Finite Group Databasebeta

← #9027

Subgroups — #9027

Order 256.

330 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 83 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×#398C211yesmaximal, characteristic
2#2316C211yesmaximal, characteristic
3#1823C212yesmaximal
4C2×#324C211yesmaximal, characteristic
5#1294C212yesmaximal
6C2×#117—21no—
7C22×C4⋊C4—21no—
8C22×C4⋊C4C2211yescharacteristic
9C22×C4⋊C4C2211yescharacteristic
10C2×C22⋊C8C2212yes—
11#398—22no—
12#398C414yescomplemented
13#389—24no—
14#380—21no—
15#380C2212yes—
16C2×#95—21no—
17C2×#95C2211yescharacteristic
18#340—24no—
19#324—22no—
20C23×C4—21no—
21C23×C4—21no—
22C23×C4D411yescharacteristic
23C23×C4D411yescharacteristic
24C23×C4C2311yescharacteristic
25C22×C8—22no—
26#117—42no—
27#117—42no—
28C2×C4⋊C4—22no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4—22no—
33C2×C4⋊C4—24no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—22no—
36C2×C4⋊C4—42no—
37C2×C4⋊C4—42no—
38C2×C4⋊C4C2×C412yes—
39C2×C4⋊C4C2×C412yes—
40C2×C4⋊C4D412yes—
41C2×C4⋊C4D414yes—
42C2×C22⋊C4—22no—
43C2×C42—22no—
44C22⋊C8—22no—
45C22⋊C8—22no—
46#95—22no—
47#95—24no—
48#95—21no—
49#95—22no—
50#95C2×C412yes—
51C24C2×D411yescharacteristic
52C22×C4—21no—
53C22×C4—21no—
54C22×C4—21no—
55C22×C4—21no—
56C22×C4—24no—
57C22×C4—22no—
58C22×C4—21no—
59C22×C4—21no—
60C22×C4—22no—
61C22×C4—24no—
62C22×C4—42no—
63C22×C4—42no—
64C22×C4—41no—
65C22×C4—41no—
66C22×C4—42no—
67C22×C4—41no—
68C22×C4—41no—
69C22×C4—41no—
70C22×C4—41no—
71C22×C4—41no—
72C22×C4—41no—
73C22×C4C22⋊C412yes—
74C22×C4C22⋊C412yes—
75C22×C4D812yes—
76C22×C4SD1612yes—
77C22×C4C22×C411yescharacteristic
78C22×C4C2×D412yes—
79C22×C4C2×D411yescharacteristic
80C22×C4#4112yes—
81C2×C8—44no—
82C2×C8—42no—
83C4⋊C4—24no—
84C4⋊C4—24no—
85C4⋊C4—24no—
86C4⋊C4—44no—
87C4⋊C4—44no—
88C4⋊C4—48no—
89C4⋊C4—42no—
90C4⋊C4—82no—
91C22⋊C4—42no—
92C22⋊C4—42no—
93C42—44no—
94C23—22no—
95C23—22no—
96C23C2×C22⋊C411yescharacteristic
97C23#12011yescharacteristic
98C23#12311yescharacteristic
99C23C2×D811yescharacteristic
100C23C2×SD1611yescharacteristic
101C23AGL(1,ℤ/8)11yescharacteristic
102C23#13711yescharacteristic
103C2×C4—22no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—22no—
107C2×C4—24no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—24no—
111C2×C4—22no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—42no—
115C2×C4—44no—
116C2×C4—44no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—44no—
125C2×C4—44no—
126C2×C4—44no—
127C2×C4—81no—
128C2×C4—81no—
129C2×C4—81no—
130C2×C4—81no—
131C2×C4—81no—
132C2×C4#10214yes—
133C2×C4C4×D412yes—
134C2×C4#12112yes—
135C8—82no—
136C22—22no—
137C22—22no—
138C22—22no—
139C22—22no—
140C22—21no—
141C22—21no—
142C22—21no—
143C22—21no—
144C22#38611yescharacteristic
145C22C2×#10211yescharacteristic
146C22#41711yescharacteristic
147C22#44711yescharacteristic
148C22#44811yescharacteristic
149C22#45612yes—
150C22#45812yes—
151C22#48011yescharacteristic
152C22#48111yescharacteristic
153C4—24no—
154C4—22no—
155C4—22no—
156C4—44no—
157C4—42no—
158C4—84no—
159C4—82no—
160C2—21no—
161C2—21no—
162C2—21no—
163C2—21no—
164C2#161012yes—
165C2#161112yes—
166C2#187411yescharacteristic
167C2#188611yescharacteristic
168C2#189011yescharacteristic
subgroupG/H
center, socleC23#120
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #398C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3982
#23162
#18232
C2×#3242
#12942