Finite Group Databasebeta

← #13579

Subgroups — #13579

Order 256.

227 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 227 conjugacy classes of subgroups — too many to draw. Only its 45 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#3151C211yesmaximal, characteristic
2#2132C211yesmaximal, characteristic, complemented
3#2126C211yesmaximal, characteristic, complemented
4#1406C214yesmaximal, complemented
5C2×AGL(1,ℤ/8)—21no—
6C2×AGL(1,ℤ/8)—21no—
7#532C2211yescharacteristic
8C2×#127C411yescharacteristic, complemented
9#496C412yescomplemented
10#496C2212yes—
11#493—21no—
12#492—21no—
13#413—21no—
14#410—21no—
15C4×M16C411yescharacteristic, complemented
16#356—21no—
17#355C2212yescomplemented
18#354C2212yescomplemented
19#353—21no—
20C2×#41D411yescharacteristic
21C2×#41D411yescharacteristic
22C22×D4—21no—
23C22×D4—41no—
24AGL(1,ℤ/8)—42no—
25AGL(1,ℤ/8)—42no—
26AGL(1,ℤ/8)—42no—
27AGL(1,ℤ/8)—42no—
28C2×SD16—22no—
29C2×SD16—22no—
30C2×D8—22no—
31C2×D8—22no—
32C2×M16—21no—
33C2×M16—41no—
34C4⋊Q8D411yescharacteristic
35#127—21no—
36#127—41no—
37#127C2×C412yes—
38#127D411yescharacteristic
39#124C2×C411yescharacteristic
40#124C2311yescharacteristic
41#121—41no—
42C4×D4—41no—
43#117—41no—
44C2×C42C2×C411yescharacteristic
45#101—22no—
46#100—41no—
47#99—22no—
48#99—41no—
49#97C2×C412yes—
50C4×C8—21no—
51C24—41no—
52#41—41no—
53#41—41no—
54#41—41no—
55#41—41no—
56C2×Q8C22⋊C411yescharacteristic
57C2×Q8C2×D411yescharacteristic
58C2×D4—21no—
59C2×D4—21no—
60C2×D4—22no—
61C2×D4—41no—
62C2×D4—41no—
63C2×D4—41no—
64C2×D4—41no—
65C2×D4—41no—
66C2×D4—42no—
67C2×D4—42no—
68C2×D4—41no—
69C2×D4—84no—
70C2×D4C22⋊C411yescharacteristic
71C2×D4C2×D411yescharacteristic
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4C22⋊C411yescharacteristic
76SD16—44no—
77SD16—44no—
78D8—44no—
79D8—44no—
80M16—22no—
81M16—22no—
82M16—42no—
83M16—81no—
84C2×C8—22no—
85C2×C8—81no—
86C4⋊C4—41no—
87C4⋊C4—81no—
88C22⋊C4—41no—
89C22⋊C4—41no—
90C22⋊C4—42no—
91C42—21no—
92C42—81no—
93C42C22⋊C411yescharacteristic
94C42C22×C411yescharacteristic
95C23—21no—
96C23—21no—
97C23—42no—
98C23—81no—
99C23—81no—
100C23—84no—
101C23#9911yescharacteristic
102Q8—41no—
103Q8—41no—
104D4—41no—
105D4—41no—
106D4—42no—
107D4—42no—
108D4—82no—
109D4—84no—
110D4—81no—
111D4—81no—
112D4—84no—
113D4—81no—
114D4—81no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—42no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—82no—
127C2×C4#9911yescharacteristic
128C2×C4C2×C22⋊C411yescharacteristic
129C8—24no—
130C8—82no—
131C22—21no—
132C22—41no—
133C22—41no—
134C22—41no—
135C22—81no—
136C22—81no—
137C22—84no—
138C22—82no—
139C22—162no—
140C22—162no—
141C22C2×#9911yescharacteristic
142C4—22no—
143C4—21no—
144C4—41no—
145C4—41no—
146C4—41no—
147C4—162no—
148C4#35311yescharacteristic
149C4#35311yescharacteristic
150C2—21no—
151C2—41no—
152C2—81no—
153C2—81no—
154C2—162no—
155C2C2×#35311yescharacteristic
subgroupG/H
center, socleC2C2×#353
commutatorC42C22×C4
Frattini, cosocle#124C23
prime radical#3151C2
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G
LECSG ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2132C2semidirect
3. #2126C2semidirect
4. #1406C2semidirect
8. C2×#127C4semidirect
9. #496C4semidirect
15. C4×M16C4semidirect
17. #355C22semidirect
18. #354C22semidirect

Maximal subgroups

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

subgroupindex
#31512
#21322
#21262
#14062