Finite Group Databasebeta

← #18682

Subgroups — #18682

Order 256.

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

normal subgroups only

This group has 567 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#3543C211yesmaximal, characteristic, complemented
2#3217C212yesmaximal, complemented
3C2×#493C211yesmaximal, characteristic, complemented
4#1621C214yesmaximal, complemented
5#1620C212yesmaximal, complemented
6#1619C212yesmaximal, complemented
7#1555C211yesmaximal, characteristic, complemented
8#1494C212yesmaximal, complemented
9C22×D8—22no—
10#560—22no—
11#559C2211yescharacteristic, complemented
12#550C2212yescomplemented
13#546—22no—
14D42—22no—
15D42C2211yescharacteristic, complemented
16#540—22no—
17#535—22no—
18#534—22no—
19C2×#127C2211yescharacteristic
20C2×#127C2211yescharacteristic, complemented
21#529C2212yes—
22#496—22no—
23#493—22no—
24#493C2212yes—
25#493C2212yescomplemented
26#490—22no—
27#460C2214yescomplemented
28#459C2214yescomplemented
29#450—22no—
30#447—22no—
31#432C2212yescomplemented
32C2×C4×C8C2211yescharacteristic, complemented
33#331C2212yescomplemented
34#331C2212yescomplemented
35Q8⋊C8C2214yescomplemented
36D4⋊C8C2214yescomplemented
37C2×#41—22no—
38C22×D4—22no—
39C22×D4—22no—
40C22×D4—22no—
41C22×D4—21no—
42C22×D4—22no—
43C22×D4D412yes—
44C2×SD16—24no—
45C2×D8—24no—
46C2×D8—24no—
47C2×D8—24no—
48C2×D8—48no—
49C22×C8—21no—
50#127—22no—
51#127—22no—
52#127—22no—
53#127—21no—
54#127D412yes—
55#127C2312yes—
56#127C2311yescharacteristic
57#127C2311yescharacteristic
58#126—22no—
59#125D412yes—
60#124—22no—
61#124D412yes—
62#123—22no—
63#123—24no—
64#123—22no—
65#123—22no—
66C22⋊Q8D412yes—
67#121—24no—
68#121—24no—
69#121—24no—
70#121—24no—
71#121—22no—
72#121—22no—
73#121D412yes—
74#120—22no—
75#120—24no—
76#120—24no—
77#120—22no—
78C4×Q8C2312yes—
79C4×D4—22no—
80C4×D4—22no—
81C4×D4—22no—
82C4×D4C2312yes—
83#117—22no—
84C2×C22⋊C4—22no—
85C2×C42C2311yescharacteristic
86C4⋊C8C2314yes—
87#102—24no—
88#102—24no—
89C22⋊C8—22no—
90#97—22no—
91C4×C8—21no—
92C4×C8C2312yes—
93C24—22no—
94C24—22no—
95C24—41no—
96#41—84no—
97C2×Q8C2×D412yes—
98C2×D4—22no—
99C2×D4—24no—
100C2×D4—24no—
101C2×D4—24no—
102C2×D4—22no—
103C2×D4—24no—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4—24no—
107C2×D4—22no—
108C2×D4—44no—
109C2×D4—44no—
110C2×D4—41no—
111C2×D4—42no—
112C2×D4—84no—
113C2×D4—84no—
114C2×D4—84no—
115C2×D4—84no—
116C2×D4C2×D412yes—
117C2×D4C2×D412yes—
118C2×D4C2×D412yes—
119C22×C4—24no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4C2×D412yes—
123C22×C4C2×D411yescharacteristic
124SD16—84no—
125D8—48no—
126D8—48no—
127D8—84no—
128C2×C8—24no—
129C2×C8—22no—
130C2×C8—24no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4C2×D412yes—
136C4⋊C4C2×D412yes—
137C22⋊C4—22no—
138C22⋊C4—22no—
139C22⋊C4—24no—
140C22⋊C4—24no—
141C22⋊C4—22no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—22no—
145C22⋊C4—22no—
146C42—22no—
147C42C2×D411yescharacteristic
148C42C2×D412yes—
149C42C2411yescharacteristic
150C23—22no—
151C23—22no—
152C23—22no—
153C23—44no—
154C23—42no—
155C23—84no—
156C23—84no—
157C23—84no—
158C23—82no—
159C23—82no—
160C23#12011yescharacteristic
161Q8—24no—
162D4—24no—
163D4—42no—
164D4—42no—
165D4—44no—
166D4—44no—
167D4—44no—
168D4—44no—
169D4—44no—
170D4—44no—
171D4—82no—
172D4—84no—
173D4—82no—
174D4—84no—
175D4—84no—
176D4—84no—
177D4—84no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—44no—
184C2×C4—82no—
185C2×C4—84no—
186C2×C4#12012yes—
187C2×C4#12011yescharacteristic
188C2×C4C22×D411yescharacteristic
189C2×C4C22×D412yes—
190C8—24no—
191C8—44no—
192C22—22no—
193C22—42no—
194C22—44no—
195C22—42no—
196C22—44no—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—84no—
201C22—82no—
202C22—164no—
203C22—164no—
204C22—164no—
205C22#45312yes—
206C22C2×#12011yescharacteristic
207C4—22no—
208C4—21no—
209C4—21no—
210C4—42no—
211C4—42no—
212C4—42no—
213C4#57614yes—
214C2—22no—
215C2—82no—
216C2—82no—
217C2—82no—
218C2—162no—
219C2#300812yes—
220C2C2×#45311yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
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. #3543C2semidirect
2. #3217C2semidirect
3. C2×#493C2semidirect
4. #1621C2semidirect
5. #1620C2semidirect
6. #1619C2semidirect
7. #1555C2semidirect
8. #1494C2semidirect
11. #559C22semidirect
12. #550C22semidirect
15. D42C22semidirect
20. C2×#127C22semidirect
25. #493C22semidirect
27. #460C22semidirect
28. #459C22semidirect
31. #432C22semidirect
32. C2×C4×C8C22semidirect
33. #331C22semidirect
34. #331C22semidirect
35. Q8⋊C8C22semidirect
36. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35432
#32172
C2×#4932
#16212
#16202
#16192
#15552
#14942