Finite Group Databasebeta

← #61339

Subgroups — #61339

Order 256.

956 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3508C211yesmaximal, characteristic, complemented
2#3501C211yesmaximal, characteristic, complemented
3#3373C212yesmaximal, complemented
4#3370C214yesmaximal, complemented
5#3366C212yesmaximal, complemented
6#3366C214yesmaximal, complemented
7#3361C212yesmaximal, complemented
8#3358C212yesmaximal, complemented
9#3138C211yesmaximal, characteristic, complemented
10#3136C212yesmaximal, complemented
11#3081C212yesmaximal, complemented
12#3076C211yesmaximal, characteristic, complemented
13#3041C212yesmaximal, complemented
14C2×#459C211yesmaximal, characteristic, complemented
15#2962C211yesmaximal, characteristic, complemented
16#2948C212yesmaximal, complemented
17C4×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
18C2×AGL(1,ℤ/8)C2211yescharacteristic
19C22×D8C2212yes—
20#559C2212yescomplemented
21#559C2212yescomplemented
22#554—21no—
23#554—22no—
24#554C2212yescomplemented
25#554C2214yescomplemented
26#550C2212yescomplemented
27#550C2212yescomplemented
28D4×Q8—21no—
29D4×Q8C2212yescomplemented
30#546—22no—
31#546—22no—
32#544C2212yes—
33#543—22no—
34#540—22no—
35#535C2212yes—
36#533C2211yescharacteristic
37#533C2211yescharacteristic
38#532—21no—
39#532—21no—
40C2×#127C2211yescharacteristic
41C2×C22⋊Q8—22no—
42#518C2212yes—
43#518C2212yes—
44C4×#41C2211yescharacteristic
45C2×C4×D4C2211yescharacteristic
46#490C2214yescomplemented
47#490C2214yescomplemented
48#489C2214yescomplemented
49#488C2214yescomplemented
50#486C2214yescomplemented
51#486C2212yescomplemented
52#486C2212yescomplemented
53#479C2212yescomplemented
54#478C2212yescomplemented
55#478C2214yescomplemented
56#478C2214yescomplemented
57#477C2211yescharacteristic, complemented
58#476C2211yescharacteristic, complemented
59#476C2214yescomplemented
60#475C2211yescharacteristic, complemented
61#474C2211yescharacteristic, complemented
62#474C2214yescomplemented
63#463C2214yescomplemented
64#463C2214yescomplemented
65#459C2212yes—
66#459C2214yes—
67#459C2212yes—
68#459C2214yescomplemented
69#459C2214yescomplemented
70#446C2214yescomplemented
71#446C2212yescomplemented
72C8×Q8C2212yescomplemented
73#442C2212yescomplemented
74#442C2214yescomplemented
75#442C2214yescomplemented
76#440C2212yescomplemented
77C4×SD16C2212yescomplemented
78C4×D8C2212yescomplemented
79C4×D8C2214yescomplemented
80C4×D8C2214yescomplemented
81#428C2211yescharacteristic
82#427C2212yes—
83#424C2212yes—
84C2×C4⋊C8C2211yescharacteristic
85#418C2212yes—
86#418C2212yes—
87#418C2211yescharacteristic
88#417C2211yescharacteristic
89#416C2212yes—
90C2×#102C2212yes—
91#405C2212yes—
92C4×M16C2211yescharacteristic
93C2×#41—22no—
94C2×#41—22no—
95C2×#41C2311yescharacteristic
96C22×Q8—22no—
97C22×D4—21no—
98C22×D4—21no—
99C22×D4C2312yes—
100C22×D4C2311yescharacteristic
101C23×C4—21no—
102AGL(1,ℤ/8)—24no—
103C2×SD16C2312yes—
104C2×D8—28no—
105C2×D8C2314yes—
106C2×D8C2314yes—
107C2×D8C2312yes—
108C2×M16C2311yescharacteristic
109C2×M16C2311yescharacteristic
110C22×C8C2312yes—
111C4⋊Q8—22no—
112C4⋊Q8D411yescharacteristic
113C4⋊Q8D411yescharacteristic
114C4⋊Q8C2311yescharacteristic
115C4⋊Q8C2311yescharacteristic
