Finite Group Databasebeta

← #18624

Subgroups — #18624

Order 256.

576 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 123 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
1#3508C211yesmaximal, characteristic, complemented
2#3208C211yesmaximal, characteristic, complemented
3C2×#493C211yesmaximal, characteristic, complemented
4#3038C211yesmaximal, characteristic, complemented
5#1621C214yesmaximal, complemented
6#1620C214yesmaximal, complemented
7#1545C211yesmaximal, characteristic, complemented
8#1493C211yesmaximal, characteristic, complemented
9#1476C211yesmaximal, characteristic, complemented
10C2×AGL(1,ℤ/8)—21no—
11C22×D8—22no—
12#559—21no—
13#559C2212yescomplemented
14#550—21no—
15#550C2212yescomplemented
16#546—22no—
17#546—22no—
18#540—22no—
19#535—21no—
20#533C2211yescharacteristic
21#532—21no—
22#532—21no—
23C2×#127C2211yescharacteristic
24C2×#127C2211yescharacteristic
25#518—21no—
26C4×#41C2211yescharacteristic
27#493—22no—
28#493C2214yes—
29#492—21no—
30#486—21no—
31#460—21no—
32#460C2212yescomplemented
33#460C2214yescomplemented
34#459—21no—
35#459C2212yescomplemented
36#450—22no—
37#431C2211yescharacteristic
38#423C2211yescharacteristic
39#418—21no—
40C2×C4×C8C2211yescharacteristic
41#331C2214yescomplemented
42Q8⋊C8C2212yescomplemented
43Q8⋊C8C2214yescomplemented
44D4⋊C8C2212yescomplemented
45C2×#41—22no—
46C2×#41—22no—
47C2×#41D411yescharacteristic
48C22×D4—21no—
49C22×D4—21no—
50C22×D4—21no—
51C22×D4—21no—
52C22×D4—21no—
53C22×D4D412yes—
54AGL(1,ℤ/8)—44no—
55C2×SD16—22no—
56C2×SD16—24no—
57C2×D8—28no—
58C2×D8—22no—
59C2×D8—48no—
60C2×M16—21no—
61C22×C8—21no—
62C4⋊Q8D411yescharacteristic
63C4⋊Q8D411yescharacteristic
64#127—22no—
65#127—21no—
66#127—21no—
67#127—21no—
68#127—21no—
69#127—21no—
70#127D412yes—
71#127C2312yes—
72#127C2312yes—
73#125—21no—
74#124—22no—
75#124—22no—
76#124—22no—
77#123—24no—
78#123—24no—
79C22⋊Q8—21no—
80C22⋊Q8D412yes—
81#121—22no—
82#121—22no—
83#121—22no—
84#121—24no—
85#121—24no—
86#121—22no—
87#120—24no—
88C4×Q8—21no—
89C4×Q8D411yescharacteristic
90C4×Q8C2311yescharacteristic
91C4×Q8C2312yes—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—21no—
96C4×D4—21no—
97C4×D4—22no—
98C4×D4—22no—
99C4×D4—22no—
100C4×D4D411yescharacteristic
101C4×D4C2311yescharacteristic
102#117—21no—
103#117—21no—
104#117D411yescharacteristic
105C2×C22⋊C4—24no—
106C2×C42—21no—
107C2×C42C2311yescharacteristic
108C4⋊C8—21no—
109C4⋊C8C2312yes—
110C4⋊C8C2312yes—
111#102—22no—
112#102—22no—
113#102—24no—
114C22⋊C8—21no—
115C4×C8—21no—
116C4×C8—21no—
117C4×C8C2312yes—
118C24—22no—
119C24—41no—
120#41—22no—
121#41—22no—
122#41—84no—
123#41—84no—
124C2×Q8—21no—
125C2×Q8C2×D411yescharacteristic
126C2×Q8C2×D412yes—
127C2×D4—21no—
128C2×D4—24no—
