Finite Group Databasebeta

← #63005

Subgroups — #63005

Order 256.

3512 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3524C214yesmaximal, complemented
2#3524C214yesmaximal, complemented
3#3522C214yesmaximal, complemented
4#3518C214yesmaximal, complemented
5#3516C214yesmaximal, complemented
6#3516C214yesmaximal, complemented
7#3510C216yesmaximal, complemented
8#3502C211yesmaximal, characteristic, complemented
9#3495C216yesmaximal, complemented
10#3494C216yesmaximal, complemented
11#3487C211yesmaximal, characteristic, complemented
12#3486C211yesmaximal, characteristic, complemented
13#3482C214yesmaximal, complemented
14#3480C214yesmaximal, complemented
15#3475C216yesmaximal, complemented
16C2×#548C211yesmaximal, characteristic, complemented
17C4×2−1+4C211yesmaximal, characteristic, complemented
18#3428C211yesmaximal, characteristic, complemented
19C2×#519C211yesmaximal, characteristic, complemented
20C2×2−1+4C2211yescharacteristic
21C22×#41C2211yescharacteristic
22C23×Q8C2211yescharacteristic
23#559C22112yescomplemented
24#559C22112yescomplemented
25Q82C2214yescomplemented
26#557C2214yescomplemented
27#557C2214yescomplemented
28#557C2214yescomplemented
29#556C2216yescomplemented
30#556C22112yescomplemented
31#556C22112yescomplemented
32#556C22112yescomplemented
33#555C2216yescomplemented
34#555C22112yescomplemented
35#554C2216yescomplemented
36#554C22112yescomplemented
37#553C2216yescomplemented
38#553C2216yescomplemented
39#553C22112yescomplemented
40#553C22112yescomplemented
41#552C22112yes—
42#552C2216yescomplemented
43#552C2216yescomplemented
44#552C22112yescomplemented
45#552C22112yescomplemented
46#551C2214yes—
47#551C22112yes—
48#551C22124yes—
49#550C2214yescomplemented
50#550C2214yescomplemented
51D4×Q8C2214yescomplemented
52D4×Q8C2214yescomplemented
53#548C2214yes—
54#548C2214yes—
55#548C2214yes—
56#548C2214yes—
57#548C2214yescomplemented
58#548C2214yescomplemented
59#548C2214yescomplemented
60#548C2214yescomplemented
61#547C2216yescomplemented
62#547C22112yescomplemented
63#546C22112yes—
64#544C2216yes—
65#542C22112yescomplemented
66#542C22112yescomplemented
67#542C22112yescomplemented
68#542C22112yescomplemented
69#541C22112yescomplemented
70#540C22112yescomplemented
71#536C2216yes—
72#535C2216yes—
73#533C2216yes—
74#533C22112yescomplemented
75#532C22112yescomplemented
76#529C22112yescomplemented
77#529C22112yescomplemented
78#529C22112yescomplemented
79#529C22112yescomplemented
80#529C22112yescomplemented
81#529C22112yescomplemented
82#529C22112yescomplemented
83C2×#124C2213yes—
84#525C2216yes—
85#525C2216yes—
86C2×C22⋊Q8C2213yes—
87C2×#121C2213yes—
88#520C2216yes—
89#520C2214yescomplemented
90#520C2214yescomplemented
91#520C2214yescomplemented
92#519C2211yescharacteristic
93#519C2211yescharacteristic
94#519C2216yes—
95#519C2211yescharacteristic
96#519C2214yescomplemented
97#519C2214yescomplemented
98#519C2214yescomplemented
99#518C2216yes—
100C4×#41C2216yes—
101C4×#41C2214yescomplemented
