Finite Group Databasebeta

← #18747

Subgroups — #18747

Order 256.

817 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 817 conjugacy classes of subgroups — too many to draw. Only its 133 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×D42C211yesmaximal, characteristic
2#3208C211yesmaximal, characteristic, complemented
3#3101C211yesmaximal, characteristic, complemented
4#3077C211yesmaximal, characteristic, complemented
5#1639C214yesmaximal, complemented
6#1629C214yesmaximal, complemented
7#1545C211yesmaximal, characteristic, complemented
8#1486C211yesmaximal, characteristic, complemented
9#1478C211yesmaximal, characteristic, complemented
10C23×D4—21no—
11C23×D4—21no—
12D42—22no—
13D42—22no—
14D42—22no—
15D42C2214yes—
16#533C2211yescharacteristic, complemented
17C2×#127C2211yescharacteristic
18C2×#121—21no—
19C2×#121—21no—
20C2×#120—22no—
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23#492C2212yescomplemented
24#486C2212yescomplemented
25#483—21no—
26#478—22no—
27#476C2212yescomplemented
28#476C2212yescomplemented
29#474C2212yescomplemented
30#461C2212yescomplemented
31#460—22no—
32#450—21no—
33#450—22no—
34C4×SD16—22no—
35#431C2211yescharacteristic, complemented
36#431C2211yescharacteristic, complemented
37#428—21no—
38#423C2211yescharacteristic, complemented
39#418—21no—
40#331C2214yescomplemented
41D4⋊C8C2214yescomplemented
42D4⋊C8C2214yescomplemented
43C25—22no—
44C22×D4—22no—
45C22×D4—22no—
46C22×D4—24no—
47C22×D4—24no—
48C22×D4—24no—
49C22×D4—24no—
50C22×D4—21no—
51C22×D4—21no—
52C22×D4—48no—
53C22×D4—44no—
54C22×D4—44no—
55C22×D4D411yescharacteristic
56C22×D4D411yescharacteristic
57C22×D4D412yes—
58C23×C4—21no—
59C23×C4—21no—
60C2×SD16—22no—
61C2×SD16—24no—
62C2×M16—21no—
63C4⋊Q8C2311yescharacteristic, complemented
64C4⋊Q8C2311yescharacteristic, complemented
65#127—21no—
66#127—21no—
67#127D412yes—
68#127C2312yes—
69#125—21no—
70C22⋊Q8—21no—
71C22⋊Q8—21no—
72#121—24no—
73#121—24no—
74#121—24no—
75#121—24no—
76#120—28no—
77#120—28no—
78C4×Q8—21no—
79C4×Q8—21no—
80C4×D4—21no—
81C4×D4—21no—
82C4×D4—21no—
83C4×D4—21no—
84C4×D4D412yes—
85C4×D4D412yes—
86C4×D4C2312yes—
87C4×D4C2312yes—
88#117—21no—
89C2×C4⋊C4D411yescharacteristic
90C2×C4⋊C4D411yescharacteristic
91C2×C22⋊C4—22no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C42C2311yescharacteristic
95#107—22no—
96#106—22no—
97#106—22no—
98C4⋊C8—21no—
99C4⋊C8—21no—
100C4⋊C8C2312yescomplemented
101C4⋊C8C2312yescomplemented
102#103—22no—
103#102—22no—
104#102—22no—
105#102—22no—
106#102—24no—
107C22⋊C8—21no—
108C22⋊C8—21no—
109C4×C8—21no—
110C4×C8C2312yescomplemented
111C24—21no—
112C24—21no—
113C24—28no—
114C24—22no—
115C24—48no—
116C24—48no—
117C24—84no—
118C2×Q8—21no—
119C2×Q8—21no—
120C2×D4—24no—
121C2×D4—28no—
122C2×D4—28no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—44no—
132C2×D4—416no—
133C2×D4—48no—
134C2×D4—48no—
135C2×D4—416no—
136C2×D4—48no—
137C2×D4—48no—
138C2×D4—48no—
139C2×D4—48no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4—82no—
143C2×D4SD1614yescomplemented
144C2×D4C2×D412yes—
145C2×D4C2×D412yes—
146C2×D4C2×D414yes—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—44no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159SD16—44no—
160SD16—84no—
161M16—42no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—22no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4C2×D412yes—
174C4⋊C4C2×D412yes—
175C22⋊C4—28no—
176C22⋊C4—22no—
177C22⋊C4—22no—
178C22⋊C4—22no—
179C22⋊C4—22no—
180C22⋊C4—41no—
181C42—41no—
182C42C2×D411yescharacteristic
183C42C2×D411yescharacteristic
184C42C2×D411yescharacteristic
185C42C2411yescharacteristic
186C23—24no—
187C23—22no—
188C23—22no—
189C23—24no—
190C23—24no—
191C23—42no—
192C23—42no—
193C23—42no—
194C23—44no—
195C23—416no—
196C23—416no—
197C23—44no—
198C23—42no—
199C23—44no—
200C23—88no—
201C23—832no—
202C23#12011yescharacteristic
203Q8—42no—
204Q8—42no—
205D4—24no—
206D4—22no—
207D4—22no—
208D4—24no—
209D4—44no—
210D4—44no—
211D4—416no—
212D4—416no—
213D4—42no—
214D4—42no—
215D4—44no—
216D4—44no—
217D4—82no—
218D4C2×SD1614yes—
219D4AGL(1,ℤ/8)14yes—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—44no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4#12011yescharacteristic
234C2×C4#12011yescharacteristic
235C2×C4#12011yescharacteristic
236C2×C4C2×SD1612yes—
237C2×C4AGL(1,ℤ/8)12yes—
238C2×C4C22×D411yescharacteristic
239C2×C4C22×D411yescharacteristic
240C2×C4C22×D411yescharacteristic
241C8—42no—
242C8—42no—
243C8—42no—
244C22—21no—
245C22—21no—
246C22—24no—
247C22—24no—
248C22—48no—
249C22—48no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—44no—
255C22—44no—
256C22—44no—
257C22—88no—
258C22—832no—
259C22#45012yes—
260C22C2×#12011yescharacteristic
261C4—21no—
262C4—21no—
263C4—21no—
264C4—42no—
265C4—42no—
266C4—81no—
267C4—81no—
268C4#45012yes—
269C4C22×SD1611yescharacteristic
270C4C2×AGL(1,ℤ/8)11yescharacteristic
271C4C2×AGL(1,ℤ/8)12yes—
272C2—22no—
273C2—44no—
274C2—44no—
275C2—84no—
276C2C2×#45011yescharacteristic
277C2#300311yescharacteristic
278C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radicalC2×D42C2
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. #3077C2semidirect
5. #1639C2semidirect
6. #1629C2semidirect
7. #1545C2semidirect
8. #1486C2semidirect
9. #1478C2semidirect
16. #533C22semidirect
23. #492C22semidirect
24. #486C22semidirect
27. #476C22semidirect
28. #476C22semidirect
29. #474C22semidirect
30. #461C22semidirect
35. #431C22semidirect
36. #431C22semidirect
38. #423C22semidirect
40. #331C22semidirect
41. D4⋊C8C22semidirect
42. D4⋊C8C22semidirect
63. C4⋊Q8C23semidirect
64. C4⋊Q8C23semidirect
100. C4⋊C8C23semidirect
101. C4⋊C8C23semidirect
110. C4×C8C23semidirect
143. C2×D4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×D422
#32082
#31012
#30772
#16392
#16292
#15452
#14862
#14782