Finite Group Databasebeta

← #60430

Subgroups — #60430

Order 256.

1381 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#577C211yesmaximal, characteristic, complemented
2C2×#576C211yesmaximal, characteristic, complemented
3#3500C211yesmaximal, characteristic, complemented
4C4×H(5,𝔽2)C211yesmaximal, characteristic, complemented
5#3046C213yesmaximal, complemented
6#3045C213yesmaximal, complemented
7#3042C213yesmaximal, complemented
8#3041C213yesmaximal, complemented
9#3040C213yesmaximal, complemented
10#3036C213yesmaximal, complemented
11#3035C213yesmaximal, complemented
12C2×#463C213yesmaximal, complemented
13#2906C211yesmaximal, characteristic, complemented
14#2898C211yesmaximal, characteristic, complemented
15#2897C211yesmaximal, characteristic, complemented
16C2×2−1+4C2211yescharacteristic
17C2×H(5,𝔽2)C2211yescharacteristic
18C2×H(5,𝔽2)C2211yescharacteristic
19#577—28no—
20#576—28no—
21C2×#137C2213yescomplemented
22C2×AGL(1,ℤ/8)C2213yes—
23C2×AGL(1,ℤ/8)C2213yes—
24C2×AGL(1,ℤ/8)C2213yescomplemented
25C2×#135C2213yes—
26C2×#135C2213yescomplemented
27C22×SD16C2213yescomplemented
28C22×D8C2213yes—
29C2×#131C2211yescharacteristic
30C2×#131C2211yescharacteristic
31#548—21no—
32#548—21no—
33#546—23no—
34#546—23no—
35#536C2213yescomplemented
36#535C2213yescomplemented
37#532C2213yescomplemented
38C2×#124C2213yescomplemented
39#520—21no—
40#520C2211yescharacteristic
41#520C2213yescomplemented
42#518—23no—
43#518C2213yescomplemented
44C4×#41—21no—
45C4×#41C2211yescharacteristic
46C4×#41C2213yescomplemented
47C2×C4×D4—23no—
48C2×C4×D4C2213yescomplemented
49#464C2211yescharacteristic, complemented
50#464C2213yescomplemented
51#464C2216yescomplemented
52#463C2213yescomplemented
53#463C2216yescomplemented
54#463C2216yescomplemented
55#463C2216yescomplemented
56#463C2213yescomplemented
57#463C2216yescomplemented
58#462C2213yescomplemented
59#461C2213yescomplemented
60#461C2216yescomplemented
61#460C2213yescomplemented
62#459C2213yescomplemented
63#459C2216yescomplemented
64#424C2213yescomplemented
65#424C2213yescomplemented
66#423C2213yescomplemented
67C2×C4⋊C8C2213yescomplemented
68#419C2213yescomplemented
69#418C2213yescomplemented
70#417C2213yescomplemented
71#417C2213yescomplemented
72#416C2213yescomplemented
73#416C2213yescomplemented
74C2×#103C2213yescomplemented
75C2×#102C2213yescomplemented
762−1+4—24no—
772−1+4—24no—
78H(5,𝔽2)—24no—
79H(5,𝔽2)—24no—
80H(5,𝔽2)—24no—
81H(5,𝔽2)D418yescomplemented
82C2×#41—21no—
83C2×#41—21no—
84C2×#41—23no—
85C2×#41C2313yes—
86C2×#41C2313yes—
87C2×#41C2313yes—
88C2×#41C2311yescharacteristic
89C22×Q8—21no—
90C22×Q8C2313yes—
91C22×D4—23no—
92C22×D4—23no—
93C22×D4C2313yes—
94C22×D4C2313yes—
95C23×C4—23no—
96#137—212no—
97AGL(1,ℤ/8)—212no—
98AGL(1,ℤ/8)—212no—
99AGL(1,ℤ/8)—212no—
100#135—212no—
101#135—212no—
102C2×Q16C2311yescharacteristic
103C2×Q16C2313yes—
104C2×SD16—212no—
105C2×SD16C2313yes—
106C2×SD16C2316yes—
107C2×SD16C2313yes—
