Finite Group Databasebeta

← #19456

Subgroups — #19456

Order 256.

361 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 361 conjugacy classes of subgroups — too many to draw. Only its 109 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#3208C211yesmaximal, characteristic, complemented
2#3110C211yesmaximal, characteristic, complemented
3C2×#459C211yesmaximal, characteristic
4#2974C211yesmaximal, characteristic, complemented
5#1701C212yesmaximal, complemented
6#1685C212yesmaximal, complemented
7#1681C212yesmaximal, complemented
8#1665C212yesmaximal, complemented
9#1573C211yesmaximal, characteristic, complemented
10#1555C211yesmaximal, characteristic, complemented
11#1552C211yesmaximal, characteristic, complemented
12C22×D8—21no—
13#533C2211yescharacteristic
14C2×#127C2211yescharacteristic
15C2×C4×D4C2211yescharacteristic
16#492C2212yescomplemented
17#486C2212yescomplemented
18#484—21no—
19#474C2211yescharacteristic, complemented
20#474C2211yescharacteristic, complemented
21#461C2211yescharacteristic, complemented
22#461C2211yescharacteristic, complemented
23#460—22no—
24#459—21no—
25#459—21no—
26#459C2212yes—
27#459C2212yes—
28#450—21no—
29#450—21no—
30#450—21no—
31#440—21no—
32#440—21no—
33#436C2212yescomplemented
34#435—22no—
35C8×D4C2212yescomplemented
36#432C2211yescharacteristic
37#432C2211yescharacteristic
38#431C2211yescharacteristic
39C2×C4⋊C8C2211yescharacteristic
40C2×#102—21no—
41#407—21no—
42C2×C22⋊C8—21no—
43#335C2212yescomplemented
44#334C2212yescomplemented
45#332C2212yescomplemented
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×SD16—22no—
57C2×SD16—22no—
58C2×D8—22no—
59C2×D8—22no—
60C2×D8—44no—
61C2×M16—22no—
62C22×C8—21no—
63C22×C8—22no—
64C4⋊Q8C2311yescharacteristic
65C4⋊Q8C2311yescharacteristic
66#127—21no—
67#127—21no—
68#127—21no—
69#127C2312yes—
70#125—21no—
71C22⋊Q8—21no—
72C22⋊Q8—21no—
73C4×Q8—21no—
74C4×Q8—21no—
75C4×D4—21no—
76C4×D4—21no—
77C4×D4D411yescharacteristic
78C4×D4D411yescharacteristic
79C4×D4C2311yescharacteristic
80C4×D4C2311yescharacteristic
81#117—21no—
82C2×C4⋊C4D411yescharacteristic
83C2×C22⋊C4—21no—
84C2×C42C2311yescharacteristic
85#107—21no—
86#107—21no—
87C4⋊C8—21no—
88C4⋊C8D412yes—
89C4⋊C8C2312yes—
90C4⋊C8C2311yescharacteristic
91C4⋊C8C2311yescharacteristic
92C4⋊C8C2312yes—
93#103—21no—
94#103—21no—
95#102—21no—
96#102—21no—
97#102—22no—
98#102—22no—
99#102—22no—
100#102—22no—
101C22⋊C8—21no—
102C22⋊C8—21no—
103C22⋊C8—22no—
104C22⋊C8—22no—
105C22⋊C8D411yescharacteristic
106C22⋊C8D411yescharacteristic
107#97—21no—
108#97—21no—
109C4×C8C2312yes—
110C24—21no—
111C24—41no—
112C2×Q8—21no—
113C2×Q8—21no—
114C2×D4—22no—
115C2×D4—21no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—41no—
120C2×D4—42no—
121C2×D4—42no—
122C2×D4—41no—
123C2×D4—41no—
124C2×D4—41no—
125C2×D4—82no—
126C2×D4—82no—
127C2×D4C2×D411yescharacteristic
128C2×D4C2×D411yescharacteristic
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—82no—
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4#4111yescharacteristic
138SD16—42no—
139SD16—42no—
140SD16—82no—
141SD16—82no—
142D8—44no—
143D8—44no—
144M16—82no—
145C2×C8—22no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—22no—
149C2×C8—42no—
150C2×C8—82no—
151C2×C8C2×D412yes—
152C2×C8C2×D412yes—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4C2×D411yescharacteristic
161C4⋊C4C2×D411yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—21no—
166C22⋊C4—41no—
167C42—41no—
168C42C2×D411yescharacteristic
169C42C2×D411yescharacteristic
170C42#4111yescharacteristic
171C42C2411yescharacteristic
172C23—21no—
173C23—21no—
174C23—42no—
175C23—42no—
176C23—42no—
177C23—81no—
178C23—82no—
179C23—82no—
180C23—82no—
181C23#12111yescharacteristic
182Q8—41no—
183Q8—41no—
184Q8—42no—
185D4—21no—
186D4—21no—
187D4—21no—
188D4—42no—
189D4—42no—
190D4—42no—
191D4—42no—
192D4—42no—
193D4—82no—
194D4—82no—
195D4—82no—
196D4—82no—
197D4AGL(1,ℤ/8)12yes—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4#12111yescharacteristic
212C2×C4#12111yescharacteristic
213C2×C4#12111yescharacteristic
214C2×C4C22×D411yescharacteristic
215C2×C4C22×D411yescharacteristic
216C2×C4C2×#4111yescharacteristic
217C8—22no—
218C8—22no—
219C8—42no—
220C8—41no—
221C8—41no—
222C8—42no—
223C22—21no—
224C22—21no—
225C22—22no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—84no—
232C22—82no—
233C22—82no—
234C22—82no—
235C22—164no—
236C22#47112yes—
237C22C2×#12111yescharacteristic
238C4—21no—
239C4—21no—
240C4—21no—
241C4—21no—
242C4—41no—
243C4—41no—
244C4—81no—
245C4—81no—
246C4C2×AGL(1,ℤ/8)11yescharacteristic
247C4#57511yescharacteristic
248C4#57612yes—
249C2—22no—
250C2—42no—
251C2—81no—
252C2—162no—
253C2#304111yescharacteristic
254C2#305511yescharacteristic
255C2C2×#47111yescharacteristic
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. #3208C2semidirect
2. #3110C2semidirect
4. #2974C2semidirect
5. #1701C2semidirect
6. #1685C2semidirect
7. #1681C2semidirect
8. #1665C2semidirect
9. #1573C2semidirect
10. #1555C2semidirect
11. #1552C2semidirect
16. #492C22semidirect
17. #486C22semidirect
19. #474C22semidirect
20. #474C22semidirect
21. #461C22semidirect
22. #461C22semidirect
33. #436C22semidirect
35. C8×D4C22semidirect
43. #335C22semidirect
44. #334C22semidirect
45. #332C22semidirect
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
#32082
#31102
C2×#4592
#29742
#17012
#16852
#16812
#16652
#15732
#15552
#15522