Finite Group Databasebeta

← #60565

Subgroups — #60565

Order 256.

1039 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3455C211yesmaximal, characteristic, complemented
2C2×#520C211yesmaximal, characteristic, complemented
3#3125C214yesmaximal, complemented
4#3124C214yesmaximal, complemented
5#3120C214yesmaximal, complemented
6#3119C214yesmaximal, complemented
7#3092C212yesmaximal, complemented
8#3078C211yesmaximal, characteristic, complemented
9#3077C211yesmaximal, characteristic, complemented
10#3037C211yesmaximal, characteristic, complemented
11#3036C211yesmaximal, characteristic, complemented
12#3005C212yesmaximal, complemented
13C4×#137C212yesmaximal, complemented
14C4×AGL(1,ℤ/8)C212yesmaximal, complemented
15#2921C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C2×#137C2212yes—
18C2×AGL(1,ℤ/8)C2212yes—
19#542C2212yescomplemented
20#542C2212yescomplemented
21#541C2212yescomplemented
22#540C2212yescomplemented
23#539—21no—
24#539—21no—
25#538—21no—
26#538—21no—
27#533C2211yescharacteristic
28#532C2211yescharacteristic
29#529—22no—
30#529—22no—
31#525C2212yes—
32#520—22no—
33#520—22no—
34#520C2214yes—
35#520C2214yes—
36C4×#41C2212yes—
37C2×C4×Q8C2211yescharacteristic
38C2×C4×Q8C2211yescharacteristic
39C2×C4×D4—21no—
40C2×C4×D4—21no—
41C2×C4×D4C2211yescharacteristic
42C2×C4×D4C2211yescharacteristic
43C2×#117—21no—
44C2×#117C2211yescharacteristic
45C22×C4⋊C4—21no—
46C22×C4⋊C4C2211yescharacteristic
47#484C2214yescomplemented
48#483C2214yescomplemented
49#481C2214yescomplemented
50#480C2214yescomplemented
51#479C2214yescomplemented
52#478C2214yescomplemented
53#477C2212yescomplemented
54#476C2212yescomplemented
55#475C2212yescomplemented
56#474C2212yescomplemented
57#464C2214yescomplemented
58#463C2214yescomplemented
59#462C2212yescomplemented
60#461C2212yescomplemented
61#460C2212yescomplemented
62#459C2212yescomplemented
63#452C2214yes—
64#452C2214yes—
65#449C2214yes—
66#449C2214yes—
67#442C2214yescomplemented
68#441C2214yescomplemented
69#440C2214yescomplemented
70#440C2214yescomplemented
71C4×Q16C2214yescomplemented
72C4×SD16C2214yescomplemented
73C4×SD16C2214yescomplemented
74C4×D8C2214yescomplemented
75#433C2214yescomplemented
76#433C2214yescomplemented
77#428C2212yes—
78#428C2212yes—
79#423C2211yescharacteristic
80#423C2211yescharacteristic
81#419C2212yes—
82#418C2212yes—
83#417C2212yes—
84#417C2212yes—
85#407C2212yes—
86C4×M16C2212yes—
87C2×#41—24no—
88C2×#41—24no—
89C2×#41—22no—
90C2×#41C2312yes—
91C2×#41C2314yes—
92C22×Q8C2311yescharacteristic
93C22×D4—21no—
94C22×D4C2311yescharacteristic
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4C2311yescharacteristic
100#137—28no—
101AGL(1,ℤ/8)—28no—
102C2×Q16C2314yes—
103C2×SD16C2314yes—
104C2×SD16C2314yes—
105C2×D8C2314yes—
106C2×M16C2312yes—
107C2×M16C2312yes—
108C4⋊Q8C2311yescharacteristic
109C4⋊Q8C2311yescharacteristic
110C4⋊Q8C2311yescharacteristic
111#127C2311yescharacteristic
112#126—22no—
113#126—22no—
114#126—22no—
115#126—22no—
