Finite Group Databasebeta

← #86247

Subgroups — #86247

Order 384.

1252 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#532C211yesmaximal, characteristic, complemented
2#5829C214yesmaximal, complemented
3#5734C211yesmaximal, characteristic, complemented
4#5730C211yesmaximal, characteristic, complemented
5#5718C211yesmaximal, characteristic, complemented
6#5717C211yesmaximal, characteristic, complemented
7#5678C214yesmaximal, complemented
8#5618C212yesmaximal, complemented
9#5609C212yesmaximal, complemented
10#5606C212yesmaximal, complemented
11D3×#121C212yesmaximal, complemented
12#5562C212yesmaximal, complemented
13#5558C212yesmaximal, complemented
14#5554C212yesmaximal, complemented
15#5545C212yesmaximal, complemented
16#5479C211yesmaximal, characteristic, complemented
17C2×C4×D12C211yesmaximal, characteristic, complemented
18#3519—31nomaximal, Sylow
19C2×C3×#41C2212yescomplemented
20C2×#237C2212yescomplemented
21C2×D3×D4C2212yes—
22C2×D3×D4C2212yescomplemented
23C2×D3×D4C2214yescomplemented
24C22×D12C2211yescharacteristic
25C22×C4×D3C2212yescomplemented
26C3×C4⋊Q8C2211yescharacteristic, complemented
27C3×#127C2211yescharacteristic, complemented
28C3×#124C2212yescomplemented
29C3×#121C2212yescomplemented
30C3×#121C2212yescomplemented
31C3×C4×D4C2212yescomplemented
32C3×C4×D4C2212yescomplemented
33C2×C3×C42C2211yescharacteristic, complemented
34#947C2214yescomplemented
35#946C2214yescomplemented
36#940C2212yescomplemented
37#939C2214yescomplemented
38#939C2214yescomplemented
39#938C2212yescomplemented
40#938C2212yescomplemented
41#938C2212yescomplemented
42#937C2214yescomplemented
43#937C2214yescomplemented
44D4×Dic3C2212yescomplemented
45D4×Dic3C2214yescomplemented
46#930C2211yescharacteristic
47#930C2214yescomplemented
48#929C2212yescomplemented
49#929C2214yescomplemented
50C4×#67C2212yescomplemented
51C4×#67C2214yescomplemented
52C2×(C2×D3)⋊C4C2212yescomplemented
53C2×(C2×D3)⋊C4C2214yescomplemented
54C2×C4⋊Dic3C2211yescharacteristic, complemented
55#924C2211yescharacteristic, complemented
56#897C2212yescomplemented
57#895C2212yescomplemented
58#895C2212yescomplemented
59#893C2212yes—
60#892C2212yescomplemented
61#892C2212yescomplemented
62D3×C4⋊C4C2212yescomplemented
63#890C2212yescomplemented
64#886C2214yescomplemented
65#884C2214yescomplemented
66#884C2214yescomplemented
67#883C2214yescomplemented
68#882C2214yescomplemented
69#67⋊C4C2214yescomplemented
70D3×C22⋊C4C2214yescomplemented
71#879C2214yescomplemented
72#876C2212yescomplemented
73#875C2211yescharacteristic, complemented
74C4×D12C2211yescharacteristic
75C4×D12C2211yescharacteristic
76C4×D12C2212yescomplemented
77C4×D12C2211yescharacteristic, complemented
78C4×D12C2211yescharacteristic, complemented
79C4×D12C2212yescomplemented
80C4×D12C2211yescharacteristic, complemented
81#872C2212yescomplemented
82#870C2211yescharacteristic, complemented
83C4×Dic6C2211yescharacteristic, complemented
84#555—31no—
85#554—31no—
86#553—32no—
87#553—32no—
88#550—31no—
89#546—34no—
90#546—32no—
91D42—31no—
92D42—32no—
93#540—32no—
94#538—34no—
95#532D311yescharacteristic, complemented
96#529—31no—
97#525—32no—
98C2×#121—32no—
99#518—32no—
100C2×C4×D4—31no—
101C23×D3C2312yes—
102C23×D3C2314yes—
