Finite Group Databasebeta

← #18326

Subgroups — #18326

Order 256.

488 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 117 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×#544C211yesmaximal, characteristic, complemented
2#3193C212yesmaximal
3#2591C211yesmaximal, characteristic
4#1617C214yesmaximal, complemented
5#1599C214yesmaximal, complemented
6#1508C212yesmaximal, complemented
7#1463C211yesmaximal, characteristic, complemented
8#544—24no—
9#544—22no—
10#544C2214yes—
11C2×C4⋊Q8—22no—
12C2×#125—22no—
13C2×C22⋊Q8—21no—
14C2×C22⋊Q8C2212yes—
15C2×#121C2211yescharacteristic
16C2×#117—21no—
17C22×C4⋊C4C2211yescharacteristic
18#484C2214yes—
19#480—22no—
20#468—24no—
21#450C2214yes—
22#448—22no—
23#407C2212yes—
24#400C2212yes—
25#394C2212yes—
26#388—22no—
27#386—21no—
28#386—21no—
29C4×C22⋊C4C2211yescharacteristic
30#328C2218yes—
31#324C2214yes—
32C22×Q8—21no—
33C22×Q8D412yes—
34C22×D4—21no—
35C22×D4C2311yescharacteristic
36C23×C4D411yescharacteristic
37C23×C4D411yescharacteristic
38C23×C4C2311yescharacteristic
39C2×SD16—24no—
40C2×M16—22no—
41C4⋊Q8—28no—
42C4⋊Q8—28no—
43#125—28no—
44#125—28no—
45C22⋊Q8—24no—
46C22⋊Q8—24no—
47C22⋊Q8—22no—
48C22⋊Q8—22no—
49C22⋊Q8D414yes—
50C22⋊Q8C2314yes—
51#121—22no—
52#121—22no—
53#117—24no—
54#117—24no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4D412yes—
63C2×C4⋊C4D412yes—
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2312yes—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C42—22no—
72C2×C42—21no—
73#107—24no—
74#103—24no—
75#102—24no—
76C22⋊C8—22no—
77C22⋊C8C2314yes—
78#95—22no—
79#95—22no—
80#95C2312yes—
81C24—21no—
82C24C2×D411yescharacteristic
83C2×Q8—24no—
84C2×Q8—22no—
85C2×Q8—82no—
86C2×Q8—82no—
87C2×Q8C2×D414yes—
88C2×D4—22no—
89C2×D4—44no—
90C2×D4—41no—
91C2×D4—41no—
92C2×D4—81no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—24no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—81no—
103C22×C4—81no—
104C22×C4C2×D412yes—
105C22×C4C2×D412yes—
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D412yes—
108C22×C4C2×D412yes—
109C22×C4C2411yescharacteristic
110SD16—84no—
111M16—82no—
112C2×C8—24no—
113C4⋊C4—24no—
114C4⋊C4—24no—
115C4⋊C4—24no—
116C4⋊C4—28no—
117C4⋊C4—24no—
118C4⋊C4—24no—
119C4⋊C4—22no—
120C4⋊C4—24no—
121C4⋊C4—24no—
122C4⋊C4—22no—
123C4⋊C4C2×D414yes—
124C22⋊C4—24no—
125C22⋊C4—24no—
126C22⋊C4—42no—
127C22⋊C4—42no—
128C22⋊C4—44no—
129C22⋊C4—82no—
130C42—28no—
131C42—81no—
132C23—22no—
133C23—22no—
134C23—41no—
135C23—42no—
136C23—42no—
137C23—41no—
138C23—42no—
139C23#12011yescharacteristic
140C23#12011yescharacteristic
141C23C22×D411yescharacteristic
142Q8—44no—
143Q8—82no—
144Q8—84no—
145D4—44no—
146D4—82no—
147D4—82no—
148C2×C4—24no—
149C2×C4—24no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—24no—
153C2×C4—24no—
154C2×C4—22no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—44no—
161C2×C4—44no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—81no—
169C2×C4#12012yes—
170C2×C4#13714yes—
171C2×C4C22×D412yes—
172C8—44no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—21no—
179C22—21no—
180C22—44no—
181C22—84no—
182C22—81no—
183C22—81no—
184C22—81no—
185C22#45412yes—
186C22C2×#12011yescharacteristic
187C22C2×#13712yes—
188C4—24no—
189C4—22no—
190C4—22no—
191C4—44no—
192C4—42no—
193C4—44no—
194C4—82no—
195C4—81no—
196C2—22no—
197C2—22no—
198C2—41no—
199C2—161no—
200C2#300411yescharacteristic
201C2C2×#45411yescharacteristic
202C2#302611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#544C2semidirect
4. #1617C2semidirect
5. #1599C2semidirect
6. #1508C2semidirect
7. #1463C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5442
#31932
#25912
#16172
#15992
#15082
#14632