Finite Group Databasebeta

← #61995

Subgroups — #61995

Order 256.

989 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3508C211yesmaximal, characteristic, complemented
2#3481C211yesmaximal, characteristic, complemented
3#3396C212yesmaximal, complemented
4#3361C212yesmaximal, complemented
5#3333C212yesmaximal, complemented
6#3322C214yesmaximal, complemented
7#3274C212yesmaximal, complemented
8#3273C214yesmaximal, complemented
9#3254C211yesmaximal, characteristic, complemented
10#3251C212yesmaximal, complemented
11#3212C212yesmaximal, complemented
12#3205C211yesmaximal, characteristic, complemented
13#3107C211yesmaximal, characteristic, complemented
14#3100C212yesmaximal, complemented
15#3038C211yesmaximal, characteristic, complemented
16#3036C212yesmaximal, complemented
17#2991C211yesmaximal, characteristic, complemented
18C2×AGL(1,ℤ/8)C2212yes—
19C2×AGL(1,ℤ/8)C2214yescomplemented
20#559—21no—
21#559C2212yescomplemented
22#559C2211yescharacteristic, complemented
23#554C2211yescharacteristic, complemented
24#554C2211yescharacteristic, complemented
25#553—21no—
26#553C2212yes—
27#551—22no—
28#550—21no—
29#550C2212yes—
30#546—21no—
31#546—22no—
32#546C2212yes—
33#546C2214yescomplemented
34D42C2211yescharacteristic, complemented
35#540—22no—
36#535—21no—
37#533C2211yescharacteristic, complemented
38#532C2211yescharacteristic
39#532C2212yescomplemented
40#532C2212yescomplemented
41C2×#127C2211yescharacteristic, complemented
42#529—21no—
43#529C2212yescomplemented
44#525—22no—
45#518—21no—
46C4×#41C2211yescharacteristic
47C4×#41C2211yescharacteristic
48C2×C4×D4—21no—
49C2×C4×D4C2212yescomplemented
50#495C2214yescomplemented
51#493C2212yes—
52#488C2214yescomplemented
53#486C2212yescomplemented
54#485C2212yescomplemented
55#480C2214yescomplemented
56#478C2214yescomplemented
57#477C2212yescomplemented
58#475C2211yescharacteristic, complemented
59#474C2211yescharacteristic, complemented
60#474C2212yescomplemented
61#469C2214yescomplemented
62#468C2214yescomplemented
63#468C2214yescomplemented
64#463C2214yescomplemented
65#463C2214yescomplemented
66#463C2214yescomplemented
67#462C2211yescharacteristic, complemented
68#461C2211yescharacteristic, complemented
69#461C2212yescomplemented
70#460C2212yescomplemented
71#460C2212yescomplemented
72#460C2212yescomplemented
73#459C2212yes—
74#459C2212yes—
75#459C2212yescomplemented
76#459C2212yescomplemented
77#452C2212yes—
78#452C2214yescomplemented
79#449C2214yescomplemented
80#447C2214yescomplemented
81#447C2214yescomplemented
82C8×Q8C2212yescomplemented
83C4×Q16C2212yescomplemented
84C4×SD16C2212yescomplemented
85C4×D8C2214yescomplemented
86C4×D8C2214yescomplemented
87#436C2212yescomplemented
88#436C2214yescomplemented
89#431C2212yescomplemented
90#431C2211yescharacteristic, complemented
91#423C2212yescomplemented
92#423C2211yescharacteristic, complemented
93#418C2212yescomplemented
94#417C2214yescomplemented
95C2×#41—21no—
96C2×#41—22no—
97C2×#41D411yescharacteristic
98C2×#41C2311yescharacteristic
99C2×#41C2314yes—
100C22×D4—21no—
101C22×D4—21no—
102C22×D4—21no—
103C22×D4C2312yes—
104C22×D4C2312yes—
105C23×C4—22no—
106AGL(1,ℤ/8)—44no—
107AGL(1,ℤ/8)—48no—
108C2×Q16C2312yes—
109C2×SD16C2312yes—
110C2×SD16C2314yes—
111C2×SD16C2314yes—
112C2×D8C2314yes—
113C2×D8C2314yes—
114C2×D8C2314yes—
115C2×M16C2312yes—
116C2×M16C2314yes—
117C4⋊Q8C2311yescharacteristic
118C4⋊Q8C2311yescharacteristic
119C4⋊Q8C2311yescharacteristic
120#127—22no—
121#127—21no—
122#127—21no—
123#127C2312yes—
124#127C2312yes—
