Finite Group Databasebeta

← #7645

Subgroups — #7645

Order 256.

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

normal subgroups only

This group has 315 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#2297C211yesmaximal, characteristic
2#1817C212yesmaximal, complemented
3#1272C212yesmaximal, complemented
4#1270C212yesmaximal, complemented
5C22×#41C411yescharacteristic
6C2×#117C411yescharacteristic
7C22×C42C2211yescharacteristic
8C2×C22⋊C8—22no—
9#391C411yescharacteristic
10#391C2211yescharacteristic
11#390C411yescharacteristic
12#390C2211yescharacteristic
13#386—22no—
14C4×C22⋊C4—22no—
15#336—22no—
16#336C2214yes—
17C2×#41—22no—
18C2×#41—22no—
19C2×#41—42no—
20C2×#41—42no—
21C22×Q8C2×C411yescharacteristic
22C22×D4—21no—
23C22×D4C2×C411yescharacteristic
24C23×C4—21no—
25C23×C4C2×C411yescharacteristic
26C23×C4D412yes—
27C22×C8—24no—
28#117—42no—
29C2×C4⋊C4C2×C411yescharacteristic
30C2×C4⋊C4C2×C411yescharacteristic
31C2×C22⋊C4—21no—
32C2×C22⋊C4—22no—
33C2×C22⋊C4—22no—
34C2×C42—21no—
35C2×C42—42no—
36C2×C42C2×C411yescharacteristic
37C2×C42D412yes—
38C2×C42C2311yescharacteristic
39C22⋊C8—44no—
40#95—22no—
41#95—22no—
42C24—21no—
43C24C22⋊C411yescharacteristic
44#41—48no—
45#41—48no—
46C2×Q8—21no—
47C2×Q8—41no—
48C2×Q8—41no—
49C2×Q8M1612yes—
50C2×D4—24no—
51C2×D4—21no—
52C2×D4—41no—
53C2×D4—41no—
54C2×D4—42no—
55C2×D4—42no—
56C2×D4M1612yes—
57C22×C4—22no—
58C22×C4—22no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—24no—
62C22×C4—24no—
63C22×C4—21no—
64C22×C4—24no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—42no—
68C22×C4—42no—
69C22×C4—42no—
70C22×C4C22⋊C412yes—
71C22×C4C22⋊C411yescharacteristic
72C22×C4C22×C411yescharacteristic
73C22×C4C2×D412yes—
74C2×C8—44no—
75C2×C8—44no—
76C2×C8—44no—
77C4⋊C4—41no—
78C4⋊C4—41no—
79C22⋊C4—44no—
80C22⋊C4—42no—
81C22⋊C4—44no—
82C42—41no—
83C42—42no—
84C42—41no—
85C42—42no—
86C23—21no—
87C23—21no—
88C23—22no—
89C23—22no—
90C23—22no—
91C23—41no—
92C23—41no—
93C23—42no—
94C23—42no—
95C23#9912yes—
96C23C2×C22⋊C411yescharacteristic
97Q8—42no—
98Q8—42no—
99D4—42no—
100D4—42no—
101D4—48no—
102C2×C4—24no—
103C2×C4—24no—
104C2×C4—21no—
105C2×C4—22no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—22no—
113C2×C4—22no—
114C2×C4—48no—
115C2×C4—48no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4#10012yes—
123C2×C4C2×M1611yescharacteristic
124C2×C4C2×M1611yescharacteristic
125C2×C4#13514yes—
126C8—84no—
127C22—22no—
128C22—24no—
129C22—41no—
130C22—41no—
131C22—42no—
132C22—42no—
133C22—44no—
134C22—81no—
135C22—81no—
136C22—82no—
137C22#40711yescharacteristic
138C22C2×#9911yescharacteristic
139C22C2×#10011yescharacteristic
140C22#41612yes—
141C22#42112yes—
142C4—22no—
143C4—21no—
144C4—21no—
145C4—24no—
146C4—42no—
147C4—42no—
148C4—81no—
149C4—81no—
150C2—42no—
151C2—82no—
152C2#146211yescharacteristic
153C2#148812yes—
154C2#151612yes—
155C2#153611yescharacteristic
156C2#154311yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×#41C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1817C2semidirect
3. #1272C2semidirect
4. #1270C2semidirect

Maximal subgroups

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

subgroupindex
#22972
#18172
#12722
#12702