103C3×#41—44no—
104C2×C3×Q8C2312yes—
105C2×C3×D4C2314yes—
106C2×C3×D4C2312yes—
107C2×C3×D4C2312yes—
108C2×C3×D4C2312yes—
109C22×C3×C4C2311yescharacteristic
110C22×C3×C4C2311yescharacteristic
111C22×C3×C4C2311yescharacteristic
112C22×C3×C4C2314yes—
113C2×#67C2314yes—
114C2×#67C2312yes—
115C2×#67C2314yes—
116C2×#67C2314yes—
117C2×#67C2314yes—
118C22×Dic3C2314yes—
119C22×Dic3C2312yes—
120#237—44no—
121D3×D4—28no—
122D3×D4—44no—
123D3×D4—48no—
124C2×D12—24no—
125C2×D12C2312yes—
126C2×D12C2311yescharacteristic
127C2×D12C2311yescharacteristic
128C2×D12C2314yes—
129C2×D12C2311yescharacteristic
130C2×C4×D3—44no—
131C2×C4×D3C2312yes—
132C2×C4×D3C2312yes—
133C2×C4×D3C2312yes—
134C2×C4×D3C2312yes—
135C2×C4×D3C2312yes—
136C2×C4×D3C2314yes—
137C2×Dic6C2311yescharacteristic
138C3×C4⋊C4C2312yes—
139C3×C4⋊C4C2312yes—
140C3×C22⋊C4C2314yes—
141C3×C22⋊C4C2314yes—
142C3×C42C2311yescharacteristic
143C3×C42C2311yescharacteristic
144C3×C42C2311yescharacteristic
145C3×C42C2311yescharacteristic
146#217C2314yes—
147#217C2314yes—
148#217C2312yes—
149(C2×D3)⋊C4C2312yes—
150(C2×D3)⋊C4C2312yes—
151(C2×D3)⋊C4C2312yes—
152(C2×D3)⋊C4C2312yes—
153(C2×D3)⋊C4C2312yes—
154(C2×D3)⋊C4C2312yes—
155(C2×D3)⋊C4C2312yes—
156(C2×D3)⋊C4C2312yes—
157(C2×D3)⋊C4C2314yes—
158(C2×D3)⋊C4C2314yes—
159(C2×D3)⋊C4C2314yes—
160C4⋊Dic3C2311yescharacteristic
161C4⋊Dic3C2311yescharacteristic
162C4⋊Dic3C2311yescharacteristic
163C4⋊Dic3C2311yescharacteristic
164C4⋊Dic3C2312yes—
165C4⋊Dic3C2314yes—
166Dic3⋊C4C2312yes—
167Dic3⋊C4C2312yes—
168Dic3⋊C4C2312yes—
169Dic3⋊C4C2312yes—
170Dic3⋊C4C2314yes—
171C4×Dic3C2312yes—
172C4×Dic3C2314yes—
173C2×#41—32no—
174C2×#41C2×D312yescomplemented
175C22×D4—32no—
176C22×D4—32no—
177C22×D4—34no—
178C22×D4—31no—
179C23×C4—32no—
180C4⋊Q8C2×D311yescharacteristic, complemented
181#127—32no—
182#127C2×D311yescharacteristic, complemented
183#126—34no—
184#126—32no—
185#125—32no—
186#125—31no—
187#124—34no—
188#124—31no—
189#124C2×D312yescomplemented
190#123—32no—
191#123—34no—
192#123—34no—
193#123—34no—
194C22⋊Q8—34no—
195C22⋊Q8—31no—
196C22⋊Q8—32no—
197#121—34no—
198#121—32no—
199#121—32no—
200#121—34no—
201#121—32no—
202#121—31no—
203#121—34no—
204#121—32no—
205#121—32no—
206#121—34no—
207#121—34no—
208#121C2×D312yescomplemented
209#121C2×D312yescomplemented
210#120—34no—
211#120—34no—
212#120—34no—
213C4×Q8—31no—
214C4×D4—32no—
215C4×D4—34no—
216C4×D4—34no—
217C4×D4—32no—
218C4×D4—32no—
219C4×D4—34no—
220C4×D4—31no—
221C4×D4—32no—
222C4×D4—31no—
223C4×D4—31no—
224C4×D4—32no—
225C4×D4—31no—
226C4×D4—31no—
227C4×D4C2×D312yescomplemented
228C4×D4C2×D312yescomplemented
229#117—32no—
230#117—32no—
231#117—32no—
232C2×C4⋊C4—31no—
233C2×C4⋊C4—32no—
234C2×C22⋊C4—32no—
235C2×C22⋊C4—34no—
236C2×C22⋊C4—34no—
237C2×C42C2×D311yescharacteristic, complemented
238C23×C3C2414yes—
239C23×C3C2411yescharacteristic
240C22×D3—24no—
241C22×D3—24no—
242C22×D3—28no—
243C22×D3—42no—
244C22×D3—44no—
245C22×D3—44no—
246C22×D3C2412yes—
247C22×D3C2412yes—
248C22×D3C2412yes—
249C3×Q8—42no—
250C3×D4—28no—
251C3×D4—24no—
252C3×D4—24no—
253C3×D4—42no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—44no—
258C2×C3×C4C2414yes—
259C2×C3×C4C2411yescharacteristic
260C2×C3×C4C2411yescharacteristic
