Finite Group Databasebeta

← #12126

Subgroups — #12126

Order 256.

261 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 261 conjugacy classes of subgroups — too many to draw. Only its 37 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#2683C211yesmaximal, characteristic, complemented
2#2006C211yesmaximal, characteristic
3#1604C212yesmaximal, complemented
4#1354C212yesmaximal, complemented
5#1318C211yesmaximal, characteristic, complemented
6C2×AGL(1,ℤ/8)—21no—
7#537C2211yescharacteristic
8C2×C22⋊Q8—22no—
9#448—22no—
10C2×#104—21no—
11#396—22no—
12#395C2212yes—
13#390C2211yescharacteristic
14#389—21no—
15#386—21no—
16#343C2211yescharacteristic
17#327—22no—
18#324C2212yes—
19C2×#41—21no—
20C22×Q8D412yes—
21C22×D4D411yescharacteristic
22C23×C4—21no—
23AGL(1,ℤ/8)—42no—
24AGL(1,ℤ/8)—42no—
25C2×SD16—41no—
26C2×SD16—42no—
27C2×D8—41no—
28C2×M16—21no—
29C22⋊Q8—41no—
30C22⋊Q8—48no—
31#121—22no—
32#121—41no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—22no—
35C2×C4⋊C4—22no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4C2311yescharacteristic
38C2×C22⋊C4—22no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—21no—
41C2×C42D411yescharacteristic
42#104—42no—
43#103—42no—
44C22⋊C8—22no—
45#95—22no—
46#95D412yes—
47C24—21no—
48#41—41no—
49#41—41no—
50#41—41no—
51#41—41no—
52C2×Q8—21no—
53C2×Q8—44no—
54C2×Q8—42no—
55C2×D4—21no—
56C2×D4—22no—
57C2×D4—21no—
58C2×D4—41no—
59C2×D4—41no—
60C2×D4—41no—
61C2×D4SD1612yescomplemented
62C22×C4—22no—
63C22×C4—21no—
64C22×C4—22no—
65C22×C4—41no—
66C22×C4—44no—
67C22×C4—81no—
68C22×C4C2×D411yescharacteristic
69C22×C4C2×D412yes—
70SD16—44no—
71SD16—84no—
72D8—44no—
73M16—22no—
74M16—41no—
75C2×C8—41no—
76C2×C8—42no—
77C4⋊C4—24no—
78C4⋊C4—22no—
79C4⋊C4—41no—
80C4⋊C4—44no—
81C4⋊C4—44no—
82C4⋊C4—81no—
83C22⋊C4—44no—
84C22⋊C4—44no—
85C22⋊C4—42no—
86C22⋊C4—81no—
87C42—41no—
88C23—22no—
89C23—42no—
90C23—42no—
91C23—41no—
92C23—41no—
93C23—41no—
94C23#12011yescharacteristic
95Q8—41no—
96Q8—41no—
97Q8—44no—
98Q8—82no—
99D4—42no—
100D4—41no—
101D4—41no—
102D4—81no—
103D4—82no—
104D4—81no—
105D4—81no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—22no—
109C2×C4—24no—
110C2×C4—22no—
111C2×C4—42no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—41no—
120C2×C4—81no—
121C2×C4—81no—
122C2×C4—84no—
123C2×C4C2×SD1611yescharacteristic
124C2×C4#13711yescharacteristic
125C8—42no—
126C8—82no—
127C22—21no—
128C22—21no—
129C22—44no—
130C22—42no—
131C22—82no—
132C22—81no—
133C22—81no—
134C22—81no—
135C22—81no—
136C22—81no—
137C22#44811yescharacteristic
138C22#45311yescharacteristic
139C22B(4,𝔽2)11yescharacteristic
140C4—21no—
141C4—21no—
142C4—42no—
143C4—42no—
144C4—44no—
145C4—82no—
146C4—81no—
147C2—21no—
148C2—21no—
149C2—82no—
150C2—161no—
151C2#159611yescharacteristic
152C2#219611yescharacteristic
153C2#220311yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23
prime radical#2006C2
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2683C2semidirect
3. #1604C2semidirect
4. #1354C2semidirect
5. #1318C2semidirect
61. C2×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#26832
#20062
#16042
#13542
#13182