Finite Group Databasebeta

← #21120

Subgroups — #21120

Order 256.

467 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 467 conjugacy classes of subgroups — too many to draw. Only its 183 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×C4×SD16C211yesmaximal, characteristic
2#2293C211yesmaximal, characteristic
3C42×Q8C211yesmaximal, characteristic, complemented
4#1925C212yesmaximal
5#1920C212yesmaximal, complemented
6#1880C212yesmaximal
7#1863C212yesmaximal, complemented
8C4×#107C211yesmaximal, characteristic, complemented
9C4×#103C211yesmaximal, characteristic, complemented
10C4×#102C211yesmaximal, characteristic
11#1744C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13C2×C4×Q8C2212yes—
14C2×C4×Q8C2211yescharacteristic
15C2×C4×D4C2211yescharacteristic
16C2×#117—21no—
17C4×SD16C418yescomplemented
18C2×#107C2212yes—
19C2×#106C2211yescharacteristic
20C2×#103C2212yes—
21C2×#103C2211yescharacteristic
22C2×#102C2212yes—
23C2×#102C2211yescharacteristic
24C2×C4×C8C2211yescharacteristic
25C2×C4×C8C2211yescharacteristic
26#390C2212yes—
27#389C2212yes—
28#388—22no—
29C4×C4⋊C4—22no—
30C4×C4⋊C4—21no—
31C4×C4⋊C4C2212yes—
32C4×C4⋊C4C2211yescharacteristic
33C4×C4⋊C4C2211yescharacteristic
34C4×C22⋊C4—21no—
35C43—21no—
36C43C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#336C2212yes—
41#336C2212yes—
42C22×Q8C2311yescharacteristic
43C22×D4C2311yescharacteristic
44C23×C4—21no—
45C2×SD16—24no—
46C2×SD16C2×C414yes—
47C22×C8C2312yes—
48C22×C8C2311yescharacteristic
49C22×C8C2311yescharacteristic
50C4×Q8—24no—
51C4×Q8—24no—
52C4×Q8—22no—
53C4×Q8C2×C414yes—
54C4×D4—22no—
55C4×D4C2×C414yes—
56#117—44no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2312yes—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—22no—
67C2×C42—22no—
68C2×C42—21no—
69C2×C42—22no—
70C2×C42—21no—
71C2×C42—21no—
72C2×C42D412yes—
73C2×C42D411yescharacteristic
74C2×C42D411yescharacteristic
75C2×C42C2312yes—
76C2×C42C2311yescharacteristic
77#107—24no—
78#106C2×C414yes—
79#103—24no—
80#103C2×C414yes—
81#102—24no—
82#102C2×C414yes—
83C4×C8—22no—
84C4×C8C2×C414yes—
85#95—22no—
86#95—22no—
87#95—21no—
88#95—21no—
89C24—21no—
90C2×Q8—21no—
91C2×Q8—22no—
92C2×Q8C22×C412yes—
93C2×Q8#4114yes—
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—21no—
97C2×D4C22×C412yes—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—24no—
103C22×C4—22no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4C2×D412yes—
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D412yes—
109C22×C4C2×D411yescharacteristic
110C22×C4C2411yescharacteristic
111SD16—28no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8C22×C412yes—
118C2×C8C22×C412yes—
119C4⋊C4—28no—
120C4⋊C4—24no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—44no—
127C4⋊C4C22×C412yes—
128C4⋊C4C22×C412yes—
129C4⋊C4#4114yes—
130C22⋊C4—24no—
131C22⋊C4—44no—
132C22⋊C4—44no—
133C42—28no—
134C42—24no—
135C42—24no—
136C42—22no—
137C42—22no—
138C42—22no—
139C42—22no—
140C42—44no—
141C42—42no—
142C42—42no—
143C42C22×C412yes—
144C42#4112yes—
145C23—22no—
146C23—42no—
147C23—42no—
148C23—41no—
149C23—41no—
150C23C22×D411yescharacteristic
151Q8—22no—
152Q8—24no—
153Q8#11714yescomplemented
154D4—22no—
155D4—22no—
156D4—42no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—24no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—24no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—48no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4C4×D412yes—
178C2×C4C4×D412yes—
179C2×C4#13514yes—
180C2×C4AGL(1,ℤ/8)12yes—
181C2×C4#13712yes—
182C2×C4C23×C411yescharacteristic
183C2×C4C2×#4112yes—
184C2×C4C2×#4112yes—
185C2×C4C2×#4111yescharacteristic
186C8—22no—
187C8—22no—
188C8—42no—
189C22—42no—
190C22—42no—
191C22—42no—
192C22—82no—
193C22—82no—
194C22C2×C4×D411yescharacteristic
195C22#52512yes—
196C22C2×#13512yes—
197C22C2×AGL(1,ℤ/8)11yescharacteristic
198C22C2×#13711yescharacteristic
199C4—24no—
200C4—22no—
201C4—24no—
202C4—42no—
203C4—44no—
204C4—42no—
205C4—42no—
206C4—82no—
207C4#44014yes—
208C4C2×#11711yescharacteristic
209C4C4×#4111yescharacteristic
210C4#52912yes—
211C2—82no—
212C2#229011yescharacteristic
213C2C4×#13511yescharacteristic
214C2C2×#44011yescharacteristic
215C2#310612yes—
216C2#310812yes—
subgroupG/H
centerC22×C4C2×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
socleC23C22×D4
prime radicalC22×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C42×Q8C2semidirect
5. #1920C2semidirect
7. #1863C2semidirect
8. C4×#107C2semidirect
9. C4×#103C2semidirect
11. #1744C2semidirect
17. C4×SD16C4semidirect
153. Q8#117semidirect

Maximal subgroups

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

subgroupindex
C2×C4×SD162
#22932
C42×Q82
#19252
#19202
#18802
#18632
C4×#1072
C4×#1032
C4×#1022
#17442