Finite Group Databasebeta

← #85787

Subgroups — #85787

Order 384.

1152 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D4C211yesmaximal, characteristic, complemented
2#5804C212yesmaximal, complemented
3#5791C212yesmaximal, complemented
4C2×#930C211yesmaximal, characteristic, complemented
5C2×#924C211yesmaximal, characteristic, complemented
6#5565C212yesmaximal, complemented
7#5556C214yesmaximal, complemented
8#5552C212yesmaximal, complemented
9#5550C212yesmaximal, complemented
10#5541C212yesmaximal, complemented
11#5540C214yesmaximal, complemented
12#5515C212yesmaximal, complemented
13#5494C212yesmaximal, complemented
14#5486C212yesmaximal, complemented
15#5479C212yesmaximal, complemented
16#3505—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C23×C3×C4C2211yescharacteristic, complemented
20C22×#67C2212yes—
21C2×#235C2212yescomplemented
22C22×D12C2211yescharacteristic, complemented
23C22×Dic6C2211yescharacteristic, complemented
24C3×C4×D4C2212yes—
25C3×C4×D4C2212yes—
26C3×C4×D4C2212yescomplemented
27C3×C4×D4C2212yescomplemented
28C2×C3×C4⋊C4C2211yescharacteristic, complemented
29C2×C3×C22⋊C4C2211yescharacteristic
30C2×C3×C22⋊C4C2211yescharacteristic, complemented
31C2×C3×C42C2211yescharacteristic, complemented
32#953C2214yescomplemented
33C2×#217C2212yes—
34#939C2214yescomplemented
35#939C2214yescomplemented
36#937C2214yes—
37#935C2214yes—
38#930C2212yes—
39#930C2212yes—
40#930C2212yescomplemented
41#930C2212yescomplemented
42#930C2212yescomplemented
43#929C2214yescomplemented
44#929C2214yescomplemented
45C4×#67C2214yescomplemented
46C4×#67C2214yescomplemented
47C4×#67C2214yescomplemented
48C2×(C2×D3)⋊C4C2212yescomplemented
49#926C2212yescomplemented
50C2×C4⋊Dic3C2211yescharacteristic, complemented
51C2×C4⋊Dic3C2211yescharacteristic, complemented
52#924C2212yescomplemented
53#924C2212yescomplemented
54#924C2212yescomplemented
55#924C2212yescomplemented
56#924C2212yescomplemented
57C2×Dic3⋊C4C2212yescomplemented
58#898C2214yescomplemented
59#896C2214yescomplemented
60#894C2214yescomplemented
61#889C2214yescomplemented
62#885C2214yescomplemented
63#884C2214yescomplemented
64#883C2214yescomplemented
65#67⋊C4C2214yescomplemented
66#67⋊C4C2214yescomplemented
67D3×C22⋊C4C2214yescomplemented
68#879C2214yescomplemented
69#877C2214yescomplemented
70#876C2214yescomplemented
71#875C2212yescomplemented
72C4×D12C2212yescomplemented
73#872C2214yescomplemented
74#870C2212yescomplemented
75C4×Dic6C2212yescomplemented
76#553—34no—
77#553—32no—
78#552—32no—
79#551—34no—
80#548—32no—
81#546—32no—
82#539—32no—
83#538—32no—
84#534—32no—
85#529—32no—
86#525—32no—
87C2×C22⋊Q8—31no—
88C2×#121—31no—
89#518—32no—
90C2×C4×D4D311yescharacteristic, complemented
91C24×C3C2311yescharacteristic
92C24×C3C2311yescharacteristic
93C23×D3C2312yes—
94C2×C3×D4—24no—
95C2×C3×D4C2312yes—
96C2×C3×D4C2312yes—
97C22×C3×C4—42no—
98C22×C3×C4—42no—
