Finite Group Databasebeta

← #61583

Subgroups — #61583

Order 256.

1059 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3505C211yesmaximal, characteristic, complemented
2C2×#548C211yesmaximal, characteristic, complemented
3#3327C214yesmaximal, complemented
4#3326C214yesmaximal, complemented
5#3277C214yesmaximal, complemented
6#3275C214yesmaximal, complemented
7#3211C212yesmaximal, complemented
8#3195C211yesmaximal, characteristic, complemented
9#3194C212yesmaximal, complemented
10#3193C211yesmaximal, characteristic, complemented
11#3108C212yesmaximal, complemented
12#3007C211yesmaximal, characteristic, complemented
13#3006C211yesmaximal, characteristic, complemented
14#3004C212yesmaximal, complemented
15#2975C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×Q8C2211yescharacteristic, complemented
18C2×#137C2214yescomplemented
19C2×#137C2212yescomplemented
20C2×AGL(1,ℤ/8)C2212yescomplemented
21#553—22no—
22#553C2212yescomplemented
23#552C2212yescomplemented
24#551—22no—
25#548—22no—
26#548—22no—
27#548C2214yes—
28#548C2214yescomplemented
29#548C2212yescomplemented
30#546C2212yescomplemented
31#539—21no—
32#538—21no—
33#534—21no—
34#529C2212yes—
35C2×#124—21no—
36C2×#124C2211yescharacteristic, complemented
37#525C2212yes—
38C2×C22⋊Q8—21no—
39C2×C22⋊Q8C2211yescharacteristic
40C2×C22⋊Q8C2211yescharacteristic
41C2×#121—21no—
42C2×#121C2211yescharacteristic
43C2×#121C2211yescharacteristic, complemented
44#518—21no—
45C2×C4×Q8C2211yescharacteristic, complemented
46C2×C4×D4—21no—
47C2×C4×D4C2211yescharacteristic
48#495C2214yescomplemented
49#488C2214yescomplemented
50#484C2212yescomplemented
51#483C2212yescomplemented
52#483C2212yescomplemented
53#482C2214yescomplemented
54#480C2212yes—
55#479C2214yescomplemented
56#470C2214yescomplemented
57#469C2214yescomplemented
58#468C2214yescomplemented
59#468C2214yescomplemented
60#464C2214yescomplemented
61#464C2214yescomplemented
62#463C2214yescomplemented
63#452C2214yescomplemented
64#451C2212yescomplemented
65#451C2212yescomplemented
66#451C2212yescomplemented
67#451C2214yescomplemented
68#450C2212yescomplemented
69#450C2212yescomplemented
70#449C2214yescomplemented
71#449C2214yescomplemented
72#448C2212yes—
73#448C2212yes—
74#448C2214yescomplemented
75#447C2212yes—
76C4×Q16C2214yescomplemented
77C4×SD16C2214yescomplemented
78#436C2214yescomplemented
79#436C2214yescomplemented
80#431C2212yescomplemented
81#419C2214yescomplemented
82#417C2212yescomplemented
83#417C2212yescomplemented
84#407C2211yescharacteristic
85#407C2211yescharacteristic
86#407C2212yescomplemented
87C2×#41—21no—
88C2×#41—21no—
89C2×#41—44no—
90C2×#41C2312yes—
91C2×#41C2312yes—
92C2×#41C2312yes—
93C22×Q8—21no—
94C22×Q8—22no—
95C22×Q8—44no—
96C22×Q8C2311yescharacteristic
97C22×Q8C2314yes—
98C22×Q8C2311yescharacteristic
99C22×Q8C2311yescharacteristic
100C22×D4—21no—
101C22×D4—21no—
102C22×D4—21no—
103C22×D4D411yescharacteristic
104C22×D4C2311yescharacteristic
105C22×D4C2311yescharacteristic
106C23×C4—21no—
107C23×C4—21no—
108C23×C4C2311yescharacteristic
109C23×C4C2311yescharacteristic
110#137—48no—
111#137—44no—
112AGL(1,ℤ/8)—44no—
113C2×Q16C2314yes—
114C2×Q16C2314yes—
115C2×SD16C2314yes—
116C2×SD16C2314yes—
117C2×SD16C2314yes—
