Finite Group Databasebeta

← #60457

Subgroups — #60457

Order 256.

1378 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#577C211yesmaximal, characteristic, complemented
2C22×#131C211yesmaximal, characteristic, complemented
3#3489C211yesmaximal, characteristic, complemented
4#3488C211yesmaximal, characteristic, complemented
5#3063C213yesmaximal, complemented
6#3062C213yesmaximal, complemented
7#3060C213yesmaximal, complemented
8#3057C213yesmaximal, complemented
9#3056C213yesmaximal, complemented
10#3055C213yesmaximal, complemented
11#3050C213yesmaximal, complemented
12C2×#465C213yesmaximal, complemented
13#2912C211yesmaximal, characteristic, complemented
14#2898C211yesmaximal, characteristic, complemented
15#2897C211yesmaximal, characteristic, complemented
16C2×2−1+4C2211yescharacteristic
17C2×H(5,𝔽2)C2211yescharacteristic
18C22×#41C2211yescharacteristic
19#577—28no—
20C2×#137C2213yescomplemented
21C2×AGL(1,ℤ/8)C2213yescomplemented
22C2×#135C2213yescomplemented
23C22×SD16C2213yescomplemented
24C2×#131—28no—
25C2×#131C2211yescharacteristic
26C2×#131C2211yescharacteristic
27C2×#131C2213yescomplemented
28C2×#131C2213yescomplemented
29C22×M16C2213yescomplemented
30C23×C8C2213yescomplemented
31#550—21no—
32D4×Q8—21no—
33#548—21no—
34#548—21no—
35#547—23no—
36#546—23no—
37#546—23no—
38D42—23no—
39#537C2213yes—
40#537C2213yescomplemented
41#536C2213yescomplemented
42#535C2213yes—
43#525C2213yes—
44#525C2213yescomplemented
45C2×C22⋊Q8C2213yescomplemented
46C2×#121C2213yes—
47#520C2211yescharacteristic
48#520C2211yescharacteristic
49#470C2216yescomplemented
50#469C2216yescomplemented
51#468C2216yescomplemented
52#468C2216yescomplemented
53#467C2213yescomplemented
54#467C2213yescomplemented
55#466C2213yescomplemented
56#466C2213yescomplemented
57#465C2213yescomplemented
58#465C2216yescomplemented
59#465C2216yescomplemented
60#465C2216yescomplemented
61#465C2211yescharacteristic, complemented
62#465C2213yescomplemented
63#465C2213yescomplemented
64#428C2213yescomplemented
65#428C2213yescomplemented
66#427C2213yescomplemented
67C2×#106C2213yescomplemented
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—
80C2×#41—24no—
81C2×#41—24no—
82C2×#41—21no—
83C2×#41—23no—
84C2×#41D413yes—
85C2×#41D411yescharacteristic
86C2×#41C2313yes—
87C2×#41C2313yes—
88C2×#41C2313yes—
89C2×#41C2311yescharacteristic
90C22×Q8—21no—
91C22×Q8D411yescharacteristic
92C22×Q8C2313yes—
93C22×D4—23no—
94C22×D4—23no—
95C22×D4D413yes—
96C22×D4C2313yes—
97C23×C4C2313yes—
98#137—212no—
99AGL(1,ℤ/8)—212no—
100#135—212no—
101C2×Q16C2313yes—
102C2×SD16—212no—
103C2×SD16C2313yes—
104C2×SD16C2316yes—
105C2×SD16C2311yescharacteristic
106C2×D8C2313yes—
107#131—24no—
108#131—212no—
109#131—212no—
110#131D418yescomplemented
111C2×M16—212no—
112C2×M16C2313yes—
113C2×M16C2316yes—
114C2×M16C2313yes—
115C22×C8—212no—
116C22×C8C2311yescharacteristic
117C22×C8C2313yes—
118C22×C8C2313yes—
119C22×C8C2313yes—
120C22×C8C2313yes—
121C22×C8C2313yes—
