Finite Group Databasebeta

← #18327

Subgroups — #18327

Order 256.

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

normal subgroups only

This group has 475 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
2#3195C212yesmaximal
3#2583C211yesmaximal, characteristic
4#1611C214yesmaximal, complemented
5#1605C214yesmaximal, complemented
6#1508C212yesmaximal
7#1463C211yesmaximal, characteristic
8#544—24no—
9#544—22no—
10#544C2214yes—
11C2×C4⋊Q8—22no—
12C2×#125—22no—
13C2×C22⋊Q8—21no—
14C2×C22⋊Q8C2212yes—
15C2×C22⋊Q8C2211yescharacteristic
16C2×#117—21no—
17C22×C4⋊C4C2211yescharacteristic
18#483—22no—
19#483C2214yes—
20#470—24no—
21#451—22no—
22#451C2214yes—
23#407C2212yes—
24#398C2212yes—
25#395C2212yes—
26#389—22no—
27#385—21no—
28#385—21no—
29C4×C22⋊C4C2211yescharacteristic
30#328C2218yes—
31#324C2214yes—
32C22×Q8—21no—
33C22×Q8D412yes—
34C22×Q8C2311yescharacteristic
35C23×C4D411yescharacteristic
36C23×C4D411yescharacteristic
37C23×C4C2311yescharacteristic
38C2×Q16—24no—
39C2×M16—22no—
40C4⋊Q8—28no—
41C4⋊Q8—28no—
42#125—28no—
43#125—28no—
44C22⋊Q8—24no—
45C22⋊Q8—24no—
46C22⋊Q8—22no—
47C22⋊Q8—22no—
48C22⋊Q8—22no—
49C22⋊Q8—22no—
50C22⋊Q8D414yes—
51C22⋊Q8C2314yes—
52#117—24no—
53#117—24no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4D412yes—
65C2×C4⋊C4D412yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2312yes—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C42—22no—
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—
81C24C2×D411yescharacteristic
82C2×Q8—24no—
83C2×Q8—22no—
84C2×Q8—22no—
85C2×Q8—81no—
86C2×Q8—82no—
87C2×Q8—82no—
88C2×Q8C2×D414yes—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—24no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—81no—
100C22×C4—81no—
101C22×C4C2×D412yes—
102C22×C4C2×D412yes—
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D412yes—
105C22×C4C2×D412yes—
106C22×C4C2411yescharacteristic
107Q16—84no—
108M16—82no—
109C2×C8—24no—
110C4⋊C4—24no—
111C4⋊C4—24no—
112C4⋊C4—24no—
113C4⋊C4—28no—
114C4⋊C4—24no—
115C4⋊C4—24no—
116C4⋊C4—22no—
117C4⋊C4—24no—
118C4⋊C4—24no—
119C4⋊C4—22no—
120C4⋊C4—42no—
121C4⋊C4—44no—
122C4⋊C4—82no—
123C4⋊C4C2×D414yes—
124C22⋊C4—24no—
125C22⋊C4—24no—
126C22⋊C4—24no—
127C22⋊C4—41no—
128C22⋊C4—41no—
129C42—28no—
130C42—81no—
131C23—22no—
132C23—22no—
133C23—41no—
134C23#12011yescharacteristic
135C23#12011yescharacteristic
136C23C22×D411yescharacteristic
137Q8—44no—
138Q8—44no—
139Q8—82no—
140Q8—84no—
141C2×C4—24no—
142C2×C4—24no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—24no—
146C2×C4—24no—
147C2×C4—24no—
148C2×C4—22no—
149C2×C4—41no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—44no—
156C2×C4—42no—
157C2×C4—44no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—81no—
164C2×C4#12012yes—
165C2×C4#13714yes—
166C2×C4C22×D412yes—
167C8—44no—
168C22—21no—
169C22—21no—
170C22—21no—
171C22—21no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—44no—
176C22#45612yes—
177C22C2×#12011yescharacteristic
178C22C2×#13712yes—
179C4—24no—
180C4—22no—
181C4—22no—
182C4—44no—
183C4—44no—
184C4—42no—
185C4—44no—
186C4—81no—
187C4—81no—
188C2—22no—
189C2—22no—
190C2—41no—
191C2#300411yescharacteristic
192C2C2×#45611yescharacteristic
193C2#302611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
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
4. #1611C2semidirect
5. #1605C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5442
#31952
#25832
#16112
#16052
#15082
#14632