Finite Group Databasebeta

← #9455

Subgroups — #9455

Order 256.

277 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 277 conjugacy classes of subgroups — too many to draw. Only its 63 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#2542C211yesmaximal, characteristic
3#1556C212yesmaximal, complemented
4#1466C211yesmaximal, characteristic
5#1285C212yesmaximal, complemented
6C2×C4⋊Q8C2211yescharacteristic
7C2×C22⋊Q8—21no—
8C2×C22⋊Q8—21no—
9C2×C4×D4C2211yescharacteristic
10#435—24no—
11#432C2212yes—
12#432C2212yes—
13#407—21no—
14C2×C22⋊C8—21no—
15#399—21no—
16#399—21no—
17#390C412yes—
18#384C412yes—
19C4×C4⋊C4C2211yescharacteristic
20#332—24no—
21#323—21no—
22#323—21no—
23C22×Q8—21no—
24C22×D4C2×C411yescharacteristic
25C23×C4D411yescharacteristic
26C23×C4D411yescharacteristic
27C2×M16—22no—
28C22×C8—22no—
29C4⋊Q8—22no—
30C4⋊Q8—22no—
31C22⋊Q8—44no—
32C22⋊Q8—44no—
33C4×D4—22no—
34C4×D4—22no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—22no—
37C2×C4⋊C4—22no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4C2×C412yes—
40C2×C4⋊C4C2×C411yescharacteristic
41C2×C22⋊C4—22no—
42C2×C22⋊C4—22no—
43C2×C22⋊C4D411yescharacteristic
44C2×C22⋊C4D411yescharacteristic
45C2×C42C2×C412yes—
46C2×C42C2311yescharacteristic
47C4⋊C8—22no—
48C4⋊C8D414yes—
49C22⋊C8—22no—
50C22⋊C8—22no—
51C22⋊C8—22no—
52C22⋊C8—22no—
53#97—22no—
54#97—22no—
55#95—21no—
56C24C22⋊C411yescharacteristic
57C24C22⋊C411yescharacteristic
58C2×Q8—42no—
59C2×Q8—42no—
60C2×Q8—41no—
61C2×Q8—41no—
62C2×D4—22no—
63C2×D4—22no—
64C2×D4—41no—
65C22×C4—22no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—22no—
71C22×C4—42no—
72C22×C4—42no—
73C22×C4C22⋊C411yescharacteristic
74C22×C4C22⋊C411yescharacteristic
75C22×C4C22×C411yescharacteristic
76C22×C4C2×D411yescharacteristic
77C22×C4C2×D411yescharacteristic
78M16—82no—
79C2×C8—24no—
80C2×C8—42no—
81C2×C8—82no—
82C4⋊C4—24no—
83C4⋊C4—22no—
84C4⋊C4—24no—
85C4⋊C4—44no—
86C4⋊C4—44no—
87C4⋊C4—42no—
88C22⋊C4—22no—
89C22⋊C4—22no—
90C22⋊C4—44no—
91C22⋊C4—44no—
92C42—42no—
93C42C2×D412yes—
94C42#4112yes—
95C23—21no—
96C23—21no—
97C23—21no—
98C23—21no—
99C23—21no—
100C23—41no—
101C23—41no—
102C23—41no—
103C23—41no—
104C23#10112yes—
105C23C2×C22⋊C411yescharacteristic
106Q8—82no—
107Q8—82no—
108D4—44no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—22no—
112C2×C4—24no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—41no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—81no—
126C2×C4—81no—
127C2×C4C4×D412yes—
128C2×C4#12011yescharacteristic
129C2×C4#12112yes—
130C2×C4#12311yescharacteristic
131C8—44no—
132C8—82no—
133C22—21no—
134C22—21no—
135C22—22no—
136C22—42no—
137C22—42no—
138C22—41no—
139C22—41no—
140C22—41no—
141C22—41no—
142C22—41no—
143C22—81no—
144C22—81no—
145C22#38611yescharacteristic
146C22C2×#10111yescharacteristic
147C22#41311yescharacteristic
148C4—21no—
149C4—22no—
150C4—22no—
151C4—44no—
152C4—41no—
153C4—41no—
154C4—82no—
155C4#45312yes—
156C2—21no—
157C2—21no—
158C2—42no—
159C2—81no—
160C2#188711yescharacteristic
161C2#190111yescharacteristic
162C2#190911yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4C2×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. #1556C2semidirect
5. #1285C2semidirect

Maximal subgroups

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

subgroupindex
#29732
#25422
#15562
#14662
#12852