Finite Group Databasebeta

← #61576

Subgroups — #61576

Order 256.

1127 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3506C211yesmaximal, characteristic, complemented
2C2×#546C211yesmaximal, characteristic, complemented
3#3319C214yesmaximal, complemented
4#3311C214yesmaximal, complemented
5#3268C214yesmaximal, complemented
6#3267C214yesmaximal, complemented
7#3215C211yesmaximal, characteristic, complemented
8#3213C211yesmaximal, characteristic, complemented
9#3196C212yesmaximal, complemented
10#3188C212yesmaximal, complemented
11#3110C211yesmaximal, characteristic, complemented
12#3109C211yesmaximal, characteristic, complemented
13#3003C212yesmaximal, complemented
14#3002C212yesmaximal, complemented
15#2973C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×D4C2211yescharacteristic
18C2×AGL(1,ℤ/8)C2214yescomplemented
19C2×#135C2214yescomplemented
20#559—22no—
21#556C2212yescomplemented
22#555C2212yescomplemented
23#553—22no—
24#550C2212yescomplemented
25#547C2212yescomplemented
26#546—22no—
27#546—22no—
28#546C2214yes—
29#546C2214yes—
30#538—21no—
31#538—21no—
32#535—21no—
33#533C2211yescharacteristic, complemented
34#532C2211yescharacteristic, complemented
35C2×#124C2211yescharacteristic
36#525C2212yes—
37#525C2212yescomplemented
38C2×#123—21no—
39C2×C22⋊Q8C2211yescharacteristic
40C2×#121—21no—
41C2×#121C2211yescharacteristic
42C2×#120—21no—
43#520—21no—
44C2×C4×D4C2211yescharacteristic
45C2×C4×D4C2211yescharacteristic
46C22×C22⋊C4—21no—
47#497C2212yescomplemented
48#496C2212yescomplemented
49#489C2212yescomplemented
50#489C2212yescomplemented
51#485C2214yescomplemented
52#484C2212yescomplemented
53#482C2214yescomplemented
54#481C2212yescomplemented
55#476C2212yescomplemented
56#474C2212yescomplemented
57#470C2214yescomplemented
58#468C2214yescomplemented
59#466C2214yescomplemented
60#465C2214yescomplemented
61#464C2214yescomplemented
62#463C2214yescomplemented
63#461C2212yescomplemented
64#459C2212yescomplemented
65#452C2214yescomplemented
66#452C2214yescomplemented
67#451C2212yescomplemented
68#450C2212yes—
69#450C2212yes—
70#450C2214yescomplemented
71#449C2214yescomplemented
72#449C2214yescomplemented
73#448C2212yescomplemented
74#447C2212yes—
75#447C2212yes—
76#447C2214yescomplemented
77#442C2214yescomplemented
78#440C2214yescomplemented
79#435C2214yescomplemented
80#435C2214yescomplemented
81#432C2211yescharacteristic, complemented
82#432C2211yescharacteristic, complemented
83#418C2214yescomplemented
84#416C2214yescomplemented
85#407C2212yes—
86C2×C22⋊C8C2212yescomplemented
87C25—21no—
88C2×#41—21no—
89C2×#41—22no—
90C2×#41—44no—
91C2×#41C2314yes—
92C2×#41C2312yes—
93C2×#41C2312yescomplemented
94C22×Q8C2311yescharacteristic
95C22×D4—21no—
96C22×D4—21no—
97C22×D4—21no—
98C22×D4—22no—
99C22×D4—44no—
100C22×D4D411yescharacteristic
101C22×D4C2311yescharacteristic
102C22×D4C2314yes—
103C22×D4C2311yescharacteristic
104C23×C4—21no—
105C23×C4—21no—
106C23×C4C2311yescharacteristic
107C23×C4C2311yescharacteristic
108AGL(1,ℤ/8)—48no—
109#135—48no—
110C2×Q16C2314yescomplemented
111C2×SD16C2314yes—
112C2×SD16C2314yes—
113C2×SD16C2314yescomplemented
114C2×D8C2314yes—
115C2×D8C2314yes—
116C2×M16C2314yes—
117C22×C8C2314yescomplemented
118C4⋊Q8C2311yescharacteristic, complemented
119C4⋊Q8C2311yescharacteristic, complemented
120C4⋊Q8C2311yescharacteristic, complemented
121#127—22no—
122#127C2311yescharacteristic, complemented
123#126—22no—
124#126—22no—
125#125—21no—
126#124—22no—
127#124—22no—
128#124—22no—
129#124—21no—
130#124—21no—
