Finite Group Databasebeta

← #11563

Subgroups — #11563

Order 256.

308 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 53 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#2541C211yesmaximal, characteristic
2#1465C211yesmaximal, characteristic, complemented
3#1459C211yesmaximal, characteristic, complemented
4#1318C214yesmaximal, complemented
5C2×#127C411yescharacteristic
6C2×#127C412yes—
7C2×#121—21no—
8C2×#121—21no—
9C2×C4×D4C2211yescharacteristic
10#407—21no—
11C2×C22⋊C8—21no—
12#392—21no—
13#392—21no—
14#390C2212yes—
15C4×C4⋊C4C411yescharacteristic
16#324C2212yes—
17#324C2212yes—
18#323—21no—
19#323—21no—
20C22×D4—22no—
21C22×D4—22no—
22C22×D4—21no—
23C22×D4—21no—
24C22×D4C2×C411yescharacteristic
25C22×D4C2×C412yes—
26C23×C4D411yescharacteristic
27C23×C4D411yescharacteristic
28C2×M16—41no—
29C22×C8—41no—
30#127—41no—
31#127—41no—
32#127—44no—
33#121—44no—
34#121—44no—
35C4×D4—41no—
36C4×D4—41no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4C2311yescharacteristic
39C2×C22⋊C4—22no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4D411yescharacteristic
42C2×C22⋊C4D411yescharacteristic
43C2×C42C2×C411yescharacteristic
44C2×C42C2×C412yes—
45C22⋊C8—22no—
46C22⋊C8—22no—
47C22⋊C8—41no—
48C22⋊C8—41no—
49#95—21no—
50C24—22no—
51C24C22⋊C411yescharacteristic
52C24C22⋊C411yescharacteristic
53C2×D4—22no—
54C2×D4—22no—
55C2×D4—24no—
56C2×D4—41no—
57C2×D4—42no—
58C2×D4—42no—
59C2×D4—42no—
60C2×D4—42no—
61C2×D4—44no—
62C2×D4—44no—
63C2×D4—42no—
64C2×D4—44no—
65C2×D4—44no—
66C2×D4—41no—
67C2×D4—41no—
68C2×D4—82no—
69C2×D4—82no—
70C2×D4—82no—
71C22×C4—22no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—42no—
77C22×C4—42no—
78C22×C4C22⋊C411yescharacteristic
79C22×C4C22⋊C411yescharacteristic
80C22×C4C22×C411yescharacteristic
81C22×C4C2×D411yescharacteristic
82C22×C4C2×D411yescharacteristic
83M16—82no—
84C2×C8—42no—
85C2×C8—42no—
86C2×C8—82no—
87C4⋊C4—24no—
88C4⋊C4—41no—
89C22⋊C4—41no—
90C22⋊C4—41no—
91C22⋊C4—44no—
92C22⋊C4—44no—
93C42—21no—
94C42—21no—
95C42—42no—
96C23—21no—
97C23—21no—
98C23—21no—
99C23—21no—
100C23—21no—
101C23—41no—
102C23—41no—
103C23—41no—
104C23—41no—
105C23—42no—
106C23—42no—
107C23—42no—
108C23—42no—
109C23—42no—
110C23—42no—
111C23—42no—
112C23#10012yes—
113C23C2×C22⋊C411yescharacteristic
114D4—44no—
115D4—84no—
116D4—82no—
117D4—88no—
118D4—88no—
119D4—84no—
120D4—82no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—22no—
124C2×C4—24no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—81no—
136C2×C4—81no—
137C2×C4#9911yescharacteristic
138C2×C4#9911yescharacteristic
139C2×C4#13112yes—
140C8—82no—
141C8—82no—
142C22—21no—
143C22—21no—
144C22—22no—
145C22—42no—
146C22—42no—
147C22—41no—
148C22—41no—
149C22—41no—
150C22—41no—
151C22—41no—
152C22—81no—
153C22—81no—
154C22—82no—
155C22—82no—
156C22—82no—
157C22—82no—
158C22—82no—
159C22—82no—
160C22—82no—
161C22#40811yescharacteristic
162C22C2×#9911yescharacteristic
163C22C2×#10011yescharacteristic
164C4—21no—
165C4—22no—
166C4—44no—
167C4—41no—
168C4—41no—
169C4—41no—
170C4#35312yes—
171C2—21no—
172C2—21no—
173C2—42no—
174C2—81no—
175C2—162no—
176C2#147111yescharacteristic
177C2C2×#35311yescharacteristic
178C2#213411yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radical#2541C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1465C2semidirect
3. #1459C2semidirect
4. #1318C2semidirect

Maximal subgroups

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

subgroupindex
#25412
#14652
#14592
#13182