Finite Group Databasebeta

← #19166

Subgroups — #19166

Order 256.

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

normal subgroups only

This group has 379 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#3138C211yesmaximal, characteristic, complemented
2#3101C211yesmaximal, characteristic, complemented
3C2×#459C211yesmaximal, characteristic
4D4×M16C211yesmaximal, characteristic, complemented
5#1701C212yesmaximal, complemented
6#1697C212yesmaximal, complemented
7#1681C212yesmaximal, complemented
8#1677C212yesmaximal, complemented
9#1565C211yesmaximal, characteristic, complemented
10#1550C211yesmaximal, characteristic, complemented
11#1545C211yesmaximal, characteristic, complemented
12C22×D8—21no—
13C22×M16—21no—
14#533C2211yescharacteristic
15C2×#127C2211yescharacteristic
16C2×C4×D4C2211yescharacteristic
17#490—22no—
18#486C2212yescomplemented
19#486C2212yescomplemented
20#483—21no—
21#476C2211yescharacteristic, complemented
22#476C2211yescharacteristic, complemented
23#461C2211yescharacteristic, complemented
24#461C2211yescharacteristic, complemented
25#459—21no—
26#459—21no—
27#459C2212yes—
28#459C2212yes—
29#450—21no—
30C4×SD16—21no—
31C4×SD16—21no—
32#436C2212yescomplemented
33#435—22no—
34C8×D4C2212yescomplemented
35#431C2211yescharacteristic
36#423C2211yescharacteristic
37C2×C4⋊C8C2211yescharacteristic
38#418—21no—
39#418—21no—
40C2×#102—21no—
41#407—21no—
42C4×M16C2211yescharacteristic
43#335C2212yescomplemented
44#334C2212yescomplemented
45#333C2212yescomplemented
46#332C2212yescomplemented
47#331C2212yescomplemented
48#331C2212yescomplemented
49C22×D4—21no—
50C22×D4—21no—
51C22×D4—21no—
52C22×D4D411yescharacteristic
53C23×C4—21no—
54C2×SD16—21no—
55C2×SD16—21no—
56C2×D8—22no—
57C2×D8—22no—
58C2×D8—44no—
59C2×M16—21no—
60C2×M16—22no—
61C2×M16—44no—
62C2×M16D411yescharacteristic
63C2×M16D411yescharacteristic
64C22×C8—21no—
65C22×C8—22no—
66C4⋊Q8C2311yescharacteristic
67C4⋊Q8C2311yescharacteristic
68#127—21no—
69#127—21no—
70#127—21no—
71#127C2312yes—
72#125—21no—
73C22⋊Q8—21no—
74C22⋊Q8—21no—
75C4×Q8—21no—
76C4×Q8—21no—
77C4×D4—21no—
78C4×D4—21no—
79C4×D4D411yescharacteristic
80C4×D4D411yescharacteristic
81C4×D4C2311yescharacteristic
82C4×D4C2311yescharacteristic
83#117—21no—
84C2×C4⋊C4D411yescharacteristic
85C2×C22⋊C4—21no—
86C2×C42C2311yescharacteristic
87#106—21no—
88#106—21no—
89C4⋊C8—21no—
90C4⋊C8—21no—
91C4⋊C8C2312yes—
92C4⋊C8C2311yescharacteristic
93C4⋊C8C2311yescharacteristic
94C4⋊C8C2312yes—
95#103—21no—
96#103—21no—
97#102—21no—
98#102—21no—
99#102—22no—
100#102—22no—
101#102—22no—
102#102—22no—
103#102—22no—
104#102—22no—
105C22⋊C8—21no—
106C22⋊C8—22no—
107C22⋊C8—22no—
108#97D412yes—
109C4×C8—21no—
110C4×C8C2312yes—
111C24—21no—
112C24—41no—
113C2×Q8—21no—
114C2×Q8—21no—
115C2×D4—22no—
116C2×D4—21no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4—41no—
121C2×D4—42no—
122C2×D4—42no—
