Finite Group Databasebeta

← #19883

Subgroups — #19883

Order 256.

399 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 399 conjugacy classes of subgroups — too many to draw. Only its 111 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#3334C211yesmaximal, characteristic, complemented
2D4×D8C211yesmaximal, characteristic, complemented
3#3272C212yesmaximal, complemented
4#1706C211yesmaximal, characteristic, complemented
5#1685C212yesmaximal, complemented
6#1673C211yesmaximal, characteristic, complemented
7#1667C211yesmaximal, characteristic, complemented
8#1639C212yesmaximal, complemented
9#1625C212yesmaximal, complemented
10#1589C211yesmaximal, characteristic, complemented
11C8×D8C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)—22no—
13C22×D8—21no—
14#547C2212yescomplemented
15D42C2212yes—
16#500C2211yescharacteristic, complemented
17#493C2211yescharacteristic
18#492C2212yescomplemented
19#486C2212yescomplemented
20#480—22no—
21#474C2212yescomplemented
22#468—22no—
23#466—21no—
24#466—21no—
25#461C2212yes—
26#459C2212yes—
27#459C2212yescomplemented
28#450—22no—
29#449—22no—
30#447—22no—
31C4×D8C2211yescharacteristic
32C4×D8C2211yescharacteristic
33#436C2212yescomplemented
34C8×D4C2211yescharacteristic
35C8×D4C2211yescharacteristic, complemented
36C2×#107—21no—
37#335C2211yescharacteristic, complemented
38#335C2211yescharacteristic, complemented
39#332C2212yescomplemented
40#331C2212yescomplemented
41D4⋊C8C2211yescharacteristic
42D4⋊C8C2211yescharacteristic, complemented
43D4⋊C8C2212yescomplemented
44C82C2211yescharacteristic, complemented
45C2×#41—22no—
46C22×D4—22no—
47C22×D4—22no—
48AGL(1,ℤ/8)—48no—
49C2×SD16—22no—
50C2×SD16—22no—
51C2×SD16—22no—
52C2×D8—22no—
53C2×D8—22no—
54C2×D8—21no—
55C2×D8—21no—
56C2×D8—44no—
57C2×D8D411yescharacteristic
58C2×D8D411yescharacteristic
59C2×M16—22no—
60C22×C8—21no—
61C22×C8—21no—
62C4⋊Q8C2312yes—
63#127C2312yes—
64#123—42no—
65C22⋊Q8—22no—
66#121—22no—
67#121—22no—
68#121—22no—
69#120—42no—
70C4×D4C2311yescharacteristic
71C4×D4C2312yes—
72C4×D4C2311yescharacteristic
73C2×C4⋊C4—22no—
74#107—21no—
75#107—21no—
76#107—22no—
77#107D411yescharacteristic
78#107D411yescharacteristic
79C4⋊C8C2311yescharacteristic
80C4⋊C8C2312yes—
81C4⋊C8C2311yescharacteristic
82#103—22no—
83#102—22no—
84#102—22no—
85#102—22no—
86#102—42no—
87#102D412yes—
88#102D411yescharacteristic
89#102D411yescharacteristic
90C22⋊C8—21no—
91C22⋊C8—21no—
92C22⋊C8—22no—
93C4×C8C2311yescharacteristic
94C4×C8C2311yescharacteristic
95C4×C8C2311yescharacteristic
96C24—42no—
97#41—44no—
98#41—82no—
99C2×Q8—22no—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—42no—
104C2×D4—42no—
105C2×D4—44no—
106C2×D4—44no—
107C2×D4—42no—
108C2×D4—82no—
109C2×D4—82no—
110C2×D4C2×D412yes—
111C2×D4C2×D411yescharacteristic
112C2×D4C2×D411yescharacteristic
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—42no—
117SD16—44no—
118SD16—44no—
119SD16—82no—
120D8—22no—
121D8—44no—
122D8—44no—
123D8—42no—
124D8—81no—
125D8D814yescomplemented
126M16—44no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—22no—
130C2×C8—21no—
131C2×C8—42no—
132C2×C8—42no—
133C2×C8C2×D411yescharacteristic
134C2×C8C2×D411yescharacteristic
135C2×C8C2×D411yescharacteristic
136C2×C8C2×D411yescharacteristic
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—42no—
140C4⋊C4C2×D411yescharacteristic
141C4⋊C4C2×D412yes—
142C4⋊C4C2×D411yescharacteristic
143C22⋊C4—21no—
144C22⋊C4—22no—
145C22⋊C4—21no—
146C22⋊C4—42no—
147C22⋊C4—42no—
148C42C2411yescharacteristic
149C23—21no—
150C23—21no—
151C23—22no—
152C23—42no—
153C23—84no—
154C23—84no—
155C23—162no—
156Q8—42no—
157Q8—42no—
158D4—22no—
159D4—21no—
160D4—21no—
161D4—22no—
162D4—42no—
163D4—42no—
164D4—42no—
165D4—42no—
166D4—82no—
167D4—84no—
168D4—84no—
169D4—84no—
170D4C2×D812yes—
171D4C2×D812yescomplemented
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—84no—
183C2×C4C22×D411yescharacteristic
184C2×C4C22×D411yescharacteristic
185C2×C4H(5,𝔽2)11yescharacteristic
186C8—21no—
187C8—21no—
188C8—41no—
189C8—42no—
190C8—41no—
191C8—41no—
192C8C2×D812yescomplemented
193C8AGL(1,ℤ/8)12yescomplemented
194C22—22no—
195C22—22no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22—41no—
200C22—42no—
201C22—42no—
202C22—42no—
203C22—82no—
204C22—82no—
205C22—82no—
206C22—168no—
207C22D4211yescharacteristic
208C4—21no—
209C4—41no—
210C4—41no—
211C4—42no—
212C4—82no—
213C4C22×D811yescharacteristic
214C4C2×AGL(1,ℤ/8)11yescharacteristic
215C4#57611yescharacteristic
216C4#57711yescharacteristic
217C2—42no—
218C2—42no—
219C2—82no—
220C2—162no—
221C2#327211yescharacteristic
222C2D4×D811yescharacteristic
223C2#329011yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3334C2semidirect
2. D4×D8C2semidirect
3. #3272C2semidirect
4. #1706C2semidirect
5. #1685C2semidirect
6. #1673C2semidirect
7. #1667C2semidirect
8. #1639C2semidirect
9. #1625C2semidirect
10. #1589C2semidirect
11. C8×D8C2semidirect
14. #547C22semidirect
16. #500C22semidirect
18. #492C22semidirect
19. #486C22semidirect
21. #474C22semidirect
27. #459C22semidirect
33. #436C22semidirect
35. C8×D4C22semidirect
37. #335C22semidirect
38. #335C22semidirect
39. #332C22semidirect
40. #331C22semidirect
42. D4⋊C8C22semidirect
43. D4⋊C8C22semidirect
44. C82C22semidirect
125. D8D8semidirect
171. D4C2×D8semidirect
192. C8C2×D8semidirect
193. C8AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
#33342
D4×D82
#32722
#17062
#16852
#16732
#16672
#16392
#16252
#15892
C8×D82