122C4⋊Q8—23no—
123#127—23no—
124#124—23no—
125#124—23no—
126#123—26no—
127#123—23no—
128#123—23no—
129C22⋊Q8—23no—
130C22⋊Q8—23no—
131C22⋊Q8—23no—
132C22⋊Q8—23no—
133C22⋊Q8C2313yes—
134C22⋊Q8C2316yes—
135C22⋊Q8C2311yescharacteristic
136C22⋊Q8C2313yes—
137C22⋊Q8C2313yes—
138#121—26no—
139#121—23no—
140#121—26no—
141#121—23no—
142#121—23no—
143#121—23no—
144#121C2313yes—
145#121C2316yes—
146#121C2313yes—
147#121C2311yescharacteristic
148#121C2313yes—
149#120—23no—
150#120—23no—
151#120—26no—
152C4×Q8—21no—
153C4×Q8—21no—
154C4×D4—23no—
155C4×D4—23no—
156C4×D4—23no—
157C4×D4—23no—
158C4×D4—23no—
159C4×D4—23no—
160#117C2313yes—
161#117C2313yes—
162C2×C4⋊C4—23no—
163C2×C4⋊C4C2313yes—
164C2×C4⋊C4C2313yes—
165C2×C22⋊C4—23no—
166C2×C22⋊C4—23no—
167#107C2313yes—
168#107C2313yes—
169#106C2313yes—
170#106C2316yes—
171#106C2311yescharacteristic
172#103C2313yes—
173#103C2316yes—
174#103C2311yescharacteristic
175#103C2313yes—
176#103C2313yes—
177#102C2313yes—
178#102C2316yes—
179#102C2311yescharacteristic
180#102C2313yes—
181#102C2313yes—
182C24—23no—
183C24C2×D413yes—
184#41—24no—
185#41—212no—
186#41—26no—
187#41—26no—
188#41—26no—
189#41—26no—
190#41—26no—
191#41—26no—
192#41—44no—
193#41—412no—
194#41C2×D414yes—
195#41#4114yes—
196C2×Q8—24no—
197C2×Q8—23no—
198C2×Q8—26no—
199C2×Q8—26no—
200C2×Q8—44no—
201C2×Q8C2×D413yes—
202C2×Q8C2×D411yescharacteristic
203C2×Q8C2413yes—
204C2×Q8C2411yescharacteristic
205C2×D4—212no—
206C2×D4—21no—
207C2×D4—23no—
208C2×D4—23no—
209C2×D4—23no—
210C2×D4—23no—
211C2×D4—23no—
212C2×D4—26no—
213C2×D4—26no—
214C2×D4—23no—
215C2×D4—26no—
216C2×D4—412no—
217C2×D4—86no—
218C2×D4C2×D416yes—
219C2×D4C2×D413yes—
220C2×D4C2×D413yes—
221C2×D4C2411yescharacteristic
222C2×D4C2413yes—
223C22×C4—26no—
224C22×C4—26no—
225C22×C4—23no—
226C22×C4—23no—
227C22×C4—23no—
228C22×C4—23no—
229C22×C4—23no—
230C22×C4C2×D413yes—
231C22×C4C2×D413yes—
232C22×C4C2×D413yes—
233C22×C4C2411yescharacteristic
234C22×C4C2413yes—
235C22×C4C2413yes—
236Q16—26no—
237SD16—26no—
238SD16—212no—
239SD16—22no—
240D8—26no—
241M16—26no—
242M16—212no—
243M16C2×D4112yescomplemented
244C2×C8—26no—
245C2×C8—22no—
246C2×C8—26no—
247C2×C8—26no—
248C2×C8—26no—
249C2×C8C2×D4112yescomplemented
250C2×C8C2411yescharacteristic
251C2×C8C2413yes—
252C2×C8C2411yescharacteristic
253C2×C8C2413yes—
254C4⋊C4—23no—
255C4⋊C4—26no—
256C4⋊C4—23no—
257C4⋊C4—23no—
258C4⋊C4—21no—
259C4⋊C4—23no—
260C4⋊C4—23no—
261C4⋊C4C2413yes—
262C4⋊C4C2411yescharacteristic
263C4⋊C4C2413yes—
264C4⋊C4C2411yescharacteristic
265C22⋊C4—23no—
266C22⋊C4—26no—
267C22⋊C4—23no—
268C22⋊C4—23no—
269C22⋊C4—21no—
270C22⋊C4—23no—
271C22⋊C4—23no—
272C22⋊C4—23no—
273C22⋊C4—26no—
274C22⋊C4—21no—
275C22⋊C4—23no—
276C22⋊C4—23no—
277C42—23no—
278C42—23no—
279C23—26no—
280C23—26no—
281C23—21no—
282C23—23no—
283C23—43no—
284C23—46no—
285C23—83no—
286C23—83no—
287C23C22×D413yes—
288C23C22×D413yes—
289C23C22×D411yescharacteristic
290Q8—26no—
291Q8—23no—
292Q8—21no—
293Q8—23no—