125#127C2311yescharacteristic
126#126—22no—
127#126—22no—
128#125—21no—
129#125—21no—
130#125C2312yes—
131#124—21no—
132#124—22no—
133#124—21no—
134#124—22no—
135#124C2312yes—
136#124C2314yes—
137#123—22no—
138#123—22no—
139#123—22no—
140#123—24no—
141#123—24no—
142#123—22no—
143C22⋊Q8—21no—
144C22⋊Q8—21no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8C2312yes—
148C22⋊Q8C2312yes—
149#121—22no—
150#121—21no—
151#121—22no—
152#121—22no—
153#121—22no—
154#121—21no—
155#121—22no—
156#121—24no—
157#121—24no—
158#121C2312yes—
159#121C2312yes—
160#121C2314yes—
161#120—22no—
162#120—24no—
163#120—22no—
164C4×Q8—21no—
165C4×Q8D411yescharacteristic
166C4×Q8C2311yescharacteristic
167C4×Q8C2311yescharacteristic
168C4×Q8C2312yes—
169C4×Q8C2311yescharacteristic
170C4×D4—21no—
171C4×D4—22no—
172C4×D4—21no—
173C4×D4—21no—
174C4×D4—21no—
175C4×D4—22no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—21no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4D411yescharacteristic
185C4×D4D412yes—
186C4×D4C2311yescharacteristic
187C4×D4C2312yes—
188C4×D4C2312yes—
189C4×D4C2312yes—
190C4×D4C2311yescharacteristic
191C4×D4C2314yes—
192C4×D4C2311yescharacteristic
193C4×D4C2312yes—
194#117—21no—
195#117—21no—
196#117—22no—
197#117D411yescharacteristic
198#117D412yes—
199#117C2311yescharacteristic
200C2×C4⋊C4—21no—
201C2×C4⋊C4C2312yes—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—24no—
205C2×C42—21no—
206C2×C42C2312yes—
207C2×C42C2311yescharacteristic
208#107C2312yes—
209#107C2314yes—
210#106C2312yes—
211C4⋊C8C2312yes—
212C4⋊C8C2311yescharacteristic
213C4⋊C8C2312yes—
214C4⋊C8C2314yes—
215C4⋊C8C2311yescharacteristic
216#103C2312yes—
217#103C2314yes—
218#103C2312yes—
219#103C2312yes—
220#102C2314yes—
221#102C2314yes—
222#102C2314yes—
223#102C2312yes—
224#102C2314yes—
225#102C2314yes—
226C22⋊C8C2312yes—
227C22⋊C8C2314yes—
228C4×C8C2314yes—
229C4×C8C2312yes—
230C24—22no—
231C24—22no—
232#41—22no—
233#41—42no—
234#41—41no—
235#41—44no—
236#41—44no—
237#41—82no—
238#41—84no—
239C2×Q8—21no—
240C2×Q8—21no—
241C2×Q8C2×D411yescharacteristic
242C2×Q8C2412yes—
243C2×Q8C2411yescharacteristic
244C2×D4—24no—
245C2×D4—21no—
246C2×D4—22no—
247C2×D4—24no—
248C2×D4—22no—
249C2×D4—24no—
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—21no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—44no—
262C2×D4—44no—
263C2×D4—82no—
264C2×D4C2×D411yescharacteristic
265C2×D4C2×D412yes—
266C2×D4C2414yes—
267C2×D4C2412yes—
268C2×D4C2411yescharacteristic
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—84no—
279C22×C4C2×D411yescharacteristic
280C22×C4C2×D412yes—
281C22×C4C2×D411yescharacteristic
282C22×C4C2×D412yes—
283C22×C4C2412yes—
284C22×C4C2411yescharacteristic
285Q16—42no—
286SD16—24no—
287SD16—44no—
288SD16—44no—
289D8—28no—
290D8—44no—
291D8—44no—
292M16—42no—
293M16—44no—
294C2×C8C2412yes—
295C2×C8C2414yes—
296C2×C8C2412yes—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4C2×D411yescharacteristic
310C4⋊C4C2×D411yescharacteristic
311C4⋊C4C2×D412yes—
312C4⋊C4C2×D412yes—
313C4⋊C4C2411yescharacteristic
314C4⋊C4C2411yescharacteristic
315C4⋊C4C2412yes—
316C4⋊C4C2412yes—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—24no—
322C22⋊C4—24no—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—21no—
329C22⋊C4—22no—
330C22⋊C4—22no—
331C22⋊C4—24no—
332C22⋊C4—24no—
333C22⋊C4C2×D412yes—
334C22⋊C4C2×D412yes—
335C22⋊C4C2×D412yes—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—22no—
340C42C2×D411yescharacteristic
341C42C2×D411yescharacteristic