116C4⋊Q8C2314yes—
117#127—21no—
118#127—21no—
119#127C2312yes—
120#127C2312yes—
121#127C2314yes—
122#125—22no—
123#125D412yes—
124#125C2312yes—
125#125C2314yes—
126#124—22no—
127#124—22no—
128#124C2314yes—
129#123—24no—
130#123—24no—
131C22⋊Q8—22no—
132C22⋊Q8—24no—
133C22⋊Q8—21no—
134C22⋊Q8—21no—
135C22⋊Q8—22no—
136C22⋊Q8—24no—
137C22⋊Q8—24no—
138C22⋊Q8—24no—
139C22⋊Q8—24no—
140C22⋊Q8—24no—
141C22⋊Q8D412yes—
142C22⋊Q8D412yes—
143#121—22no—
144#121—22no—
145#121—22no—
146#121—22no—
147#121—24no—
148#121—24no—
149#120—24no—
150C4×Q8—21no—
151C4×Q8—21no—
152C4×Q8C2311yescharacteristic
153C4×Q8C2311yescharacteristic
154C4×Q8C2312yes—
155C4×Q8C2312yes—
156C4×D4—21no—
157C4×D4—21no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—22no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—22no—
164C4×D4—21no—
165C4×D4—21no—
166C4×D4C2312yes—
167C4×D4C2312yes—
168C4×D4C2311yescharacteristic
169C4×D4C2311yescharacteristic
170C4×D4C2314yes—
171C4×D4C2314yes—
172C4×D4C2314yes—
173C4×D4C2314yes—
174#117—21no—
175#117C2311yescharacteristic
176#117C2312yes—
177#117C2312yes—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4C2311yescharacteristic
181C2×C22⋊C4—21no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C42—21no—
185C2×C42C2311yescharacteristic
186#107C2312yes—
187#107C2312yes—
188#107C2312yes—
189#106C2312yes—
190#106C2312yes—
191#106C2312yes—
192C4⋊C8C2311yescharacteristic
193C4⋊C8C2311yescharacteristic
194C4⋊C8C2312yes—
195C4⋊C8C2314yes—
196C4⋊C8C2314yes—
197#103C2312yes—
198#103C2312yes—
199#103C2312yes—
200#102C2314yes—
201#102C2314yes—
202#102C2312yes—
203#102C2312yes—
204#102C2312yes—
205#102C2312yes—
206#102C2312yes—
207#102C2314yes—
208#102C2314yes—
209#102C2314yes—
210#102C2314yes—
211#97C2312yes—
212#97C2314yes—
213C4×C8C2312yes—
214C4×C8C2314yes—
215C24—21no—
216C24—22no—
217#41—22no—
218#41—22no—
219#41—84no—
220#41—84no—
221C2×Q8—21no—
222C2×Q8—21no—
223C2×Q8—24no—
224C2×Q8—84no—
225C2×Q8C2×D412yes—
226C2×Q8C2×D412yes—
227C2×Q8C2411yescharacteristic
228C2×D4—21no—
229C2×D4—21no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—24no—
234C2×D4—24no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—24no—
238C2×D4—22no—
239C2×D4—24no—
240C2×D4—82no—
241C2×D4—82no—
242C2×D4#4114yescomplemented
243C2×D4C2414yes—
244C2×D4C2411yescharacteristic
245C2×D4C2412yes—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—82no—
257C22×C4C2×D411yescharacteristic
258C22×C4C2×D411yescharacteristic
259C22×C4C2411yescharacteristic
260SD16—24no—
261D8—28no—
262D8—28no—
263D8—24no—
264M16—22no—
265M16—22no—
266C2×C8—24no—
267C2×C8C2412yes—
268C2×C8C2412yes—
269C2×C8C2412yes—
270C2×C8C2412yes—
271C4⋊C4—22no—
272C4⋊C4—21no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—22no—
280C4⋊C4—21no—
281C4⋊C4C2×D412yes—
282C4⋊C4C2×D414yes—
283C4⋊C4C2×D412yes—
284C4⋊C4C2×D412yes—
285C4⋊C4C2×D412yes—
286C4⋊C4C2×D414yes—
287C4⋊C4C2412yes—
288C4⋊C4C2412yes—
289C4⋊C4C2412yes—
290C4⋊C4C2411yescharacteristic
291C4⋊C4C2411yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—22no—
294C22⋊C4—24no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4C2×D414yes—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—22no—
