Finite Group Databasebeta

← #18485

Subgroups — #18485

Order 256.

498 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 119 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#3481C211yesmaximal, characteristic
2#3131C211yesmaximal, characteristic, complemented
3#3107C212yesmaximal, complemented
4#1644C212yesmaximal, complemented
5#1643C212yesmaximal, complemented
6#1640C212yesmaximal, complemented
7#1639C212yesmaximal, complemented
8#1543C211yesmaximal, characteristic, complemented
9#1476C212yesmaximal, complemented
10#559C2211yescharacteristic
11#554C2212yes—
12#553—22no—
13#551—22no—
14#546—22no—
15D42C2211yescharacteristic
16#533C2211yescharacteristic, complemented
17#532C2211yescharacteristic
18#529—22no—
19#525—22no—
20C4×#41C2212yes—
21C2×C4×D4—22no—
22#488—22no—
23#487C2212yescomplemented
24#486C2212yes—
25#485—22no—
26#475C2212yescomplemented
27#474C2212yes—
28#462C2212yescomplemented
29#461C2212yes—
30#452—22no—
31C4×Q16—22no—
32C4×SD16—22no—
33#431C2212yescomplemented
34#416—22no—
35C2×C4×C8C2211yescharacteristic, complemented
36#333C2211yescharacteristic, complemented
37#333C2211yescharacteristic, complemented
38#331C2212yes—
39Q8⋊C8C2212yescomplemented
40Q8⋊C8C2212yescomplemented
41D4⋊C8C2212yes—
42D4⋊C8C2212yes—
43C2×#41D412yes—
44C2×#41D412yes—
45C22×D4—22no—
46C23×C4—22no—
47C2×Q16—22no—
48C2×SD16—22no—
49C22×C8—21no—
50C4⋊Q8C2312yes—
51C4⋊Q8C2311yescharacteristic
52#127C2311yescharacteristic
53#126—24no—
54#125—22no—
55#125—21no—
56#124—22no—
57#124D412yes—
58#123—22no—
59#123—24no—
60#123—22no—
61C22⋊Q8—24no—
62C22⋊Q8—22no—
63C22⋊Q8—22no—
64#121—24no—
65#121—22no—
66#121—22no—
67#120—22no—
68#120—22no—
69C4×Q8—22no—
70C4×Q8D412yes—
71C4×Q8C2312yes—
72C4×D4—24no—
73C4×D4—24no—
74C4×D4—22no—
75C4×D4—22no—
76C4×D4—22no—
77C4×D4—22no—
78C4×D4—22no—
79C4×D4D412yes—
80C4×D4C2312yes—
81#117—22no—
82#117—22no—
83#117—21no—
84#117D412yes—
85C2×C4⋊C4—22no—
86C2×C22⋊C4—24no—
87C2×C42—22no—
88C2×C42C2311yescharacteristic
89#107—22no—
90#106—22no—
91C4⋊C8C2312yes—
92C4⋊C8C2312yes—
93#103—24no—
94#103—22no—
95#103—22no—
96#103—22no—
97#102—24no—
98#102—22no—
99C22⋊C8—22no—
100C4×C8—21no—
101C4×C8—22no—
102C4×C8C2312yes—
103C24—22no—
104#41—24no—
105#41—42no—
106#41—44no—
107C2×Q8—22no—
108C2×Q8C2×D412yes—
109C2×Q8C2×D412yes—
110C2×D4—24no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—44no—
115C2×D4—44no—
116C2×D4C2×D412yes—
117C2×D4C2×D412yes—
118C22×C4—24no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—84no—
125C22×C4C2×D412yes—
126C22×C4C2×D411yescharacteristic
127Q16—44no—
128SD16—44no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—42no—
133C4⋊C4—24no—
134C4⋊C4—24no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—42no—
141C4⋊C4C2×D412yes—
142C4⋊C4C2×D412yes—
143C22⋊C4—24no—
144C22⋊C4—24no—
145C22⋊C4—22no—
146C22⋊C4—22no—
147C22⋊C4—22no—
148C22⋊C4—22no—
149C22⋊C4—24no—
150C22⋊C4—22no—
151C22⋊C4—41no—
152C42—22no—
153C42—22no—
154C42—22no—
155C42—41no—
156C42C2×D411yescharacteristic
157C42C2×D412yes—
158C42C2411yescharacteristic
159C23—22no—
160C23—22no—
161C23—48no—
162C23—82no—
163C23#12011yescharacteristic
164Q8—22no—
165Q8—22no—
166Q8—42no—
167Q8—42no—
168Q8—42no—
169D4—22no—
170D4—42no—
171D4—44no—
172D4—48no—
173D4—42no—
174D4—42no—
175D4—82no—
176D4#13514yes—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—44no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—44no—
188C2×C4—44no—
189C2×C4—42no—
190C2×C4—82no—
191C2×C4—84no—
192C2×C4#12012yes—
193C2×C4#12011yescharacteristic
194C2×C4C22×D411yescharacteristic
195C2×C4C22×D412yes—
196C8—42no—
197C8—42no—
198C8—42no—
199C22—22no—
200C22—24no—
201C22—21no—
202C22—42no—
203C22—42no—
204C22—44no—
205C22—42no—
206C22—88no—
207C22C2×#12011yescharacteristic
208C4—22no—
209C4—21no—
210C4—42no—
211C4—42no—
212C4—42no—
213C4—42no—
214C4—82no—
215C4#45512yes—
216C4C2×#13512yes—
217C4#57512yes—
218C2—41no—
219C2—44no—
220C2—82no—
221C2#300212yes—
222C2C2×#45511yescharacteristic
subgroupG/H
centerC2×C4#120
commutator, Frattini, cosocleC42C24
socleC22C2×#120
prime radical#3481C2
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
2. #3131C2semidirect
3. #3107C2semidirect
4. #1644C2semidirect
5. #1643C2semidirect
6. #1640C2semidirect
7. #1639C2semidirect
8. #1543C2semidirect
9. #1476C2semidirect
16. #533C22semidirect
23. #487C22semidirect
26. #475C22semidirect
28. #462C22semidirect
33. #431C22semidirect
35. C2×C4×C8C22semidirect
36. #333C22semidirect
37. #333C22semidirect
39. Q8⋊C8C22semidirect
40. Q8⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#34812
#31312
#31072
#16442
#16432
#16402
#16392
#15432
#14762