Finite Group Databasebeta

← #60433

Subgroups — #60433

Order 256.

1339 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#578C211yesmaximal, characteristic, complemented
2C2×#577C211yesmaximal, characteristic, complemented
3#3500C211yesmaximal, characteristic, complemented
4C4×2−1+4C211yesmaximal, characteristic, complemented
5#3046C213yesmaximal, complemented
6#3045C213yesmaximal, complemented
7#3044C213yesmaximal, complemented
8#3043C213yesmaximal, complemented
9#3040C213yesmaximal, complemented
10#3037C213yesmaximal, complemented
11#3035C213yesmaximal, complemented
12C2×#464C213yesmaximal, complemented
13#2906C211yesmaximal, characteristic, complemented
14#2898C211yesmaximal, characteristic, complemented
15#2897C211yesmaximal, characteristic, complemented
16C2×2−1+4C2211yescharacteristic
17C2×2−1+4C2211yescharacteristic
18C2×H(5,𝔽2)C2211yescharacteristic
19#578—28no—
20#577—28no—
21C2×#137C2213yes—
22C2×#137C2213yescomplemented
23C2×#137C2213yescomplemented
24C2×AGL(1,ℤ/8)C2213yes—
25C2×#135C2213yes—
26C2×#135C2213yescomplemented
27C22×Q16C2213yescomplemented
28C22×SD16C2213yes—
29C2×#131C2211yescharacteristic
30C2×#131C2211yescharacteristic
31#548—21no—
32#548—21no—
33#546—23no—
34#546—23no—
35#536C2213yes—
36#535C2213yes—
37#532C2213yes—
38C2×#124C2213yes—
39#520—23no—
40#520C2211yescharacteristic
41#520C2213yescomplemented
42#519—21no—
43#519C2213yescomplemented
44C4×#41—23no—
45C4×#41C2211yescharacteristic
46C4×#41C2213yescomplemented
47C2×C4×Q8—21no—
48C2×C4×Q8C2213yescomplemented
49#464C2213yescomplemented
50#464C2216yescomplemented
51#464C2216yescomplemented
52#464C2216yescomplemented
53#464C2213yescomplemented
54#464C2216yescomplemented
55#463C2211yescharacteristic, complemented
56#463C2213yescomplemented
57#463C2216yescomplemented
58#462C2213yescomplemented
59#462C2216yescomplemented
60#461C2213yescomplemented
61#460C2213yescomplemented
62#460C2216yescomplemented
63#459C2213yescomplemented
64#424C2213yescomplemented
65#424C2213yescomplemented
66#423C2213yescomplemented
67C2×C4⋊C8C2213yescomplemented
68#419C2213yescomplemented
69#418C2213yes—
70#417C2213yes—
71#417C2213yescomplemented
72#416C2213yes—
73#416C2213yescomplemented
74C2×#103C2213yescomplemented
75C2×#102C2213yes—
762−1+4—24no—
772−1+4—24no—
782−1+4—24no—
792−1+4D418yescomplemented
80H(5,𝔽2)—24no—
81H(5,𝔽2)—24no—
82C2×#41—21no—
83C2×#41—23no—
84C2×#41—23no—
85C2×#41C2313yes—
86C2×#41C2313yes—
87C2×#41C2313yes—
88C2×#41C2311yescharacteristic
89C22×Q8—21no—
90C22×Q8—21no—
91C22×Q8C2313yes—
92C22×Q8C2313yes—
93C22×D4—23no—
94C22×D4C2313yes—
95#137—212no—
96#137—212no—
97#137—212no—
98AGL(1,ℤ/8)—212no—
99#135—212no—
100#135—212no—
101C2×Q16—212no—
102C2×Q16C2313yes—
103C2×Q16C2316yes—
104C2×Q16C2313yes—
