Finite Group Databasebeta

← #11570

Subgroups — #11570

Order 256.

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

normal subgroups only

This group has 240 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#2542C211yesmaximal, characteristic
2#1465C211yesmaximal, characteristic
3#1459C211yesmaximal, characteristic
4#1325C214yesmaximal, complemented
5C2×C4⋊Q8C411yescharacteristic
6C2×C22⋊Q8—21no—
7C2×C22⋊Q8—21no—
8C2×C4×D4C2211yescharacteristic
9#407—21no—
10C2×C22⋊C8—21no—
11#399—21no—
12#399—21no—
13#390C412yes—
14#384C2212yes—
15C4×C4⋊C4C411yescharacteristic
16#324C2212yes—
17#324C2212yes—
18#323—21no—
19#323—21no—
20C22×Q8—21no—
21C22×D4C2×C411yescharacteristic
22C23×C4D411yescharacteristic
23C23×C4D411yescharacteristic
24C2×M16—41no—
25C22×C8—41no—
26C4⋊Q8—41no—
27C4⋊Q8—41no—
28C22⋊Q8—44no—
29C22⋊Q8—44no—
30C4×D4—41no—
31C4×D4—41no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—22no—
35C2×C4⋊C4—22no—
36C2×C4⋊C4C2×C412yes—
37C2×C4⋊C4C2311yescharacteristic
38C2×C22⋊C4—22no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4D411yescharacteristic
41C2×C22⋊C4D411yescharacteristic
42C2×C42C2×C412yes—
43C2×C42C2×C411yescharacteristic
44C22⋊C8—22no—
45C22⋊C8—22no—
46C22⋊C8—41no—
47C22⋊C8—41no—
48#95—21no—
49C24C22⋊C411yescharacteristic
50C24C22⋊C411yescharacteristic
51C2×Q8—42no—
52C2×Q8—42no—
53C2×Q8—41no—
54C2×Q8—41no—
55C2×D4—22no—
56C2×D4—22no—
57C2×D4—41no—
58C22×C4—22no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—22no—
64C22×C4—42no—
65C22×C4—42no—
66C22×C4C22⋊C411yescharacteristic
67C22×C4C22⋊C411yescharacteristic
68C22×C4C22×C411yescharacteristic
69C22×C4C2×D411yescharacteristic
70C22×C4C2×D411yescharacteristic
71M16—82no—
72C2×C8—42no—
73C2×C8—42no—
74C2×C8—82no—
75C4⋊C4—24no—
76C4⋊C4—41no—
77C4⋊C4—42no—
78C4⋊C4—44no—
79C4⋊C4—44no—
80C4⋊C4—42no—
81C22⋊C4—41no—
82C22⋊C4—41no—
83C22⋊C4—44no—
84C22⋊C4—44no—
85C42—21no—
86C42—21no—
87C42—24no—
88C23—21no—
89C23—21no—
90C23—21no—
91C23—21no—
92C23—21no—
93C23—41no—
94C23—41no—
95C23—41no—
96C23—41no—
97C23#10112yes—
98C23C2×C22⋊C411yescharacteristic
99Q8—82no—
100Q8—82no—
101D4—44no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—22no—
105C2×C4—24no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—42no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4—42no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—81no—
120C2×C4—81no—
121C2×C4#9911yescharacteristic
122C2×C4#9911yescharacteristic
123C2×C4#13112yes—
124C8—82no—
125C8—82no—
126C22—21no—
127C22—21no—
128C22—22no—
129C22—42no—
130C22—42no—
131C22—41no—
132C22—41no—
133C22—41no—
134C22—41no—
135C22—41no—
136C22—81no—
137C22—81no—
138C22#40811yescharacteristic
139C22C2×#9911yescharacteristic
140C22C2×#10111yescharacteristic
141C4—21no—
142C4—22no—
143C4—44no—
144C4—41no—
145C4—41no—
146C4—41no—
147C4—82no—
148C4#35312yes—
149C2—21no—
150C2—21no—
151C2—42no—
152C2—81no—
153C2#147211yescharacteristic
154C2C2×#35311yescharacteristic
155C2#213511yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC22×D4C2×C4
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
4. #1325C2semidirect

Maximal subgroups

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

subgroupindex
#25422
#14652
#14592
#13252