Finite Group Databasebeta

← #9469

Subgroups — #9469

Order 256.

289 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 289 conjugacy classes of subgroups — too many to draw. Only its 77 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#2972C211yesmaximal, characteristic
2#2487C211yesmaximal, characteristic
3#1824C212yesmaximal, complemented
4#1799C211yesmaximal, characteristic
5#1289C212yesmaximal, complemented
6C2×C4×D4C2211yescharacteristic
7C22×C4⋊C4C2211yescharacteristic
8C8×D4—24no—
9#431—21no—
10C2×#99—21no—
11C2×C22⋊C8—21no—
12C2×C22⋊C8C2212yes—
13#400C412yescomplemented
14#399C412yescomplemented
15#389—21no—
16#386—21no—
17#385—21no—
18#380—21no—
19#380C2212yes—
20#380C2211yescharacteristic
21#342—21no—
22#340—24no—
23C22×D4D411yescharacteristic
24C23×C4—21no—
25C23×C4D411yescharacteristic
26C23×C4C2311yescharacteristic
27C22×C8—22no—
28C22×C8D414yes—
29C4×D4—22no—
30C4×D4—22no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4—22no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—22no—
36C2×C4⋊C4—22no—
37C2×C4⋊C4C2×C412yes—
38C2×C4⋊C4C2×C411yescharacteristic
39C2×C22⋊C4—21no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—22no—
42C2×C22⋊C4C2×C411yescharacteristic
43C2×C22⋊C4D411yescharacteristic
44C2×C42D411yescharacteristic
45C4⋊C8—22no—
46#99—42no—
47C22⋊C8—22no—
48C22⋊C8—22no—
49C22⋊C8—22no—
50C4×C8—22no—
51#95—22no—
52#95—22no—
53#95—22no—
54#95C2×C412yes—
55C24C22⋊C411yescharacteristic
56C24C2×D411yescharacteristic
57C2×D4—22no—
58C2×D4—22no—
59C2×D4SD1612yes—
60C2×D4Q1612yes—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—22no—
64C22×C4—22no—
65C22×C4—42no—
66C22×C4—42no—
67C22×C4—42no—
68C22×C4—42no—
69C22×C4—82no—
70C22×C4C22⋊C411yescharacteristic
71C22×C4C22⋊C411yescharacteristic
72C22×C4C22⋊C411yescharacteristic
73C22×C4C22×C411yescharacteristic
74C22×C4C2×D412yes—
75C22×C4C2×D411yescharacteristic
76C22×C4#4112yes—
77C2×C8—24no—
78C2×C8—44no—
79C2×C8—82no—
80C4⋊C4—22no—
81C4⋊C4—24no—
82C4⋊C4—42no—
83C4⋊C4—44no—
84C4⋊C4—41no—
85C4⋊C4—41no—
86C4⋊C4—82no—
87C22⋊C4—22no—
88C22⋊C4—22no—
89C22⋊C4—42no—
90C22⋊C4—42no—
91C22⋊C4—42no—
92C22⋊C4—82no—
93C42—21no—
94C42—21no—
95C23—21no—
96C23—21no—
97C23—41no—
98C23—41no—
99C23—41no—
100C23—41no—
101C23#10312yes—
102C23C2×C22⋊C411yescharacteristic
103C23#12112yes—
104C23C2×SD1611yescharacteristic
105C23C2×Q1611yescharacteristic
106C23#13512yes—
107D4—24no—
108D4—24no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—24no—
112C2×C4—22no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—41no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—81no—
127C2×C4—84no—
128C2×C4—82no—
129C2×C4—82no—
130C2×C4—82no—
131C2×C4#10312yes—
132C2×C4C4×D412yes—
133C2×C4#12011yescharacteristic
134C2×C4#12311yescharacteristic
135C8—44no—
136C22—21no—
137C22—22no—
138C22—21no—
139C22—22no—
140C22—22no—
141C22—41no—
142C22—41no—
143C22—42no—
144C22—42no—
145C22—41no—
146C22—41no—
147C22—81no—
148C22#38611yescharacteristic
149C22C2×#10311yescharacteristic
150C22#41611yescharacteristic
151C22#46512yes—
152C22#46712yes—
153C4—21no—
154C4—22no—
155C4—22no—
156C4—41no—
157C4—41no—
158C4—82no—
159C4—82no—
160C4—162no—
161C4#45812yes—
162C2—21no—
163C2—22no—
164C2—21no—
165C2—22no—
166C2—42no—
167C2#189011yescharacteristic
168C2#189211yescharacteristic
169C2#191011yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1824C2semidirect
5. #1289C2semidirect
13. #400C4semidirect
14. #399C4semidirect

Maximal subgroups

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

subgroupindex
#29722
#24872
#18242
#17992
#12892