99C22×C3×C4C2311yescharacteristic
100C22×C3×C4C2311yescharacteristic
101C22×C3×C4C2312yes—
102C22×C3×C4C2311yescharacteristic
103C22×C3×C4C2311yescharacteristic
104C22×C3×C4C2311yescharacteristic
105C22×C3×C4C2311yescharacteristic
106C22×C3×C4C2311yescharacteristic
107C22×C3×C4C2311yescharacteristic
108C22×C3×C4C2311yescharacteristic
109C22×C3×C4C2312yes—
110C2×#67—28no—
111C2×#67C2314yes—
112C2×#67C2314yes—
113C2×#67C2314yes—
114C2×#67C2314yes—
115C22×Dic3C2312yes—
116C22×Dic3C2312yes—
117C22×Dic3C2312yes—
118#235—44no—
119C2×D12—42no—
120C2×D12C2312yes—
121C2×D12C2312yes—
122C2×C4×D3C2314yes—
123C2×C4×D3C2314yes—
124C2×Dic6—42no—
125C2×Dic6C2312yes—
126C2×Dic6C2312yes—
127C3×C4⋊C4C2312yes—
128C3×C4⋊C4C2312yes—
129C3×C22⋊C4C2312yes—
130C3×C22⋊C4C2312yes—
131C3×C22⋊C4C2312yes—
132C3×C22⋊C4C2312yes—
133C3×C42C2312yes—
134C3×C42C2312yes—
135#217C2314yes—
136#217C2314yes—
137#217C2314yes—
138#217C2314yes—
139(C2×D3)⋊C4C2314yes—
140(C2×D3)⋊C4C2314yes—
141(C2×D3)⋊C4C2314yes—
142(C2×D3)⋊C4C2314yes—
143(C2×D3)⋊C4C2314yes—
144(C2×D3)⋊C4C2314yes—
145C4⋊Dic3C2312yes—
146C4⋊Dic3C2312yes—
147C4⋊Dic3C2312yes—
148C4⋊Dic3C2312yes—
149Dic3⋊C4C2314yes—
150Dic3⋊C4C2314yes—
151Dic3⋊C4C2314yes—
152Dic3⋊C4C2314yes—
153Dic3⋊C4C2314yes—
154Dic3⋊C4C2314yes—
155C4×Dic3C2314yes—
156C4×Dic3C2314yes—
157C2×#41—32no—
158C22×Q8—31no—
159C22×D4—32no—
160C22×D4—31no—
161C22×D4C2×D311yescharacteristic
162C23×C4C2×D311yescharacteristic
163C23×C4C2×D311yescharacteristic, complemented
164#126—34no—
165#126—34no—
166#126—34no—
167#125—34no—
168#125—32no—
169#124—34no—
170#124—32no—
171#123—34no—
172#123—34no—
173#123—34no—
174#123—34no—
175#123—34no—
176C22⋊Q8—32no—
177C22⋊Q8—32no—
178C22⋊Q8—32no—
179C22⋊Q8—32no—
180C22⋊Q8—32no—
181C22⋊Q8—34no—
182#121—34no—
183#121—34no—
184#121—32no—
185#121—32no—
186#121—32no—
187#121—32no—
188#121—32no—
189#121—34no—
190#120—34no—
191#120—34no—
192C4×Q8—32no—
193C4×D4—34no—
194C4×D4—34no—
195C4×D4—34no—
196C4×D4—34no—
197C4×D4—34no—
198C4×D4—32no—
199C4×D4C2×D312yes—
200C4×D4C2×D312yes—
201C4×D4C2×D312yescomplemented
202C4×D4C2×D312yescomplemented
203#117—32no—
204#117—34no—
205#117—34no—
206C2×C4⋊C4—31no—
207C2×C4⋊C4—32no—
208C2×C4⋊C4—31no—
209C2×C4⋊C4C2×D311yescharacteristic, complemented
210C2×C22⋊C4—32no—
211C2×C22⋊C4—32no—
212C2×C22⋊C4—34no—
213C2×C22⋊C4C2×D311yescharacteristic
214C2×C22⋊C4C2×D311yescharacteristic, complemented
215C2×C42C2×D311yescharacteristic, complemented
216C23×C3—22no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3#4114yescomplemented
222C23×C3C2412yes—
223C23×C3C2411yescharacteristic
224C23×C3C2411yescharacteristic
225C23×C3C2411yescharacteristic
226C22×D3—24no—
227C22×D3—42no—
228C22×D3—42no—
229C22×D3C2414yes—
230C3×D4—24no—
231C3×D4—24no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—24no—
236C2×C3×C4—22no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—42no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4C2412yes—
