Finite Group Databasebeta

← #18319

Subgroups — #18319

Order 256.

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

normal subgroups only

This group has 529 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×#543C211yesmaximal, characteristic
2#3195C212yesmaximal, complemented
3#2561C211yesmaximal, characteristic
4#1608C214yesmaximal, complemented
5#1602C214yesmaximal, complemented
6#1508C212yesmaximal, complemented
7#1462C211yesmaximal, characteristic
8#543—24no—
9#543—22no—
10#543C2214yes—
11C2×C22⋊Q8—22no—
12C2×C22⋊Q8—21no—
13C2×C22⋊Q8—22no—
14C2×C22⋊Q8C2212yes—
15C2×C22⋊Q8C2211yescharacteristic
16C22×C22⋊C4—21no—
17C22×C22⋊C4C2211yescharacteristic
18#483—22no—
19#483C2214yescomplemented
20#470—24no—
21#451—22no—
22#451C2214yescomplemented
23#407C2212yes—
24#397C2212yes—
25#393C2212yes—
26#388—22no—
27#381—21no—
28#379—21no—
29C4×C22⋊C4C2211yescharacteristic
30#328C2218yescomplemented
31#323C2214yes—
32C25D411yescharacteristic
33C22×Q8—21no—
34C22×Q8D412yes—
35C22×Q8C2311yescharacteristic
36C23×C4—21no—
37C23×C4—21no—
38C23×C4D411yescharacteristic
39C23×C4C2311yescharacteristic
40C2×Q16—24no—
41C2×M16—22no—
42C22⋊Q8—28no—
43C22⋊Q8—28no—
44C22⋊Q8—24no—
45C22⋊Q8—28no—
46C22⋊Q8—24no—
47C22⋊Q8—22no—
48C22⋊Q8—22no—
49C22⋊Q8—28no—
50C22⋊Q8—22no—
51C22⋊Q8—22no—
52C22⋊Q8D414yes—
53C22⋊Q8C2314yes—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4D412yes—
59C2×C4⋊C4C2311yescharacteristic
60C2×C22⋊C4—22no—
61C2×C22⋊C4—24no—
62C2×C22⋊C4—22no—
63C2×C22⋊C4—24no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4D412yes—
71C2×C22⋊C4C2312yes—
72C2×C42—21no—
73#106—24no—
74#103—24no—
75#103—24no—
76C22⋊C8—22no—
77C22⋊C8C2314yes—
78#95—22no—
79#95—22no—
80#95C2312yes—
81C24—21no—
82C24—21no—
83C24—24no—
84C24—42no—
85C24—81no—
86C24C2×D411yescharacteristic
87C24C2×D412yes—
88C2×Q8—24no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—81no—
92C2×Q8—82no—
93C2×Q8—82no—
94C2×Q8C2×D414yes—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—22no—
98C22×C4—24no—
99C22×C4—24no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—42no—
105C22×C4—81no—
106C22×C4—82no—
107C22×C4—82no—
108C22×C4C2×D412yes—
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D412yes—
111C22×C4C2×D412yes—
112C22×C4C2411yescharacteristic
113Q16—84no—
114M16—82no—
115C2×C8—24no—
116C4⋊C4—24no—
117C4⋊C4—28no—
118C4⋊C4—24no—
119C4⋊C4—22no—
120C4⋊C4—24no—
121C4⋊C4—22no—
122C4⋊C4—42no—
123C4⋊C4C2×D414yes—
124C22⋊C4—28no—
125C22⋊C4—24no—
126C22⋊C4—24no—
127C22⋊C4—24no—
128C22⋊C4—24no—
129C22⋊C4—42no—
130C22⋊C4—41no—
131C22⋊C4—41no—
132C22⋊C4—44no—
133C22⋊C4—82no—
134C42—81no—
135C23—24no—
136C23—24no—
137C23—22no—
138C23—22no—
139C23—22no—
140C23—22no—
141C23—41no—
142C23—42no—
143C23—42no—
144C23—42no—
145C23—42no—
146C23—81no—
147C23—81no—
148C23—88no—
149C23#12011yescharacteristic
150C23#12011yescharacteristic
151C23#13714yes—
152C23C22×D411yescharacteristic
153Q8—44no—
154Q8—44no—
155Q8—82no—
156Q8—84no—
157C2×C4—24no—
158C2×C4—22no—
159C2×C4—24no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4—44no—
165C2×C4—42no—
166C2×C4—44no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—81no—
176C2×C4—84no—
177C2×C4—84no—
178C2×C4#12012yes—
179C2×C4C22×D412yes—
180C8—44no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—24no—
187C22—21no—
188C22—21no—
189C22—44no—
190C22—44no—
191C22—48no—
192C22—42no—
193C22—42no—
194C22—82no—
195C22—84no—
196C22#45512yes—
197C22C2×#12011yescharacteristic
198C22C2×#13712yes—
199C4—22no—
200C4—22no—
201C4—42no—
202C4—44no—
203C4—44no—
204C4—82no—
205C4—81no—
206C4—81no—
207C2—22no—
208C2—22no—
209C2—41no—
210C2—44no—
211C2#300411yescharacteristic
212C2C2×#45511yescharacteristic
213C2#302611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC25D4
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
2. #3195C2semidirect
4. #1608C2semidirect
5. #1602C2semidirect
6. #1508C2semidirect
19. #483C22semidirect
22. #451C22semidirect
30. #328C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5432
#31952
#25612
#16082
#16022
#15082
#14622