118C2×D8C2314yes—
119C2×M16C2312yes—
120C2×M16C2312yes—
121C2×M16C2314yes—
122#126—22no—
123#126—22no—
124#126—22no—
125#125—22no—
126#125C2312yes—
127#124—24no—
128#124—24no—
129#124—22no—
130#124—22no—
131#124C2314yes—
132#124C2312yes—
133#123—22no—
134#123—22no—
135#123—22no—
136#123—22no—
137#123—22no—
138C22⋊Q8—21no—
139C22⋊Q8—21no—
140C22⋊Q8—21no—
141C22⋊Q8—21no—
142C22⋊Q8—24no—
143C22⋊Q8—24no—
144C22⋊Q8—22no—
145C22⋊Q8C2312yes—
146C22⋊Q8C2312yes—
147C22⋊Q8C2312yes—
148C22⋊Q8C2312yes—
149C22⋊Q8C2312yes—
150#121—24no—
151#121—24no—
152#121—22no—
153#121—22no—
154#121—21no—
155#121—21no—
156#121—22no—
157#121—22no—
158#121C2312yes—
159#121C2312yes—
160#121C2314yes—
161#121C2312yes—
162#120—22no—
163#120—22no—
164C4×Q8—22no—
165C4×Q8C2314yes—
166C4×Q8C2312yes—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—24no—
171C4×D4—24no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4D412yes—
175C4×D4D412yes—
176C4×D4C2312yes—
177C4×D4C2312yes—
178C4×D4C2312yes—
179#117—22no—
180#117—22no—
181#117C2312yes—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4D411yescharacteristic
186C2×C4⋊C4C2311yescharacteristic
187C2×C4⋊C4C2311yescharacteristic
188C2×C4⋊C4C2311yescharacteristic
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4D411yescharacteristic
197C2×C22⋊C4D411yescharacteristic
198C2×C42—21no—
199C2×C42C2311yescharacteristic
200#107C2314yes—
201#106C2314yes—
202C4⋊C8C2314yes—
203#103C2314yes—
204#103C2314yes—
205#103C2314yes—
206#103C2314yes—
207#103C2314yes—
208#102C2314yes—
209#102C2314yes—
210#102C2314yes—
211C22⋊C8C2312yes—
212C22⋊C8C2312yes—
213C22⋊C8C2312yes—
214C22⋊C8C2312yes—
215C22⋊C8C2314yes—
216C4×C8C2314yes—
217C24—21no—
218C24—21no—
219C24C2×D411yescharacteristic
220C24C2×D411yescharacteristic
221#41—24no—
222#41—24no—
223#41—42no—
224#41—42no—
225#41—44no—
226#41—44no—
227#41—44no—
228C2×Q8—24no—
229C2×Q8—21no—
230C2×Q8—24no—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—28no—
234C2×Q8—44no—
235C2×Q8—42no—
236C2×Q8—44no—
237C2×Q8—42no—
238C2×Q8—48no—
239C2×Q8C2412yes—
240C2×Q8C2412yes—
241C2×Q8C2412yes—
242C2×D4—22no—
243C2×D4—24no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—44no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4C2×D412yes—
258C2×D4C2×D412yes—
259C2×D4C2412yes—
260C2×D4C2412yes—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—24no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4C2×D411yescharacteristic
277C22×C4C2×D411yescharacteristic
278C22×C4C2×D411yescharacteristic
279C22×C4C2×D411yescharacteristic
280C22×C4C2×D411yescharacteristic
281C22×C4C2×D411yescharacteristic
282C22×C4C2×D412yes—
283C22×C4C2411yescharacteristic
284C22×C4C2411yescharacteristic
285C22×C4C2411yescharacteristic
286C22×C4C2412yes—
287Q16—28no—
288Q16—44no—
289SD16—28no—
290SD16—44no—
291SD16—44no—
292D8—44no—
293M16—42no—
294M16—44no—
295M16—42no—
296C2×C8C2414yes—
297C2×C8C2414yes—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—24no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4C2×D412yes—
311C4⋊C4C2×D412yes—
312C4⋊C4C2412yes—
313C4⋊C4C2412yes—
314C4⋊C4C2412yes—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—24no—
325C22⋊C4—24no—
326C22⋊C4—24no—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C22⋊C4—22no—