105C2×SD16—212no—
106C2×SD16C2313yes—
107C2×SD16C2316yes—
108C2×SD16C2311yescharacteristic
109C2×SD16C2313yes—
110C2×SD16C2313yes—
111C2×D8C2313yes—
112C2×D8C2311yescharacteristic
113#131—24no—
114#131—24no—
115C2×M16C2313yes—
116C2×M16C2313yes—
117C22×C8C2313yes—
118C22×C8C2313yes—
119C4⋊Q8C2313yes—
120#127C2313yes—
121#124—23no—
122#124—23no—
123#124C2313yes—
124#124C2316yes—
125#124C2311yescharacteristic
126#123—26no—
127C22⋊Q8—23no—
128C22⋊Q8—23no—
129#121—23no—
130#121—23no—
131#121—23no—
132#121—23no—
133#120—26no—
134C4×Q8—21no—
135C4×Q8—23no—
136C4×Q8—23no—
137C4×Q8—23no—
138C4×Q8—23no—
139C4×Q8D411yescharacteristic
140C4×Q8D411yescharacteristic
141C4×Q8C2313yes—
142C4×Q8C2316yes—
143C4×Q8C2313yes—
144C4×D4—23no—
145C4×D4—23no—
146C4×D4—23no—
147C4×D4—26no—
148C4×D4—23no—
149C4×D4—23no—
150C4×D4D413yes—
151C4×D4D413yes—
152C4×D4C2311yescharacteristic
153C4×D4C2313yes—
154#117—23no—
155#117—26no—
156#117—23no—
157#117C2313yes—
158#117C2313yes—
159C2×C4⋊C4—23no—
160C2×C4⋊C4—23no—
161C2×C4⋊C4C2313yes—
162C2×C22⋊C4—23no—
163C2×C22⋊C4—23no—
164C2×C42—23no—
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#102C2313yes—
180#102C2311yescharacteristic
181#102C2313yes—
182C24—23no—
183#41—224no—
184#41—26no—
185#41—26no—
186#41—26no—
187#41—26no—
188#41—26no—
189#41—44no—
190#41—412no—
191#41C2×D4112yes—
192#41C2×D414yes—
193#41#4114yes—
194C2×Q8—28no—
195C2×Q8—23no—
196C2×Q8—23no—
197C2×Q8—26no—
198C2×Q8—26no—
199C2×Q8—26no—
200C2×Q8—44no—
201C2×Q8C2×D4112yes—
202C2×Q8C2×D411yescharacteristic
203C2×Q8C2413yes—
204C2×Q8C2411yescharacteristic
205C2×Q8C2413yes—
206C2×D4—23no—
207C2×D4—23no—
208C2×D4—23no—
209C2×D4—26no—
210C2×D4—23no—
211C2×D4—23no—
212C2×D4—23no—
213C2×D4—26no—
214C2×D4—412no—
215C2×D4C2×D413yes—
216C2×D4C2413yes—
217C2×D4C2411yescharacteristic
218C2×D4C2411yescharacteristic
219C22×C4—23no—
220C22×C4—23no—
221C22×C4—23no—
222C22×C4—23no—
223C22×C4—21no—
224C22×C4—23no—
225C22×C4—23no—
226C22×C4—23no—
227C22×C4C2×D413yes—
228C22×C4C2×D413yes—
229C22×C4C2413yes—
230Q16—26no—
231Q16—212no—
232Q16—26no—
233SD16—26no—
234SD16—212no—
235SD16—26no—
236SD16—22no—
237SD16—26no—
238D8—22no—
239D8—26no—
240M16—26no—
241M16—26no—
242C2×C8—26no—
243C2×C8—26no—
244C2×C8C2411yescharacteristic
245C2×C8C2413yes—
246C2×C8C2411yescharacteristic
247C2×C8C2413yes—
248C4⋊C4—26no—
249C4⋊C4—23no—
250C4⋊C4—23no—
251C4⋊C4—23no—
252C4⋊C4—23no—
253C4⋊C4—23no—
254C4⋊C4C2×D413yes—
255C4⋊C4C2×D413yes—
256C4⋊C4C2413yes—
257C4⋊C4C2411yescharacteristic
258C22⋊C4—23no—
259C22⋊C4—23no—
260C22⋊C4—23no—
261C22⋊C4—26no—
262C22⋊C4—26no—
263C22⋊C4—21no—
264C22⋊C4—23no—
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4C2×D416yes—
268C42—23no—
269C42—23no—
270C42—26no—
271C42—23no—
272C42C2×D413yes—