116#125—22no—
117#125—22no—
118#125C2312yes—
119#124—22no—
120#124—22no—
121#124C2312yes—
122#123—22no—
123#123—22no—
124#123—22no—
125#123—22no—
126#123—22no—
127C22⋊Q8—21no—
128C22⋊Q8—21no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8C2312yes—
132C22⋊Q8C2312yes—
133#121—21no—
134#121—21no—
135#121—22no—
136#121—22no—
137#121C2312yes—
138#121C2312yes—
139#120—22no—
140C4×Q8—22no—
141C4×Q8—21no—
142C4×Q8—21no—
143C4×Q8—21no—
144C4×Q8—21no—
145C4×Q8C2314yes—
146C4×Q8C2312yes—
147C4×Q8C2312yes—
148C4×Q8C2312yes—
149C4×Q8C2312yes—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4—22no—
153C4×D4—22no—
154C4×D4—24no—
155C4×D4—24no—
156C4×D4—24no—
157C4×D4—24no—
158C4×D4—22no—
159C4×D4—21no—
160C4×D4—21no—
161C4×D4—21no—
162C4×D4—21no—
163C4×D4C2314yes—
164C4×D4C2312yes—
165C4×D4C2312yes—
166C4×D4C2312yes—
167C4×D4C2312yes—
168#117—24no—
169#117—24no—
170#117—22no—
171#117D412yes—
172#117D412yes—
173#117C2312yes—
174#117C2312yes—
175#117C2312yes—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4D411yescharacteristic
183C2×C4⋊C4D411yescharacteristic
184C2×C4⋊C4C2311yescharacteristic
185C2×C4⋊C4C2311yescharacteristic
186C2×C4⋊C4C2314yes—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4D411yescharacteristic
190C2×C22⋊C4D411yescharacteristic
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42C2311yescharacteristic
195C2×C42C2311yescharacteristic
196#107C2314yes—
197#107C2314yes—
198#106C2314yes—
199#106C2314yes—
200C4⋊C8C2312yes—
201C4⋊C8C2312yes—
202C4⋊C8C2312yes—
203C4⋊C8C2312yes—
204#103C2314yes—
205#103C2314yes—
206#103C2314yes—
207#103C2314yes—
208#102C2314yes—
209#102C2314yes—
210#102C2314yes—
211#102C2314yes—
212C22⋊C8C2314yes—
213C22⋊C8C2314yes—
214#97C2314yes—
215C4×C8C2314yes—
216C24—21no—
217C24C2×D411yescharacteristic
218#41—216no—
219#41—28no—
220#41—28no—
221#41—24no—
222#41—24no—
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8—22no—
227C2×Q8#4114yes—
228C2×Q8C2412yes—
229C2×Q8C2412yes—
230C2×D4—28no—
231C2×D4—24no—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—22no—
239C2×D4#4114yes—
240C2×D4C2412yes—
241C2×D4C2412yes—
242C22×C4—28no—
243C22×C4—21no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4C2×D411yescharacteristic
260C22×C4C2×D412yes—
261C22×C4C2×D411yescharacteristic
262C22×C4C2×D411yescharacteristic
263C22×C4C2×D411yescharacteristic
264C22×C4C2×D411yescharacteristic
265C22×C4C2412yes—
266C22×C4C2411yescharacteristic
267Q16—28no—
268SD16—28no—
269SD16—28no—
270D8—28no—
271M16—24no—
272M16—24no—
273C2×C8C2414yes—
274C2×C8C2414yes—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4C2×D412yes—
292C4⋊C4C2×D412yes—
293C4⋊C4C2×D412yes—
294C4⋊C4C2×D412yes—
295C4⋊C4C2412yes—
296C4⋊C4C2412yes—
297C4⋊C4C2412yes—
298C4⋊C4C2412yes—
299C22⋊C4—22no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—24no—
303C22⋊C4—22no—
304C22⋊C4—22no—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4C2×D412yes—
312C22⋊C4C2×D412yes—
313C22⋊C4C2×D412yes—
314C22⋊C4C2×D412yes—