309C42C2×D411yescharacteristic
310C42C2×D411yescharacteristic
311C42C2411yescharacteristic
312C42C2414yes—
313C42C2411yescharacteristic
314C23—22no—
315C23—24no—
316C23—24no—
317C23—21no—
318C23—44no—
319C23—81no—
320C23—81no—
321C23—82no—
322C23—82no—
323C23C22×D411yescharacteristic
324Q8—24no—
325Q8—21no—
326Q8—21no—
327Q8—42no—
328Q8—42no—
329Q8—81no—
330Q8—84no—
331D4—24no—
332D4—24no—
333D4—21no—
334D4—22no—
335D4—24no—
336D4—21no—
337D4—44no—
338D4—42no—
339D4—44no—
340D4—44no—
341D4—82no—
342D4—82no—
343D4—84no—
344D4—84no—
345D4C2×#4114yes—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—84no—
359C2×C4—82no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4—84no—
363C2×C4—81no—
364C2×C4C22×D411yescharacteristic
365C2×C4C22×D411yescharacteristic
366C2×C4C22×D411yescharacteristic
367C2×C4C22×D414yes—
368C2×C4C22×D414yes—
369C2×C4C22×D411yescharacteristic
370C2×C4C22×D411yescharacteristic
371C2×C4C2×#4112yes—
372C2×C42−1+412yes—
373C2×C4C2511yescharacteristic
374C8—22no—
375C8—22no—
376C8—22no—
377C8—22no—
378C22—21no—
379C22—21no—
380C22—24no—
381C22—42no—
382C22—42no—
383C22—41no—
384C22—44no—
385C22—41no—
386C22—44no—
387C22—41no—
388C22—44no—
389C22—84no—
390C22—164no—
391C22#54812yes—
392C22C23×D411yescharacteristic
393C4—21no—
394C4—21no—
395C4—24no—
396C4—21no—
397C4—24no—
398C4—41no—
399C4—42no—
400C4—41no—
401C4—41no—
402C4—42no—
403C4—42no—
404C4#54812yes—
405C4#57612yes—
406C4C22×#4111yescharacteristic
407C4C2×2−1+411yescharacteristic
408C2—22no—
409C2—44no—
410C2—81no—
411C2—84no—
412C2C2×#54811yescharacteristic
413C2C2×#57611yescharacteristic
414C2#358511yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ #533 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3508C2semidirect
2. #3501C2semidirect
3. #3373C2semidirect
4. #3370C2semidirect
5. #3366C2semidirect
6. #3366C2semidirect
7. #3361C2semidirect
8. #3358C2semidirect
9. #3138C2semidirect
10. #3136C2semidirect
11. #3081C2semidirect
12. #3076C2semidirect
13. #3041C2semidirect
14. C2×#459C2semidirect
15. #2962C2semidirect
16. #2948C2semidirect
17. C4×AGL(1,ℤ/8)C2semidirect
20. #559C22semidirect
21. #559C22semidirect
24. #554C22semidirect
25. #554C22semidirect
26. #550C22semidirect
27. #550C22semidirect
29. D4×Q8C22semidirect
46. #490C22semidirect
47. #490C22semidirect
48. #489C22semidirect
49. #488C22semidirect
50. #486C22semidirect
51. #486C22semidirect
52. #486C22semidirect
53. #479C22semidirect
54. #478C22semidirect
55. #478C22semidirect
56. #478C22semidirect
57. #477C22semidirect
58. #476C22semidirect
59. #476C22semidirect
60. #475C22semidirect
61. #474C22semidirect
62. #474C22semidirect
63. #463C22semidirect
64. #463C22semidirect
68. #459C22semidirect
69. #459C22semidirect
70. #446C22semidirect
71. #446C22semidirect
72. C8×Q8C22semidirect
73. #442C22semidirect
74. #442C22semidirect
75. #442C22semidirect
76. #440C22semidirect
77. C4×SD16C22semidirect
78. C4×D8C22semidirect
79. C4×D8C22semidirect
80. C4×D8C22semidirect
242. C2×D4#41semidirect

Maximal subgroups

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

subgroupindex
#35082
#35012
#33732
#33702
#33662
#33612
#33582
#31382
#31362
#30812
#30762
#30412
C2×#4592
#29622
#29482
C4×AGL(1,ℤ/8)2