108C2×SD16C2313yes—
109C2×SD16C2311yescharacteristic
110C2×D8—212no—
111C2×D8C2313yes—
112C2×D8C2316yes—
113C2×D8C2313yes—
114#131—24no—
115#131—24no—
116C2×M16C2313yes—
117C2×M16C2313yes—
118C22×C8C2313yes—
119C22×C8C2313yes—
120C4⋊Q8C2313yes—
121#127C2313yes—
122#124—23no—
123#124—23no—
124#124C2313yes—
125#124C2316yes—
126#124C2311yescharacteristic
127#123—26no—
128C22⋊Q8—23no—
129C22⋊Q8—23no—
130#121—23no—
131#121—23no—
132#121—23no—
133#121—23no—
134#120—26no—
135C4×Q8—21no—
136C4×Q8D411yescharacteristic
137C4×Q8D411yescharacteristic
138C4×Q8C2311yescharacteristic
139C4×Q8C2313yes—
140C4×D4—23no—
141C4×D4—23no—
142C4×D4—23no—
143C4×D4—23no—
144C4×D4—23no—
145C4×D4—26no—
146C4×D4—23no—
147C4×D4—23no—
148C4×D4—23no—
149C4×D4—23no—
150C4×D4D413yes—
151C4×D4D413yes—
152C4×D4C2313yes—
153C4×D4C2316yes—
154C4×D4C2313yes—
155#117—23no—
156#117—23no—
157#117C2313yes—
158#117C2313yes—
159C2×C4⋊C4—23no—
160C2×C4⋊C4C2313yes—
161C2×C22⋊C4—23no—
162C2×C22⋊C4—23no—
163C2×C22⋊C4—23no—
164C2×C22⋊C4—26no—
165C2×C42—23no—
166C2×C42C2313yes—
167C4⋊C8C2311yescharacteristic
168C4⋊C8C2313yes—
169C4⋊C8C2313yes—
170C4⋊C8C2313yes—
171C4⋊C8C2316yes—
172#103C2313yes—
173#103C2316yes—
174#103C2311yescharacteristic
175#103C2313yes—
176#103C2313yes—
177#102C2313yes—
178#102C2316yes—
179#102C2311yescharacteristic
180#102C2313yes—
181#102C2313yes—
182C24—23no—
183C24—23no—
184#41—28no—
185#41—26no—
186#41—26no—
187#41—26no—
188#41—26no—
189#41—26no—
190#41—44no—
191#41—412no—
192#41C2×D4112yes—
193#41C2×D414yes—
194#41#4114yes—
195C2×Q8—23no—
196C2×Q8—26no—
197C2×Q8—26no—
198C2×Q8—44no—
199C2×Q8C2×D411yescharacteristic
200C2×Q8C2413yes—
201C2×Q8C2411yescharacteristic
202C2×Q8C2411yescharacteristic
203C2×D4—23no—
204C2×D4—224no—
205C2×D4—23no—
206C2×D4—23no—
207C2×D4—23no—
208C2×D4—26no—
209C2×D4—23no—
210C2×D4—23no—
211C2×D4—23no—
212C2×D4—26no—
213C2×D4—26no—
214C2×D4—412no—
215C2×D4C2×D4112yes—
216C2×D4C2×D413yes—
217C2×D4C2413yes—
218C2×D4C2411yescharacteristic
219C2×D4C2413yes—
220C22×C4—23no—
221C22×C4—23no—
222C22×C4—23no—
223C22×C4—23no—
224C22×C4—23no—
225C22×C4—23no—
226C22×C4—23no—
227C22×C4—26no—
228C22×C4—86no—
229C22×C4C2×D413yes—
230C22×C4C2×D413yes—
231C22×C4C2413yes—
232Q16—22no—
233Q16—26no—
234SD16—22no—
235SD16—26no—
236SD16—26no—
237SD16—212no—
238SD16—26no—
239D8—26no—
240D8—212no—
241D8—26no—
242M16—26no—
243M16—26no—
244C2×C8—26no—
245C2×C8—26no—
246C2×C8C2411yescharacteristic
247C2×C8C2413yes—
248C2×C8C2411yescharacteristic
249C2×C8C2413yes—
250C4⋊C4—23no—
251C4⋊C4—23no—
252C4⋊C4—21no—
253C4⋊C4—23no—
254C4⋊C4—23no—
255C4⋊C4C2×D413yes—
256C4⋊C4C2×D413yes—
257C4⋊C4C2413yes—
258C4⋊C4C2411yescharacteristic
259C22⋊C4—23no—
260C22⋊C4—23no—
261C22⋊C4—26no—
262C22⋊C4—23no—
263C22⋊C4—23no—
264C22⋊C4—23no—
265C22⋊C4—26no—
266C22⋊C4—26no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4C2×D416yes—
270C42—21no—
271C42—23no—
272C42—23no—
273C42C2×D413yes—
274C42C2×D413yes—
