Finite Group Databasebeta

← #18550

Subgroups — #18550

Order 256.

615 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 615 conjugacy classes of subgroups — too many to draw. Only its 119 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#3138C211yesmaximal, characteristic, complemented
3#3113C211yesmaximal, characteristic, complemented
4#3110C211yesmaximal, characteristic, complemented
5#1642C214yesmaximal, complemented
6#1641C214yesmaximal, complemented
7#1555C211yesmaximal, characteristic, complemented
8#1491C211yesmaximal, characteristic, complemented
9#1490C211yesmaximal, 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#490—22no—
24#486C2212yescomplemented
25#486C2212yescomplemented
26#484—21no—
27#483—21no—
28#475C2212yescomplemented
29#474C2212yescomplemented
30#462C2212yescomplemented
31#461C2212yescomplemented
32#451—21no—
33#450—21no—
34#441—22no—
35#440—22no—
36#432C2211yescharacteristic
37#432C2211yescharacteristic
38#418—22no—
39C4×M16C2211yescharacteristic
40#331C2214yescomplemented
41Q8⋊C8C2214yescomplemented
42D4⋊C8C2214yescomplemented
43C2×#41—24no—
44C2×#41—24no—
45C2×#41—48no—
46C22×Q8D411yescharacteristic
47C22×D4—22no—
48C22×D4—22no—
49C22×D4—21no—
50C22×D4—21no—
51C22×D4D411yescharacteristic
52C22×D4D412yes—
53C23×C4—22no—
54C23×C4—21no—
55C2×Q16—22no—
56C2×SD16—22no—
57C2×M16—21no—
58C4⋊Q8C2311yescharacteristic
59C4⋊Q8C2311yescharacteristic
60#127—24no—
61#127—24no—
62#127—21no—
63#127—21no—
64#127D412yes—
65#127C2312yes—
66#125—21no—
67C22⋊Q8—21no—
68C22⋊Q8—21no—
69#121—24no—
70#121—28no—
71#121—28no—
72#121—24no—
73C4×Q8—21no—
74C4×Q8—21no—
75C4×Q8—21no—
76C4×Q8—21no—
77C4×Q8D412yes—
78C4×Q8C2312yes—
79C4×D4—24no—
80C4×D4—24no—
81C4×D4—21no—
82C4×D4—21no—
83C4×D4D412yes—
84C4×D4C2312yes—
85#117—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4D411yescharacteristic
88C2×C4⋊C4D411yescharacteristic
89C2×C22⋊C4—22no—
90C2×C22⋊C4—21no—
91C2×C42—21no—
92C2×C42C2311yescharacteristic
93#107—22no—
94#106—22no—
95C4⋊C8C2312yes—
96C4⋊C8C2312yes—
97#103—22no—
98#103—22no—
99#103—22no—
100#102—24no—
101#102—24no—
102#102—22no—
103C22⋊C8—21no—
104C22⋊C8—21no—
105#97—21no—
106#97—21no—
107#97—21no—
108C4×C8C2312yes—
109C24—22no—
110C24—21no—
111#41—416no—
112#41—48no—
113#41—48no—
114C2×Q8—24no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—21no—
118C2×Q8C2×D412yes—
119C2×D4—28no—
120C2×D4—28no—
121C2×D4—21no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—42no—
128C2×D4—44no—
129C2×D4—48no—
130C2×D4—44no—
131C2×D4—44no—
132C2×D4—84no—
133C2×D4C2×D412yes—
134C2×D4C2×D414yes—
135C22×C4—21no—
136C22×C4—28no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—48no—
142C22×C4—44no—
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146Q16—44no—
147SD16—44no—
148M16—42no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—22no—
152C4⋊C4—22no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—22no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4C2×D412yes—
163C4⋊C4C2×D412yes—
164C22⋊C4—28no—
165C22⋊C4—22no—
166C22⋊C4—22no—
167C22⋊C4—41no—
168C42—22no—
169C42—22no—
170C42—41no—
171C42C2×D411yescharacteristic
172C42C2×D411yescharacteristic
173C42C2×D411yescharacteristic
174C42C2411yescharacteristic
175C23—22no—
176C23—24no—
177C23—44no—
178C23—42no—
179C23—42no—
180C23—44no—
181C23—44no—
182C23—42no—
183C23#12011yescharacteristic
184Q8—24no—
185Q8—22no—
186Q8—42no—
187Q8—42no—
188Q8AGL(1,ℤ/8)14yes—
189D4—22no—
190D4—22no—
191D4—44no—
192D4—42no—
193D4—44no—
194D4—416no—
195D4—44no—
196D4—44no—
197D4—44no—
198D4—82no—
199D4—88no—
200C2×C4—24no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—416no—
208C2×C4—44no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—44no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4#12011yescharacteristic
215C2×C4#12011yescharacteristic
216C2×C4#12011yescharacteristic
217C2×C4C22×D411yescharacteristic
218C2×C4C22×D411yescharacteristic
219C2×C4C22×D411yescharacteristic
220C8—42no—
221C8—42no—
222C8—42no—
223C22—21no—
224C22—21no—
225C22—44no—
226C22—42no—
227C22—42no—
228C22—44no—
229C22—44no—
230C22—42no—
231C22—88no—
232C22—84no—
233C22#45512yes—
234C22C2×#12011yescharacteristic
235C4—24no—
236C4—21no—
237C4—21no—
238C4—21no—
239C4—21no—
240C4—42no—
241C4—42no—
242C4—81no—
243C4—81no—
244C4C2×AGL(1,ℤ/8)12yes—
245C4#57512yes—
246C2—22no—
247C2—82no—
248C2—84no—
249C2#300311yescharacteristic
250C2#300511yescharacteristic
251C2C2×#45511yescharacteristic
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. #3138C2semidirect
3. #3113C2semidirect
4. #3110C2semidirect
5. #1642C2semidirect
6. #1641C2semidirect
7. #1555C2semidirect
8. #1491C2semidirect
9. #1490C2semidirect
24. #486C22semidirect
25. #486C22semidirect
28. #475C22semidirect
29. #474C22semidirect
30. #462C22semidirect
31. #461C22semidirect
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
C2×#5502
#31382
#31132
#31102
#16422
#16412
#15552
#14912
#14902