261C2×C3×C4C2411yescharacteristic
262C2×C3×C4C2411yescharacteristic
263C2×C3×C4C2411yescharacteristic
264C2×C3×C4C2411yescharacteristic
265#67—24no—
266#67—28no—
267#67—44no—
268#67—44no—
269#67—44no—
270C2×Dic3—42no—
271C2×Dic3—44no—
272C2×Dic3C2412yes—
273C2×Dic3C2412yes—
274C2×Dic3C2412yes—
275C2×Dic3C2412yes—
276C2×Dic3C2412yes—
277D12—24no—
278D12—22no—
279D12—22no—
280D12—44no—
281D12#4114yescomplemented
282C4×D3—24no—
283C4×D3—24no—
284C4×D3—24no—
285C4×D3—42no—
286C4×D3—42no—
287C4×D3—44no—
288Dic6—22no—
289C24—32no—
290C24—34no—
291#41—44no—
292#41—124no—
293C2×Q8—31no—
294C2×Q8C22×D312yes—
295C2×D4—34no—
296C2×D4—32no—
297C2×D4—34no—
298C2×D4—34no—
299C2×D4—34no—
300C2×D4—31no—
301C2×D4—32no—
302C2×D4—31no—
303C2×D4—31no—
304C2×D4—34no—
305C2×D4—64no—
306C2×D4—68no—
307C2×D4—124no—
308C2×D4—128no—
309C2×D4C22×D312yes—
310C2×D4C22×D314yes—
311C2×D4C22×D312yes—
312C2×D4C22×D312yes—
313C22×C4—32no—
314C22×C4—34no—
315C22×C4—32no—
316C22×C4—32no—
317C22×C4—32no—
318C22×C4—32no—
319C22×C4—34no—
320C22×C4—32no—
321C22×C4—124no—
322C22×C4C22×D311yescharacteristic
323C22×C4C22×D311yescharacteristic
324C22×C4C22×D311yescharacteristic
325C22×C4C22×D314yes—
326C4⋊C4—31no—
327C4⋊C4—31no—
328C4⋊C4—31no—
329C4⋊C4—32no—
330C4⋊C4—34no—
331C4⋊C4—31no—
332C4⋊C4—32no—
333C4⋊C4—32no—
334C4⋊C4—32no—
335C4⋊C4—34no—
336C4⋊C4—32no—
337C4⋊C4C22×D312yes—
338C4⋊C4C22×D312yes—
339C22⋊C4—34no—
340C22⋊C4—32no—
341C22⋊C4—34no—
342C22⋊C4—32no—
343C22⋊C4—32no—
344C22⋊C4—34no—
345C22⋊C4—32no—
346C22⋊C4—34no—
347C22⋊C4—32no—
348C22⋊C4—32no—
349C22⋊C4—32no—
350C22⋊C4—34no—
351C22⋊C4—32no—
352C22⋊C4—32no—
353C22⋊C4C22×D314yes—
354C22⋊C4C22×D314yes—
355C42—34no—
356C42—32no—
357C42C22×D311yescharacteristic
358C42C22×D311yescharacteristic
359C42C22×D311yescharacteristic
360C42C22×D311yescharacteristic
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—24no—
364C22×C3—24no—
365C22×C3—24no—
366C22×C3H(5,𝔽2)12yes—
367C22×C3C2511yescharacteristic
368C2×D3—22no—
369C2×D3—22no—
370C2×D3—22no—
371C2×D3—22no—
372C2×D3—22no—
373C2×D3—22no—
374C2×D3—22no—
375C2×D3—22no—
376C2×D3—48no—
377C2×D3—88no—
378C2×D3C2×#4114yes—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—24no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4C2×#4112yes—
385C3×C4H(5,𝔽2)12yes—
386Dic3—22no—
387Dic3—22no—
388Dic3—22no—
389Dic3—22no—
390Dic3—22no—
391C23—32no—
392C23—32no—
393C23—32no—
394C23—64no—
395C23—68no—
396C23—64no—
397C23—122no—
398C23—124no—
399C23—124no—
400C23C23×D314yes—
401C23C23×D311yescharacteristic
402Q8—42no—
403Q8—62no—
404D4—28no—
405D4—24no—
406D4—24no—
407D4—34no—
408D4—42no—
409D4—64no—
410D4—68no—
411D4—64no—
412D4—62no—
413D4—62no—
414D4—124no—
415D4—124no—
416D4—124no—
417D4—124no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—32no—
422C2×C4—32no—
423C2×C4—32no—
424C2×C4—32no—
425C2×C4—32no—
426C2×C4—44no—
427C2×C4—64no—
428C2×C4—64no—
429C2×C4—64no—
430C2×C4—122no—
431C2×C4—122no—
432C2×C4—124no—
433C2×C4—122no—
434C2×C4—124no—
435C2×C4C23×D314yes—
436C2×C4C23×D311yescharacteristic
437C2×C4C23×D311yescharacteristic
438C2×C4C23×D311yescharacteristic
439C2×C4C23×D311yescharacteristic
440C2×C4C23×D311yescharacteristic
441C2×C4C23×D311yescharacteristic
442C2×C3—22no—
