Finite Group Databasebeta

← #11727

Subgroups — #11727

Order 256.

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

normal subgroups only

This group has 228 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#2395C211yesmaximal, characteristic
2C4×#99C211yesmaximal, characteristic
3#1460C211yesmaximal, characteristic
4#1352C212yesmaximal, complemented
5#1344C212yesmaximal, complemented
6C2×C4×D4C2211yescharacteristic
7C2×#117C411yescharacteristic, complemented
8#431—21no—
9C2×#99—21no—
10#400C411yescharacteristic, complemented
11#399C2211yescharacteristic
12#398C411yescharacteristic, complemented
13#397—21no—
14#397C2211yescharacteristic
15#394—21no—
16C4×C4⋊C4—21no—
17C4×C22⋊C4—21no—
18C4×C22⋊C4—21no—
19C4×C22⋊C4C411yescharacteristic, complemented
20#342C2212yes—
21#323—21no—
22#323C2212yes—
23C22×D4D411yescharacteristic
24C23×C4C2×C411yescharacteristic
25C23×C4D411yescharacteristic
26C4×D4—41no—
27C4×D4—41no—
28#117—41no—
29#117—41no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4C2×C411yescharacteristic
33C2×C4⋊C4C2×C411yescharacteristic
34C2×C4⋊C4C2×C411yescharacteristic
35C2×C22⋊C4—22no—
36C2×C22⋊C4—21no—
37C2×C22⋊C4—21no—
38C2×C22⋊C4—21no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4D411yescharacteristic
41C2×C22⋊C4C2311yescharacteristic
42C2×C42—21no—
43C2×C42—21no—
44C2×C42—41no—
45C2×C42D411yescharacteristic
46C4⋊C8—41no—
47#99—24no—
48C22⋊C8—22no—
49C4×C8—41no—
50#95—21no—
51#95—21no—
52#95—21no—
53#95—21no—
54#95—41no—
55#95C2×C411yescharacteristic
56#95C2×C411yescharacteristic
57C24C22⋊C411yescharacteristic
58C24C2×D411yescharacteristic
59C2×D4—22no—
60C2×D4—22no—
61C2×D4—41no—
62C22×C4—21no—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—42no—
71C22×C4—42no—
72C22×C4—42no—
73C22×C4C22⋊C411yescharacteristic
74C22×C4C22⋊C411yescharacteristic
75C22×C4C22⋊C411yescharacteristic
76C22×C4C22×C411yescharacteristic
77C22×C4C2×D411yescharacteristic
78C2×C8—42no—
79C4⋊C4—41no—
80C4⋊C4—41no—
81C4⋊C4—41no—
82C4⋊C4—42no—
83C4⋊C4—42no—
84C22⋊C4—24no—
85C22⋊C4—41no—
86C22⋊C4—41no—
87C22⋊C4—42no—
88C22⋊C4—42no—
89C22⋊C4—42no—
90C22⋊C4—42no—
91C22⋊C4—42no—
92C42—21no—
93C42—21no—
94C42—42no—
95C42—41no—
96C42—41no—
97C42—82no—
98C23—21no—
99C23—21no—
100C23—21no—
101C23—21no—
102C23—21no—
103C23—41no—
104C23—41no—
105C23—41no—
106C23—41no—
107C23C2×C22⋊C411yescharacteristic
108C23#13512yes—
109D4—44no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—22no—
113C2×C4—22no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—42no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—44no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—81no—
136C2×C4—82no—
137C2×C4—82no—
138C2×C4#9911yescharacteristic
139C2×C4#9911yescharacteristic
140C2×C4#10412yes—
141C8—82no—
142C22—21no—
143C22—21no—
144C22—22no—
145C22—42no—
146C22—41no—
147C22—41no—
148C22—41no—
149C22—41no—
150C22—41no—
151C22—81no—
152C22—81no—
153C22—81no—
154C22C2×#9911yescharacteristic
155C22#41611yescharacteristic
156C22C2×#10411yescharacteristic
157C4—21no—
158C4—22no—
159C4—42no—
160C4—41no—
161C4—41no—
162C4—41no—
163C4—81no—
164C4—84no—
165C4—81no—
166C4—81no—
167C4#35212yes—
168C2—21no—
169C2—21no—
170C2—42no—
171C2—81no—
172C2#150111yescharacteristic
173C2C2×#35211yescharacteristic
174C2#212611yescharacteristic
subgroupG/H
centerC2×C4#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
socleC22C2×#99
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1352C2semidirect
5. #1344C2semidirect
7. C2×#117C4semidirect
10. #400C4semidirect
12. #398C4semidirect
19. C4×C22⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#23952
C4×#992
#14602
#13522
#13442