Finite Group Databasebeta

← #9473

Subgroups — #9473

Order 256.

278 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 69 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#2973C211yesmaximal, characteristic
2#2487C211yesmaximal, characteristic
3#1824C212yesmaximal, complemented
4#1799C211yesmaximal, characteristic
5#1289C212yesmaximal, complemented
6C2×C4×D4C2211yescharacteristic
7C22×C4⋊C4C2211yescharacteristic
8#435—24no—
9#432—21no—
10C2×#99—21no—
11#407—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
27C2×M16—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—
50#97—22no—
51#95—22no—
52#95—22no—
53#95—22no—
54#95C2×C412yes—
55C24C22⋊C411yescharacteristic
56C24C2×D411yescharacteristic
57C2×D4—22no—
58C2×D4—21no—
59C2×D4—21no—
60C2×D4—22no—
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—
77M16—82no—
78C2×C8—24no—
79C2×C8—44no—
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—41no—
90C22⋊C4—41no—
91C22⋊C4—42no—
92C22⋊C4—42no—
93C22⋊C4—82no—
94C42—21no—
95C42—21no—
96C23—21no—
97C23—21no—
98C23—21no—
99C23—41no—
100C23—41no—
101C23—41no—
102C23—41no—
103C23C2×C22⋊C411yescharacteristic
104C23#12112yes—
105C23AGL(1,ℤ/8)11yescharacteristic
106C23#13711yescharacteristic
107C23#13712yes—
108D4—42no—
109D4—42no—
110C2×C4—24no—
111C2×C4—22no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—41no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—42no—
128C2×C4—81no—
129C2×C4—84no—
130C2×C4—82no—
131C2×C4—82no—
132C2×C4—82no—
133C2×C4C4×D412yes—
134C2×C4#12011yescharacteristic
135C2×C4#12311yescharacteristic
136C8—44no—
137C22—21no—
138C22—22no—
139C22—21no—
140C22—22no—
141C22—41no—
142C22—41no—
143C22—42no—
144C22—42no—
145C22—41no—
146C22—41no—
147C22—41no—
148C22—81no—
149C22#38611yescharacteristic
150C22#41711yescharacteristic
151C22#41911yescharacteristic
152C22#46812yes—
153C22#47012yes—
154C4—21no—
155C4—22no—
156C4—41no—
157C4—41no—
158C4—41no—
159C4—82no—
160C4—82no—
161C4—162no—
162C4#45812yes—
163C2—21no—
164C2—22no—
165C2—21no—
166C2—22no—
167C2—81no—
168C2#189011yescharacteristic
169C2#189411yescharacteristic
170C2#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
#29732
#24872
#18242
#17992
#12892