Finite Group Databasebeta

← #18598

Subgroups — #18598

Order 256.

612 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3496C211yesmaximal, characteristic, complemented
2#3214C211yesmaximal, characteristic, complemented
3#3184C211yesmaximal, characteristic, complemented
4#3100C211yesmaximal, characteristic, complemented
5#1654C212yesmaximal, complemented
6#1639C212yesmaximal, complemented
7#1625C212yesmaximal, complemented
8#1619C212yesmaximal, complemented
9#1553C211yesmaximal, characteristic, complemented
10#1490C211yesmaximal, characteristic, complemented
11#1484C211yesmaximal, characteristic, complemented
12C2×H(5,𝔽2)—21no—
13#547—21no—
14#547C2212yes—
15#546—22no—
16#546—22no—
17D42—21no—
18D42C2212yes—
19#537—21no—
20#534—22no—
21#532C2211yescharacteristic
22#532C2211yescharacteristic
23#532C2211yescharacteristic, complemented
24C2×#121—22no—
25C2×C4×D4C2211yescharacteristic
26#498C2211yescharacteristic, complemented
27#496—21no—
28#493C2211yescharacteristic, complemented
29#489—21no—
30#486C2211yescharacteristic, complemented
31#486C2211yescharacteristic, complemented
32#480—21no—
33#474C2212yescomplemented
34#469—22no—
35#461C2212yescomplemented
36#459C2212yescomplemented
37#459C2212yescomplemented
38#449—22no—
39#447—21no—
40#442—22no—
41C4×D8—22no—
42#432C2211yescharacteristic, complemented
43#431C2211yescharacteristic, complemented
44C4×M16C2211yescharacteristic, complemented
45#332C2212yescomplemented
46#331C2212yescomplemented
47D4⋊C8C2212yescomplemented
48D4⋊C8C2212yescomplemented
49D4⋊C8C2212yescomplemented
50D4⋊C8C2212yescomplemented
51H(5,𝔽2)—48no—
52C2×#41—22no—
53C2×#41—22no—
54C2×#41—21no—
55C2×#41—21no—
56C2×#41—21no—
57C22×D4—24no—
58C22×D4—22no—
59C22×D4—22no—
60C22×D4—22no—
61C22×D4D411yescharacteristic
62C23×C4—21no—
63C2×SD16—22no—
64C2×D8—22no—
65C2×D8—22no—
66C2×D8—22no—
67C2×M16—21no—
68C4⋊Q8D411yescharacteristic
69C4⋊Q8C2311yescharacteristic
70C4⋊Q8C2311yescharacteristic
71#127D411yescharacteristic
72#127C2311yescharacteristic
73#127C2311yescharacteristic
74#124—21no—
75#124—21no—
76#124—21no—
77#123—24no—
78#123—24no—
79C22⋊Q8—22no—
80C22⋊Q8—22no—
81#121—24no—
82#121—24no—
83#121—22no—
84#121—22no—
85#121—21no—
86#121—21no—
87#121—22no—
88#121—22no—
89#121—21no—
90#121—21no—
91#121D412yes—
92#121D412yes—
93#120—24no—
94#120—24no—
95C4×D4—21no—
96C4×D4—21no—
97C4×D4—21no—
98C4×D4—21no—
99C4×D4—21no—
100C4×D4—21no—
101C4×D4D412yes—
102C4×D4D412yes—
103C4×D4C2312yes—
104C4×D4C2312yes—
105C2×C4⋊C4—22no—
106C2×C4⋊C4D411yescharacteristic
107C2×C22⋊C4—24no—
108C2×C22⋊C4—21no—
109C2×C42C2311yescharacteristic
110#107—22no—
111#106—22no—
112C4⋊C8C2312yes—
113C4⋊C8C2312yes—
114#103—22no—
115#102—22no—
116#102—22no—
117#102—22no—
118#102—22no—
119#102—22no—
120C22⋊C8—21no—
121C22⋊C8—21no—
122#97—21no—
123#97—21no—
124C4×C8—21no—
125C4×C8C2311yescharacteristic
126C4×C8C2311yescharacteristic
127C24—21no—
128C24—24no—
129#41—44no—
130#41—44no—
131#41—48no—
132#41—44no—
133#41—44no—
134#41—82no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×D4—24no—