123C2×D4—41no—
124C2×D4—41no—
125C2×D4—41no—
126C2×D4—42no—
127C2×D4—42no—
128C2×D4—82no—
129C2×D4C2×D411yescharacteristic
130C2×D4C2×D411yescharacteristic
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4#4111yescharacteristic
141SD16—42no—
142SD16—42no—
143D8—44no—
144D8—44no—
145M16—22no—
146M16—22no—
147M16—44no—
148M16—42no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—44no—
154C2×C8—42no—
155C2×C8C2×D412yes—
156C2×C8C2×D412yes—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4C2×D411yescharacteristic
165C4⋊C4C2×D411yescharacteristic
166C22⋊C4—21no—
167C22⋊C4—21no—
168C22⋊C4—21no—
169C22⋊C4—21no—
170C22⋊C4—41no—
171C42—41no—
172C42C2×D411yescharacteristic
173C42C2×D411yescharacteristic
174C42#4111yescharacteristic
175C42C2411yescharacteristic
176C23—21no—
177C23—21no—
178C23—42no—
179C23—42no—
180C23—42no—
181C23—81no—
182C23—82no—
183C23—82no—
184C23—82no—
185C23#12111yescharacteristic
186Q8—41no—
187Q8—41no—
188Q8—81no—
189D4—21no—
190D4—21no—
191D4—21no—
192D4—42no—
193D4—42no—
194D4—42no—
195D4—42no—
196D4—42no—
197D4—82no—
198D4—82no—
199D4—82no—
200D4—82no—
201D4AGL(1,ℤ/8)12yes—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4#12111yescharacteristic
217C2×C4#12111yescharacteristic
218C2×C4#12111yescharacteristic
219C2×C4#13512yes—
220C2×C4AGL(1,ℤ/8)12yes—
221C2×C4C22×D411yescharacteristic
222C2×C4C22×D411yescharacteristic
223C2×C4C2×#4111yescharacteristic
224C8—22no—
225C8—22no—
226C8—42no—
227C8—42no—
228C8—41no—
229C8—41no—
230C22—21no—
231C22—21no—
232C22—22no—
233C22—41no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—84no—
239C22—82no—
240C22—82no—
241C22—82no—
242C22—164no—
243C22#46312yes—
244C22C2×#12111yescharacteristic
245C4—21no—
246C4—21no—
247C4—21no—
248C4—41no—
249C4—41no—
250C4—81no—
251C4—81no—
252C4#46312yes—
253C4C2×#13511yescharacteristic
254C4C2×AGL(1,ℤ/8)11yescharacteristic
255C4C2×AGL(1,ℤ/8)11yescharacteristic
256C4#57511yescharacteristic
257C2—22no—
258C2—42no—
259C2—81no—
260C2—162no—
261C2C2×#46311yescharacteristic
262C2#305511yescharacteristic
263C2#306811yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radicalC2×#459C2
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. #3138C2semidirect
2. #3101C2semidirect
4. D4×M16C2semidirect
5. #1701C2semidirect
6. #1697C2semidirect
7. #1681C2semidirect
8. #1677C2semidirect
9. #1565C2semidirect
10. #1550C2semidirect
11. #1545C2semidirect
18. #486C22semidirect
19. #486C22semidirect
21. #476C22semidirect
22. #476C22semidirect
23. #461C22semidirect
24. #461C22semidirect
32. #436C22semidirect
34. C8×D4C22semidirect
43. #335C22semidirect
44. #334C22semidirect
45. #333C22semidirect
46. #332C22semidirect
47. #331C22semidirect
48. #331C22semidirect

Maximal subgroups

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

subgroupindex
#31382
#31012
C2×#4592
D4×M162
#17012
#16972
#16812
#16772
#15652
#15502
#15452