Finite Group Databasebeta

← #19142

Subgroups — #19142

Order 256.

342 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 113 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#3131C211yesmaximal, characteristic, complemented
2#3101C211yesmaximal, characteristic, complemented
3#3100C211yesmaximal, characteristic
4C2×#436C211yesmaximal, characteristic, complemented
5#1703C212yesmaximal, complemented
6#1701C212yesmaximal, complemented
7#1699C212yesmaximal, complemented
8#1697C212yesmaximal, complemented
9#1564C211yesmaximal, characteristic, complemented
10#1542C211yesmaximal, characteristic, complemented
11#1540C211yesmaximal, characteristic, complemented
12C22×M16—21no—
13#533C2211yescharacteristic
14#532C2211yescharacteristic
15C2×C4×D4C2211yescharacteristic
16#488—21no—
17#488—21no—
18#487C2211yescharacteristic, complemented
19#487C2211yescharacteristic, complemented
20#486C2211yescharacteristic, complemented
21#486C2211yescharacteristic, complemented
22#483—21no—
23#480—21no—
24#476C2212yes—
25#474C2212yes—
26#461C2212yes—
27#459C2212yes—
28#450—21no—
29#447—21no—
30C4×SD16—22no—
31C4×D8—22no—
32#436—21no—
33#436—21no—
34#436C2212yes—
35#436C2212yes—
36#431C2211yescharacteristic
37#431C2211yescharacteristic
38C2×C4⋊C8C2211yescharacteristic
39#416—21no—
40#416—21no—
41C2×C22⋊C8—21no—
42C2×C4×C8C2211yescharacteristic
43#335C2212yescomplemented
44#334C2212yescomplemented
45#333C2212yescomplemented
46#332C2212yescomplemented
47#332C2212yescomplemented
48#331C2212yescomplemented
49C2×#41—21no—
50C22×D4D411yescharacteristic
51C23×C4—21no—
52C2×SD16—22no—
53C2×D8—22no—
54C2×M16—22no—
55C2×M16—22no—
56C2×M16—44no—
57C22×C8—21no—
58C22×C8D411yescharacteristic
59C22×C8D411yescharacteristic
60C4⋊Q8C2311yescharacteristic
61C4⋊Q8C2311yescharacteristic
62C4⋊Q8C2311yescharacteristic
63#127C2311yescharacteristic
64#125—21no—
65#124—21no—
66C22⋊Q8—21no—
67C22⋊Q8—21no—
68#121—21no—
69#121—21no—
70C4×Q8—21no—
71C4×Q8—21no—
72C4×D4—21no—
73C4×D4—21no—
74C4×D4—21no—
75C4×D4—21no—
76C4×D4D412yes—
77C4×D4C2312yes—
78#117—21no—
79C2×C4⋊C4D411yescharacteristic
80C2×C22⋊C4—21no—
81C2×C42C2311yescharacteristic
82#107—22no—
83#106—22no—
84C4⋊C8—21no—
85C4⋊C8C2312yes—
86C4⋊C8C2312yes—
87C4⋊C8C2312yes—
88#103—21no—
89#103—21no—
90#103—21no—
91#103—21no—
92#103—22no—
93#102—22no—
94#102—22no—
95#102—21no—
96#102—21no—
97#102—21no—
98#102—21no—
99#102—22no—
100C22⋊C8—21no—
101C22⋊C8—21no—
102C22⋊C8—22no—
103C22⋊C8—22no—
104C4×C8—21no—
105C4×C8—21no—
106C4×C8D411yescharacteristic
107C4×C8D411yescharacteristic
108C4×C8C2311yescharacteristic
109C4×C8C2311yescharacteristic
110C24—21no—
111#41—42no—
112#41—42no—
113C2×Q8—21no—
114C2×Q8—21no—
115C2×Q8—21no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4—42no—
121C2×D4—41no—
122C2×D4C2×D412yes—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—41no—
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4#4111yescharacteristic
132SD16—44no—
133D8—44no—
134M16—44no—
135M16—44no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—42no—
142C2×C8C2×D411yescharacteristic
143C2×C8C2×D411yescharacteristic
144C2×C8C2×D411yescharacteristic
145C2×C8C2×D411yescharacteristic
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4C2×D412yes—
156C22⋊C4—22no—
157C22⋊C4—22no—
158C22⋊C4—41no—
159C22⋊C4—41no—
160C22⋊C4—41no—
161C42—41no—
162C42C2×D411yescharacteristic
163C42C2×D411yescharacteristic
164C42#4111yescharacteristic
165C42C2411yescharacteristic
166C23—22no—
167C23—42no—
168C23—42no—
169C23—42no—
170C23—41no—
171C23#12111yescharacteristic
172Q8—41no—
173Q8—41no—
174Q8—42no—
175Q8—81no—
176D4—22no—
177D4—22no—
178D4—42no—
179D4—41no—
180D4—41no—
181D4—42no—
182D4—81no—
183D4—82no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—44no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—82no—
195C2×C4#12111yescharacteristic
196C2×C4#12111yescharacteristic
197C2×C4#12111yescharacteristic
198C2×C4#13512yes—
199C2×C4AGL(1,ℤ/8)12yes—
200C2×C4C22×D411yescharacteristic
201C2×C4C22×D411yescharacteristic
202C2×C4C2×#4111yescharacteristic
203C8—21no—
204C8—21no—
205C8—21no—
206C8—21no—
207C8—42no—
208C8—42no—
209C8—42no—
210C22—21no—
211C22—21no—
212C22—42no—
213C22—42no—
214C22—42no—
215C22—84no—
216C22—81no—
217C22—81no—
218C22—81no—
219C22#46312yes—
220C22C2×#12111yescharacteristic
221C4—21no—
222C4—21no—
223C4—21no—
224C4—42no—
225C4—81no—
226C4—81no—
227C4—81no—
228C4#46312yes—
229C4C2×#13511yescharacteristic
230C4C2×AGL(1,ℤ/8)11yescharacteristic
231C4#57512yes—
232C2—22no—
233C2—82no—
234C2—161no—
235C2C2×#46311yescharacteristic
236C2#305411yescharacteristic
237C2#306711yescharacteristic
subgroupG/H
centerC2×C4#121
commutator, Frattini, cosocleC42C24
socleC22C2×#121
prime radical#3100C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #3131C2semidirect
2. #3101C2semidirect
4. C2×#436C2semidirect
5. #1703C2semidirect
6. #1701C2semidirect
7. #1699C2semidirect
8. #1697C2semidirect
9. #1564C2semidirect
10. #1542C2semidirect
11. #1540C2semidirect
18. #487C22semidirect
19. #487C22semidirect
20. #486C22semidirect
21. #486C22semidirect
43. #335C22semidirect
44. #334C22semidirect
45. #333C22semidirect
46. #332C22semidirect
47. #332C22semidirect
48. #331C22semidirect

Maximal subgroups

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

subgroupindex
#31312
#31012
#31002
C2×#4362
#17032
#17012
#16992
#16972
#15642
#15422
#15402