330C22⋊C4C2×D412yes—
331C22⋊C4C2×D412yes—
332C22⋊C4C2×D412yes—
333C22⋊C4C2×D412yes—
334C42—24no—
335C42—22no—
336C42—22no—
337C42C2×D412yes—
338C42C2412yes—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—81no—
352C23—81no—
353C23C22×D411yescharacteristic
354C23C22×D411yescharacteristic
355C23C22×D412yes—
356C23C22×D411yescharacteristic
357Q8—24no—
358Q8—22no—
359Q8—24no—
360Q8—24no—
361Q8—48no—
362Q8—42no—
363Q8—81no—
364Q8H(5,𝔽2)14yes—
365D4—22no—
366D4—24no—
367D4—22no—
368D4—42no—
369D4—42no—
370D4—44no—
371D4—44no—
372D4—42no—
373D4—44no—
374D4—48no—
375D4—44no—
376D4—81no—
377D4—82no—
378D4—82no—
379C2×C4—24no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—48no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4C22×D412yes—
408C2×C4C22×D412yes—
409C2×C4C22×D411yescharacteristic
410C2×C4C22×D412yes—
411C2×C4C22×D411yescharacteristic
412C2×C4C22×D411yescharacteristic
413C2×C4C2511yescharacteristic
414C8—24no—
415C8—24no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—22no—
420C22—21no—
421C22—22no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—21no—
427C22—44no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—82no—
435C22—84no—
436C22—162no—
437C22#57512yes—
438C22C23×D411yescharacteristic
439C4—22no—
440C4—24no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—22no—
445C4—22no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4—42no—
450C4—42no—
451C4C2×H(5,𝔽2)12yes—
452C2—22no—
453C2—42no—
454C2—41no—
455C2—41no—
456C2—82no—
457C2—82no—
458C2#348411yescharacteristic
459C2C2×#57511yescharacteristic
460C2#358611yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3505C2semidirect
2. C2×#548C2semidirect
3. #3327C2semidirect
4. #3326C2semidirect
5. #3277C2semidirect
6. #3275C2semidirect
7. #3211C2semidirect
8. #3195C2semidirect
9. #3194C2semidirect
10. #3193C2semidirect
11. #3108C2semidirect
12. #3007C2semidirect
13. #3006C2semidirect
14. #3004C2semidirect
15. #2975C2semidirect
17. C23×Q8C22semidirect
18. C2×#137C22semidirect
19. C2×#137C22semidirect
20. C2×AGL(1,ℤ/8)C22semidirect
22. #553C22semidirect
23. #552C22semidirect
28. #548C22semidirect
29. #548C22semidirect
30. #546C22semidirect
36. C2×#124C22semidirect
43. C2×#121C22semidirect
45. C2×C4×Q8C22semidirect
48. #495C22semidirect
49. #488C22semidirect
50. #484C22semidirect
51. #483C22semidirect
52. #483C22semidirect
53. #482C22semidirect
55. #479C22semidirect
56. #470C22semidirect
57. #469C22semidirect
58. #468C22semidirect
59. #468C22semidirect
60. #464C22semidirect
61. #464C22semidirect
62. #463C22semidirect
63. #452C22semidirect
64. #451C22semidirect
65. #451C22semidirect
66. #451C22semidirect
67. #451C22semidirect
68. #450C22semidirect
69. #450C22semidirect
70. #449C22semidirect
71. #449C22semidirect
74. #448C22semidirect
76. C4×Q16C22semidirect
77. C4×SD16C22semidirect
78. #436C22semidirect
79. #436C22semidirect
80. #431C22semidirect
81. #419C22semidirect
82. #417C22semidirect
83. #417C22semidirect
86. #407C22semidirect

Maximal subgroups

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

subgroupindex
#35052
C2×#5482
#33272
#33262
#32772
#32752
#32112
#31952
#31942
#31932
#31082
#30072
#30062
#30042
#29752