244C2×C3×C4C2412yes—
245C2×C3×C4C2411yescharacteristic
246C2×C3×C4C2412yes—
247C2×C3×C4C2411yescharacteristic
248C2×C3×C4C2411yescharacteristic
249C2×C3×C4C2411yescharacteristic
250#67—28no—
251#67—28no—
252#67—44no—
253#67—44no—
254C2×Dic3—24no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3C2414yes—
258C2×Dic3C2414yes—
259C2×Dic3C2414yes—
260D12—24no—
261D12—42no—
262C4×D3—44no—
263C4×D3—44no—
264Dic6—24no—
265Dic6—42no—
266C24—32no—
267C24C22×D311yescharacteristic
268C24C22×D311yescharacteristic
269#41—124no—
270C2×Q8—32no—
271C2×Q8—32no—
272C2×Q8—122no—
273C2×D4—24no—
274C2×D4—34no—
275C2×D4—34no—
276C2×D4—34no—
277C2×D4—34no—
278C2×D4—32no—
279C2×D4—32no—
280C2×D4—68no—
281C2×D4—122no—
282C2×D4C22×D312yes—
283C2×D4C22×D312yes—
284C22×C4—32no—
285C22×C4—32no—
286C22×C4—32no—
287C22×C4—34no—
288C22×C4—34no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4C22×D311yescharacteristic
292C22×C4C22×D311yescharacteristic
293C22×C4C22×D311yescharacteristic
294C22×C4C22×D311yescharacteristic
295C22×C4C22×D312yes—
296C22×C4C22×D311yescharacteristic
297C22×C4C22×D311yescharacteristic
298C22×C4C22×D311yescharacteristic
299C22×C4C22×D311yescharacteristic
300C22×C4C22×D311yescharacteristic
301C22×C4C22×D312yes—
302C4⋊C4—32no—
303C4⋊C4—34no—
304C4⋊C4—32no—
305C4⋊C4—32no—
306C4⋊C4—34no—
307C4⋊C4—32no—
308C4⋊C4—34no—
309C4⋊C4—34no—
310C4⋊C4—34no—
311C4⋊C4—34no—
312C4⋊C4C22×D312yes—
313C4⋊C4C22×D312yes—
314C22⋊C4—34no—
315C22⋊C4—34no—
316C22⋊C4—34no—
317C22⋊C4—34no—
318C22⋊C4—34no—
319C22⋊C4—34no—
320C22⋊C4—34no—
321C22⋊C4—34no—
322C22⋊C4—34no—
323C22⋊C4—34no—
324C22⋊C4C22×D312yes—
325C22⋊C4C22×D312yes—
326C22⋊C4C22×D312yes—
327C22⋊C4C22×D312yes—
328C42—34no—
329C42—34no—
330C42C22×D312yes—
331C42C22×D312yes—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—22no—
339C22×C3—22no—
340C22×C3—22no—
341C22×C3—44no—
342C22×C3—44no—
343C22×C3C2×#4112yes—
344C22×C3C2×#4112yes—
345C22×C3C2×#4112yes—
346C22×C3C2511yescharacteristic
347C2×D3—24no—
348C2×D3—24no—
349C2×D3—24no—
350C2×D3—84no—
351C3×C4—22no—
352C3×C4—21no—
353C3×C4—22no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—22no—
357C3×C4H(5,𝔽2)12yes—
358Dic3—24no—
359Dic3—24no—
360Dic3—24no—
361C23—22no—
362C23—34no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—64no—
368C23—122no—
369C23—122no—
370C23#23514yescomplemented
371C23C23×D312yes—
372C23C23×D311yescharacteristic
373C23C23×D311yescharacteristic
374C23C23×D311yescharacteristic
375Q8—64no—
376Q8—122no—
377D4—24no—
378D4—24no—
379D4—68no—
380D4—68no—
381D4—64no—
382D4—122no—
383D4—124no—
384D4—124no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—34no—
391C2×C4—34no—
392C2×C4—34no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—64no—
400C2×C4—124no—
401C2×C4—124no—
402C2×C4—122no—
403C2×C4—122no—
404C2×C4C23×D312yes—
405C2×C4C23×D312yes—
406C2×C4C23×D312yes—
