Finite Group Databasebeta

← #9195

Subgroups — #9195

Order 256.

262 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 75 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#2905C211yesmaximal, characteristic
2#2621C211yesmaximal, characteristic
3#1546C212yesmaximal, complemented
4#1458C211yesmaximal, characteristic
5#1286C212yesmaximal, complemented
6C22×M16—21no—
7C2×C4⋊Q8C2211yescharacteristic
8C2×C22⋊Q8—21no—
9C2×#117C2211yescharacteristic
10#431C2212yes—
11#424—24no—
12#423—21no—
13#423C2212yes—
14#398—21no—
15#398C412yes—
16#397—21no—
17#397C412yes—
18#391—21no—
19#382C2211yescharacteristic
20#332—24no—
21#324—21no—
22#323—21no—
23C22×Q8—21no—
24C23×C4D411yescharacteristic
25C2×M16—22no—
26C2×M16—22no—
27C2×M16—42no—
28C2×M16—42no—
29C22×C8—22no—
30C4⋊Q8—22no—
31C4⋊Q8—22no—
32C22⋊Q8—42no—
33C22⋊Q8—42no—
34#117—22no—
35#117—22no—
36C2×C4⋊C4—22no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4C2×C412yes—
40C2×C4⋊C4C2×C411yescharacteristic
41C2×C4⋊C4D411yescharacteristic
42C2×C22⋊C4—22no—
43C2×C22⋊C4C2×C411yescharacteristic
44C2×C22⋊C4D411yescharacteristic
45C2×C42D411yescharacteristic
46C2×C42C2311yescharacteristic
47C4⋊C8—24no—
48C4⋊C8—22no—
49C4⋊C8—22no—
50C4⋊C8D414yes—
51C22⋊C8—22no—
52C4×C8—22no—
53#95—22no—
54#95—22no—
55#95—21no—
56#95C2×C412yes—
57C24C22⋊C411yescharacteristic
58C2×Q8—41no—
59C2×Q8—41no—
60C2×Q8—41no—
61C2×Q8—41no—
62C2×Q8—41no—
63C2×Q8—41no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—22no—
69C22×C4—22no—
70C22×C4C22⋊C411yescharacteristic
71C22×C4C22⋊C411yescharacteristic
72C22×C4C22⋊C411yescharacteristic
73C22×C4D812yes—
74C22×C4SD1612yes—
75C22×C4C22×C411yescharacteristic
76C22×C4C2×D411yescharacteristic
77C22×C4C2×D411yescharacteristic
78M16—44no—
79M16—42no—
80M16—42no—
81C2×C8—24no—
82C2×C8—44no—
83C2×C8—42no—
84C4⋊C4—22no—
85C4⋊C4—22no—
86C4⋊C4—24no—
87C4⋊C4—44no—
88C4⋊C4—42no—
89C4⋊C4—41no—
90C4⋊C4—41no—
91C22⋊C4—22no—
92C22⋊C4—22no—
93C22⋊C4—44no—
94C42—21no—
95C42—21no—
96C42C2×D412yes—
97C42#4112yes—
98C23—21no—
99C23—21no—
100C23—41no—
101C23—41no—
102C23#10212yes—
103C23C2×C22⋊C411yescharacteristic
104Q8—81no—
105Q8—81no—
106Q8—81no—
107Q8—81no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—22no—
113C2×C4—22no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—41no—
117C2×C4—41no—
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#10212yes—
127C2×C4C4×D412yes—
128C2×C4#12011yescharacteristic
129C2×C4#12112yes—
130C2×C4#12311yescharacteristic
131C2×C4C2×D811yescharacteristic
132C2×C4C2×SD1611yescharacteristic
133C2×C4AGL(1,ℤ/8)11yescharacteristic
134C2×C4#13711yescharacteristic
135C8—44no—
136C8—82no—
137C22—21no—
138C22—21no—
139C22—22no—
140C22—42no—
141C22—41no—
142C22—41no—
143C22—81no—
144C22#38611yescharacteristic
145C22C2×#10211yescharacteristic
146C22#41711yescharacteristic
147C4—22no—
148C4—22no—
149C4—22no—
150C4—41no—
151C4—41no—
152C4—82no—
153C4—82no—
154C4#44711yescharacteristic
155C4#44811yescharacteristic
156C4#48011yescharacteristic
157C4#48111yescharacteristic
158C2—21no—
159C2—21no—
160C2—42no—
161C2#187411yescharacteristic
162C2#190911yescharacteristic
163C2#191511yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1546C2semidirect
5. #1286C2semidirect

Maximal subgroups

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

subgroupindex
#29052
#26212
#15462
#14582
#12862