplemented
76#882C2214yes—
77#882C2214yescomplemented
78D3×C22⋊C4C2214yescomplemented
79#879C2212yescomplemented
80#874C2211yescharacteristic
81C4×D12C2211yescharacteristic
82C42×D3C2211yescharacteristic, complemented
83C4×Dic6C2211yescharacteristic, complemented
84#560—34no—
85#556—31no—
86#553—32no—
87#550—32no—
88#550—31no—
89#547D311yescharacteristic, complemented
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—
102C3×#41—44no—
103C2×C3×Q8C2312yes—
104C2×C3×D4C2311yescharacteristic
105C2×C3×D4C2311yescharacteristic
106C2×C3×D4C2314yes—
107C22×C3×C4C2312yes—
108C22×C3×C4C2312yes—
109C22×C3×C4C2314yes—
110C2×#67C2314yes—
111C2×#67C2312yes—
112C2×#67C2312yes—
113C2×#67C2314yes—
114C2×#67C2314yes—
115C2×#67C2312yes—
116C2×#67C2312yes—
117#239—44no—
118D3×D4—44no—
119#235—44no—
120#235—44no—
121C2×D12—44no—
122C2×D12C2311yescharacteristic
123C2×D12C2312yes—
124C2×D12C2312yes—
125C2×D12C2312yes—
126C2×D12C2314yes—
127C2×D12C2312yes—
128C2×D12C2314yes—
129C2×C4×D3C2311yescharacteristic
130C2×C4×D3C2311yescharacteristic
131C2×C4×D3C2312yes—
132C2×C4×D3C2311yescharacteristic
133C2×C4×D3C2314yes—
134C2×C4×D3C2314yes—
135C2×Dic6C2311yescharacteristic
136C2×Dic6C2312yes—
137C3×C4⋊C4C2311yescharacteristic
138C3×C4⋊C4C2311yescharacteristic
139C3×C4⋊C4C2312yes—
140C3×C4⋊C4C2312yes—
141C3×C4⋊C4C2314yes—
142C3×C22⋊C4C2312yes—
143C3×C22⋊C4C2314yes—
144C3×C22⋊C4C2312yes—
145C3×C42C2311yescharacteristic
146#217C2312yes—
147#217C2312yes—
148#217C2312yes—
149#217C2312yes—
150(C2×D3)⋊C4C2312yes—
151(C2×D3)⋊C4C2312yes—
152(C2×D3)⋊C4C2314yes—
153(C2×D3)⋊C4C2314yes—
154(C2×D3)⋊C4C2314yes—
155(C2×D3)⋊C4C2314yes—
156(C2×D3)⋊C4C2314yes—
157(C2×D3)⋊C4C2312yes—
158(C2×D3)⋊C4C2314yes—
159C4⋊Dic3C2311yescharacteristic
160C4⋊Dic3C2311yescharacteristic
161C4⋊Dic3C2312yes—
162C4⋊Dic3C2312yes—
163C4⋊Dic3C2314yes—
164Dic3⋊C4C2312yes—
165Dic3⋊C4C2312yes—
166Dic3⋊C4C2312yes—
167C4×Dic3C2312yes—
168C4×Dic3C2311yescharacteristic
169C4×Dic3C2311yescharacteristic
170C4×Dic3C2311yescharacteristic
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—31no—
181#127—32no—
182#127—34no—
183#127—31no—
184#126—34no—
185#126—34no—
186#126—32no—
187#125—31no—
188#124—32no—
189#124—32no—
190#124—31no—
191#124—32no—
192#124—34no—
193#123—34no—
194#123—32no—
195#123—34no—
196#123C2×D314yescomplemented
197C22⋊Q8—31no—
198C22⋊Q8—34no—
199C22⋊Q8—32no—
200C22⋊Q8C2×D312yescomplemented
201#121—31no—
202#121—31no—
203#121—34no—
204#121—34no—
205#121—32no—
206#121—34no—
207#121—34no—
208#121—32no—
209#121—32no—
210#121—31no—
211#121—32no—
212#121—32no—
213#121—34no—
214#121—32no—
215#121—34no—
216#121C2×D312yes—
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—32no—
227C4×D4—32no—
228C4×D4—32no—
229C4×D4—31no—
230C4×D4C2×D311yescharacteristic
231C4×D4C2×D311yescharacteristic, complemented
232#117—32no—
233#117—32no—
234C2×C4⋊C4C2×D312yescomplemented
235C2×C22⋊C4—34no—
236C2×C42—31no—
237C23×C3C2412yes—
238C23×C3C2412yes—
239C22×D3—44no—
240C22×D3—44no—
241C22×D3—44no—
