Finite Group Databasebeta

← #8720

Subgroups — #8720

Order 256.

241 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 241 conjugacy classes of subgroups — too many to draw. Only its 67 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
1#2283C211yesmaximal, characteristic
2#1749C211yesmaximal, characteristic
3#1749C211yesmaximal, characteristic
4#1295C212yesmaximal, complemented
5#1295C212yesmaximal, complemented
6C22×C42C2211yescharacteristic
7C2×C22⋊C8—21no—
8C2×C22⋊C8—21no—
9C2×#97C2212yes—
10C2×#97C2212yes—
11#383—21no—
12#382C2211yescharacteristic
13#382C2211yescharacteristic
14#381—21no—
15#380C411yescharacteristic
16#380C411yescharacteristic
17C4×C4⋊C4C411yescharacteristic
18C4×C4⋊C4C411yescharacteristic
19C4×C22⋊C4—21no—
20#376—21no—
21#336—21no—
22#336—21no—
23C23×C4C2×C411yescharacteristic
24C23×C4D411yescharacteristic
25C23×C4D411yescharacteristic
26C22×C8—22no—
27C22×C8—22no—
28C2×C4⋊C4C2×C411yescharacteristic
29C2×C4⋊C4C2×C411yescharacteristic
30C2×C4⋊C4C2×C411yescharacteristic
31C2×C4⋊C4C2×C411yescharacteristic
32C2×C22⋊C4—21no—
33C2×C22⋊C4—21no—
34C2×C42—24no—
35C2×C42—21no—
36C2×C42—21no—
37C2×C42C2×C411yescharacteristic
38C2×C42D411yescharacteristic
39C2×C42D411yescharacteristic
40C2×C42C2311yescharacteristic
41C22⋊C8—24no—
42C22⋊C8—24no—
43#97—24no—
44#97—24no—
45#95—21no—
46#95—21no—
47#95—21no—
48#95—21no—
49#95—21no—
50#95—21no—
51#95—21no—
52#95—21no—
53C24C22⋊C411yescharacteristic
54C22×C4—22no—
55C22×C4—22no—
56C22×C4—22no—
57C22×C4—22no—
58C22×C4—22no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—41no—
64C22×C4—41no—
65C22×C4—41no—
66C22×C4—41no—
67C22×C4C22⋊C411yescharacteristic
68C22×C4C22⋊C411yescharacteristic
69C22×C4C22⋊C411yescharacteristic
70C22×C4C22×C411yescharacteristic
71C22×C4C2×D411yescharacteristic
72C22×C4C2×D411yescharacteristic
73C2×C8—24no—
74C2×C8—24no—
75C2×C8—42no—
76C2×C8—42no—
77C2×C8—42no—
78C2×C8—42no—
79C4⋊C4—41no—
80C4⋊C4—41no—
81C4⋊C4—41no—
82C4⋊C4—41no—
83C22⋊C4—42no—
84C22⋊C4—42no—
85C42—22no—
86C42—22no—
87C42—22no—
88C42—42no—
89C42—42no—
90C42#4114yes—
91C23—21no—
92C23—21no—
93C23—21no—
94C23—21no—
95C23—21no—
96C23—21no—
97C23#10412yes—
98C23C2×C22⋊C411yescharacteristic
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—44no—
114C2×C4—44no—
115C2×C4—44no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4#10412yes—
129C2×C4#11712yes—
130C2×C4#12312yes—
131C2×C4#12312yes—
132C8—44no—
133C8—44no—
134C22—22no—
135C22—22no—
136C22—22no—
137C22—41no—
138C22—41no—
139C22—41no—
140C22—41no—
141C22#38111yescharacteristic
142C22#41311yescharacteristic
143C22#41311yescharacteristic
144C22C2×#10411yescharacteristic
145C22C2×#10411yescharacteristic
146C22#42111yescharacteristic
147C22#42111yescharacteristic
148C4—22no—
149C4—22no—
150C4—22no—
151C4—22no—
152C4—41no—
153C4—41no—
154C4—81no—
155C4—81no—
156C4—81no—
157C4—81no—
158C2—42no—
159C2#152411yescharacteristic
160C2#152911yescharacteristic
161C2#153211yescharacteristic
162C2#153511yescharacteristic
163C2#182211yescharacteristic
164C2#182611yescharacteristic
165C2#182611yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1295C2semidirect
5. #1295C2semidirect

Maximal subgroups

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

subgroupindex
#22832
#17492
#12952