Finite Group Databasebeta

← #18746

Subgroups — #18746

Order 256.

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

normal subgroups only

This group has 844 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, complemented
2#3205C211yesmaximal, characteristic, complemented
3#3100C211yesmaximal, characteristic, complemented
4#3036C211yesmaximal, characteristic, complemented
5#1639C214yesmaximal, complemented
6#1619C214yesmaximal, complemented
7#1545C211yesmaximal, characteristic, complemented
8#1486C211yesmaximal, characteristic, complemented
9#1478C211yesmaximal, characteristic, complemented
10C23×D4—21no—
11C23×D4—21no—
12C2×AGL(1,ℤ/8)—21no—
13D42—22no—
14D42—22no—
15D42—22no—
16D42C2214yes—
17#532C2211yescharacteristic, complemented
18C2×#127C2211yescharacteristic
19C2×#121—21no—
20C2×#121—21no—
21C2×#120—22no—
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24#493C2212yescomplemented
25#486C2212yescomplemented
26#480—21no—
27#474C2212yescomplemented
28#463—22no—
29#461C2212yescomplemented
30#459—22no—
31#459C2212yescomplemented
32#459C2212yescomplemented
33#447—21no—
34#447—22no—
35C4×D8—22no—
36#431C2211yescharacteristic, complemented
37#431C2211yescharacteristic, complemented
38#423C2211yescharacteristic, complemented
39#417—21no—
40#331C2214yescomplemented
41D4⋊C8C2214yescomplemented
42D4⋊C8C2214yescomplemented
43C25—22no—
44C2×#41—21no—
45C22×D4—22no—
46C22×D4—22no—
47C22×D4—24no—
48C22×D4—24no—
49C22×D4—24no—
50C22×D4—24no—
51C22×D4—21no—
52C22×D4—21no—
53C22×D4—48no—
54C22×D4—44no—
55C22×D4—44no—
56C22×D4D411yescharacteristic
57C22×D4D411yescharacteristic
58C22×D4D412yes—
59C23×C4—21no—
60C23×C4—21no—
61AGL(1,ℤ/8)—44no—
62C2×SD16—22no—
63C2×D8—22no—
64C2×D8—22no—
65C2×D8—24no—
66C2×M16—21no—
67C4⋊Q8C2311yescharacteristic, complemented
68#127—21no—
69#127—21no—
70#127D412yes—
71#127C2312yes—
72#127C2311yescharacteristic, complemented
73#124—21no—
74#121—24no—
75#121—24no—
76#121—24no—
77#121—24no—
78#121—21no—
79#121—21no—
80#120—28no—
81#120—28no—
82C4×D4—21no—
83C4×D4—21no—
84C4×D4—21no—
85C4×D4—21no—
86C4×D4—21no—
87C4×D4—21no—
88C4×D4D412yes—
89C4×D4D412yes—
90C4×D4C2312yes—
91C4×D4C2312yes—
92C2×C4⋊C4D411yescharacteristic
93C2×C4⋊C4D411yescharacteristic
94C2×C22⋊C4—22no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C42C2311yescharacteristic
98#107—22no—
99C4⋊C8—21no—
100C4⋊C8—21no—
101C4⋊C8C2312yescomplemented
102C4⋊C8C2312yescomplemented
103#103—22no—
104#102—22no—
105#102—24no—
106#102—22no—
107#102—22no—
108C22⋊C8—21no—
109C22⋊C8—21no—
110C4×C8—21no—
111C4×C8C2312yescomplemented
112C24—21no—
113C24—21no—
114C24—28no—
115C24—22no—
116C24—48no—
117C24—48no—
118C24—84no—
119#41—42no—
120#41—42no—
121C2×Q8—21no—
122C2×D4—24no—
123C2×D4—28no—
124C2×D4—28no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—22no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—44no—
137C2×D4—416no—
138C2×D4—48no—
139C2×D4—48no—
140C2×D4—416no—
141C2×D4—48no—
142C2×D4—48no—
143C2×D4—48no—
144C2×D4—48no—
145C2×D4—41no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4—82no—