443C2×C3—44no—
444C2×C3C22×#4111yescharacteristic
445C2×C3C2×H(5,𝔽2)11yescharacteristic
446C2×C3C2×H(5,𝔽2)11yescharacteristic
447D3—24no—
448D3—42no—
449D3—42no—
450C22—21no—
451C22—21no—
452C22—24no—
453C22—24no—
454C22—24no—
455C22—34no—
456C22—62no—
457C22—62no—
458C22—62no—
459C22—62no—
460C22—62no—
461C22—62no—
462C22—62no—
463C22—62no—
464C22—128no—
465C22—248no—
466C22#100412yes—
467C22C24×D311yescharacteristic
468C4—21no—
469C4—21no—
470C4—24no—
471C4—21no—
472C4—21no—
473C4—62no—
474C4—62no—
475C4—62no—
476C4—62no—
477C4—62no—
478C4D3×#4112yes—
479C4#100912yes—
480C3#351911yescharacteristic, complemented, Sylow
481C2—22no—
482C2—44no—
483C2—64no—
484C2—122no—
485C2—122no—
486C2C2×#100411yescharacteristic
487C2C2×D3×#4111yescharacteristic
488C2C2×#100911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#532C2
hypercenter#532D3
hypocenterC3#3519
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. #5829C2semidirect
3. #5734C2semidirect
4. #5730C2semidirect
5. #5718C2semidirect
6. #5717C2semidirect
7. #5678C2semidirect
8. #5618C2semidirect
9. #5609C2semidirect
10. #5606C2semidirect
11. D3×#121C2semidirect
12. #5562C2semidirect
13. #5558C2semidirect
14. #5554C2semidirect
15. #5545C2semidirect
16. #5479C2semidirect
17. C2×C4×D12C2semidirect
19. C2×C3×#41C22semidirect
20. C2×#237C22semidirect
22. C2×D3×D4C22semidirect
23. C2×D3×D4C22semidirect
25. C22×C4×D3C22semidirect
26. C3×C4⋊Q8C22semidirect
27. C3×#127C22semidirect
28. C3×#124C22semidirect
29. C3×#121C22semidirect
30. C3×#121C22semidirect
31. C3×C4×D4C22semidirect
32. C3×C4×D4C22semidirect
33. C2×C3×C42C22semidirect
34. #947C22semidirect
35. #946C22semidirect
36. #940C22semidirect
37. #939C22semidirect
38. #939C22semidirect
39. #938C22semidirect
40. #938C22semidirect
41. #938C22semidirect
42. #937C22semidirect
43. #937C22semidirect
44. D4×Dic3C22semidirect
45. D4×Dic3C22semidirect
47. #930C22semidirect
48. #929C22semidirect
49. #929C22semidirect
50. C4×#67C22semidirect
51. C4×#67C22semidirect
52. C2×(C2×D3)⋊C4C22semidirect
53. C2×(C2×D3)⋊C4C22semidirect
54. C2×C4⋊Dic3C22semidirect
55. #924C22semidirect
56. #897C22semidirect
57. #895C22semidirect
58. #895C22semidirect
60. #892C22semidirect
61. #892C22semidirect
62. D3×C4⋊C4C22semidirect
63. #890C22semidirect
64. #886C22semidirect
65. #884C22semidirect
66. #884C22semidirect
67. #883C22semidirect
68. #882C22semidirect
69. #67⋊C4C22semidirect
70. D3×C22⋊C4C22semidirect
71. #879C22semidirect
72. #876C22semidirect
73. #875C22semidirect
76. C4×D12C22semidirect
77. C4×D12C22semidirect
78. C4×D12C22semidirect
79. C4×D12C22semidirect
80. C4×D12C22semidirect
81. #872C22semidirect
82. #870C22semidirect
83. C4×Dic6C22semidirect
95. #532D3semidirect
174. C2×#41C2×D3semidirect
180. C4⋊Q8C2×D3semidirect
182. #127C2×D3semidirect
189. #124C2×D3semidirect
208. #121C2×D3semidirect
209. #121C2×D3semidirect
227. C4×D4C2×D3semidirect
228. C4×D4C2×D3semidirect
237. C2×C42C2×D3semidirect
281. D12#41semidirect
480. C3#3519semidirect

Maximal subgroups

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

subgroupindex
C3×#5322
#58292
#57342
#57302
#57182
#57172
#56782
#56182
#56092
#56062
D3×#1212
#55622
#55582
#55542
#55452
#54792
C2×C4×D122
#35193