129C2×D4—24no—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—44no—
139C2×D4—44no—
140C2×D4—41no—
141C2×D4—41no—
142C2×D4—41no—
143C2×D4—42no—
144C2×D4—42no—
145C2×D4—42no—
146C2×D4—42no—
147C2×D4—82no—
148C2×D4—82no—
149C2×D4C2×D411yescharacteristic
150C2×D4C2×D414yes—
151C22×C4—24no—
152C22×C4—24no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159SD16—44no—
160SD16—84no—
161D8—416no—
162D8—44no—
163M16—42no—
164C2×C8—24no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—22no—
173C4⋊C4—21no—
174C4⋊C4—22no—
175C4⋊C4C2×D411yescharacteristic
176C4⋊C4C2×D411yescharacteristic
177C4⋊C4C2×D412yes—
178C22⋊C4—24no—
179C22⋊C4—24no—
180C22⋊C4—24no—
181C22⋊C4—24no—
182C22⋊C4—21no—
183C22⋊C4—22no—
184C22⋊C4—21no—
185C22⋊C4—21no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—22no—
190C42C2×D411yescharacteristic
191C42C2×D411yescharacteristic
192C42C2×D411yescharacteristic
193C42C2411yescharacteristic
194C23—24no—
195C23—21no—
196C23—44no—
197C23—42no—
198C23—82no—
199C23—82no—
200C23—82no—
201C23—82no—
202C23—82no—
203C23#12011yescharacteristic
204Q8—21no—
205Q8—21no—
206Q8—24no—
207Q8—42no—
208D4—21no—
209D4—21no—
210D4—44no—
211D4—44no—
212D4—42no—
213D4—42no—
214D4—42no—
215D4—42no—
216D4—42no—
217D4—44no—
218D4—44no—
219D4—44no—
220D4—44no—
221D4—82no—
222D4—82no—
223D4—84no—
224D4—84no—
225D4—82no—
226D4—82no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—84no—
238C2×C4—84no—
239C2×C4#12011yescharacteristic
240C2×C4#12011yescharacteristic
241C2×C4#12011yescharacteristic
242C2×C4#13512yes—
243C2×C4AGL(1,ℤ/8)12yes—
244C2×C4C22×D411yescharacteristic
245C2×C4C22×D411yescharacteristic
246C2×C4C22×D411yescharacteristic
247C8—24no—
248C8—42no—
249C8—42no—
250C22—21no—
251C22—21no—
252C22—41no—
253C22—44no—
254C22—41no—
255C22—44no—
256C22—41no—
257C22—44no—
258C22—82no—
259C22—82no—
260C22—82no—
261C22—164no—
262C22—164no—
263C22#44912yes—
264C22C2×#12011yescharacteristic
265C4—21no—
266C4—21no—
267C4—41no—
268C4—41no—
269C4—42no—
270C4—41no—
271C4—42no—
272C4#44912yes—
273C4#45312yes—
274C4C2×#13511yescharacteristic
275C4C2×AGL(1,ℤ/8)11yescharacteristic
276C4#57612yes—
277C2—22no—
278C2—81no—
279C2—84no—
280C2—162no—
281C2C2×#44911yescharacteristic
282C2#300811yescharacteristic
283C2C2×#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. #3508C2semidirect
2. #3208C2semidirect
3. C2×#493C2semidirect
4. #3038C2semidirect
5. #1621C2semidirect
6. #1620C2semidirect
7. #1545C2semidirect
8. #1493C2semidirect
9. #1476C2semidirect
13. #559C22semidirect
15. #550C22semidirect
32. #460C22semidirect
33. #460C22semidirect
35. #459C22semidirect
41. #331C22semidirect
42. Q8⋊C8C22semidirect
43. Q8⋊C8C22semidirect
44. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35082
#32082
C2×#4932
#30382
#16212
#16202
#15452
#14932
#14762