315C42—24no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42C2×D412yes—
322C42C2×D411yescharacteristic
323C42C2×D411yescharacteristic
324C42C2412yes—
325C42C2411yescharacteristic
326C42C2411yescharacteristic
327C23—22no—
328C23—24no—
329C23—22no—
330C23—22no—
331C23—41no—
332C23—41no—
333C23—42no—
334C23—42no—
335C23C22×D412yes—
336C23C22×D411yescharacteristic
337Q8—24no—
338Q8—22no—
339Q8—22no—
340Q8—22no—
341Q8—81no—
342Q8—81no—
343Q8C2×#4114yes—
344D4—216no—
345D4—24no—
346D4—22no—
347D4—22no—
348D4—22no—
349D4—44no—
350D4—81no—
351D4—81no—
352D4—82no—
353D4—82no—
354D4C2×#4114yes—
355C2×C4—216no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—84no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—84no—
380C2×C4C22×D412yes—
381C2×C4C22×D411yescharacteristic
382C2×C4C22×D411yescharacteristic
383C2×C4C22×D412yes—
384C2×C4C22×D411yescharacteristic
385C2×C4C22×D411yescharacteristic
386C2×C4C22×D412yes—
387C2×C4C22×D411yescharacteristic
388C2×C4C2×#4112yes—
389C2×C4C2×#4112yes—
390C2×C4C2511yescharacteristic
391C8—24no—
392C8—24no—
393C22—21no—
394C22—21no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—24no—
399C22—44no—
400C22—48no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—42no—
406C22#52512yes—
407C22C23×D411yescharacteristic
408C4—22no—
409C4—21no—
410C4—21no—
411C4—22no—
412C4—21no—
413C4—21no—
414C4—22no—
415C4—24no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—42no—
420C4—42no—
421C4#52512yes—
422C4C22×#4111yescharacteristic
423C4C22×#4111yescharacteristic
424C2—22no—
425C2—42no—
426C2—44no—
427C2—82no—
428C2C2×#52511yescharacteristic
429C2#358511yescharacteristic
430C2#358611yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#117 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3455C2semidirect
2. C2×#520C2semidirect
3. #3125C2semidirect
4. #3124C2semidirect
5. #3120C2semidirect
6. #3119C2semidirect
7. #3092C2semidirect
8. #3078C2semidirect
9. #3077C2semidirect
10. #3037C2semidirect
11. #3036C2semidirect
12. #3005C2semidirect
13. C4×#137C2semidirect
14. C4×AGL(1,ℤ/8)C2semidirect
15. #2921C2semidirect
19. #542C22semidirect
20. #542C22semidirect
21. #541C22semidirect
22. #540C22semidirect
47. #484C22semidirect
48. #483C22semidirect
49. #481C22semidirect
50. #480C22semidirect
51. #479C22semidirect
52. #478C22semidirect
53. #477C22semidirect
54. #476C22semidirect
55. #475C22semidirect
56. #474C22semidirect
57. #464C22semidirect
58. #463C22semidirect
59. #462C22semidirect
60. #461C22semidirect
61. #460C22semidirect
62. #459C22semidirect
67. #442C22semidirect
68. #441C22semidirect
69. #440C22semidirect
70. #440C22semidirect
71. C4×Q16C22semidirect
72. C4×SD16C22semidirect
73. C4×SD16C22semidirect
74. C4×D8C22semidirect
75. #433C22semidirect
76. #433C22semidirect

Maximal subgroups

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

subgroupindex
#34552
C2×#5202
#31252
#31242
#31202
#31192
#30922
#30782
#30772
#30372
#30362
#30052
C4×#1372
C4×AGL(1,ℤ/8)2
#29212