Finite Group Databasebeta

← #12097

Subgroups — #12097

Order 256.

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

normal subgroups only

This group has 233 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#2724C211yesmaximal, characteristic, complemented
2#2006C211yesmaximal, characteristic
3#1610C212yesmaximal, complemented
4#1346C212yesmaximal, complemented
5#1318C211yesmaximal, characteristic, complemented
6C2×AGL(1,ℤ/8)—21no—
7#537C2211yescharacteristic
8C2×#123—21no—
9#480—22no—
10C2×#104—21no—
11#398C2212yes—
12#397—22no—
13#390C2211yescharacteristic
14#389—21no—
15#385—22no—
16#343C2211yescharacteristic
17#327—22no—
18#324C2212yes—
19C2×#41—21no—
20C22×D4D411yescharacteristic
21C23×C4—21no—
22AGL(1,ℤ/8)—42no—
23AGL(1,ℤ/8)—42no—
24C2×SD16—41no—
25C2×D8—41no—
26C2×M16—21no—
27#123—42no—
28#123—42no—
29C22⋊Q8—41no—
30#121—22no—
31#121—41no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—22no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4D412yes—
36C2×C4⋊C4C2311yescharacteristic
37C2×C22⋊C4—22no—
38C2×C22⋊C4—21no—
39C2×C22⋊C4—22no—
40C2×C42D411yescharacteristic
41#107—42no—
42#104—42no—
43#102—42no—
44C22⋊C8—22no—
45#95—22no—
46#95—22no—
47#95D412yes—
48C24—21no—
49#41—41no—
50#41—41no—
51#41—41no—
52#41—41no—
53C2×Q8—21no—
54C2×D4—21no—
55C2×D4—22no—
56C2×D4—21no—
57C2×D4—41no—
58C2×D4—41no—
59C2×D4—41no—
60C2×D4D812yes—
61C22×C4—21no—
62C22×C4—22no—
63C22×C4—22no—
64C22×C4—41no—
65C22×C4—42no—
66C22×C4—81no—
67C22×C4—81no—
68C22×C4C2×D411yescharacteristic
69C22×C4C2×D412yes—
70SD16—44no—
71D8—44no—
72M16—22no—
73M16—41no—
74C2×C8—41no—
75C2×C8—42no—
76C4⋊C4—22no—
77C4⋊C4—41no—
78C4⋊C4—42no—
79C4⋊C4—42no—
80C4⋊C4—42no—
81C4⋊C4—42no—
82C4⋊C4—81no—
83C22⋊C4—44no—
84C22⋊C4—42no—
85C22⋊C4—42no—
86C22⋊C4—44no—
87C42—41no—
88C23—22no—
89C23—42no—
90C23—42no—
91C23—41no—
92C23—41no—
93C23—41no—
94C23#12011yescharacteristic
95Q8—41no—
96Q8—41no—
97D4—42no—
98D4—41no—
99D4—41no—
100D4—81no—
101D4—82no—
102D4—81no—
103D4—81no—
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—41no—
117C2×C4—42no—
118C2×C4—81no—
119C2×C4—81no—
120C2×C4—84no—
121C2×C4C2×D811yescharacteristic
122C2×C4AGL(1,ℤ/8)11yescharacteristic
123C8—42no—
124C8—82no—
125C22—21no—
126C22—21no—
127C22—44no—
128C22—42no—
129C22—82no—
130C22—81no—
131C22—81no—
132C22—81no—
133C22—81no—
134C22—81no—
135C22#44711yescharacteristic
136C22#45311yescharacteristic
137C22B(4,𝔽2)11yescharacteristic
138C4—21no—
139C4—21no—
140C4—42no—
141C4—42no—
142C4—82no—
143C4—82no—
144C4—81no—
145C2—21no—
146C2—21no—
147C2—82no—
148C2—161no—
149C2#159511yescharacteristic
150C2#219811yescharacteristic
151C2#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. #2724C2semidirect
3. #1610C2semidirect
4. #1346C2semidirect
5. #1318C2semidirect

Maximal subgroups

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

subgroupindex
#27242
#20062
#16102
#13462
#13182