Finite Group Databasebeta

← #20749

Subgroups — #20749

Order 256.

629 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 629 conjugacy classes of subgroups — too many to draw. Only its 199 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#3428C211yesmaximal, characteristic
2#2921C211yesmaximal, characteristic, complemented
3#2298C211yesmaximal, characteristic, complemented
4#1826C212yesmaximal, complemented
5#1806C214yesmaximal, complemented
6#1789C212yesmaximal
7#1759C212yesmaximal, complemented
8C4×#104C212yesmaximal, complemented
9C22×#41C2211yescharacteristic
10#520—22no—
11#520C414yescomplemented
12#519C412yescomplemented
13#518—22no—
14#518C412yescomplemented
15C4×#41—22no—
16C4×#41C2212yes—
17C4×#41C2212yes—
18C2×#117—22no—
19C2×#117C2211yescharacteristic
20C2×#117C2211yescharacteristic
21#433—22no—
22#433—22no—
23#423C2212yescomplemented
24C2×#104C2214yes—
25C2×#104C2214yes—
26#407C2212yes—
27C4×M16C2212yescomplemented
28#388—22no—
29#387—22no—
30C4×C4⋊C4C2212yescomplemented
31C4×C22⋊C4C2212yes—
32#376C2211yescharacteristic, complemented
33C43C2211yescharacteristic, complemented
34#339C2214yescomplemented
35#339C2214yescomplemented
36C2×#41—28no—
37C2×#41—21no—
38C2×#41—21no—
39C2×#41C2×C412yes—
40C2×#41C2312yes—
41C22×Q8C2×C411yescharacteristic
42C22×D4—21no—
43C22×D4C2×C411yescharacteristic
44C23×C4—21no—
45C23×C4C2311yescharacteristic
46C2×M16C2314yes—
47C4×Q8—24no—
48C4×Q8C2×C414yes—
49C4×Q8C2×C414yes—
50C4×D4—24no—
51C4×D4—24no—
52C4×D4—24no—
53C4×D4—24no—
54C4×D4—24no—
55C4×D4C2×C414yes—
56C4×D4C2×C414yes—
57#117—21no—
58#117—21no—
59#117—24no—
60#117—24no—
61#117—22no—
62#117—22no—
63#117C2×C412yes—
64#117C2×C412yes—
65#117C2×C412yes—
66#117D412yes—
67#117D412yes—
68#117C2312yes—
69C2×C4⋊C4—24no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4C2×C412yes—
72C2×C4⋊C4D412yes—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4D412yes—
77C2×C42—21no—
78C2×C42—22no—
79C2×C42—22no—
80C2×C42—22no—
81C2×C42C2312yes—
82C2×C42C2312yes—
83C2×C42C2311yescharacteristic
84C2×C42C2311yescharacteristic
85C4⋊C8—24no—
86#104—28no—
87#104—28no—
88C22⋊C8—22no—
89C22⋊C8—22no—
90#97—22no—
91C4×C8—22no—
92#95—22no—
93#95—22no—
94#95—22no—
95C24—21no—
96C24C2×D411yescharacteristic
97#41—28no—
98#41—28no—
99#41—24no—
100#41—24no—
101#41—24no—
102#41#4118yes—
103C2×Q8—24no—
104C2×Q8—21no—
105C2×Q8C22×C412yes—
106C2×D4—28no—
107C2×D4—22no—
108C2×D4—22no—
109C2×D4—24no—
110C2×D4—21no—
111C2×D4C22×C412yes—
112C22×C4—28no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—24no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—24no—
120C22×C4—24no—
121C22×C4—22no—
122C22×C4C22×C411yescharacteristic
123C22×C4C22×C411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D412yes—
128C22×C4C2×D412yes—
129C22×C4C2411yescharacteristic
130M16—24no—
131M16—24no—
132C2×C8—24no—
133C4⋊C4—22no—
134C4⋊C4—24no—
135C4⋊C4—24no—
136C4⋊C4—44no—
137C4⋊C4C22⋊C414yes—
138C4⋊C4C22⋊C414yes—
139C4⋊C4C22×C414yes—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—24no—
143C22⋊C4—24no—
144C22⋊C4—44no—
145C22⋊C4C22⋊C414yes—
146C22⋊C4C22⋊C414yes—
147C42—24no—
148C42—24no—
149C42—22no—
150C42—22no—
151C42—24no—
152C42—24no—
153C42—44no—
154C42C22×C412yes—
155C42C22×C412yes—
156C42C2×D412yes—
157C42C2×D412yes—
158C23—24no—
159C23—22no—
160C23—22no—
161C23—42no—
162C23—82no—
163C23C2×C22⋊C411yescharacteristic
164C23C2×C22⋊C411yescharacteristic
165C23C22×D411yescharacteristic
166Q8—24no—
167Q8—22no—
168Q8#11714yes—
169D4—28no—
170D4—24no—
171D4—28no—
172D4—22no—
173D4#11714yes—
174C2×C4—24no—
175C2×C4—28no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—28no—
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—24no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—84no—
193C2×C4—84no—
194C2×C4—82no—
195C2×C4C2×C22⋊C411yescharacteristic
196C2×C4C2×C22⋊C414yes—
197C2×C4C2×C22⋊C414yes—
198C2×C4C2×C22⋊C411yescharacteristic
199C2×C4C23×C411yescharacteristic
200C2×C4C22×D411yescharacteristic
201C2×C4C2×#4114yes—
202C8—44no—
203C22—22no—
204C22—22no—
205C22—22no—
206C22—21no—
207C22—21no—
208C22—24no—
209C22—44no—
210C22—82no—
211C22—84no—
212C22C22×C22⋊C411yescharacteristic
213C22#52512yes—
214C4—24no—
215C4—24no—
216C4—21no—
217C4—24no—
218C4—21no—
219C4—44no—
220C4—44no—
221C4—82no—
222C4C2×#11712yes—
223C4#52512yes—
224C2—22no—
225C2—41no—
226C2—41no—
227C2—44no—
228C2#228011yescharacteristic
229C2#290112yes—
subgroupG/H
centerC2×C4C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
socleC22C22×C22⋊C4
prime radicalC22×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2921C2semidirect
3. #2298C2semidirect
4. #1826C2semidirect
5. #1806C2semidirect
7. #1759C2semidirect
8. C4×#104C2semidirect
11. #520C4semidirect
12. #519C4semidirect
14. #518C4semidirect
23. #423C22semidirect
27. C4×M16C22semidirect
30. C4×C4⋊C4C22semidirect
32. #376C22semidirect
33. C43C22semidirect
34. #339C22semidirect
35. #339C22semidirect

Maximal subgroups

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

subgroupindex
#34282
#29212
#22982
#18262
#18062
#17892
#17592
C4×#1042