273C42C2×D413yes—
274C42C2411yescharacteristic
275C42C2413yes—
276C23—21no—
277C23—23no—
278C23—21no—
279C23—46no—
280C23—83no—
281C23—83no—
282C23C22×D413yes—
283Q8—212no—
284Q8—23no—
285Q8—23no—
286Q8—21no—
287Q8—23no—
288Q8—21no—
289Q8—46no—
290Q8#12114yes—
291Q8C22×D416yes—
292D4—212no—
293D4—23no—
294D4—21no—
295D4—21no—
296D4—23no—
297D4—21no—
298D4—46no—
299D4—46no—
300D4—46no—
301D4—83no—
302D4—83no—
303D4#121112yes—
304D4C22×D412yes—
305C2×C4—212no—
306C2×C4—23no—
307C2×C4—212no—
308C2×C4—21no—
309C2×C4—23no—
310C2×C4—23no—
311C2×C4—21no—
312C2×C4—43no—
313C2×C4—43no—
314C2×C4—42no—
315C2×C4—46no—
316C2×C4—46no—
317C2×C4—46no—
318C2×C4—83no—
319C2×C4—83no—
320C2×C4C22×D411yescharacteristic
321C2×C4C22×D413yes—
322C2×C4C22×D413yes—
323C2×C4C22×D411yescharacteristic
324C2×C4C22×D413yes—
325C2×C4C22×D416yes—
326C2×C4C2×#4116yes—
327C2×C4C2511yescharacteristic
328C8—21no—
329C8—21no—
330C8—23no—
331C8—23no—
332C22—23no—
333C22—23no—
334C22—22no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—43no—
339C22—41no—
340C22—43no—
341C22—41no—
342C22—43no—
343C22—166no—
344C22C2×#12116yes—
345C22C23×D411yescharacteristic
346C4—21no—
347C4—23no—
348C4—21no—
349C4—23no—
350C4—26no—
351C4—43no—
352C4—41no—
353C4—41no—
354C4—43no—
355C4C2×#12116yes—
356C4C23×D411yescharacteristic
357C4C22×#4111yescharacteristic
358C2—26no—
359C2—42no—
360C2—81no—
361C2—83no—
362C2C22×#12111yescharacteristic
363C2#358411yescharacteristic
364C2#358611yescharacteristic
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×#578C2semidirect
2. C2×#577C2semidirect
3. #3500C2semidirect
4. C4×2−1+4C2semidirect
5. #3046C2semidirect
6. #3045C2semidirect
7. #3044C2semidirect
8. #3043C2semidirect
9. #3040C2semidirect
10. #3037C2semidirect
11. #3035C2semidirect
12. C2×#464C2semidirect
13. #2906C2semidirect
14. #2898C2semidirect
15. #2897C2semidirect
22. C2×#137C22semidirect
23. C2×#137C22semidirect
26. C2×#135C22semidirect
27. C22×Q16C22semidirect
41. #520C22semidirect
43. #519C22semidirect
46. C4×#41C22semidirect
48. C2×C4×Q8C22semidirect
49. #464C22semidirect
50. #464C22semidirect
51. #464C22semidirect
52. #464C22semidirect
53. #464C22semidirect
54. #464C22semidirect
55. #463C22semidirect
56. #463C22semidirect
57. #463C22semidirect
58. #462C22semidirect
59. #462C22semidirect
60. #461C22semidirect
61. #460C22semidirect
62. #460C22semidirect
63. #459C22semidirect
64. #424C22semidirect
65. #424C22semidirect
66. #423C22semidirect
67. C2×C4⋊C8C22semidirect
68. #419C22semidirect
71. #417C22semidirect
73. #416C22semidirect
74. C2×#103C22semidirect
79. 2−1+4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5782
C2×#5772
#35002
C4×2−1+42
#30462
#30452
#30442
#30432
#30402
#30372
#30352
C2×#4642
#29062
#28982
#28972