Finite Group Databasebeta

← #8776

Subgroups — #8776

Order 256.

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

normal subgroups only

This group has 240 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×#99C211yesmaximal, characteristic
3#1465C211yesmaximal, characteristic
4#1297C212yesmaximal, complemented
5#1284C212yesmaximal, complemented
6C2×C4×D4C2211yescharacteristic
7C2×#99—21no—
8#407C2212yes—
9#388—21no—
10#387—21no—
11#386—21no—
12#383C411yescharacteristic, complemented
13#381C2211yescharacteristic
14#380C411yescharacteristic, complemented
15C4×C22⋊C4—21no—
16C4×C22⋊C4C2212yes—
17C4×C22⋊C4C2211yescharacteristic
18#376C411yescharacteristic, complemented
19C2×#95C411yescharacteristic, complemented
20#342—21no—
21#324—21no—
22#323—21no—
23C22×D4D411yescharacteristic
24C23×C4—21no—
25C23×C4C2×C411yescharacteristic
26C23×C4C2311yescharacteristic
27C2×M16—22no—
28C4×D4—22no—
29C4×D4—22no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4D411yescharacteristic
32C2×C22⋊C4—22no—
33C2×C22⋊C4—21no—
34C2×C22⋊C4—21no—
35C2×C22⋊C4—21no—
36C2×C22⋊C4D411yescharacteristic
37C2×C22⋊C4D411yescharacteristic
38C2×C42—21no—
39C2×C42—21no—
40C2×C42—22no—
41C2×C42C2×C411yescharacteristic
42#99—24no—
43C22⋊C8—22no—
44C22⋊C8—22no—
45#95—21no—
46#95—21no—
47#95—21no—
48#95—21no—
49#95—21no—
50#95—22no—
51#95C2×C411yescharacteristic
52#95C2×C411yescharacteristic
53#95C2×C411yescharacteristic
54#95C2×C411yescharacteristic
55C24C22⋊C411yescharacteristic
56C24C2×D411yescharacteristic
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—42no—
67C22×C4—42no—
68C22×C4—42no—
69C22×C4—81no—
70C22×C4—81no—
71C22×C4—81no—
72C22×C4C22⋊C411yescharacteristic
73C22×C4C22⋊C411yescharacteristic
74C22×C4C22⋊C411yescharacteristic
75C22×C4C22×C411yescharacteristic
76C22×C4C2×D411yescharacteristic
77C22×C4#4112yes—
78C22×C4#4112yes—
79M16—82no—
80C2×C8—42no—
81C4⋊C4—22no—
82C4⋊C4—81no—
83C22⋊C4—22no—
84C22⋊C4—22no—
85C22⋊C4—24no—
86C22⋊C4—42no—
87C22⋊C4—42no—
88C22⋊C4—42no—
89C22⋊C4—81no—
90C42—21no—
91C42—21no—
92C42—42no—
93C42—81no—
94C42—82no—
95C23—21no—
96C23—21no—
97C23—21no—
98C23—21no—
99C23—41no—
100C23—41no—
101C23—41no—
102C23—41no—
103C23#10412yes—
104C23C2×C22⋊C411yescharacteristic
105C23#12312yes—
106D4—44no—
107C2×C4—22no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—41no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—44no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—81no—
132C2×C4—81no—
133C2×C4—84no—
134C2×C4—82no—
135C2×C4—82no—
136C2×C4#11711yescharacteristic
137C2×C4#11711yescharacteristic
138C2×C4#12312yes—
139C8—82no—
140C22—21no—
141C22—21no—
142C22—22no—
143C22—22no—
144C22—22no—
145C22—22no—
146C22—42no—
147C22—42no—
148C22—41no—
149C22—41no—
150C22—81no—
151C22—81no—
152C22#38111yescharacteristic
153C22C2×#10411yescharacteristic
154C22#42111yescharacteristic
155C4—22no—
156C4—21no—
157C4—21no—
158C4—42no—
159C4—41no—
160C4—41no—
161C4—41no—
162C4—81no—
163C4—81no—
164C4—81no—
165C4—84no—
166C2—21no—
167C2—21no—
168C2—42no—
169C2—42no—
170C2#182611yescharacteristic
171C2#182711yescharacteristic
172C2#182811yescharacteristic
subgroupG/H
center, socleC22#381
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1297C2semidirect
5. #1284C2semidirect
12. #383C4semidirect
14. #380C4semidirect
18. #376C4semidirect
19. C2×#95C4semidirect

Maximal subgroups

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

subgroupindex
#23532
C4×#992
#14652
#12972
#12842