407C2×C4C23×D311yescharacteristic
408C2×C4C23×D311yescharacteristic
409C2×C4C23×D311yescharacteristic
410C2×C4C23×D311yescharacteristic
411C2×C3—22no—
412C2×C3—22no—
413C2×C3—22no—
414C2×C3—42no—
415C2×C3C22×#4111yescharacteristic
416C2×C3C2×H(5,𝔽2)11yescharacteristic
417C2×C3#58511yescharacteristic
418D3—44no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—44no—
429C22—44no—
430C22—64no—
431C22—64no—
432C22—64no—
433C22—244no—
434C22C2×#23512yes—
435C22C2×#23512yes—
436C22C2×#23512yes—
437C22C24×D311yescharacteristic
438C4—22no—
439C4—21no—
440C4—22no—
441C4—21no—
442C4—21no—
443C4—22no—
444C4—64no—
445C4—64no—
446C4—64no—
447C4#100412yes—
448C3#350511yescharacteristic, complemented, Sylow
449C2—22no—
450C2—22no—
451C2—22no—
452C2—42no—
453C2—124no—
454C2C22×#23511yescharacteristic
455C2C2×#100411yescharacteristic
456C2#596411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC2×C3×C4×D4C2
hypercenterC2×C4×D4D3
hypocenterC3#3505
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×D4C2semidirect
2. #5804C2semidirect
3. #5791C2semidirect
4. C2×#930C2semidirect
5. C2×#924C2semidirect
6. #5565C2semidirect
7. #5556C2semidirect
8. #5552C2semidirect
9. #5550C2semidirect
10. #5541C2semidirect
11. #5540C2semidirect
12. #5515C2semidirect
13. #5494C2semidirect
14. #5486C2semidirect
15. #5479C2semidirect
19. C23×C3×C4C22semidirect
21. C2×#235C22semidirect
22. C22×D12C22semidirect
23. C22×Dic6C22semidirect
26. C3×C4×D4C22semidirect
27. C3×C4×D4C22semidirect
28. C2×C3×C4⋊C4C22semidirect
30. C2×C3×C22⋊C4C22semidirect
31. C2×C3×C42C22semidirect
32. #953C22semidirect
34. #939C22semidirect
35. #939C22semidirect
40. #930C22semidirect
41. #930C22semidirect
42. #930C22semidirect
43. #929C22semidirect
44. #929C22semidirect
45. C4×#67C22semidirect
46. C4×#67C22semidirect
47. C4×#67C22semidirect
48. C2×(C2×D3)⋊C4C22semidirect
49. #926C22semidirect
50. C2×C4⋊Dic3C22semidirect
51. C2×C4⋊Dic3C22semidirect
52. #924C22semidirect
53. #924C22semidirect
54. #924C22semidirect
55. #924C22semidirect
56. #924C22semidirect
57. C2×Dic3⋊C4C22semidirect
58. #898C22semidirect
59. #896C22semidirect
60. #894C22semidirect
61. #889C22semidirect
62. #885C22semidirect
63. #884C22semidirect
64. #883C22semidirect
65. #67⋊C4C22semidirect
66. #67⋊C4C22semidirect
67. D3×C22⋊C4C22semidirect
68. #879C22semidirect
69. #877C22semidirect
70. #876C22semidirect
71. #875C22semidirect
72. C4×D12C22semidirect
73. #872C22semidirect
74. #870C22semidirect
75. C4×Dic6C22semidirect
90. C2×C4×D4D3semidirect
163. C23×C4C2×D3semidirect
201. C4×D4C2×D3semidirect
202. C4×D4C2×D3semidirect
209. C2×C4⋊C4C2×D3semidirect
214. C2×C22⋊C4C2×D3semidirect
215. C2×C42C2×D3semidirect
221. C23×C3#41semidirect
370. C23#235semidirect
448. C3#3505semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D42
#58042
#57912
C2×#9302
C2×#9242
#55652
#55562
#55522
#55502
#55412
#55402
#55152
#54942
#54862
#54792
#35053