Finite Group Databasebeta

← #9154

Subgroups — #9154

Order 256.

451 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 451 conjugacy classes of subgroups — too many to draw. Only its 79 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#2848C211yesmaximal, characteristic
2C2×#381C211yesmaximal, characteristic
3#1779C211yesmaximal, characteristic
4C2×#328C212yesmaximal, complemented
5#1294C212yesmaximal, complemented
6C24×C4C2211yescharacteristic
7C2×C22⋊Q8C412yescomplemented
8C2×C22⋊Q8C2211yescharacteristic
9C22×C22⋊C4—21no—
10C2×C22⋊C8—22no—
11C2×C22⋊C8C2212yes—
12#385—21no—
13#385—22no—
14#381—24no—
15#380C412yescomplemented
16#380C2211yescharacteristic
17#379—21no—
18C2×#95—21no—
19C2×#95C2212yes—
20#328—24no—
21C25D411yescharacteristic
22C22×Q8C2×C411yescharacteristic
23C23×C4—22no—
24C23×C4—22no—
25C23×C4—22no—
26C23×C4—44no—
27C23×C4C2×C412yes—
28C23×C4D412yes—
29C23×C4D411yescharacteristic
30C23×C4C2311yescharacteristic
31C22×C8—22no—
32C22⋊Q8—22no—
33C22⋊Q8—22no—
34C22⋊Q8—44no—
35C2×C4⋊C4—22no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4C2×C411yescharacteristic
38C2×C4⋊C4C2×C412yes—
39C2×C22⋊C4—24no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—22no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—44no—
45C22⋊C8—22no—
46C22⋊C8—22no—
47C22⋊C8—44no—
48#95—24no—
49#95—22no—
50#95—22no—
51#95—21no—
52#95D414yes—
53C24—24no—
54C24—21no—
55C24—21no—
56C24—22no—
57C24—22no—
58C24—42no—
59C24C22⋊C412yes—
60C24C2×D411yescharacteristic
61C2×Q8—22no—
62C2×Q8—42no—
63C22×C4—24no—
64C22×C4—24no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—24no—
68C22×C4—24no—
69C22×C4—21no—
70C22×C4—22no—
71C22×C4—21no—
72C22×C4—24no—
73C22×C4—42no—
74C22×C4—42no—
75C22×C4—42no—
76C22×C4—42no—
77C22×C4—48no—
78C22×C4—48no—
79C22×C4—44no—
80C22×C4—44no—
81C22×C4—42no—
82C22×C4—42no—
83C22×C4C22⋊C412yes—
84C22×C4C22×C411yescharacteristic
85C22×C4C2×D411yescharacteristic
86C22×C4C2×D412yes—
87C22×C4#4112yes—
88C2×C8—44no—
89C2×C8—42no—
90C4⋊C4—22no—
91C4⋊C4—42no—
92C4⋊C4—44no—
93C4⋊C4—42no—
94C22⋊C4—44no—
95C22⋊C4—48no—
96C22⋊C4—44no—
97C22⋊C4—41no—
98C22⋊C4—41no—
99C22⋊C4—42no—
100C23—22no—
101C23—22no—
102C23—24no—
103C23—24no—
104C23—22no—
105C23—22no—
106C23—22no—
107C23—22no—
108C23—22no—
109C23—22no—
110C23—48no—
111C23—48no—
112C23—42no—
113C23—42no—
114C23—42no—
115C23—42no—
116C23#9914yes—
117C23C2×C22⋊C411yescharacteristic
118C23#12011yescharacteristic
119C23#12311yescharacteristic
120C23#13512yes—
121C23#13712yes—
122Q8—82no—
123C2×C4—22no—
124C2×C4—22no—
125C2×C4—22no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—48no—
131C2×C4—48no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—44no—
135C2×C4—44no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—44no—
144C2×C4—84no—
145C2×C4—84no—
146C2×C4C4×D412yes—
147C2×C4#12112yes—
148C8—82no—
149C22—21no—
150C22—21no—
151C22—21no—
152C22—21no—
153C22—24no—
154C22—24no—
155C22—24no—
156C22—24no—
157C22—24no—
158C22—48no—
159C22—48no—
160C22—42no—
161C22—42no—
162C22—42no—
163C22—42no—
164C22#38611yescharacteristic
165C22C2×#9912yes—
166C22#41611yescharacteristic
167C22#41911yescharacteristic
168C22#42112yes—
169C22#45212yes—
170C22#48512yes—
171C4—22no—
172C4—22no—
173C4—42no—
174C4—84no—
175C4—81no—
176C4—81no—
177C4—82no—
178C2—22no—
179C2—22no—
180C2—44no—
181C2#151112yes—
182C2#151612yes—
183C2#187311yescharacteristic
184C2#189711yescharacteristic
185C2#189811yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC25D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#328C2semidirect
5. #1294C2semidirect
7. C2×C22⋊Q8C4semidirect
15. #380C4semidirect

Maximal subgroups

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

subgroupindex
#28482
C2×#3812
#17792
C2×#3282
#12942