294Q8—21no—
295Q8—46no—
296Q8#12114yes—
297D4—212no—
298D4—26no—
299D4—21no—
300D4—23no—
301D4—23no—
302D4—21no—
303D4—46no—
304D4—46no—
305D4—42no—
306D4—46no—
307D4—46no—
308D4—83no—
309D4—83no—
310D4—86no—
311D4#121112yes—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—26no—
315C2×C4—21no—
316C2×C4—23no—
317C2×C4—21no—
318C2×C4—23no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—23no—
322C2×C4—23no—
323C2×C4—21no—
324C2×C4—23no—
325C2×C4—46no—
326C2×C4—46no—
327C2×C4—46no—
328C2×C4—83no—
329C2×C4—83no—
330C2×C4C22×D411yescharacteristic
331C2×C4C22×D413yes—
332C2×C4C22×D413yes—
333C2×C4C22×D416yes—
334C2×C4C2×#4116yes—
335C2×C4C2511yescharacteristic
336C8—21no—
337C8—23no—
338C8C22×D412yes—
339C8C22×D416yes—
340C22—23no—
341C22—21no—
342C22—21no—
343C22—23no—
344C22—23no—
345C22—23no—
346C22—23no—
347C22—412no—
348C22—41no—
349C22—43no—
350C22—41no—
351C22—43no—
352C22—43no—
353C22—41no—
354C22—166no—
355C22C2×#12116yes—
356C22#57712yescomplemented
357C22C23×D411yescharacteristic
358C4—21no—
359C4—23no—
360C4—43no—
361C4—41no—
362C4—41no—
363C4—43no—
364C4—43no—
365C4—41no—
366C4C2×#12116yes—
367C4C23×D411yescharacteristic
368C4C22×#4111yescharacteristic
369C2—22no—
370C2—26no—
371C2—43no—
372C2—81no—
373C2—83no—
374C2C22×#12111yescharacteristic
375C2C2×#57711yescharacteristic
376C2#358411yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×#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. C22×#131C2semidirect
3. #3489C2semidirect
4. #3488C2semidirect
5. #3063C2semidirect
6. #3062C2semidirect
7. #3060C2semidirect
8. #3057C2semidirect
9. #3056C2semidirect
10. #3055C2semidirect
11. #3050C2semidirect
12. C2×#465C2semidirect
13. #2912C2semidirect
14. #2898C2semidirect
15. #2897C2semidirect
20. C2×#137C22semidirect
21. C2×AGL(1,ℤ/8)C22semidirect
22. C2×#135C22semidirect
23. C22×SD16C22semidirect
27. C2×#131C22semidirect
28. C2×#131C22semidirect
29. C22×M16C22semidirect
30. C23×C8C22semidirect
40. #537C22semidirect
41. #536C22semidirect
44. #525C22semidirect
45. C2×C22⋊Q8C22semidirect
49. #470C22semidirect
50. #469C22semidirect
51. #468C22semidirect
52. #468C22semidirect
53. #467C22semidirect
54. #467C22semidirect
55. #466C22semidirect
56. #466C22semidirect
57. #465C22semidirect
58. #465C22semidirect
59. #465C22semidirect
60. #465C22semidirect
61. #465C22semidirect
62. #465C22semidirect
63. #465C22semidirect
64. #428C22semidirect
65. #428C22semidirect
66. #427C22semidirect
67. C2×#106C22semidirect
68. #419C22semidirect
69. #418C22semidirect
70. #417C22semidirect
71. #417C22semidirect
72. #416C22semidirect
73. #416C22semidirect
74. C2×#103C22semidirect
75. C2×#102C22semidirect
110. #131D4semidirect
243. M16C2×D4semidirect
249. C2×C8C2×D4semidirect
356. C22#577semidirect

Maximal subgroups

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

subgroupindex
C2×#5772
C22×#1312
#34892
#34882
#30632
#30622
#30602
#30572
#30562
#30552
#30502
C2×#4652
#29122
#28982
#28972