275C42C2411yescharacteristic
276C42C2413yes—
277C23—23no—
278C23—212no—
279C23—23no—
280C23—21no—
281C23—46no—
282C23—83no—
283C23—83no—
284C23—83no—
285C23—83no—
286C23C22×D413yes—
287Q8—21no—
288Q8—23no—
289Q8—21no—
290Q8—23no—
291Q8—21no—
292Q8—46no—
293Q8#12114yes—
294Q8C22×D412yes—
295D4—212no—
296D4—212no—
297D4—23no—
298D4—21no—
299D4—23no—
300D4—23no—
301D4—21no—
302D4—46no—
303D4—46no—
304D4—46no—
305D4—83no—
306D4—83no—
307D4#121112yes—
308D4C22×D416yes—
309C2×C4—21no—
310C2×C4—212no—
311C2×C4—21no—
312C2×C4—23no—
313C2×C4—23no—
314C2×C4—21no—
315C2×C4—43no—
316C2×C4—43no—
317C2×C4—46no—
318C2×C4—46no—
319C2×C4—46no—
320C2×C4—46no—
321C2×C4—83no—
322C2×C4—86no—
323C2×C4—83no—
324C2×C4C22×D411yescharacteristic
325C2×C4C22×D413yes—
326C2×C4C22×D411yescharacteristic
327C2×C4C22×D413yes—
328C2×C4C22×D413yes—
329C2×C4C22×D416yes—
330C2×C4C2×#4116yes—
331C2×C4C2511yescharacteristic
332C8—21no—
333C8—21no—
334C8—23no—
335C8—23no—
336C22—23no—
337C22—23no—
338C22—26no—
339C22—43no—
340C22—43no—
341C22—41no—
342C22—43no—
343C22—41no—
344C22—43no—
345C22—41no—
346C22—43no—
347C22—812no—
348C22—166no—
349C22C2×#12116yes—
350C22C23×D411yescharacteristic
351C4—21no—
352C4—23no—
353C4—21no—
354C4—23no—
355C4—22no—
356C4—41no—
357C4—43no—
358C4—41no—
359C4—43no—
360C4C2×#12116yes—
361C4C23×D411yescharacteristic
362C4C22×#4111yescharacteristic
363C2—26no—
364C2—46no—
365C2—81no—
366C2—83no—
367C2C22×#12111yescharacteristic
368C2#358411yescharacteristic
369C2#358511yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#577C2semidirect
2. C2×#576C2semidirect
3. #3500C2semidirect
4. C4×H(5,𝔽2)C2semidirect
5. #3046C2semidirect
6. #3045C2semidirect
7. #3042C2semidirect
8. #3041C2semidirect
9. #3040C2semidirect
10. #3036C2semidirect
11. #3035C2semidirect
12. C2×#463C2semidirect
13. #2906C2semidirect
14. #2898C2semidirect
15. #2897C2semidirect
21. C2×#137C22semidirect
24. C2×AGL(1,ℤ/8)C22semidirect
26. C2×#135C22semidirect
27. C22×SD16C22semidirect
35. #536C22semidirect
36. #535C22semidirect
37. #532C22semidirect
38. C2×#124C22semidirect
41. #520C22semidirect
43. #518C22semidirect
46. C4×#41C22semidirect
48. C2×C4×D4C22semidirect
49. #464C22semidirect
50. #464C22semidirect
51. #464C22semidirect
52. #463C22semidirect
53. #463C22semidirect
54. #463C22semidirect
55. #463C22semidirect
56. #463C22semidirect
57. #463C22semidirect
58. #462C22semidirect
59. #461C22semidirect
60. #461C22semidirect
61. #460C22semidirect
62. #459C22semidirect
63. #459C22semidirect
64. #424C22semidirect
65. #424C22semidirect
66. #423C22semidirect
67. C2×C4⋊C8C22semidirect
68. #419C22semidirect
69. #418C22semidirect
70. #417C22semidirect
71. #417C22semidirect
72. #416C22semidirect
73. #416C22semidirect
74. C2×#103C22semidirect
75. C2×#102C22semidirect
81. H(5,𝔽2)D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5772
C2×#5762
#35002
C4×H(5,𝔽2)2
#30462
#30452
#30422
#30412
#30402
#30362
#30352
C2×#4632
#29062
#28982
#28972