Finite Group Databasebeta

← #18767

Subgroups — #18767

Order 256.

617 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 617 conjugacy classes of subgroups — too many to draw. Only its 129 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
1C2×#550C211yesmaximal, characteristic
2#3208C211yesmaximal, characteristic, complemented
3#3101C211yesmaximal, characteristic, complemented
4#3078C211yesmaximal, characteristic, complemented
5#1642C214yesmaximal, complemented
6#1631C214yesmaximal, complemented
7#1545C211yesmaximal, characteristic, complemented
8#1486C211yesmaximal, characteristic, complemented
9#1479C211yesmaximal, characteristic, complemented
10C22×#41—21no—
11#550—22no—
12#550—22no—
13#550—22no—
14#550C2214yes—
15#533C2211yescharacteristic
16C2×#127—21no—
17C2×#127C2211yescharacteristic
18C2×#121—22no—
19C2×#121—21no—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4—21no—
22C2×C4×D4C2211yescharacteristic
23#492C2212yescomplemented
24#486C2212yescomplemented
25#483—21no—
26#479—22no—
27#477C2212yescomplemented
28#476C2212yescomplemented
29#475C2212yescomplemented
30#461C2212yescomplemented
31#460—22no—
32#450—21no—
33#450—22no—
34C4×SD16—22no—
35#431C2211yescharacteristic
36#431C2211yescharacteristic
37#428—21no—
38#423C2211yescharacteristic
39#419—21no—
40#331C2214yescomplemented
41Q8⋊C8C2214yescomplemented
42D4⋊C8C2214yescomplemented
43C2×#41—24no—
44C2×#41—24no—
45C2×#41—44no—
46C2×#41—44no—
47C22×Q8D411yescharacteristic
48C22×D4—22no—
49C22×D4—22no—
50C22×D4—21no—
51C22×D4—21no—
52C22×D4D411yescharacteristic
53C22×D4D412yes—
54C23×C4—22no—
55C23×C4—21no—
56C2×SD16—22no—
57C2×SD16—24no—
58C2×M16—21no—
59C4⋊Q8C2311yescharacteristic
60C4⋊Q8C2311yescharacteristic
61#127—24no—
62#127—24no—
63#127—21no—
64#127—21no—
65#127D412yes—
66#127C2312yes—
67#125—21no—
68C22⋊Q8—21no—
69C22⋊Q8—21no—
70#121—24no—
71#121—28no—
72#121—28no—
73#121—24no—
74C4×Q8—21no—
75C4×Q8—21no—
76C4×Q8—21no—
77C4×Q8—21no—
78C4×Q8D412yes—
79C4×Q8C2312yes—
80C4×D4—24no—
81C4×D4—24no—
82C4×D4—21no—
83C4×D4—21no—
84C4×D4D412yes—
85C4×D4C2312yes—
86#117—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4D411yescharacteristic
89C2×C4⋊C4D411yescharacteristic
90C2×C22⋊C4—22no—
91C2×C22⋊C4—21no—
92C2×C42—21no—
93C2×C42C2311yescharacteristic
94#107—22no—
95#106—22no—
96#106—22no—
97C4⋊C8—21no—
98C4⋊C8—21no—
99C4⋊C8C2312yes—
100C4⋊C8C2312yes—
101#103—22no—
102#103—22no—
103#103—22no—
104#102—22no—
105#102—24no—
106C22⋊C8—21no—
107C22⋊C8—21no—
108C4×C8—21no—
109C4×C8C2312yes—
110C24—22no—
111C24—21no—
112#41—48no—
113#41—416no—
114#41—48no—
115C2×Q8—21no—
116C2×Q8—22no—
117C2×Q8—21no—
118C2×Q8—21no—
119C2×Q8SD1614yescomplemented
120C2×Q8C2×D412yes—
121C2×D4—28no—
122C2×D4—28no—
123C2×D4—21no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—22no—
129C2×D4—42no—
130C2×D4—44no—
131C2×D4—48no—
132C2×D4—42no—
133C2×D4—42no—
134C2×D4—82no—
135C2×D4—84no—
136C2×D4C2×D412yes—
137C2×D4C2×D414yes—
138C22×C4—21no—
139C22×C4—28no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—48no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150SD16—44no—
151SD16—84no—
152M16—42no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—22no—
156C4⋊C4—22no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—22no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4C2×D412yes—
167C4⋊C4C2×D412yes—
168C22⋊C4—28no—
169C22⋊C4—22no—
170C22⋊C4—22no—
171C22⋊C4—41no—
172C42—22no—
173C42—22no—
174C42—41no—
175C42C2×D411yescharacteristic
176C42C2×D411yescharacteristic
177C42C2×D411yescharacteristic
178C42C2411yescharacteristic
179C23—22no—
180C23—24no—
181C23—42no—
182C23—44no—
183C23—44no—
184C23—42no—
185C23—44no—
186C23—42no—
187C23#12011yescharacteristic
188Q8—22no—
189Q8—24no—
190Q8—42no—
191Q8—42no—
192Q8C2×SD1614yes—
193D4—22no—
194D4—22no—
195D4—44no—
196D4—42no—
197D4—44no—
198D4—416no—
199D4—42no—
200D4—42no—
201D4—44no—
202D4—44no—
203D4—82no—
204D4—88no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—41no—
211C2×C4—416no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—44no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4#12011yescharacteristic
220C2×C4#12011yescharacteristic
221C2×C4#12011yescharacteristic
222C2×C4C2×SD1612yes—
223C2×C4AGL(1,ℤ/8)12yes—
224C2×C4C22×D411yescharacteristic
225C2×C4C22×D411yescharacteristic
226C2×C4C22×D411yescharacteristic
227C8—42no—
228C8—42no—
229C8—42no—
230C22—21no—
231C22—21no—
232C22—44no—
233C22—42no—
234C22—42no—
235C22—44no—
236C22—44no—
237C22—42no—
238C22—84no—
239C22—88no—
240C22#45012yes—
241C22C2×#12011yescharacteristic
242C4—24no—
243C4—21no—
244C4—21no—
245C4—21no—
246C4—42no—
247C4—42no—
248C4—81no—
249C4—81no—
250C4#45012yes—
251C4C22×SD1611yescharacteristic
252C4C2×AGL(1,ℤ/8)11yescharacteristic
253C4#57512yes—
254C2—22no—
255C2—82no—
256C2—84no—
257C2C2×#45011yescharacteristic
258C2#300511yescharacteristic
259C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radicalC2×#550C2
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
2. #3208C2semidirect
3. #3101C2semidirect
4. #3078C2semidirect
5. #1642C2semidirect
6. #1631C2semidirect
7. #1545C2semidirect
8. #1486C2semidirect
9. #1479C2semidirect
23. #492C22semidirect
24. #486C22semidirect
27. #477C22semidirect
28. #476C22semidirect
29. #475C22semidirect
30. #461C22semidirect
40. #331C22semidirect
41. Q8⋊C8C22semidirect
42. D4⋊C8C22semidirect
119. C2×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#5502
#32082
#31012
#30782
#16422
#16312
#15452
#14862
#14792