131#124—21no—
132#124C2312yes—
133#124C2312yes—
134#123—22no—
135#123—22no—
136#123—24no—
137#123—24no—
138#123—22no—
139#123—22no—
140C22⋊Q8—22no—
141C22⋊Q8—21no—
142C22⋊Q8C2314yes—
143C22⋊Q8C2312yes—
144C22⋊Q8C2312yes—
145C22⋊Q8C2312yescomplemented
146#121—22no—
147#121—22no—
148#121—22no—
149#121—21no—
150#121—24no—
151#121—24no—
152#121—22no—
153#121—22no—
154#121C2314yes—
155#121C2312yes—
156#121C2312yes—
157#121C2312yescomplemented
158#120—22no—
159#120—22no—
160#120—24no—
161#120—24no—
162C4×Q8—22no—
163C4×Q8—21no—
164C4×Q8C2312yescomplemented
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—21no—
171C4×D4—21no—
172C4×D4—21no—
173C4×D4D412yes—
174C4×D4D412yes—
175C4×D4C2312yes—
176C4×D4C2312yes—
177C4×D4C2312yes—
178C4×D4C2312yes—
179C4×D4C2312yescomplemented
180#117—21no—
181#117—22no—
182#117C2312yes—
183#117C2312yescomplemented
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4D411yescharacteristic
187C2×C4⋊C4C2311yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4D411yescharacteristic
196C2×C22⋊C4D411yescharacteristic
197C2×C42C2311yescharacteristic
198#107C2314yescomplemented
199#106C2314yescomplemented
200C4⋊C8C2312yescomplemented
201C4⋊C8C2312yescomplemented
202#103C2314yescomplemented
203#103C2314yescomplemented
204#103C2314yescomplemented
205#102C2314yes—
206#102C2314yes—
207#102C2314yes—
208#102C2314yes—
209#102C2314yescomplemented
210C22⋊C8C2312yes—
211C22⋊C8C2312yes—
212C22⋊C8C2314yes—
213C22⋊C8C2314yescomplemented
214#97C2312yescomplemented
215#97C2312yescomplemented
216C24—21no—
217C24—21no—
218C24—24no—
219C24—44no—
220C24—44no—
221C24—82no—
222C24C2×D411yescharacteristic
223C24C2×D411yescharacteristic
224#41—28no—
225#41—44no—
226#41—44no—
227#41—44no—
228#41—44no—
229#41—48no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—42no—
234C2×Q8C2412yes—
235C2×Q8C2412yes—
236C2×D4—24no—
237C2×D4—28no—
238C2×D4—22no—
239C2×D4—24no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—24no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—24no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—44no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—48no—
254C2×D4—44no—
255C2×D4—44no—
256C2×D4—82no—
257C2×D4C2×D412yes—
258C2×D4C2×D412yes—
259C2×D4C2412yes—
260C2×D4C2412yes—
261C2×D4C2412yes—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4—82no—
274C22×C4C2×D411yescharacteristic
275C22×C4C2×D411yescharacteristic
276C22×C4C2×D411yescharacteristic
277C22×C4C2×D411yescharacteristic
278C22×C4C2×D412yes—
279C22×C4C2×D412yes—
280C22×C4C2411yescharacteristic
281C22×C4C2412yes—
282C22×C4C2412yes—
283Q16—44no—
284SD16—28no—
285SD16—44no—
286SD16—44no—
287D8—28no—
288D8—44no—
289M16—44no—
290C2×C8—44no—
291C2×C8C2414yes—
292C2×C8C2414yes—
293C4⋊C4—21no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—24no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—22no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—22no—
303C4⋊C4C2×D412yes—
304C4⋊C4C2×D412yes—
305C4⋊C4C2412yes—
306C4⋊C4C2412yes—
307C4⋊C4C2412yes—
308C22⋊C4—24no—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—22no—
316C22⋊C4—24no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4C2×D412yes—
326C22⋊C4C2×D412yes—
327C22⋊C4C2×D412yes—
328C22⋊C4C2×D412yes—
329C42—22no—
330C42—22no—
331C42—22no—
332C42C2×D411yescharacteristic
333C42C2×D411yescharacteristic
334C42C2411yescharacteristic
335C42C2411yescharacteristic
336C23—24no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—48no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—48no—
