Finite Group Databasebeta

← #18770

Subgroups — #18770

Order 256.

644 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 125 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, complemented
2#3205C211yesmaximal, characteristic, complemented
3#3103C211yesmaximal, characteristic, complemented
4#3036C211yesmaximal, characteristic, complemented
5#1642C214yesmaximal, complemented
6#1621C214yesmaximal, complemented
7#1545C211yesmaximal, characteristic, complemented
8#1487C211yesmaximal, characteristic, complemented
9#1478C211yesmaximal, characteristic, complemented
10C22×#41—21no—
11C2×AGL(1,ℤ/8)—21no—
12#550—22no—
13#550—22no—
14#550—22no—
15#550C2214yes—
16#532C2211yescharacteristic
17C2×#127—21no—
18C2×#127C2211yescharacteristic
19C2×#121—22no—
20C2×#121—21no—
21C2×C4×Q8C2211yescharacteristic
22C2×C4×D4—21no—
23C2×C4×D4C2211yescharacteristic
24#493C2212yescomplemented
25#486C2212yescomplemented
26#481—21no—
27#475C2212yescomplemented
28#463—22no—
29#461C2212yescomplemented
30#460C2212yescomplemented
31#459—22no—
32#459C2212yescomplemented
33#448—21no—
34#447—22no—
35C4×SD16—22no—
36#431C2211yescharacteristic
37#431C2211yescharacteristic
38#423C2211yescharacteristic
39#417—21no—
40#331C2214yescomplemented
41Q8⋊C8C2214yescomplemented
42D4⋊C8C2214yescomplemented
43C2×#41—24no—
44C2×#41—24no—
45C2×#41—21no—
46C2×#41—48no—
47C22×Q8D411yescharacteristic
48C22×D4—22no—
49C22×D4—22no—
50C22×D4—21no—
51C22×D4—21no—
52C22×D4D411yescharacteristic
53C22×D4D412yes—
54C23×C4—22no—
55C23×C4—21no—
56AGL(1,ℤ/8)—44no—
57C2×SD16—22no—
58C2×SD16—22no—
59C2×D8—22no—
60C2×D8—24no—
61C2×M16—21no—
62C4⋊Q8C2311yescharacteristic
63#127—24no—
64#127—24no—
65#127—21no—
66#127—21no—
67#127D412yes—
68#127C2311yescharacteristic
69#127C2312yes—
70#124—21no—
71#121—24no—
72#121—28no—
73#121—28no—
74#121—24no—
75#121—21no—
76#121—21no—
77C4×Q8—21no—
78C4×Q8—21no—
79C4×Q8D412yes—
80C4×Q8C2312yes—
81C4×D4—24no—
82C4×D4—24no—
83C4×D4—21no—
84C4×D4—21no—
85C4×D4—21no—
86C4×D4—21no—
87C4×D4D412yes—
88C4×D4C2312yes—
89C2×C4⋊C4—21no—
90C2×C4⋊C4D411yescharacteristic
91C2×C4⋊C4D411yescharacteristic
92C2×C22⋊C4—22no—
93C2×C22⋊C4—21no—
94C2×C42—21no—
95C2×C42C2311yescharacteristic
96#106—22no—
97C4⋊C8—21no—
98C4⋊C8—21no—
99C4⋊C8C2312yes—
100C4⋊C8C2312yes—
101#103—22no—
102#103—22no—
103#102—24no—
104#102—22no—
105#102—22no—
106C22⋊C8—21no—
107C22⋊C8—21no—
108C4×C8—21no—
109C4×C8C2312yes—
110C24—21no—
111C24—22no—
112#41—416no—
113#41—48no—
114#41—48no—
115#41—42no—
116#41—42no—
117C2×Q8—24no—
118C2×Q8—21no—
119C2×Q8—21no—
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—22no—
130C2×D4—22no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—48no—
135C2×D4—44no—
136C2×D4—41no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4—82no—
140C2×D4—84no—
141C2×D4C2×D412yes—
142C2×D4C2×D414yes—
143C22×C4—21no—
144C22×C4—28no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—48no—
150C22×C4—44no—
151C22×C4—41no—
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155SD16—44no—
156SD16—44no—
157D8—44no—
158D8—84no—
159M16—42no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—22no—
163C4⋊C4—22no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—22no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4C2×D412yes—
170C4⋊C4C2×D412yes—
171C22⋊C4—28no—
172C22⋊C4—22no—
173C22⋊C4—22no—
174C22⋊C4—41no—
175C22⋊C4—41no—
176C42—22no—
177C42—22no—
178C42C2×D411yescharacteristic
179C42C2×D411yescharacteristic
180C42C2×D411yescharacteristic
181C42C2411yescharacteristic
182C23—22no—
183C23—24no—
184C23—44no—
185C23—42no—
186C23—44no—
187C23—42no—
188C23—42no—
189C23—44no—
190C23—41no—
191C23#12011yescharacteristic
192Q8—24no—
193Q8—22no—
194Q8—41no—
195Q8—41no—
196Q8AGL(1,ℤ/8)14yescomplemented
197D4—22no—
198D4—22no—
199D4—44no—
200D4—416no—
201D4—44no—
202D4—42no—
203D4—42no—
204D4—41no—
205D4—42no—
206D4—44no—
207D4—41no—
208D4—42no—
209D4—42no—
210D4—44no—
211D4—82no—
212D4—88no—
213D4—82no—
214C2×C4—24no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—41no—
220C2×C4—416no—
221C2×C4—44no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—44no—
225C2×C4—41no—
226C2×C4—82no—
227C2×C4#12011yescharacteristic
228C2×C4#12011yescharacteristic
229C2×C4#12011yescharacteristic
230C2×C4#13512yes—
231C2×C4AGL(1,ℤ/8)12yes—
232C2×C4C22×D411yescharacteristic
233C2×C4C22×D411yescharacteristic
234C2×C4C22×D411yescharacteristic
235C8—42no—
236C8—42no—
237C8—42no—
238C22—21no—
239C22—21no—
240C22—42no—
241C22—44no—
242C22—44no—
243C22—42no—
244C22—44no—
245C22—42no—
246C22—84no—
247C22—88no—
248C22—81no—
249C22—81no—
250C22—81no—
251C22#44912yes—
252C22C2×#12011yescharacteristic
253C4—24no—
254C4—21no—
255C4—21no—
256C4—21no—
257C4—42no—
258C4—42no—
259C4—81no—
260C4#44912yes—
261C4C2×#13511yescharacteristic
262C4C2×AGL(1,ℤ/8)11yescharacteristic
263C4C2×AGL(1,ℤ/8)12yes—
264C2—22no—
265C2—82no—
266C2—84no—
267C2—161no—
268C2C2×#44911yescharacteristic
269C2#300311yescharacteristic
270C2#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×#550C2semidirect
2. #3205C2semidirect
3. #3103C2semidirect
4. #3036C2semidirect
5. #1642C2semidirect
6. #1621C2semidirect
7. #1545C2semidirect
8. #1487C2semidirect
9. #1478C2semidirect
24. #493C22semidirect
25. #486C22semidirect
27. #475C22semidirect
29. #461C22semidirect
30. #460C22semidirect
32. #459C22semidirect
40. #331C22semidirect
41. Q8⋊C8C22semidirect
42. D4⋊C8C22semidirect
196. Q8AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
C2×#5502
#32052
#31032
#30362
#16422
#16212
#15452
#14872
#14782