102C4×#41C2214yescomplemented
103C4×#41C2214yescomplemented
104C4×#41C22112yescomplemented
105C2×C4×Q8C2211yescharacteristic
106C2×C4×Q8C2214yescomplemented
107C2×C4×Q8C2214yescomplemented
108C2×C4×Q8C2214yescomplemented
109C2×C4×D4C2213yes—
110C2×#117C2216yes—
111C2×#117C2216yes—
1122−1+4—28no—
113C2×#41—28no—
114C2×#41C2316yes—
115C2×#41C2314yes—
116C2×#41C2314yes—
117C2×#41C2314yes—
118C2×#41C23112yes—
119C22×Q8—28no—
120C22×Q8C2314yes—
121C22×Q8C2314yes—
122C22×Q8C2313yes—
123C22×Q8C2314yes—
124C22×Q8C2311yescharacteristic
125C22×D4C2313yes—
126C22×D4C2313yes—
127C23×C4C2313yes—
128C23×C4C2313yes—
129C4⋊Q8C23112yes—
130C4⋊Q8C2316yes—
131C4⋊Q8C2316yes—
132C4⋊Q8C23112yes—
133C4⋊Q8C23112yes—
134#127C23112yes—
135#126C2314yes—
136#126C23112yes—
137#126C2314yes—
138#126C23112yes—
139#126C23124yes—
140#126C23124yes—
141#126C23112yes—
142#126C23112yes—
143#125C2316yes—
144#125C2316yes—
145#125C23112yes—
146#125C23112yes—
147#125C23112yes—
148#125C23112yes—
149#125C23112yes—
150#124C23112yes—
151#124C23112yes—
152#124C2314yes—
153#124C23112yes—
154#124C23112yes—
155#124C23112yes—
156#124C23112yes—
157#124C23124yes—
158#123C2314yes—
159#123C23112yes—
160#123C23112yes—
161#123C2314yes—
162#123C2314yes—
163#123C23112yes—
164#123C23124yes—
165#123C23124yes—
166#123C23112yes—
167#123C23112yes—
168C22⋊Q8C23112yes—
169C22⋊Q8C23112yes—
170C22⋊Q8C2316yes—
171C22⋊Q8C2316yes—
172C22⋊Q8C2316yes—
173C22⋊Q8C2316yes—
174C22⋊Q8C2314yes—
175C22⋊Q8C23112yes—
176C22⋊Q8C23112yes—
177C22⋊Q8C23112yes—
178C22⋊Q8C23112yes—
179C22⋊Q8C2314yes—
180C22⋊Q8C23112yes—
181C22⋊Q8C23124yes—
182#121C23112yes—
183#121C23112yes—
184#121C2316yes—
185#121C2316yes—
186#121C2316yes—
187#121C2316yes—
188#121C23112yes—
189#121C23112yes—
190#121C23112yes—
191#121C23112yes—
192#121C23112yes—
193#120C2314yes—
194#120C23112yes—
195C4×Q8C2314yes—
196C4×Q8C2314yes—
197C4×Q8C2316yes—
198C4×Q8C2314yes—
199C4×Q8C2314yes—
200C4×Q8C23112yes—
201C4×Q8C23112yes—
202C4×Q8C2314yes—
203C4×Q8C2314yes—
204C4×Q8C23112yes—
205C4×Q8C23112yes—
206C4×Q8C23112yes—
207C4×Q8C2316yes—
208C4×Q8C2314yes—
209C4×Q8C2314yes—
210C4×Q8C23112yes—
211C4×Q8C23112yes—
212C4×Q8C23112yes—
213C4×D4C2316yes—
214C4×D4C2316yes—
215C4×D4C2316yes—
216C4×D4C2316yes—
217C4×D4C2314yes—
218C4×D4C2316yes—
219C4×D4C23112yes—
220C4×D4C2316yes—
221C4×D4C2314yes—
222C4×D4C23112yes—
223C4×D4C23112yes—
224C4×D4C23112yes—
225C4×D4C23112yes—
226C4×D4C23112yes—
227C4×D4C23112yes—
228C4×D4C23112yes—
229C4×D4C23112yes—
230C4×D4C23112yes—
231C4×D4C23112yes—
232C4×D4C23112yes—
233C4×D4C23124yes—
234#117C2314yes—
235#117C2316yes—
236#117C2316yes—
237#117C2316yes—
238#117C2316yes—
239#117C2316yes—
240#117C23112yes—
241#117C23112yes—
242#117C2316yes—
243#117C2316yes—
244#117C23112yes—
245#117C23112yes—
246#117C2314yes—
247#117C23112yes—
248#117C23112yes—
249#117C23112yes—
250#117C23124yes—
251C2×C4⋊C4C2316yes—
252C2×C4⋊C4C2316yes—
253C2×C4⋊C4C2316yes—
254C2×C4⋊C4C2316yes—
255C2×C4⋊C4C2313yes—
256C2×C4⋊C4C2313yes—
