Finite Group Databasebeta

← #18545

Subgroups — #18545

Order 256.

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

normal subgroups only

This group has 658 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, complemented
2#3151C211yesmaximal, characteristic, complemented
3#3112C211yesmaximal, characteristic, complemented
4#3109C211yesmaximal, characteristic, complemented
5#1631C214yesmaximal, complemented
6#1621C214yesmaximal, complemented
7#1555C211yesmaximal, characteristic, complemented
8#1491C211yesmaximal, characteristic, complemented
9#1490C211yesmaximal, characteristic, complemented
10C22×#41—21no—
11C2×AGL(1,ℤ/8)—22no—
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#496—22no—
25#493C2212yescomplemented
26#492C2212yescomplemented
27#481—21no—
28#480—21no—
29#477C2212yescomplemented
30#476C2212yescomplemented
31#460C2212yescomplemented
32#459C2212yescomplemented
33#448—21no—
34#447—21no—
35#442—22no—
36#440—22no—
37#432C2211yescharacteristic
38#432C2211yescharacteristic
39C4×M16C2211yescharacteristic
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)—48no—
57C2×SD16—22no—
58C2×SD16—24no—
59C2×D8—24no—
60C2×D8—22no—
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#107—22no—
97#106—22no—
98C4⋊C8C2312yes—
99C4⋊C8C2312yes—
100#103—22no—
101#102—22no—
102#102—22no—
103#102—22no—
104C22⋊C8—21no—
105C22⋊C8—21no—
106#97—21no—
107#97—21no—
108#97—21no—
109C4×C8C2312yes—
110C24—22no—
111C24—21no—
112#41—44no—
113#41—416no—
114#41—48no—
115#41—48no—
116C2×Q8—24no—
117C2×Q8—21no—
118C2×Q8—21no—
119C2×Q8C2×D412yes—
120C2×D4—28no—
121C2×D4—28no—
122C2×D4—21no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—42no—
132C2×D4—44no—
133C2×D4—48no—
134C2×D4—44no—
135C2×D4—44no—
136C2×D4—41no—
137C2×D4—84no—
138C2×D4C2×D412yes—
139C2×D4C2×D414yes—
140C22×C4—21no—
141C22×C4—28no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—48no—
147C22×C4—44no—
148C22×C4—41no—
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152SD16—48no—
153SD16—44no—
154D8—48no—
155D8—44no—
156M16—24no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C4⋊C4—22no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—22no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4C2×D412yes—
167C4⋊C4C2×D412yes—
168C22⋊C4—28no—
169C22⋊C4—22no—
170C22⋊C4—22no—
171C22⋊C4—41no—
172C22⋊C4—41no—
173C42—22no—
174C42—22no—
175C42C2×D411yescharacteristic
176C42C2×D411yescharacteristic
177C42C2×D411yescharacteristic
178C42C2411yescharacteristic
179C23—22no—
180C23—24no—
181C23—44no—
182C23—42no—
183C23—42no—
184C23—44no—
185C23—44no—
186C23—42no—
187C23—41no—
188C23#12011yescharacteristic
189Q8—24no—
190Q8—22no—
191Q8—42no—
192Q8AGL(1,ℤ/8)14yescomplemented
193D4—22no—
194D4—22no—
195D4—44no—
196D4—42no—
197D4—44no—
198D4—416no—
199D4—44no—
200D4—42no—
201D4—44no—
202D4—42no—
203D4—44no—
204D4—42no—
205D4—82no—
206D4—88no—
207D4—82no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—416no—
216C2×C4—44no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—44no—
220C2×C4—41no—
221C2×C4—82no—
222C2×C4#12011yescharacteristic
223C2×C4#12011yescharacteristic
224C2×C4#12011yescharacteristic
225C2×C4C22×D411yescharacteristic
226C2×C4C22×D411yescharacteristic
227C2×C4C22×D411yescharacteristic
228C8—24no—
229C8—42no—
230C8—42no—
231C22—21no—
232C22—21no—
233C22—44no—
234C22—42no—
235C22—42no—
236C22—44no—
237C22—44no—
238C22—42no—
239C22—88no—
240C22—84no—
241C22—81no—
242C22—81no—
243C22—81no—
244C22#45312yes—
245C22C2×#12011yescharacteristic
246C4—24no—
247C4—21no—
248C4—21no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—81no—
254C4C2×AGL(1,ℤ/8)12yes—
255C4#57512yes—
256C2—22no—
257C2—82no—
258C2—84no—
259C2—161no—
260C2#300311yescharacteristic
261C2#300511yescharacteristic
262C2C2×#45311yescharacteristic
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. #3151C2semidirect
3. #3112C2semidirect
4. #3109C2semidirect
5. #1631C2semidirect
6. #1621C2semidirect
7. #1555C2semidirect
8. #1491C2semidirect
9. #1490C2semidirect
25. #493C22semidirect
26. #492C22semidirect
29. #477C22semidirect
30. #476C22semidirect
31. #460C22semidirect
32. #459C22semidirect
40. #331C22semidirect
41. Q8⋊C8C22semidirect
42. D4⋊C8C22semidirect
192. 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
#31512
#31122
#31092
#16312
#16212
#15552
#14912
#14902