Finite Group Databasebeta

← #18722

Subgroups — #18722

Order 256.

502 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 115 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#3549C211yesmaximal, characteristic
2#3226C211yesmaximal, characteristic, complemented
3#3206C211yesmaximal, characteristic, complemented
4#3141C211yesmaximal, characteristic, complemented
5#1646C214yesmaximal, complemented
6#1639C214yesmaximal, complemented
7#1543C211yesmaximal, characteristic, complemented
8#1484C211yesmaximal, characteristic, complemented
9#1484C211yesmaximal, characteristic, complemented
10H(3,𝔽4)—22no—
11#555—21no—
12#555—21no—
13#555C2212yes—
14D42—21no—
15D42—21no—
16D42C2212yes—
17#535—22no—
18#535—22no—
19#535—22no—
20#532C2211yescharacteristic
21#532C2211yescharacteristic
22#532C2211yescharacteristic
23C2×C4⋊Q8C2211yescharacteristic, complemented
24#500—21no—
25#492—21no—
26#489—22no—
27#487—21no—
28#487C2212yescomplemented
29#486—21no—
30#486C2212yes—
31#480—22no—
32#474C2214yescomplemented
33#461C2214yescomplemented
34#448—22no—
35#431C2211yescharacteristic, complemented
36#431C2211yescharacteristic, complemented
37#417—22no—
38C4×M16C2211yescharacteristic, complemented
39#333C2212yescomplemented
40#331C2212yes—
41D4⋊C8C2214yescomplemented
42D4⋊C8C2214yescomplemented
43C2×#41—21no—
44C2×#41—21no—
45C2×#41D412yes—
46C22×Q8—21no—
47C22×D4—22no—
48C22×D4—22no—
49C22×D4—22no—
50C2×SD16—24no—
51C2×M16—21no—
52C4⋊Q8—21no—
53C4⋊Q8—21no—
54C4⋊Q8—21no—
55C4⋊Q8D411yescharacteristic
56C4⋊Q8D411yescharacteristic
57C4⋊Q8C2312yes—
58C4⋊Q8C2311yescharacteristic
59#127—24no—
60#127D411yescharacteristic
61#127D411yescharacteristic
62#127C2311yescharacteristic
63#124—24no—
64#124—24no—
65#124—24no—
66#124—21no—
67#124—21no—
68#124D412yes—
69#121—22no—
70#121—22no—
71#121—22no—
72#121—22no—
73#121—21no—
74#121—21no—
75#121—21no—
76#121—21no—
77#121—22no—
78#121—22no—
79#121D412yes—
80#121D412yes—
81#120—24no—
82#120—24no—
83#120—24no—
84#120—24no—
85C4×D4—21no—
86C4×D4—21no—
87C4×D4—21no—
88C4×D4—21no—
89C4×D4C2312yes—
90C4×D4C2312yes—
91#117—22no—
92#117—22no—
93#117—22no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C42C2311yescharacteristic
97#107—24no—
98C4⋊C8C2312yes—
99C4⋊C8C2312yes—
100#103—24no—
101#103—24no—
102#102—24no—
103#102—24no—
104C22⋊C8—21no—
105C22⋊C8—21no—
106#97—21no—
107C4×C8—21no—
108C4×C8—21no—
109C4×C8C2312yes—
110C24—24no—
111#41—44no—
112#41—82no—
113#41—82no—
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—22no—
117C2×Q8—41no—
118C2×Q8—82no—
119C2×Q8C2×D412yes—
120C2×D4—24no—
121C2×D4—24no—
122C2×D4—24no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—84no—
133C2×D4—84no—
134C2×D4—84no—
135C2×D4C2×D412yes—
136C2×D4C2×D412yes—
137C2×D4C2×D412yes—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—41no—
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145SD16—84no—
146M16—42no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4C2×D412yes—
159C4⋊C4C2×D412yes—
160C22⋊C4—24no—
161C22⋊C4—24no—
162C22⋊C4—24no—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—22no—
167C22⋊C4—22no—
168C22⋊C4—22no—
169C42—24no—
170C42C2×D411yescharacteristic
171C42C2×D411yescharacteristic
172C42C2×D411yescharacteristic
173C42C2411yescharacteristic
174C23—22no—
175C23—22no—
176C23—22no—
177C23—84no—
178C23—84no—
179C23—84no—
180C23#12011yescharacteristic
181Q8—42no—
182Q8—44no—
183Q8—81no—
184Q8—81no—
185Q8—82no—
186D4—24no—
187D4—24no—
188D4—42no—
189D4—42no—
190D4—42no—
191D4—42no—
192D4—42no—
193D4—81no—
194D4—81no—
195D4—82no—
196D4—82no—
197D4—82no—
198D4—84no—
199D4—84no—
200D4—84no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—82no—
212C2×C4#12011yescharacteristic
213C2×C4#12011yescharacteristic
214C2×C4#12011yescharacteristic
215C2×C4C22×D411yescharacteristic
216C2×C4C22×D411yescharacteristic
217C2×C4C22×D411yescharacteristic
218C8—42no—
219C8—42no—
220C8—42no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—42no—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—168no—
234C22C2×#12011yescharacteristic
235C4—21no—
236C4—21no—
237C4—21no—
238C4—42no—
239C4—42no—
240C4—42no—
241C4—82no—
242C4#45512yes—
243C4#57612yes—
244C4#57712yes—
245C2—41no—
246C2—82no—
247C2—82no—
248C2—82no—
249C2#300811yescharacteristic
250C2#300911yescharacteristic
251C2C2×#45511yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3549C2
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. #3226C2semidirect
3. #3206C2semidirect
4. #3141C2semidirect
5. #1646C2semidirect
6. #1639C2semidirect
7. #1543C2semidirect
8. #1484C2semidirect
9. #1484C2semidirect
23. C2×C4⋊Q8C22semidirect
28. #487C22semidirect
32. #474C22semidirect
33. #461C22semidirect
35. #431C22semidirect
36. #431C22semidirect
38. C4×M16C22semidirect
39. #333C22semidirect
41. D4⋊C8C22semidirect
42. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35492
#32262
#32062
#31412
#16462
#16392
#15432
#14842