Finite Group Databasebeta

← #18918

Subgroups — #18918

Order 256.

521 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 521 conjugacy classes of subgroups — too many to draw. Only its 121 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#3506C211yesmaximal, characteristic, complemented
2#3208C211yesmaximal, characteristic, complemented
3#3205C211yesmaximal, characteristic, complemented
4C2×#459C211yesmaximal, characteristic, complemented
5#1625C214yesmaximal, complemented
6#1624C214yesmaximal, complemented
7#1542C211yesmaximal, characteristic, complemented
8#1480C211yesmaximal, characteristic, complemented
9#1478C211yesmaximal, characteristic, complemented
10C22×D8—21no—
11#559—22no—
12#556—21no—
13#555C2212yescomplemented
14#553—22no—
15#550—21no—
16#547C2212yescomplemented
17#538—22no—
18#535—21no—
19#533C2211yescharacteristic
20#532C2211yescharacteristic
21C2×#127C2211yescharacteristic
22#525—22no—
23#520—21no—
24C2×C4×D4C2211yescharacteristic
25#493—21no—
26#492C2212yescomplemented
27#486—21no—
28#486C2212yescomplemented
29#460—22no—
30#459—22no—
31#459C2214yes—
32#459C2214yescomplemented
33#450—22no—
34#447—22no—
35#431C2211yescharacteristic
36#431C2211yescharacteristic
37C2×C4⋊C8C2211yescharacteristic
38C2×#102—21no—
39#333C2212yescomplemented
40#332C2212yescomplemented
41D4⋊C8C2214yescomplemented
42D4⋊C8C2214yescomplemented
43C2×#41—22no—
44C2×#41—21no—
45C22×D4—21no—
46C22×D4—21no—
47C22×D4—21no—
48C22×D4D411yescharacteristic
49C23×C4—21no—
50C2×SD16—24no—
51C2×D8—24no—
52C2×D8—24no—
53C2×D8—44no—
54C22×C8—21no—
55C4⋊Q8D411yescharacteristic
56C4⋊Q8C2311yescharacteristic
57C4⋊Q8C2311yescharacteristic
58#127—22no—
59#127—21no—
60#127—21no—
61#127—21no—
62#127D411yescharacteristic
63#127C2312yes—
64#126—24no—
65#125—21no—
66#124—22no—
67#124—24no—
68#124—21no—
69#123—24no—
70#123—24no—
71C22⋊Q8—21no—
72C22⋊Q8—22no—
73C22⋊Q8D412yes—
74#121—24no—
75#121—22no—
76#121—22no—
77#121—22no—
78#121—21no—
79#121D412yes—
80#120—24no—
81C4×Q8—22no—
82C4×Q8—21no—
83C4×Q8D412yes—
84C4×D4—24no—
85C4×D4—22no—
86C4×D4—24no—
87C4×D4—21no—
88C4×D4—21no—
89C4×D4—21no—
90C4×D4D412yes—
91C4×D4C2312yes—
92C4×D4C2312yes—
93#117—21no—
94#117—22no—
95#117—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4D411yescharacteristic
98C2×C22⋊C4—21no—
99C2×C42C2311yescharacteristic
100C4⋊C8—21no—
101C4⋊C8—21no—
102C4⋊C8C2312yes—
103C4⋊C8C2312yes—
104#102—24no—
105#102—24no—
106#102—24no—
107C22⋊C8—21no—
108C22⋊C8—21no—
109C4×C8—21no—
110C4×C8C2312yes—
111C24—21no—
112C24—41no—
113#41—42no—
114#41—42no—
115#41—84no—
116C2×Q8—21no—
117C2×Q8—21no—
118C2×Q8C2×D412yes—
119C2×D4—22no—
120C2×D4—24no—
121C2×D4—24no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—41no—
131C2×D4—41no—
132C2×D4—41no—
133C2×D4—42no—
134C2×D4—42no—
135C2×D4—82no—
136C2×D4—82no—
137C2×D4C2×D412yes—
138C2×D4C2×D412yes—
139C22×C4—24no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—82no—
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148SD16—84no—
149D8—48no—
150D8—84no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—42no—
155C4⋊C4—24no—
156C4⋊C4—21no—
157C4⋊C4—22no—
158C4⋊C4—21no—
159C4⋊C4—22no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4C2×D412yes—
164C4⋊C4C2×D412yes—
165C4⋊C4C2×D412yes—
166C22⋊C4—24no—
167C22⋊C4—24no—
168C22⋊C4—24no—
169C22⋊C4—22no—
170C22⋊C4—22no—
171C22⋊C4—22no—
172C22⋊C4—22no—
173C22⋊C4—22no—
174C42—24no—
175C42—22no—
176C42C2×D411yescharacteristic
177C42C2×D411yescharacteristic
178C42C2×D411yescharacteristic
179C42C2411yescharacteristic
180C23—22no—
181C23—22no—
182C23—42no—
183C23—42no—
184C23—81no—
185C23—81no—
186C23—82no—
187C23—82no—
188C23—82no—
189C23#12011yescharacteristic
190Q8—42no—
191Q8—42no—
192Q8—42no—
193D4—22no—
194D4—22no—
195D4—24no—
196D4—44no—
197D4—44no—
198D4—42no—
199D4—42no—
200D4—44no—
201D4—42no—
202D4—42no—
203D4—81no—
204D4—82no—
205D4—82no—
206D4—84no—
207D4—84no—
208D4—82no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—44no—
218C2×C4—82no—
219C2×C4—81no—
220C2×C4—84no—
221C2×C4#12011yescharacteristic
222C2×C4#12011yescharacteristic
223C2×C4#12011yescharacteristic
224C2×C4#13512yes—
225C2×C4AGL(1,ℤ/8)12yes—
226C2×C4C22×D411yescharacteristic
227C2×C4C22×D411yescharacteristic
228C2×C4C22×D411yescharacteristic
229C8—42no—
230C8—42no—
231C8—42no—
232C22—21no—
233C22—21no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—84no—
241C22—82no—
242C22—82no—
243C22—82no—
244C22—164no—
245C22#44912yes—
246C22C2×#12011yescharacteristic
247C4—21no—
248C4—21no—
249C4—21no—
250C4—42no—
251C4—42no—
252C4—42no—
253C4—42no—
254C4#44912yes—
255C4C2×#13511yescharacteristic
256C4C2×AGL(1,ℤ/8)11yescharacteristic
257C4#57612yes—
258C2—22no—
259C2—82no—
260C2—82no—
261C2—162no—
262C2C2×#44911yescharacteristic
263C2#300811yescharacteristic
264C2#302111yescharacteristic
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. #3506C2semidirect
2. #3208C2semidirect
3. #3205C2semidirect
4. C2×#459C2semidirect
5. #1625C2semidirect
6. #1624C2semidirect
7. #1542C2semidirect
8. #1480C2semidirect
9. #1478C2semidirect
13. #555C22semidirect
16. #547C22semidirect
26. #492C22semidirect
28. #486C22semidirect
32. #459C22semidirect
39. #333C22semidirect
40. #332C22semidirect
41. D4⋊C8C22semidirect
42. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35062
#32082
#32052
C2×#4592
#16252
#16242
#15422
#14802
#14782