Finite Group Databasebeta

← #9143

Subgroups — #9143

Order 256.

635 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 635 conjugacy classes of subgroups — too many to draw. Only its 99 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
2C22×#95C211yesmaximal, characteristic
3#1779C211yesmaximal, characteristic
4C2×#328C212yesmaximal, complemented
5#1294C212yesmaximal, complemented
6C24×C4—21no—
7C24×C4C2211yescharacteristic
8C2×C22⋊Q8C412yescomplemented
9C2×C22⋊Q8C2211yescharacteristic
10C2×C22⋊C8—22no—
11C2×C22⋊C8C2212yes—
12#385—21no—
13#385—22no—
14#380C412yescomplemented
15#380C2211yescharacteristic
16#379—21no—
17C2×#95—21no—
18C2×#95—24no—
19C2×#95C2212yes—
20#328—24no—
21C25D411yescharacteristic
22C22×Q8C2×C411yescharacteristic, complemented
23C23×C4—28no—
24C23×C4—24no—
25C23×C4—22no—
26C23×C4—22no—
27C23×C4—22no—
28C23×C4—44no—
29C23×C4—44no—
30C23×C4—42no—
31C23×C4—42no—
32C23×C4C2×C412yes—
33C23×C4D412yes—
34C23×C4D411yescharacteristic
35C23×C4C2311yescharacteristic
36C22×C8—22no—
37C22⋊Q8—22no—
38C22⋊Q8—22no—
39C22⋊Q8—44no—
40C2×C4⋊C4—22no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4C2×C411yescharacteristic, complemented
43C2×C4⋊C4C2×C412yescomplemented
44C2×C22⋊C4—22no—
45C2×C22⋊C4—22no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C22⋊C8—22no—
49C22⋊C8—22no—
50C22⋊C8—44no—
51#95—24no—
52#95—22no—
53#95—22no—
54#95—21no—
55#95D414yes—
56C24—24no—
57C24—24no—
58C24—21no—
59C24—21no—
60C24—21no—
61C24—21no—
62C24C22⋊C412yes—
63C24SD1612yes—
64C24Q1612yes—
65C24C2×D411yescharacteristic
66C2×Q8—22no—
67C2×Q8—42no—
68C22×C4—216no—
69C22×C4—28no—
70C22×C4—24no—
71C22×C4—24no—
72C22×C4—21no—
73C22×C4—24no—
74C22×C4—21no—
75C22×C4—24no—
76C22×C4—24no—
77C22×C4—21no—
78C22×C4—22no—
79C22×C4—21no—
80C22×C4—42no—
81C22×C4—42no—
82C22×C4—42no—
83C22×C4—42no—
84C22×C4—48no—
85C22×C4—48no—
86C22×C4—48no—
87C22×C4—416no—
88C22×C4—416no—
89C22×C4—44no—
90C22×C4—48no—
91C22×C4—42no—
92C22×C4—42no—
93C22×C4C22⋊C412yes—
94C22×C4C22×C411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D412yes—
97C22×C4#4112yes—
98C2×C8—44no—
99C2×C8—42no—
100C4⋊C4—22no—
101C4⋊C4—42no—
102C4⋊C4—44no—
103C4⋊C4—42no—
104C22⋊C4—44no—
105C22⋊C4—44no—
106C22⋊C4—41no—
107C22⋊C4—41no—
108C22⋊C4—41no—
109C22⋊C4—41no—
110C23—24no—
111C23—24no—
112C23—28no—
113C23—24no—
114C23—24no—
115C23—28no—
116C23—28no—
117C23—28no—
118C23—22no—
119C23—22no—
120C23—22no—
121C23—22no—
122C23—22no—
123C23—22no—
124C23—22no—
125C23—22no—
126C23—22no—
127C23—22no—
128C23#9914yes—
129C23#10314yes—
130C23#10414yes—
131C23C2×C22⋊C411yescharacteristic
132C23#12011yescharacteristic
133C23#12311yescharacteristic
134C23C2×SD1611yescharacteristic
135C23C2×Q1611yescharacteristic
136C23AGL(1,ℤ/8)11yescharacteristic
137C23#13711yescharacteristic
138Q8—82no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—28no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—48no—
148C2×C4—48no—
149C2×C4—432no—
150C2×C4—44no—
151C2×C4—48no—
152C2×C4—48no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—44no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4C4×D412yes—
164C2×C4#12112yes—
165C8—82no—
166C22—28no—
167C22—28no—
168C22—28no—
169C22—28no—
170C22—24no—
171C22—21no—
172C22—24no—
173C22—21no—
174C22—21no—
175C22—24no—
176C22—21no—
177C22—22no—
178C22—24no—
179C22—22no—
180C22—22no—
181C22—24no—
182C22—24no—
183C22—22no—
184C22#32818yescomplemented
185C22#38611yescharacteristic
186C22C2×#9912yes—
187C22C2×#10311yescharacteristic
188C22#41711yescharacteristic
189C22C2×#10412yes—
190C22#45011yescharacteristic
191C22#45111yescharacteristic
192C22#48311yescharacteristic
193C22#48411yescharacteristic
194C4—22no—
195C4—22no—
196C4—48no—
197C4—42no—
198C4—81no—
199C4—81no—
200C4—82no—
201C2—21no—
202C2—28no—
203C2—21no—
204C2—21no—
205C2—21no—
206C2C2×#32812yes—
207C2#150412yes—
208C2#187211yescharacteristic
209C2#189611yescharacteristic
210C2#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
8. C2×C22⋊Q8C4semidirect
14. #380C4semidirect
22. C22×Q8C2×C4semidirect
42. C2×C4⋊C4C2×C4semidirect
43. C2×C4⋊C4C2×C4semidirect
184. C22#328semidirect

Maximal subgroups

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

subgroupindex
#28482
C22×#952
#17792
C2×#3282
#12942