Finite Group Databasebeta

← #21402

Subgroups — #21402

Order 256.

613 conjugacy classes of subgroups; only the 215 normal subgroups are shown.

normal subgroups only

This group has 613 conjugacy classes of subgroups — too many to draw. Only its 215 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×C4×C8C211yesmaximal, characteristic, complemented
3#2283C211yesmaximal, characteristic, complemented
4#1991C212yesmaximal, complemented
5#1958C212yesmaximal, complemented
6#1893C212yesmaximal
7#1892C212yesmaximal, complemented
8C4×#103C212yesmaximal, complemented
9C4×#102C212yesmaximal, 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—
23C22×C42C2211yescharacteristic
24C2×#103C2214yes—
25C2×#102C2214yes—
26C2×C4×C8—24no—
27C2×C4×C8C2212yes—
28C2×C4×C8C2212yes—
29#383—21no—
30#382C2212yes—
31#381—21no—
32#380C2212yes—
33C4×C4⋊C4C2212yes—
34C4×C22⋊C4—21no—
35#376—21no—
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—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4C2314yes—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—22no—
77C2×C42—21no—
78C2×C42—22no—
79C2×C42C2×C414yes—
80C2×C42D411yescharacteristic
81C2×C42D411yescharacteristic
82C2×C42D414yes—
83C2×C42C2311yescharacteristic
84C2×C42C2311yescharacteristic
85#103—28no—
86#102—28no—
87C4×C8—24no—
88C4×C8—24no—
89#95—22no—
90#95—22no—
91#95—22no—
92#95—22no—
93C24—21no—
94C24C2×D411yescharacteristic
95C2×Q8—21no—
96C2×Q8—24no—
97C2×Q8C22×C412yes—
98C2×D4—24no—
99C2×D4—21no—
100C2×D4—24no—
101C2×D4—44no—
102C2×D4C22×C412yes—
103C22×C4—24no—
104C22×C4—24no—
105C22×C4—24no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—24no—
110C22×C4—24no—
111C22×C4—44no—
112C22×C4C22⋊C414yes—
113C22×C4C22⋊C414yes—
114C22×C4C22×C412yes—
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D412yes—
120C22×C4C2×D411yescharacteristic
121C22×C4C2411yescharacteristic
122C2×C8—216no—
123C2×C8—24no—
124C2×C8—24no—
125C2×C8#4118yes—
126C4⋊C4—24no—
127C4⋊C4—24no—
128C4⋊C4—24no—
129C4⋊C4—24no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—24no—
133C4⋊C4—24no—
134C4⋊C4C22×C412yes—
135C4⋊C4C22×C412yes—
136C22⋊C4—24no—
137C22⋊C4—24no—
138C22⋊C4—24no—
139C22⋊C4—24no—
140C22⋊C4—24no—
141C22⋊C4—44no—
142C42—24no—
143C42—44no—
144C42C22⋊C414yes—
145C42C22⋊C414yes—
146C42C22×C412yes—
147C42C22×C412yes—
148C42C2×D412yes—
149C42C2×D412yes—
150C23—22no—
151C23—22no—
152C23—22no—
153C23—42no—
154C23—42no—
155C23—42no—
156C23C2×C22⋊C412yes—
157C23#13514yes—
158C23C22×D411yescharacteristic
159Q8—24no—
160Q8—42no—
161D4—24no—
162D4—48no—
163D4—42no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—28no—
169C2×C4—24no—
170C2×C4—24no—
171C2×C4—28no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—48no—
179C2×C4—44no—
180C2×C4—44no—
181C2×C4—44no—
182C2×C4C2×C22⋊C412yes—
183C2×C4C2×C22⋊C412yes—
184C2×C4C2×C22⋊C412yes—
185C2×C4C2×C22⋊C412yes—
186C2×C4C2×C22⋊C412yes—
187C2×C4#13514yes—
188C2×C4C23×C411yescharacteristic
189C2×C4C22×D411yescharacteristic
190C2×C4C2×#4114yes—
191C8—28no—
192C22—22no—
193C22—28no—
194C22—22no—
195C22—42no—
196C22—42no—
197C22—42no—
198C22—84no—
199C22#41614yes—
200C22C22×C22⋊C411yescharacteristic
201C22#53212yes—
202C22C2×#13512yes—
203C22C2×#13512yes—
204C4—22no—
205C4—22no—
206C4—22no—
207C4—22no—
208C4—42no—
209C4—42no—
210C4—42no—
211C4—84no—
212C4#41614yes—
213C4C4×#4112yes—
214C4#52512yes—
215C2—24no—
216C2—82no—
217C2#229711yescharacteristic
218C2C2×#41611yescharacteristic
219C2C2×#41611yescharacteristic
220C2#316814yes—
subgroupG/H
centerC22×C4C2×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
socleC23C22×D4
prime radicalC2×#121C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×C4×C8C2semidirect
3. #2283C2semidirect
4. #1991C2semidirect
5. #1958C2semidirect
7. #1892C2semidirect
8. C4×#103C2semidirect
9. C4×#102C2semidirect
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×C4×C82
#22832
#19912
#19582
#18932
#18922
C4×#1032
C4×#1022