257C2×C4⋊C4C23112yes—
258C2×C22⋊C4C2314yes—
259C2×C22⋊C4C23112yes—
260C2×C22⋊C4C2311yescharacteristic
261C2×C22⋊C4C2313yes—
262C2×C22⋊C4C2316yes—
263C2×C42C23112yes—
264C2×C42C23112yes—
265C2×C42C2313yes—
266C2×C42C23112yes—
267C24C2411yescharacteristic
268C24C2413yes—
269#41—216no—
270#41—216no—
271#41—216no—
272#41—224no—
273#41—424no—
274C2×Q8—216no—
275C2×Q8—216no—
276C2×Q8—216no—
277C2×Q8—212no—
278C2×Q8#4118yescomplemented
279C2×Q8C24112yes—
280C2×Q8C2416yes—
281C2×Q8C2414yes—
282C2×Q8C24112yes—
283C2×Q8C2414yes—
284C2×D4—212no—
285C2×D4—46no—
286C2×D4C2414yes—
287C2×D4C2414yes—
288C2×D4C2416yes—
289C2×D4C24112yes—
290C2×D4C2416yes—
291C2×D4C2414yes—
292C2×D4C24112yes—
293C2×D4C24112yes—
294C2×D4C2416yes—
295C22×C4—212no—
296C22×C4—212no—
297C22×C4C2414yes—
298C22×C4C2416yes—
299C22×C4C2416yes—
300C22×C4C24112yes—
301C22×C4C2413yes—
302C22×C4C2411yescharacteristic
303C22×C4C2416yes—
304C22×C4C2416yes—
305C22×C4C2411yescharacteristic
306C22×C4C2414yes—
307C22×C4C2414yes—
308C22×C4C2413yes—
309C22×C4C24112yes—
310C22×C4C24112yes—
311C22×C4C24112yes—
312C22×C4C2413yes—
313C4⋊C4C2416yes—
314C4⋊C4C2414yes—
315C4⋊C4C2414yes—
316C4⋊C4C2416yes—
317C4⋊C4C24112yes—
318C4⋊C4C24112yes—
319C4⋊C4C2416yes—
320C4⋊C4C2416yes—
321C4⋊C4C2416yes—
322C4⋊C4C2416yes—
323C4⋊C4C2416yes—
324C4⋊C4C24112yes—
325C4⋊C4C24112yes—
326C4⋊C4C2416yes—
327C4⋊C4C2414yes—
328C4⋊C4C2414yes—
329C4⋊C4C24112yes—
330C4⋊C4C24112yes—
331C4⋊C4C2414yes—
332C4⋊C4C2414yes—
333C4⋊C4C24112yes—
334C4⋊C4C24112yes—
335C4⋊C4C2414yes—
336C4⋊C4C24112yes—
337C4⋊C4C2414yes—
338C22⋊C4C2414yes—
339C22⋊C4C2414yes—
340C22⋊C4C2414yes—
341C22⋊C4C2414yes—
342C22⋊C4C2414yes—
343C22⋊C4C2411yescharacteristic
344C22⋊C4C24112yes—
345C22⋊C4C24112yes—
346C22⋊C4C2416yes—
347C22⋊C4C24124yes—
348C22⋊C4C2411yescharacteristic
349C22⋊C4C2411yescharacteristic
350C22⋊C4C2411yescharacteristic
351C22⋊C4C2416yes—
352C22⋊C4C2414yes—
353C22⋊C4C24112yes—
354C22⋊C4C2414yes—
355C22⋊C4C24112yes—
356C22⋊C4C2414yes—
357C22⋊C4C24112yes—
358C22⋊C4C24112yes—
359C22⋊C4C2414yes—
360C22⋊C4C2414yes—
361C22⋊C4C24112yes—
362C42C2414yes—
363C42C24112yes—
364C42C2414yes—
365C42C2414yes—
366C42C2416yes—
367C42C24112yes—
368C42C24112yes—
369C42C24112yes—
370C42C24112yes—
371C42C2416yes—
372C42C24112yes—
373C23—22no—
374C23—22no—
375C23—26no—
376C23—41no—
377C23—43no—
378C23—43no—
379C23C2511yescharacteristic
380C23C2516yes—
381C23C2514yes—
382C23C2511yescharacteristic
383C23C2511yescharacteristic
384Q8—224no—
385Q8—212no—
386Q8—44no—
387Q8—412no—
388Q8C2×#41116yes—
389D4—28no—
390D4—212no—
391D4—28no—
392D4—224no—
393D4—212no—
394D4—46no—
395D4—44no—
396D4—412no—
397D4—412no—
398C2×C4—22no—
399C2×C4—26no—
400C2×C4—28no—
401C2×C4—212no—
402C2×C4—212no—
403C2×C4—212no—
404C2×C4—212no—
405C2×C4—224no—
406C2×C4—44no—
407C2×C4—412no—
408C2×C4—44no—
409C2×C4—41no—
410C2×C4—43no—
411C2×C4—412no—
412C2×C4—412no—
413C2×C4C2×#41112yes—
414C2×C4C2514yes—
415C2×C4C2511yescharacteristic
416C2×C4C2511yescharacteristic