139C2×D4—24no—
140C2×D4—24no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—24no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—48no—
153C2×D4—44no—
154C2×D4—416no—
155C2×D4—44no—
156C2×D4—41no—
157C2×D4—82no—
158C2×D4—82no—
159C2×D4C2×D412yes—
160C2×D4C2×D412yes—
161C2×D4C2×D412yes—
162C22×C4—24no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—41no—
169C22×C4—82no—
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173SD16—82no—
174D8—44no—
175D8—44no—
176D8—82no—
177M16—42no—
178C2×C8—21no—
179C2×C8—22no—
180C2×C8—21no—
181C2×C8—22no—
182C4⋊C4—24no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4C2×D412yes—
189C4⋊C4C2×D412yes—
190C4⋊C4C2×D412yes—
191C22⋊C4—24no—
192C22⋊C4—24no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—22no—
197C22⋊C4—22no—
198C22⋊C4—22no—
199C22⋊C4—41no—
200C22⋊C4—41no—
201C42C2×D411yescharacteristic
202C42C2×D411yescharacteristic
203C42C2×D411yescharacteristic
204C42C2411yescharacteristic
205C23—24no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—48no—
210C23—41no—
211C23—81no—
212C23—81no—
213C23—84no—
214C23—84no—
215C23#12011yescharacteristic
216Q8—42no—
217Q8—42no—
218Q8—42no—
219D4—24no—
220D4—22no—
221D4—22no—
222D4—22no—
223D4—42no—
224D4—42no—
225D4—42no—
226D4—44no—
227D4—44no—
228D4—48no—
229D4—48no—
230D4—42no—
231D4—42no—
232D4—42no—
233D4—42no—
234D4—81no—
235D4—84no—
236D4—82no—
237D4AGL(1,ℤ/8)14yes—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—44no—
245C2×C4—48no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—81no—
254C2×C4#12011yescharacteristic
255C2×C4#12011yescharacteristic
256C2×C4#12011yescharacteristic
257C2×C4C22×D411yescharacteristic
258C2×C4C22×D411yescharacteristic
259C2×C4C22×D411yescharacteristic
260C8—22no—
261C8—41no—
262C8—42no—
263C8—42no—
264C22—21no—
265C22—24no—
266C22—21no—
267C22—21no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—84no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—168no—
281C22C2×#12011yescharacteristic
282C4—21no—
283C4—21no—
284C4—21no—
285C4—21no—
286C4—42no—
287C4—42no—
288C4—42no—
289C4—81no—
290C4C2×AGL(1,ℤ/8)12yes—
291C4#57612yes—
292C2—41no—
293C2—44no—
294C2—82no—
295C2—82no—
296C2—161no—
297C2#300311yescharacteristic
298C2#300811yescharacteristic
299C2#301811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3496C2semidirect
2. #3214C2semidirect
3. #3184C2semidirect
4. #3100C2semidirect
5. #1654C2semidirect
6. #1639C2semidirect
7. #1625C2semidirect
8. #1619C2semidirect
9. #1553C2semidirect
10. #1490C2semidirect
11. #1484C2semidirect
23. #532C22semidirect
26. #498C22semidirect
28. #493C22semidirect
30. #486C22semidirect
31. #486C22semidirect
33. #474C22semidirect
35. #461C22semidirect
36. #459C22semidirect
37. #459C22semidirect
42. #432C22semidirect
43. #431C22semidirect
44. C4×M16C22semidirect
45. #332C22semidirect
46. #331C22semidirect
47. D4⋊C8C22semidirect
48. D4⋊C8C22semidirect
49. D4⋊C8C22semidirect
50. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#34962
#32142
#31842
#31002
#16542
#16392
#16252
#16192
#15532
#14902
#14842