242C22×D3C2412yes—
243C22×D3C2414yes—
244C22×D3C2411yescharacteristic
245C3×Q8—24no—
246C3×D4—22no—
247C3×D4—22no—
248C3×D4—44no—
249C2×C3×C4—24no—
250C2×C3×C4—24no—
251C2×C3×C4—44no—
252C2×C3×C4C2411yescharacteristic
253C2×C3×C4C2414yes—
254C2×C3×C4C2412yes—
255C2×C3×C4C2412yes—
256C2×C3×C4C2411yescharacteristic
257C2×C3×C4C2411yescharacteristic
258#67—44no—
259#67—42no—
260#67—42no—
261#67—44no—
262#67—42no—
263#67—42no—
264#67—44no—
265C2×Dic3C2412yes—
266C2×Dic3C2411yescharacteristic
267C2×Dic3C2411yescharacteristic
268C2×Dic3C2412yes—
269C2×Dic3C2411yescharacteristic
270C2×Dic3C2412yes—
271D12—22no—
272D12—24no—
273D12—24no—
274D12—44no—
275D12—42no—
276D12—42no—
277D12—44no—
278C4×D3—22no—
279C4×D3—24no—
280C4×D3—22no—
281C4×D3—41no—
282C4×D3—44no—
283C4×D3—44no—
284Dic6—22no—
285Dic6—42no—
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—32no—
296C2×D4—34no—
297C2×D4—32no—
298C2×D4—34no—
299C2×D4—34no—
300C2×D4—32no—
301C2×D4—32no—
302C2×D4—31no—
303C2×D4—32no—
304C2×D4—32no—
305C2×D4—32no—
306C2×D4—34no—
307C2×D4—34no—
308C2×D4—124no—
309C2×D4—124no—
310C2×D4C22×D311yescharacteristic
311C2×D4C22×D311yescharacteristic
312C2×D4C22×D314yes—
313C22×C4—31no—
314C22×C4—31no—
315C22×C4—32no—
316C22×C4—31no—
317C22×C4—34no—
318C22×C4—34no—
319C22×C4C22×D312yes—
320C22×C4C22×D312yes—
321C22×C4C22×D314yes—
322C4⋊C4—31no—
323C4⋊C4—32no—
324C4⋊C4—32no—
325C4⋊C4—32no—
326C4⋊C4—31no—
327C4⋊C4—32no—
328C4⋊C4—34no—
329C4⋊C4—32no—
330C4⋊C4C22×D311yescharacteristic
331C4⋊C4C22×D312yes—
332C4⋊C4C22×D314yes—
333C4⋊C4C22×D311yescharacteristic
334C4⋊C4C22×D312yes—
335C22⋊C4—32no—
336C22⋊C4—32no—
337C22⋊C4—32no—
338C22⋊C4—32no—
339C22⋊C4—32no—
340C22⋊C4—34no—
341C22⋊C4—34no—
342C22⋊C4—32no—
343C22⋊C4—34no—
344C22⋊C4—34no—
345C22⋊C4—34no—
346C22⋊C4—34no—
347C22⋊C4—32no—
348C22⋊C4C22×D314yes—
349C22⋊C4C22×D312yes—
350C22⋊C4C22×D312yes—
351C42—31no—
352C42—31no—
353C42—31no—
354C42—32no—
355C42—34no—
356C42C22×D311yescharacteristic
357C22×C3—22no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3—22no—
363C22×C3C2511yescharacteristic
364C2×D3—21no—
365C2×D3—21no—
366C2×D3—22no—
367C2×D3—24no—
368C2×D3—24no—
369C2×D3—22no—
370C2×D3—21no—
371C2×D3—24no—
372C2×D3—22no—
373C2×D3—88no—
374C3×C4—22no—
375C3×C4—24no—
376C3×C4—22no—
377C3×C4—21no—
378C3×C4H(5,𝔽2)12yes—
379C3×C4H(5,𝔽2)12yes—
380Dic3—22no—
381Dic3—21no—
382Dic3—22no—
383Dic3—21no—
384Dic3—21no—
385Dic3—22no—
386C23—32no—
387C23—34no—
388C23—31no—
389C23—124no—
390C23—124no—
391C23—124no—
392C23C23×D312yes—
393C23C23×D312yes—
394Q8—24no—
395Q8—62no—
396Q8—122no—
397D4—22no—
398D4—22no—
399D4—44no—
400D4—62no—
401D4—64no—
402D4—64no—
403D4—124no—
404D4—124no—
405D4—122no—
406D4—122no—
407D4—124no—
408D4—122no—
409D4—122no—
410D4—124no—
411D4—122no—
412D4—124no—
413D4—122no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—32no—
417C2×C4—31no—
418C2×C4—31no—
419C2×C4—31no—
420C2×C4—32no—
421C2×C4—32no—
422C2×C4—44no—
423C2×C4—62no—
424C2×C4—64no—
425C2×C4—62no—
426C2×C4—124no—
