Finite Group Databasebeta

← #12127

Subgroups — #12127

Order 256.

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

normal subgroups only

This group has 227 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#2804C211yesmaximal, characteristic, complemented
2#2006C211yesmaximal, characteristic
3#1616C212yesmaximal, complemented
4#1353C212yesmaximal, complemented
5#1318C211yesmaximal, characteristic, complemented
6C2×AGL(1,ℤ/8)—21no—
7#537C2211yescharacteristic
8#481—22no—
9C2×#104—21no—
10#400C2212yes—
11#393—22no—
12#391—21no—
13#390C2211yescharacteristic
14#386—21no—
15#385—22no—
16#343C2211yescharacteristic
17#327—22no—
18#324C2212yes—
19C2×#41—21no—
20C22×Q8—21no—
21C22×D4D411yescharacteristic
22C23×C4—21no—
23AGL(1,ℤ/8)—42no—
24AGL(1,ℤ/8)—42no—
25C2×SD16—41no—
26C2×D8—41no—
27C2×M16—21no—
28C22⋊Q8—41no—
29#121—22no—
30#121—41no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4D412yes—
33C2×C4⋊C4C2311yescharacteristic
34C2×C22⋊C4—22no—
35C2×C22⋊C4—22no—
36C2×C22⋊C4—21no—
37C2×C42D411yescharacteristic
38#106—42no—
39#104—42no—
40#102—42no—
41C22⋊C8—22no—
42#95—22no—
43#95—22no—
44#95—22no—
45#95D412yes—
46C24—21no—
47#41—41no—
48#41—41no—
49#41—41no—
50#41—41no—
51C2×Q8—21no—
52C2×Q8—42no—
53C2×Q8—81no—
54C2×Q8—81no—
55C2×D4—21no—
56C2×D4—22no—
57C2×D4—21no—
58C2×D4—41no—
59C2×D4—41no—
60C2×D4—41no—
61C2×D4SD1612yes—
62C22×C4—21no—
63C22×C4—22no—
64C22×C4—22no—
65C22×C4—41no—
66C22×C4—82no—
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—81no—
80C22⋊C4—44no—
81C22⋊C4—42no—
82C22⋊C4—82no—
83C22⋊C4—81no—
84C42—41no—
85C23—22no—
86C23—42no—
87C23—42no—
88C23—41no—
89C23—41no—
90C23—41no—
91C23#12011yescharacteristic
92Q8—41no—
93Q8—41no—
94Q8—84no—
95D4—42no—
96D4—41no—
97D4—41no—
98D4—81no—
99D4—82no—
100D4—81no—
101D4—81no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—41no—
115C2×C4—42no—
116C2×C4—81no—
117C2×C4—81no—
118C2×C4—84no—
119C2×C4C2×SD1611yescharacteristic
120C2×C4#13711yescharacteristic
121C8—42no—
122C8—82no—
123C22—21no—
124C22—21no—
125C22—44no—
126C22—42no—
127C22—82no—
128C22—81no—
129C22—81no—
130C22—81no—
131C22—81no—
132C22—81no—
133C22#44811yescharacteristic
134C22#45311yescharacteristic
135C22B(4,𝔽2)11yescharacteristic
136C4—21no—
137C4—21no—
138C4—42no—
139C4—42no—
140C4—82no—
141C4—81no—
142C4—82no—
143C2—21no—
144C2—21no—
145C2—82no—
146C2—161no—
147C2#159611yescharacteristic
148C2#219811yescharacteristic
149C2#220211yescharacteristic
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. #2804C2semidirect
3. #1616C2semidirect
4. #1353C2semidirect
5. #1318C2semidirect

Maximal subgroups

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

subgroupindex
#28042
#20062
#16162
#13532
#13182