Finite Group Databasebeta

← #9151

Subgroups — #9151

Order 256.

473 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 473 conjugacy classes of subgroups — too many to draw. Only its 87 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×#380C211yesmaximal, characteristic
3#1779C211yesmaximal, characteristic
4C2×#328C212yesmaximal, complemented
5#1294C212yesmaximal, complemented
6C24×C4C2211yescharacteristic
7C2×C22⋊Q8C412yescomplemented
8C2×C22⋊Q8C2211yescharacteristic
9C22×C4⋊C4—21no—
10C22×C22⋊C4—21no—
11C2×C22⋊C8—22no—
12C2×C22⋊C8C2212yes—
13#385—21no—
14#385—22no—
15#380—24no—
16#380C412yescomplemented
17#380C2211yescharacteristic
18#379—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×C4D411yescharacteristic
29C23×C4D412yes—
30C23×C4C2311yescharacteristic
31C22×C8—22no—
32C22⋊Q8—22no—
33C22⋊Q8—22no—
34C22⋊Q8—44no—
35C2×C4⋊C4—24no—
36C2×C4⋊C4—22no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—44no—
39C2×C4⋊C4C2×C411yescharacteristic
40C2×C4⋊C4C2×C412yes—
41C2×C22⋊C4—24no—
42C2×C22⋊C4—22no—
43C2×C22⋊C4—22no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—42no—
47C2×C22⋊C4—42no—
48C22⋊C8—22no—
49C22⋊C8—22no—
50C22⋊C8—44no—
51#95—22no—
52#95—22no—
53#95—21no—
54#95D414yes—
55C24—21no—
56C24—21no—
57C24—21no—
58C24—21no—
59C24—42no—
60C24—42no—
61C24C22⋊C412yes—
62C24SD1612yes—
63C24Q1612yes—
64C24C2×D411yescharacteristic
65C2×Q8—22no—
66C2×Q8—42no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—24no—
70C22×C4—24no—
71C22×C4—24no—
72C22×C4—24no—
73C22×C4—24no—
74C22×C4—21no—
75C22×C4—22no—
76C22×C4—21no—
77C22×C4—24no—
78C22×C4—42no—
79C22×C4—42no—
80C22×C4—42no—
81C22×C4—48no—
82C22×C4—48no—
83C22×C4—44no—
84C22×C4—44no—
85C22×C4—42no—
86C22×C4—42no—
87C22×C4C22⋊C412yes—
88C22×C4C22×C411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D412yes—
91C22×C4#4112yes—
92C2×C8—44no—
93C2×C8—42no—
94C4⋊C4—22no—
95C4⋊C4—42no—
96C4⋊C4—48no—
97C4⋊C4—44no—
98C4⋊C4—42no—
99C22⋊C4—44no—
100C22⋊C4—48no—
101C22⋊C4—44no—
102C22⋊C4—41no—
103C22⋊C4—41no—
104C22⋊C4—41no—
105C22⋊C4—41no—
106C23—22no—
107C23—22no—
108C23—22no—
109C23—22no—
110C23—22no—
111C23—22no—
112C23—22no—
113C23—22no—
114C23—22no—
115C23—22no—
116C23—22no—
117C23—22no—
118C23—44no—
119C23—44no—
120C23—42no—
121C23—42no—
122C23—42no—
123C23—42no—
124C23—44no—
125C23—44no—
126C23#10314yes—
127C23C2×C22⋊C411yescharacteristic
128C23#12011yescharacteristic
129C23#12311yescharacteristic
130C23C2×SD1611yescharacteristic
131C23C2×Q1611yescharacteristic
132C23AGL(1,ℤ/8)11yescharacteristic
133C23#13711yescharacteristic
134Q8—82no—
135C2×C4—24no—
136C2×C4—24no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—48no—
144C2×C4—48no—
145C2×C4—44no—
146C2×C4—44no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—44no—
155C2×C4—84no—
156C2×C4—84no—
157C2×C4#9914yes—
158C2×C4C4×D412yes—
159C2×C4#12112yes—
160C8—82no—
161C22—21no—
162C22—21no—
163C22—21no—
164C22—21no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—22no—
169C22—24no—
170C22—24no—
171C22—24no—
172C22—44no—
173C22—44no—
174C22—44no—
175C22—44no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—42no—
180C22#38611yescharacteristic
181C22C2×#9912yes—
182C22C2×#10311yescharacteristic
183C22#41711yescharacteristic
184C22#42112yes—
185C22#45011yescharacteristic
186C22#45111yescharacteristic
187C22#48311yescharacteristic
188C22#48411yescharacteristic
189C4—22no—
190C4—24no—
191C4—22no—
192C4—84no—
193C4—81no—
194C4—81no—
195C4—82no—
196C2—21no—
197C2—21no—
198C2—21no—
199C2—21no—
200C2—44no—
201C2#151312yes—
202C2#152012yes—
203C2#187211yescharacteristic
204C2#189711yescharacteristic
205C2#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
16. #380C4semidirect

Maximal subgroups

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

subgroupindex
#28482
C2×#3802
#17792
C2×#3282
#12942