342C42C2×D412yes—
343C42C2×D412yes—
344C42C2412yes—
345C42C2411yescharacteristic
346C42C2411yescharacteristic
347C23—21no—
348C23—22no—
349C23—24no—
350C23—44no—
351C23—44no—
352C23—44no—
353C23—82no—
354C23—82no—
355C23—82no—
356C23C22×D411yescharacteristic
357C23C22×D412yes—
358Q8—22no—
359Q8—22no—
360Q8—21no—
361Q8—21no—
362Q8—22no—
363Q8—42no—
364Q8—81no—
365D4—22no—
366D4—24no—
367D4—22no—
368D4—21no—
369D4—22no—
370D4—24no—
371D4—42no—
372D4—44no—
373D4—44no—
374D4—42no—
375D4—44no—
376D4—42no—
377D4—44no—
378D4—44no—
379D4—82no—
380D4—82no—
381D4—81no—
382D4—84no—
383D4—82no—
384D4—84no—
385D4H(5,𝔽2)12yes—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—84no—
407C2×C4—84no—
408C2×C4—82no—
409C2×C4C22×D411yescharacteristic
410C2×C4C22×D411yescharacteristic
411C2×C4C22×D412yes—
412C2×C4C22×D412yes—
413C2×C4C22×D412yes—
414C2×C4C22×D411yescharacteristic
415C2×C4C22×D411yescharacteristic
416C2×C4C22×D411yescharacteristic
417C2×C4C2511yescharacteristic
418C8—24no—
419C8—22no—
420C8—22no—
421C22—21no—
422C22—22no—
423C22—22no—
424C22—21no—
425C22—22no—
426C22—22no—
427C22—21no—
428C22—41no—
429C22—41no—
430C22—44no—
431C22—42no—
432C22—42no—
433C22—44no—
434C22—42no—
435C22—44no—
436C22—88no—
437C22—164no—
438C22C23×D411yescharacteristic
439C4—21no—
440C4—22no—
441C4—21no—
442C4—22no—
443C4—22no—
444C4—21no—
445C4—41no—
446C4—42no—
447C4—41no—
448C4—41no—
449C4—42no—
450C4—42no—
451C4#57512yes—
452C4#57612yes—
453C4C2×H(5,𝔽2)11yescharacteristic
454C4#58511yescharacteristic
455C2—42no—
456C2—41no—
457C2—42no—
458C2—82no—
459C2—84no—
460C2#348811yescharacteristic
461C2C2×#57511yescharacteristic
462C2C2×#57611yescharacteristic
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. #3508C2semidirect
2. #3481C2semidirect
3. #3396C2semidirect
4. #3361C2semidirect
5. #3333C2semidirect
6. #3322C2semidirect
7. #3274C2semidirect
8. #3273C2semidirect
9. #3254C2semidirect
10. #3251C2semidirect
11. #3212C2semidirect
12. #3205C2semidirect
13. #3107C2semidirect
14. #3100C2semidirect
15. #3038C2semidirect
16. #3036C2semidirect
17. #2991C2semidirect
19. C2×AGL(1,ℤ/8)C22semidirect
21. #559C22semidirect
22. #559C22semidirect
23. #554C22semidirect
24. #554C22semidirect
33. #546C22semidirect
34. D42C22semidirect
37. #533C22semidirect
39. #532C22semidirect
40. #532C22semidirect
41. C2×#127C22semidirect
43. #529C22semidirect
49. C2×C4×D4C22semidirect
50. #495C22semidirect
52. #488C22semidirect
53. #486C22semidirect
54. #485C22semidirect
55. #480C22semidirect
56. #478C22semidirect
57. #477C22semidirect
58. #475C22semidirect
59. #474C22semidirect
60. #474C22semidirect
61. #469C22semidirect
62. #468C22semidirect
63. #468C22semidirect
64. #463C22semidirect
65. #463C22semidirect
66. #463C22semidirect
67. #462C22semidirect
68. #461C22semidirect
69. #461C22semidirect
70. #460C22semidirect
71. #460C22semidirect
72. #460C22semidirect
75. #459C22semidirect
76. #459C22semidirect
78. #452C22semidirect
79. #449C22semidirect
80. #447C22semidirect
81. #447C22semidirect
82. C8×Q8C22semidirect
83. C4×Q16C22semidirect
84. C4×SD16C22semidirect
85. C4×D8C22semidirect
86. C4×D8C22semidirect
87. #436C22semidirect
88. #436C22semidirect
89. #431C22semidirect
90. #431C22semidirect
91. #423C22semidirect
92. #423C22semidirect
93. #418C22semidirect
94. #417C22semidirect

Maximal subgroups

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

subgroupindex
#35082
#34812
#33962
#33612
#33332
#33222
#32742
#32732
#32542
#32512
#32122
#32052
#31072
#31002
#30382
#30362
#29912