351C23—42no—
352C23—42no—
353C23—84no—
354C23—816no—
355C23C22×D412yes—
356C23C22×D412yes—
357C23C22×D411yescharacteristic
358Q8—24no—
359Q8—24no—
360Q8—42no—
361Q8—42no—
362Q8—81no—
363D4—24no—
364D4—22no—
365D4—24no—
366D4—24no—
367D4—42no—
368D4—42no—
369D4—48no—
370D4—42no—
371D4—42no—
372D4—48no—
373D4—44no—
374D4—81no—
375D4—82no—
376D4—82no—
377D4—84no—
378D4—82no—
379D4—82no—
380D4H(5,𝔽2)14yes—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—48no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4C22×D411yescharacteristic
404C2×C4C22×D412yes—
405C2×C4C22×D412yes—
406C2×C4C22×D411yescharacteristic
407C2×C4C22×D411yescharacteristic
408C2×C4C22×D411yescharacteristic
409C2×C4C22×D411yescharacteristic
410C2×C4C2511yescharacteristic
411C8—24no—
412C8—24no—
413C22—21no—
414C22—21no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—24no—
420C22—22no—
421C22—22no—
422C22—48no—
423C22—44no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—82no—
433C22—84no—
434C22—816no—
435C22#57512yes—
436C22C23×D411yescharacteristic
437C4—21no—
438C4—21no—
439C4—22no—
440C4—21no—
441C4—22no—
442C4—21no—
443C4—21no—
444C4—42no—
445C4—42no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4C2×H(5,𝔽2)12yes—
450C2—22no—
451C2—42no—
452C2—42no—
453C2—44no—
454C2—82no—
455C2—82no—
456C2#348411yescharacteristic
457C2C2×#57511yescharacteristic
458C2#358511yescharacteristic
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. #3506C2semidirect
2. C2×#546C2semidirect
3. #3319C2semidirect
4. #3311C2semidirect
5. #3268C2semidirect
6. #3267C2semidirect
7. #3215C2semidirect
8. #3213C2semidirect
9. #3196C2semidirect
10. #3188C2semidirect
11. #3110C2semidirect
12. #3109C2semidirect
13. #3003C2semidirect
14. #3002C2semidirect
15. #2973C2semidirect
18. C2×AGL(1,ℤ/8)C22semidirect
19. C2×#135C22semidirect
21. #556C22semidirect
22. #555C22semidirect
24. #550C22semidirect
25. #547C22semidirect
33. #533C22semidirect
34. #532C22semidirect
37. #525C22semidirect
47. #497C22semidirect
48. #496C22semidirect
49. #489C22semidirect
50. #489C22semidirect
51. #485C22semidirect
52. #484C22semidirect
53. #482C22semidirect
54. #481C22semidirect
55. #476C22semidirect
56. #474C22semidirect
57. #470C22semidirect
58. #468C22semidirect
59. #466C22semidirect
60. #465C22semidirect
61. #464C22semidirect
62. #463C22semidirect
63. #461C22semidirect
64. #459C22semidirect
65. #452C22semidirect
66. #452C22semidirect
67. #451C22semidirect
70. #450C22semidirect
71. #449C22semidirect
72. #449C22semidirect
73. #448C22semidirect
76. #447C22semidirect
77. #442C22semidirect
78. #440C22semidirect
79. #435C22semidirect
80. #435C22semidirect
81. #432C22semidirect
82. #432C22semidirect
83. #418C22semidirect
84. #416C22semidirect
86. C2×C22⋊C8C22semidirect
93. C2×#41C23semidirect
110. C2×Q16C23semidirect
113. C2×SD16C23semidirect
117. C22×C8C23semidirect
118. C4⋊Q8C23semidirect
119. C4⋊Q8C23semidirect
120. C4⋊Q8C23semidirect
122. #127C23semidirect
145. C22⋊Q8C23semidirect
157. #121C23semidirect
164. C4×Q8C23semidirect
179. C4×D4C23semidirect
183. #117C23semidirect
198. #107C23semidirect
199. #106C23semidirect
200. C4⋊C8C23semidirect
201. C4⋊C8C23semidirect
202. #103C23semidirect
203. #103C23semidirect
204. #103C23semidirect
209. #102C23semidirect
213. C22⋊C8C23semidirect
214. #97C23semidirect
215. #97C23semidirect

Maximal subgroups

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

subgroupindex
#35062
C2×#5462
#33192
#33112
#32682
#32672
#32152
#32132
#31962
#31882
#31102
#31092
#30032
#30022
#29732