Finite Group Databasebeta

← #18916

Subgroups — #18916

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#1628C214yesmaximal, complemented
6#1621C214yesmaximal, complemented
7#1540C211yesmaximal, characteristic, complemented
8#1481C211yesmaximal, characteristic, complemented
9#1478C211yesmaximal, characteristic, complemented
10C22×D8—21no—
11#559—22no—
12#556C2212yescomplemented
13#555—21no—
14#553—22no—
15#550C2212yescomplemented
16#547—21no—
17#538—22no—
18#535—21no—
19#533C2211yescharacteristic
20#532C2211yescharacteristic
21C2×#127C2211yescharacteristic
22#525—22no—
23#520—21no—
24C2×C4×D4C2211yescharacteristic
25#493C2212yescomplemented
26#492—21no—
27#486—21no—
28#486C2212yescomplemented
29#460C2214yescomplemented
30#459—22no—
31#459—22no—
32#459C2214yes—
33#450—22no—
34#447—22no—
35#431C2211yescharacteristic
36#431C2211yescharacteristic
37C2×C4⋊C8C2211yescharacteristic
38C2×#102—21no—
39#332C2212yescomplemented
40#331C2212yescomplemented
41Q8⋊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⋊Q8D411yescharacteristic
57C4⋊Q8C2311yescharacteristic
58#127—22no—
59#127—21no—
60#127—21no—
61#127—21no—
62#127C2312yes—
63#127C2311yescharacteristic
64#126—24no—
65#125—21no—
66#124—24no—
67#124—22no—
68#124—21no—
69#123—24no—
70#123—24no—
71C22⋊Q8—22no—
72C22⋊Q8—21no—
73C22⋊Q8D412yes—
74#121—22no—
75#121—22no—
76#121—24no—
77#121—21no—
78#121—22no—
79#121D412yes—
80#120—24no—
81C4×Q8—22no—
82C4×Q8—21no—
83C4×Q8C2312yes—
84C4×D4—24no—
85C4×D4—24no—
86C4×D4—22no—
87C4×D4—21no—
88C4×D4—21no—
89C4×D4—21no—
90C4×D4D412yes—
91C4×D4D412yes—
92C4×D4C2312yes—
93#117—22no—
94#117—21no—
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—22no—
124C2×D4—22no—
125C2×D4—21no—
126C2×D4—21no—
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—21no—
158C4⋊C4—21no—
159C4⋊C4—22no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—22no—
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—24no—
191Q8—41no—
192Q8—41no—
193Q8—81no—
194D4—22no—
195D4—22no—
196D4—42no—
197D4—44no—
198D4—44no—
199D4—41no—
200D4—42no—
201D4—41no—
202D4—42no—
203D4—42no—
204D4—44no—
205D4—42no—
206D4—82no—
207D4—82no—
208D4—84no—
209D4—84no—
210D4—82no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—41no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—82no—
220C2×C4—81no—
221C2×C4—84no—
222C2×C4#12011yescharacteristic
223C2×C4#12011yescharacteristic
224C2×C4#12011yescharacteristic
225C2×C4#13512yes—
226C2×C4AGL(1,ℤ/8)12yes—
227C2×C4C22×D411yescharacteristic
228C2×C4C22×D411yescharacteristic
229C2×C4C22×D411yescharacteristic
230C8—42no—
231C8—42no—
232C8—42no—
233C22—21no—
234C22—21no—
235C22—42no—
236C22—42no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—84no—
242C22—82no—
243C22—82no—
244C22—82no—
245C22—164no—
246C22#44912yes—
247C22C2×#12011yescharacteristic
248C4—21no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—42no—
254C4—42no—
255C4#44912yes—
256C4C2×#13511yescharacteristic
257C4C2×AGL(1,ℤ/8)11yescharacteristic
258C4#57612yes—
259C2—22no—
260C2—82no—
261C2—82no—
262C2—162no—
263C2C2×#44911yescharacteristic
264C2#300811yescharacteristic
265C2#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. #1628C2semidirect
6. #1621C2semidirect
7. #1540C2semidirect
8. #1481C2semidirect
9. #1478C2semidirect
12. #556C22semidirect
15. #550C22semidirect
25. #493C22semidirect
28. #486C22semidirect
29. #460C22semidirect
39. #332C22semidirect
40. #331C22semidirect
41. Q8⋊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
#16282
#16212
#15402
#14812
#14782