427C2×C4—124no—
428C2×C4—121no—
429C2×C4C23×D311yescharacteristic
430C2×C4C23×D314yes—
431C2×C4C23×D312yes—
432C2×C4C23×D312yes—
433C2×C4C23×D311yescharacteristic
434C2×C4C23×D311yescharacteristic
435C2×C3—42no—
436C2×C3—42no—
437C2×C3C2×H(5,𝔽2)11yescharacteristic
438C2×C3C2×H(5,𝔽2)11yescharacteristic
439C2×C3#58511yescharacteristic
440D3—41no—
441D3—44no—
442D3—42no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—61no—
450C22—64no—
451C22—62no—
452C22—61no—
453C22—62no—
454C22—64no—
455C22—61no—
456C22—64no—
457C22—62no—
458C22—248no—
459C22C24×D311yescharacteristic
460C4—22no—
461C4—24no—
462C4—22no—
463C4—21no—
464C4—61no—
465C4—62no—
466C4—61no—
467C4—62no—
468C4—61no—
469C4—62no—
470C4#100412yes—
471C4#100912yes—
472C3#354611yescharacteristic, complemented, Sylow
473C2—42no—
474C2—42no—
475C2—122no—
476C2—124no—
477C2—121no—
478C2C2×#100411yescharacteristic
479C2C2×#100911yescharacteristic
480C2#596811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#547C2
hypercenter#547D3
hypocenterC3#3546
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C3×#547C2semidirect
2. #5834C2semidirect
3. #5830C2semidirect
4. #5732C2semidirect
5. #5725C2semidirect
6. #5666C2semidirect
7. #5657C2semidirect
8. #5643C2semidirect
9. #5630C2semidirect
10. #5612C2semidirect
11. #5607C2semidirect
12. #5557C2semidirect
13. #5553C2semidirect
14. D4×D12C2semidirect
15. #5548C2semidirect
16. #5514C2semidirect
17. #5510C2semidirect
19. C2×C3×#41C22semidirect
20. C2×#239C22semidirect
22. C2×#235C22semidirect
23. C2×#235C22semidirect
25. C3×C4⋊Q8C22semidirect
26. C3×#123C22semidirect
27. C3×C22⋊Q8C22semidirect
30. C3×C4×D4C22semidirect
31. C2×C3×C4⋊C4C22semidirect
33. #947C22semidirect
34. Q8×Dic3C22semidirect
36. #940C22semidirect
37. #940C22semidirect
39. #938C22semidirect
40. #938C22semidirect
41. #937C22semidirect
42. #937C22semidirect
45. #930C22semidirect
46. #930C22semidirect
47. #930C22semidirect
48. #930C22semidirect
49. C4×#67C22semidirect
50. C4×#67C22semidirect
51. C4×#67C22semidirect
52. #926C22semidirect
53. #898C22semidirect
54. #898C22semidirect
55. #897C22semidirect
56. #897C22semidirect
57. #897C22semidirect
60. #895C22semidirect
61. #895C22semidirect
62. #895C22semidirect
63. #894C22semidirect
64. #893C22semidirect
65. #893C22semidirect
66. #893C22semidirect
67. #892C22semidirect
68. #890C22semidirect
69. #888C22semidirect
70. #885C22semidirect
71. #885C22semidirect
72. #884C22semidirect
73. #884C22semidirect
74. #883C22semidirect
75. #883C22semidirect
77. #882C22semidirect
78. D3×C22⋊C4C22semidirect
79. #879C22semidirect
82. C42×D3C22semidirect
83. C4×Dic6C22semidirect
89. #547D3semidirect
175. C2×#41C2×D3semidirect
179. C4⋊Q8C2×D3semidirect
196. #123C2×D3semidirect
200. C22⋊Q8C2×D3semidirect
231. C4×D4C2×D3semidirect
234. C2×C4⋊C4C2×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×#5472
#58342
#58302
#57322
#57252
#56662
#56572
#56432
#56302
#56122
#56072
#55572
#55532
D4×D122
#55482
#55142
#55102
#35463