Finite Group Databasebeta

← #8772

Subgroups — #8772

Order 256.

223 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 223 conjugacy classes of subgroups — too many to draw. Only its 57 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#2353C211yesmaximal, characteristic
2C4×#100C211yesmaximal, characteristic
3#1466C211yesmaximal, characteristic
4#1300C212yesmaximal, complemented
5#1288C212yesmaximal, complemented
6C2×C4×D4C2211yescharacteristic
7#432C2212yes—
8C2×#100—21no—
9C4×M16C2212yes—
10#388—21no—
11#387—21no—
12#386—21no—
13#383C2211yescharacteristic
14#381C411yescharacteristic
15#380C411yescharacteristic
16C4×C22⋊C4—21no—
17C4×C22⋊C4C411yescharacteristic
18#376C2211yescharacteristic
19C2×#95C411yescharacteristic
20#343—21no—
21#323—21no—
22#323—21no—
23C22×D4D411yescharacteristic
24C23×C4—21no—
25C23×C4C2×C411yescharacteristic
26C23×C4C2×C411yescharacteristic
27C2×M16—22no—
28C4×D4—22no—
29C4×D4—22no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4D411yescharacteristic
32C2×C22⋊C4—21no—
33C2×C22⋊C4—21no—
34C2×C22⋊C4—21no—
35C2×C22⋊C4D411yescharacteristic
36C2×C22⋊C4D411yescharacteristic
37C2×C42—21no—
38C2×C42—21no—
39C2×C42C2311yescharacteristic
40C4⋊C8—22no—
41#100—24no—
42C22⋊C8—22no—
43#97—22no—
44#97—22no—
45C4×C8—22no—
46#95—21no—
47#95—21no—
48#95—21no—
49#95—21no—
50#95—21no—
51#95C2×C411yescharacteristic
52#95C2×C411yescharacteristic
53#95C2×C411yescharacteristic
54#95C2×C411yescharacteristic
55C24C22⋊C411yescharacteristic
56C24C22⋊C411yescharacteristic
57C2×D4—22no—
58C2×D4—22no—
59C2×D4—41no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—42no—
69C22×C4—42no—
70C22×C4—81no—
71C22×C4—81no—
72C22×C4—81no—
73C22×C4C22⋊C411yescharacteristic
74C22×C4C22⋊C411yescharacteristic
75C22×C4C22×C411yescharacteristic
76C22×C4C2×D411yescharacteristic
77C22×C4C2×D411yescharacteristic
78M16—24no—
79M16—42no—
80C2×C8—42no—
81C2×C8—42no—
82C4⋊C4—22no—
83C4⋊C4—81no—
84C22⋊C4—22no—
85C22⋊C4—22no—
86C22⋊C4—42no—
87C22⋊C4—42no—
88C22⋊C4—81no—
89C42—42no—
90C42—81no—
91C42#4112yes—
92C42#4112yes—
93C23—21no—
94C23—21no—
95C23—21no—
96C23—21no—
97C23—21no—
98C23—41no—
99C23—41no—
100C23—41no—
101C23—41no—
102C23#10412yes—
103C23C2×C22⋊C411yescharacteristic
104D4—44no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—22no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—41no—
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—81no—
126C2×C4—81no—
127C2×C4—81no—
128C2×C4—81no—
129C2×C4—84no—
130C2×C4#11711yescharacteristic
131C2×C4#11711yescharacteristic
132C2×C4#12312yes—
133C2×C4#12312yes—
134C8—44no—
135C8—82no—
136C22—21no—
137C22—21no—
138C22—22no—
139C22—42no—
140C22—42no—
141C22—41no—
142C22—41no—
143C22—41no—
144C22—41no—
145C22—41no—
146C22—81no—
147C22—81no—
148C22#38111yescharacteristic
149C22C2×#10411yescharacteristic
150C22#42111yescharacteristic
151C4—22no—
152C4—22no—
153C4—21no—
154C4—21no—
155C4—42no—
156C4—41no—
157C4—41no—
158C4—81no—
159C4—81no—
160C4—81no—
161C2—21no—
162C2—21no—
163C2—42no—
164C2—81no—
165C2#182611yescharacteristic
166C2#182711yescharacteristic
167C2#182911yescharacteristic
subgroupG/H
center, socleC22#381
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1300C2semidirect
5. #1288C2semidirect

Maximal subgroups

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

subgroupindex
#23532
C4×#1002
#14662
#13002
#12882