Finite Group Databasebeta

← #21406

Subgroups — #21406

Order 256.

615 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 615 conjugacy classes of subgroups — too many to draw. Only its 207 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
1C2×#529C211yesmaximal, characteristic
2C22×#97C211yesmaximal, characteristic, complemented
3#2285C211yesmaximal, characteristic, complemented
4#1992C212yesmaximal, complemented
5#1959C212yesmaximal, complemented
6#1893C212yesmaximal
7#1892C212yesmaximal, complemented
8#1763C212yesmaximal, complemented
9#1762C212yesmaximal, complemented
10C23×C8C2212yes—
11#529—24no—
12#529C418yescomplemented
13C2×#126—21no—
14C2×#125C2211yescharacteristic
15C2×#124C2211yescharacteristic
16C2×#123—21no—
17C2×C22⋊Q8C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×C4×D4—21no—
21C2×C4×D4C2211yescharacteristic
22C2×#117—21no—
23C2×#117—21no—
24C22×C42C2211yescharacteristic
25C2×#103C2214yes—
26C2×#102C2214yes—
27C2×#97—24no—
28C2×#97C2212yes—
29C2×#97C2212yes—
30#384C2212yes—
31#382—21no—
32#380—21no—
33#380C2212yes—
34C4×C4⋊C4—21no—
35C4×C4⋊C4C2212yes—
36#340C2214yes—
37#340C2214yes—
38C22×Q8C2311yescharacteristic
39C22×D4—21no—
40C22×D4C2311yescharacteristic
41C23×C4—21no—
42C23×C4D411yescharacteristic
43C23×C4D411yescharacteristic
44C23×C4C2311yescharacteristic
45C22×C8—24no—
46C22×C8—28no—
47C22×C8C2314yes—
48#126—28no—
49#125—22no—
50#125C2×C414yes—
51#124—22no—
52#124C2×C414yes—
53#123—28no—
54C22⋊Q8—22no—
55C22⋊Q8C2×C414yes—
56#121—22no—
57#121C2×C414yes—
58C4×Q8—22no—
59C4×Q8C2×C414yes—
60C4×D4—28no—
61C4×D4—22no—
62C4×D4C2×C414yes—
63#117—28no—
64#117—44no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4C2314yes—
72C2×C4⋊C4C2311yescharacteristic
73C2×C4⋊C4C2311yescharacteristic
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—22no—
80C2×C42—21no—
81C2×C42—22no—
82C2×C42C2×C414yes—
83C2×C42D411yescharacteristic
84C2×C42D411yescharacteristic
85C2×C42D414yes—
86C2×C42C2311yescharacteristic
87C2×C42C2311yescharacteristic
88#103—28no—
89#102—28no—
90#97—24no—
91#97—24no—
92#95—22no—
93#95—22no—
94C24—21no—
95C24C2×D411yescharacteristic
96C2×Q8—21no—
97C2×Q8—24no—
98C2×Q8C22×C412yes—
99C2×D4—24no—
100C2×D4—21no—
101C2×D4—24no—
102C2×D4—42no—
103C2×D4—42no—
104C2×D4C22×C412yes—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—24no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—24no—
112C22×C4—24no—
113C22×C4—42no—
114C22×C4—44no—
115C22×C4C22⋊C414yes—
116C22×C4C22⋊C414yes—
117C22×C4C22×C412yes—
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D412yes—
124C22×C4C2411yescharacteristic
125C2×C8—216no—
126C2×C8—24no—
127C2×C8—24no—
128C2×C8#4118yes—
129C4⋊C4—24no—
130C4⋊C4—24no—
131C4⋊C4—24no—
132C4⋊C4—24no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—24no—
136C4⋊C4—24no—
137C4⋊C4—44no—
138C4⋊C4—44no—
139C4⋊C4C22×C412yes—
140C4⋊C4C22×C412yes—
141C22⋊C4—24no—
142C22⋊C4—24no—
143C22⋊C4—24no—
144C22⋊C4—24no—
145C22⋊C4—24no—
146C22⋊C4—44no—
147C42—24no—
148C42—44no—
149C42C22⋊C414yes—
150C42C22⋊C414yes—
151C42C22×C412yes—
152C42C22×C412yes—
153C42C2×D412yes—
154C42C2×D412yes—
155C23—22no—
156C23—22no—
157C23—22no—
158C23—42no—
159C23—41no—
160C23—41no—
161C23—42no—
162C23C2×C22⋊C412yes—
163C23AGL(1,ℤ/8)12yes—
164C23#13712yes—
165C23C22×D411yescharacteristic
166Q8—24no—
167Q8—42no—
168D4—24no—
169D4—48no—
170D4—42no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—28no—
179C2×C4—24no—
180C2×C4—24no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—44no—
185C2×C4—48no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—44no—
191C2×C4—44no—
192C2×C4—44no—
193C2×C4C2×C22⋊C412yes—
194C2×C4C2×C22⋊C412yes—
195C2×C4C2×C22⋊C412yes—
196C2×C4C2×C22⋊C412yes—
197C2×C4C2×C22⋊C412yes—
198C2×C4C23×C411yescharacteristic
199C2×C4C22×D411yescharacteristic
200C2×C4C2×#4114yes—
201C8—28no—
202C22—22no—
203C22—24no—
204C22—24no—
205C22—22no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—82no—
210C22—82no—
211C22#41714yes—
212C22C22×C22⋊C411yescharacteristic
213C22#53212yes—
214C22C2×AGL(1,ℤ/8)11yescharacteristic
215C22C2×#13711yescharacteristic
216C22#57512yes—
217C4—22no—
218C4—22no—
219C4—22no—
220C4—22no—
221C4—22no—
222C4—42no—
223C4—42no—
224C4—42no—
225C4—84no—
226C4C4×#4112yes—
227C4#52512yes—
228C2—24no—
229C2—82no—
230C2#229711yescharacteristic
231C2C2×#41711yescharacteristic
232C2#289611yescharacteristic
233C2#318012yes—
234C2#318112yes—
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#97C2semidirect
3. #2285C2semidirect
4. #1992C2semidirect
5. #1959C2semidirect
7. #1892C2semidirect
8. #1763C2semidirect
9. #1762C2semidirect
12. #529C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5292
C22×#972
#22852
#19922
#19592
#18932
#18922
#17632
#17622