Finite Group Databasebeta

← #22250

Subgroups — #22250

Order 256.

363 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 163 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×#501C211yesmaximal, characteristic
2C4×C4⋊Q8C211yesmaximal, characteristic
3C4×#125C211yesmaximal, characteristic
4#1993C211yesmaximal, characteristic
5#1992C211yesmaximal, characteristic
6#1921C212yesmaximal
7#1919C212yesmaximal
8#1919C212yesmaximal
9#1918C212yesmaximal
10#1836C211yesmaximal, characteristic
11#1745C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×#125C2211yescharacteristic
14C2×C4×Q8—21no—
15#501C418yescomplemented
16C2×#107C2212yes—
17C2×#106C2212yes—
18C2×#97C2211yescharacteristic
19C2×#97C2211yescharacteristic
20#391—21no—
21#389—21no—
22#389C2211yescharacteristic
23#389C2211yescharacteristic
24#389C2212yes—
25#387—21no—
26#387—21no—
27#384C2211yescharacteristic
28#382C2211yescharacteristic
29C4×C4⋊C4—21no—
30C4×C4⋊C4—21no—
31C4×C4⋊C4—21no—
32C4×C4⋊C4C2211yescharacteristic
33C4×C4⋊C4C2211yescharacteristic
34C4×C4⋊C4C2212yes—
35C43C2211yescharacteristic
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43#336C2212yes—
44C22×Q8—21no—
45C22×C8C2312yes—
46C22×C8C2312yes—
47C4⋊Q8—22no—
48C4⋊Q8C2×C414yes—
49#125—22no—
50#125C2×C414yes—
51C4×Q8—44no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2312yes—
66C2×C42—22no—
67C2×C42—21no—
68C2×C42—21no—
69C2×C42D411yescharacteristic
70C2×C42D411yescharacteristic
71C2×C42D411yescharacteristic
72C2×C42D411yescharacteristic
73C2×C42C2311yescharacteristic
74C2×C42C2311yescharacteristic
75C2×C42C2311yescharacteristic
76#107C2×C418yes—
77#106C2×C418yes—
78#97—22no—
79#97C2×C414yes—
80#95—21no—
81#95—21no—
82#95—21no—
83#95—21no—
84#95—22no—
85#95—21no—
86#95—21no—
87#95—22no—
88C2×Q8—24no—
89C2×Q8—42no—
90C2×Q8—42no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—22no—
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4C2411yescharacteristic
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—22no—
108C2×C8—22no—
109C2×C8C22×C414yes—
110C4⋊C4—24no—
111C4⋊C4—24no—
112C4⋊C4—24no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—24no—
119C4⋊C4—42no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C4⋊C4—44no—
123C4⋊C4—42no—
124C4⋊C4C22×C412yes—
125C4⋊C4C22×C414yes—
126C4⋊C4C22×C412yes—
127C42—22no—
128C42—22no—
129C42—22no—
130C42—22no—
131C42—41no—
132C42—41no—
133C42—42no—
134C42—44no—
135C42—42no—
136C42C22×C412yes—
137C42#4112yes—
138C23C22×D411yescharacteristic
139Q8—48no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—24no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—42no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4C4×D412yes—
173C2×C4C4×D412yes—
174C2×C4AGL(1,ℤ/8)12yes—
175C2×C4#13712yes—
176C2×C4C23×C411yescharacteristic
177C2×C4C2×#4111yescharacteristic
178C2×C42−1+412yes—
179C2×C42−1+412yes—
180C8—24no—
181C8—42no—
182C22C2×C4×D411yescharacteristic
183C22#53611yescharacteristic
184C22#53611yescharacteristic
185C22C2×AGL(1,ℤ/8)11yescharacteristic
186C22C2×#13711yescharacteristic
187C22#57511yescharacteristic
188C22#57511yescharacteristic
189C4—22no—
190C4—22no—
191C4—22no—
192C4—22no—
193C4—42no—
194C4—44no—
195C4—42no—
196C4—41no—
197C4—41no—
198C4—82no—
199C4—81no—
200C4—81no—
201C4#51912yes—
202C4#54112yes—
203C2#232711yescharacteristic
204C2C4×AGL(1,ℤ/8)11yescharacteristic
205C2C4×#13711yescharacteristic
206C2#323411yescharacteristic
207C2#323711yescharacteristic
208C2#323911yescharacteristic
209C2#324011yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #501C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5012
C4×C4⋊Q82
C4×#1252
#19932
#19922
#19212
#19192
#19182
#18362
#17452