149C2×D4D814yescomplemented
150C2×D4C2×D412yes—
151C2×D4C2×D412yes—
152C2×D4C2×D414yes—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—44no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—41no—
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166SD16—44no—
167D8—44no—
168D8—44no—
169D8—84no—
170M16—42no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4C2×D412yes—
179C4⋊C4C2×D412yes—
180C22⋊C4—28no—
181C22⋊C4—22no—
182C22⋊C4—22no—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—41no—
186C22⋊C4—41no—
187C42C2×D411yescharacteristic
188C42C2×D411yescharacteristic
189C42C2×D411yescharacteristic
190C42C2411yescharacteristic
191C23—24no—
192C23—22no—
193C23—22no—
194C23—24no—
195C23—24no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—44no—
200C23—416no—
201C23—416no—
202C23—44no—
203C23—42no—
204C23—44no—
205C23—41no—
206C23—88no—
207C23—832no—
208C23#12011yescharacteristic
209Q8—41no—
210Q8—41no—
211D4—24no—
212D4—22no—
213D4—22no—
214D4—24no—
215D4—44no—
216D4—44no—
217D4—42no—
218D4—416no—
219D4—416no—
220D4—42no—
221D4—41no—
222D4—42no—
223D4—44no—
224D4—42no—
225D4—41no—
226D4—44no—
227D4—82no—
228D4—82no—
229D4C2×D814yes—
230D4AGL(1,ℤ/8)14yescomplemented
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—41no—
235C2×C4—44no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—41no—
243C2×C4—82no—
244C2×C4#12011yescharacteristic
245C2×C4#12011yescharacteristic
246C2×C4#12011yescharacteristic
247C2×C4C2×D812yes—
248C2×C4AGL(1,ℤ/8)12yes—
249C2×C4C22×D411yescharacteristic
250C2×C4C22×D411yescharacteristic
251C2×C4C22×D411yescharacteristic
252C8—42no—
253C8—42no—
254C8—42no—
255C22—21no—
256C22—21no—
257C22—24no—
258C22—24no—
259C22—48no—
260C22—48no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—44no—
266C22—44no—
267C22—44no—
268C22—88no—
269C22—832no—
270C22—81no—
271C22—81no—
272C22—81no—
273C22#44712yes—
274C22C2×#12011yescharacteristic
275C4—21no—
276C4—21no—
277C4—21no—
278C4—42no—
279C4—42no—
280C4—81no—
281C4#44712yes—
282C4C22×D811yescharacteristic
283C4C2×AGL(1,ℤ/8)11yescharacteristic
284C4C2×AGL(1,ℤ/8)12yes—
285C2—22no—
286C2—44no—
287C2—44no—
288C2—84no—
289C2—161no—
290C2C2×#44711yescharacteristic
291C2#300311yescharacteristic
292C2#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×D42C2semidirect
2. #3205C2semidirect
3. #3100C2semidirect
4. #3036C2semidirect
5. #1639C2semidirect
6. #1619C2semidirect
7. #1545C2semidirect
8. #1486C2semidirect
9. #1478C2semidirect
17. #532C22semidirect
24. #493C22semidirect
25. #486C22semidirect
27. #474C22semidirect
29. #461C22semidirect
31. #459C22semidirect
32. #459C22semidirect
36. #431C22semidirect
37. #431C22semidirect
38. #423C22semidirect
40. #331C22semidirect
41. D4⋊C8C22semidirect
42. D4⋊C8C22semidirect
67. C4⋊Q8C23semidirect
72. #127C23semidirect
101. C4⋊C8C23semidirect
102. C4⋊C8C23semidirect
111. C4×C8C23semidirect
149. C2×D4D8semidirect
230. D4AGL(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×D422
#32052
#31002
#30362
#16392
#16192
#15452
#14862
#14782