417C2×C4C2514yes—
418C2×C4C2511yescharacteristic
419C2×C4C2514yes—
420C2×C4C2514yes—
421C2×C4C2516yes—
422C2×C4C2514yes—
423C2×C4C2514yes—
424C2×C4C2514yes—
425C2×C4C2516yes—
426C2×C4C2511yescharacteristic
427C2×C4C2516yes—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—24no—
434C22—26no—
435C22—24no—
436C22—26no—
437C22—21no—
438C22—26no—
439C22—24no—
440C22—21no—
441C22—44no—
442C22—86no—
443C22C22×#4112yes—
444C22#58512yes—
445C22C2611yescharacteristic
446C4—21no—
447C4—21no—
448C4—24no—
449C4—26no—
450C4—24no—
451C4—24no—
452C4—26no—
453C4—24no—
454C4—21no—
455C4—24no—
456C4—24no—
457C4—24no—
458C4—21no—
459C4C22×#41112yes—
460C2—22no—
461C2—22no—
462C2—44no—
463C2—41no—
464C2—46no—
465C2C23×#4111yescharacteristic
466C2C2×#58511yescharacteristic
467C22−1+611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC22C26
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ G
LECSG ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3524C2semidirect
2. #3524C2semidirect
3. #3522C2semidirect
4. #3518C2semidirect
5. #3516C2semidirect
6. #3516C2semidirect
7. #3510C2semidirect
8. #3502C2semidirect
9. #3495C2semidirect
10. #3494C2semidirect
11. #3487C2semidirect
12. #3486C2semidirect
13. #3482C2semidirect
14. #3480C2semidirect
15. #3475C2semidirect
16. C2×#548C2semidirect
17. C4×2−1+4C2semidirect
18. #3428C2semidirect
19. C2×#519C2semidirect
23. #559C22semidirect
24. #559C22semidirect
25. Q82C22semidirect
26. #557C22semidirect
27. #557C22semidirect
28. #557C22semidirect
29. #556C22semidirect
30. #556C22semidirect
31. #556C22semidirect
32. #556C22semidirect
33. #555C22semidirect
34. #555C22semidirect
35. #554C22semidirect
36. #554C22semidirect
37. #553C22semidirect
38. #553C22semidirect
39. #553C22semidirect
40. #553C22semidirect
42. #552C22semidirect
43. #552C22semidirect
44. #552C22semidirect
45. #552C22semidirect
49. #550C22semidirect
50. #550C22semidirect
51. D4×Q8C22semidirect
52. D4×Q8C22semidirect
57. #548C22semidirect
58. #548C22semidirect
59. #548C22semidirect
60. #548C22semidirect
61. #547C22semidirect
62. #547C22semidirect
65. #542C22semidirect
66. #542C22semidirect
67. #542C22semidirect
68. #542C22semidirect
69. #541C22semidirect
70. #540C22semidirect
74. #533C22semidirect
75. #532C22semidirect
76. #529C22semidirect
77. #529C22semidirect
78. #529C22semidirect
79. #529C22semidirect
80. #529C22semidirect
81. #529C22semidirect
82. #529C22semidirect
89. #520C22semidirect
90. #520C22semidirect
91. #520C22semidirect
96. #519C22semidirect
97. #519C22semidirect
98. #519C22semidirect
101. C4×#41C22semidirect
102. C4×#41C22semidirect
103. C4×#41C22semidirect
104. C4×#41C22semidirect
106. C2×C4×Q8C22semidirect
107. C2×C4×Q8C22semidirect
108. C2×C4×Q8C22semidirect
278. C2×Q8#41semidirect

Maximal subgroups

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

subgroupindex
#35242
#35222
#35182
#35162
#35102
#35022
#34952
#34942
#34872
#34862
#34822
#34802
#34752
C2×#5482
C4×2−1+42
#34282
C2×#5192