Finite Group Databasebeta

← #18772

Subgroups — #18772

Order 256.

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

normal subgroups only

This group has 520 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
1C2×#555C211yesmaximal, characteristic, complemented
2#3206C211yesmaximal, characteristic, complemented
3#3100C211yesmaximal, characteristic, complemented
4#3036C211yesmaximal, characteristic, complemented
5#1646C214yesmaximal, complemented
6#1624C214yesmaximal, complemented
7#1548C211yesmaximal, characteristic, complemented
8#1486C211yesmaximal, characteristic, complemented
9#1478C211yesmaximal, characteristic, complemented
10C2×AGL(1,ℤ/8)—21no—
11#555—22no—
12#555—22no—
13#555—22no—
14#555C2214yes—
15#532C2211yescharacteristic
16C2×C4⋊Q8C2211yescharacteristic
17C2×#124—22no—
18C2×#121—21no—
19C2×#121—21no—
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×#117—21no—
23C2×#117—21no—
24#492C2212yescomplemented
25#487C2212yescomplemented
26#480—21no—
27#474C2212yescomplemented
28#463—22no—
29#461—22no—
30#461C2212yescomplemented
31#459C2212yescomplemented
32#459C2212yescomplemented
33#448—22no—
34#447—21no—
35C4×D8—22no—
36#431C2211yescharacteristic
37#431C2211yescharacteristic
38#423C2211yescharacteristic
39#417—21no—
40#333C2214yescomplemented
41D4⋊C8C2214yescomplemented
42D4⋊C8C2214yescomplemented
43C2×#41—21no—
44C22×Q8D412yes—
45C22×D4—22no—
46C22×D4D411yescharacteristic
47C22×D4D411yescharacteristic
48C23×C4—21no—
49C23×C4—21no—
50AGL(1,ℤ/8)—44no—
51C2×SD16—22no—
52C2×SD16—24no—
53C2×D8—22no—
54C2×D8—22no—
55C2×M16—21no—
56C4⋊Q8—21no—
57C4⋊Q8—21no—
58C4⋊Q8D412yes—
59C4⋊Q8C2311yescharacteristic
60C4⋊Q8C2312yes—
61#127C2311yescharacteristic
62#124—28no—
63#124—28no—
64#124—21no—
65#121—24no—
66#121—24no—
67#121—24no—
68#121—24no—
69#121—21no—
70#121—21no—
71C4×D4—21no—
72C4×D4—21no—
73C4×D4—21no—
74C4×D4—21no—
75C4×D4—21no—
76C4×D4—21no—
77C4×D4D412yes—
78C4×D4D412yes—
79C4×D4C2312yes—
80C4×D4C2312yes—
81#117—24no—
82#117—24no—
83#117—24no—
84#117—24no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4D411yescharacteristic
88C2×C4⋊C4D411yescharacteristic
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C42—22no—
94C2×C42C2311yescharacteristic
95#107—22no—
96C4⋊C8—21no—
97C4⋊C8—21no—
98C4⋊C8C2312yes—
99C4⋊C8C2312yes—
100#103—24no—
101#103—22no—
102#102—22no—
103#102—22no—
104#102—22no—
105C22⋊C8—21no—
106C22⋊C8—21no—
107C4×C8—21no—
108C4×C8C2312yes—
109C24—21no—
110C24—21no—
111#41—42no—
112#41—42no—
113C2×Q8—22no—
114C2×Q8—21no—
115C2×Q8—82no—
116C2×Q8C2×D414yes—
117C2×D4—28no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—42no—
126C2×D4—41no—
127C2×D4—84no—
128C2×D4C2×D412yes—
129C2×D4C2×D412yes—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—44no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—41no—
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144SD16—44no—
145SD16—84no—
146D8—44no—
147D8—44no—
148M16—42no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—22no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4C2×D412yes—
161C4⋊C4C2×D412yes—
162C22⋊C4—28no—
163C22⋊C4—28no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—22no—
167C22⋊C4—22no—
168C22⋊C4—41no—
169C22⋊C4—41no—
170C42—28no—
171C42C2×D411yescharacteristic
172C42C2×D411yescharacteristic
173C42C2×D411yescharacteristic
174C42C2411yescharacteristic
175C23—22no—
176C23—22no—
177C23—42no—
178C23—42no—
179C23—42no—
180C23—42no—
181C23—42no—
182C23—42no—
183C23—41no—
184C23#12011yescharacteristic
185Q8—44no—
186Q8—41no—
187Q8—41no—
188Q8—82no—
189D4—22no—
190D4—22no—
191D4—22no—
192D4—22no—
193D4—42no—
194D4—42no—
195D4—41no—
196D4—41no—
197D4—42no—
198D4—42no—
199D4—88no—
200D4—82no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—41no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—41no—
215C2×C4—82no—
216C2×C4#12011yescharacteristic
217C2×C4#12011yescharacteristic
218C2×C4#12011yescharacteristic
219C2×C4#13512yes—
220C2×C4AGL(1,ℤ/8)12yes—
221C2×C4C22×D411yescharacteristic
222C2×C4C22×D411yescharacteristic
223C2×C4C22×D411yescharacteristic
224C8—42no—
225C8—42no—
226C8—42no—
227C22—21no—
228C22—21no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—84no—
236C22—84no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22#44912yes—
241C22C2×#12011yescharacteristic
242C4—21no—
243C4—21no—
244C4—21no—
245C4—42no—
246C4—42no—
247C4—44no—
248C4—81no—
249C4#44912yes—
250C4C2×#13511yescharacteristic
251C4C2×AGL(1,ℤ/8)11yescharacteristic
252C4#57512yes—
253C2—22no—
254C2—82no—
255C2—82no—
256C2—161no—
257C2C2×#44911yescharacteristic
258C2#300511yescharacteristic
259C2#301911yescharacteristic
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. C2×#555C2semidirect
2. #3206C2semidirect
3. #3100C2semidirect
4. #3036C2semidirect
5. #1646C2semidirect
6. #1624C2semidirect
7. #1548C2semidirect
8. #1486C2semidirect
9. #1478C2semidirect
24. #492C22semidirect
25. #487C22semidirect
27. #474C22semidirect
30. #461C22semidirect
31. #459C22semidirect
32. #459C22semidirect
40. #333C22semidirect
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
C2×#5552
#32062
#31002
#30362
#16462
#16242
#15482
#14862
#14782