Finite Group Databasebeta

← #9191

Subgroups — #9191

Order 256.

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

normal subgroups only

This group has 252 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#2628C211yesmaximal, characteristic
3#1548C212yesmaximal, complemented
4#1458C211yesmaximal, characteristic
5#1286C212yesmaximal, complemented
6C22×M16—21no—
7C2×C4⋊Q8C2211yescharacteristic
8C2×#117C2211yescharacteristic
9#431C2212yes—
10#424—24no—
11#423—21no—
12#423C2212yes—
13#400—21no—
14#399—21no—
15#395C412yes—
16#393C412yes—
17#389—21no—
18#382C2211yescharacteristic
19#380—21no—
20#333—24no—
21#324—21no—
22#323—21no—
23C22×Q8C2×C412yes—
24C23×C4D411yescharacteristic
25C2×M16—22no—
26C2×M16—22no—
27C2×M16—42no—
28C2×M16—42no—
29C22×C8—22no—
30C4⋊Q8—22no—
31C4⋊Q8—22no—
32#117—22no—
33#117—22no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—22no—
38C2×C4⋊C4C2×C411yescharacteristic
39C2×C4⋊C4D411yescharacteristic
40C2×C22⋊C4—22no—
41C2×C22⋊C4C2×C411yescharacteristic
42C2×C22⋊C4D411yescharacteristic
43C2×C42D411yescharacteristic
44C2×C42C2311yescharacteristic
45C4⋊C8—24no—
46C4⋊C8—22no—
47C4⋊C8—22no—
48C4⋊C8D414yes—
49C22⋊C8—22no—
50C4×C8—22no—
51#95—22no—
52#95—22no—
53#95—21no—
54#95C2×C412yes—
55C24C22⋊C411yescharacteristic
56C2×Q8—24no—
57C2×Q8—82no—
58C22×C4—21no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—22no—
63C22×C4—22no—
64C22×C4C22⋊C411yescharacteristic
65C22×C4C22⋊C411yescharacteristic
66C22×C4C22⋊C411yescharacteristic
67C22×C4SD1612yes—
68C22×C4Q1612yes—
69C22×C4C22×C411yescharacteristic
70C22×C4C2×D411yescharacteristic
71C22×C4C2×D411yescharacteristic
72M16—44no—
73M16—42no—
74M16—42no—
75C2×C8—24no—
76C2×C8—44no—
77C2×C8—42no—
78C4⋊C4—22no—
79C4⋊C4—22no—
80C4⋊C4—41no—
81C4⋊C4—42no—
82C4⋊C4—41no—
83C4⋊C4—41no—
84C4⋊C4—41no—
85C4⋊C4—82no—
86C22⋊C4—22no—
87C22⋊C4—22no—
88C22⋊C4—82no—
89C42—21no—
90C42—21no—
91C42C2×D412yes—
92C42#4112yes—
93C23—21no—
94C23—21no—
95C23—41no—
96C23—41no—
97C23#10312yes—
98C23C2×C22⋊C411yescharacteristic
99Q8—84no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—22no—
103C2×C4—22no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4#10312yes—
119C2×C4C4×D412yes—
120C2×C4#12011yescharacteristic
121C2×C4#12112yes—
122C2×C4#12311yescharacteristic
123C2×C4C2×SD1611yescharacteristic
124C2×C4C2×Q1611yescharacteristic
125C2×C4AGL(1,ℤ/8)11yescharacteristic
126C2×C4#13711yescharacteristic
127C8—44no—
128C8—82no—
129C22—21no—
130C22—21no—
131C22—22no—
132C22—42no—
133C22—41no—
134C22—41no—
135C22—81no—
136C22#38611yescharacteristic
137C22C2×#10311yescharacteristic
138C22#41711yescharacteristic
139C4—22no—
140C4—22no—
141C4—22no—
142C4—41no—
143C4—41no—
144C4—82no—
145C4—82no—
146C4#45011yescharacteristic
147C4#45111yescharacteristic
148C4#48311yescharacteristic
149C4#48411yescharacteristic
150C2—21no—
151C2—21no—
152C2—42no—
153C2#187211yescharacteristic
154C2#190911yescharacteristic
155C2#191411yescharacteristic
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. #1548C2semidirect
5. #1286C2semidirect

Maximal subgroups

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

subgroupindex
#29052
#26282
#15482
#14582
#12862