Finite Group Databasebeta

← #9459

Subgroups — #9459

Order 256.

452 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 65 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#2435C211yesmaximal, characteristic
2#2308C211yesmaximal, characteristic
3#1783C211yesmaximal, characteristic
4#1511C212yesmaximal, complemented
5#1284C212yesmaximal, complemented
6C23×D4C2211yescharacteristic
7C2×#124C412yescomplemented
8C2×C22⋊Q8C2211yescharacteristic
9C22×C22⋊C4—21no—
10C2×#100—22no—
11#407C2212yes—
12#394—22no—
13#390C412yescomplemented
14#386—24no—
15#381C2212yes—
16#379—21no—
17#379—21no—
18#379—21no—
19C4×C22⋊C4C2211yescharacteristic
20#328—24no—
21C25D411yescharacteristic
22C25D411yescharacteristic
23C22×Q8C2×C411yescharacteristic, complemented
24C22×D4—22no—
25C22×D4—22no—
26C22×D4—44no—
27C22×D4C2×C412yes—
28C22×D4D412yes—
29C23×C4—22no—
30C23×C4C2311yescharacteristic
31C2×M16—22no—
32#124—44no—
33C22⋊Q8—22no—
34C22⋊Q8—22no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4C2×C411yescharacteristic, complemented
37C2×C22⋊C4—22no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—22no—
44C2×C22⋊C4—22no—
45C2×C22⋊C4—22no—
46C2×C22⋊C4—22no—
47C2×C22⋊C4—44no—
48C2×C42C2×C412yescomplemented
49#100—44no—
50C22⋊C8—22no—
51C22⋊C8—22no—
52#95—21no—
53#95—22no—
54#95D414yes—
55C24—21no—
56C24—24no—
57C24—21no—
58C24—21no—
59C24—21no—
60C24—42no—
61C24—42no—
62C24—42no—
63C24—81no—
64C24—81no—
65C24C22⋊C412yes—
66C24C22⋊C412yes—
67C24C2×D411yescharacteristic
68C2×Q8—22no—
69C2×Q8—42no—
70C2×D4—24no—
71C2×D4—24no—
72C2×D4—24no—
73C2×D4—24no—
74C2×D4—42no—
75C2×D4—42no—
76C2×D4—44no—
77C2×D4—44no—
78C2×D4—48no—
79C22×C4—22no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—24no—
83C22×C4—42no—
84C22×C4—44no—
85C22×C4—82no—
86C22×C4—82no—
87C22×C4C22×C411yescharacteristic
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D412yes—
90C22×C4#4112yes—
91M16—44no—
92C2×C8—42no—
93C4⋊C4—22no—
94C4⋊C4—42no—
95C22⋊C4—42no—
96C22⋊C4—44no—
97C22⋊C4—44no—
98C22⋊C4—44no—
99C22⋊C4—44no—
100C22⋊C4—41no—
101C22⋊C4—41no—
102C22⋊C4—82no—
103C22⋊C4—82no—
104C22⋊C4—82no—
105C22⋊C4—82no—
106C42—42no—
107C23—22no—
108C23—22no—
109C23—24no—
110C23—22no—
111C23—22no—
112C23—22no—
113C23—22no—
114C23—22no—
115C23—22no—
116C23—22no—
117C23—41no—
118C23—42no—
119C23—42no—
120C23—42no—
121C23—42no—
122C23—42no—
123C23—42no—
124C23—42no—
125C23—42no—
126C23—42no—
127C23—42no—
128C23—81no—
129C23—81no—
130C23—81no—
131C23—81no—
132C23—88no—
133C23—88no—
134C23#9914yes—
135C23C2×C22⋊C411yescharacteristic
136C23#12011yescharacteristic
137C23#12311yescharacteristic
138Q8—82no—
139D4—48no—
140D4—48no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—44no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—44no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—44no—
155C2×C4—88no—
156C2×C4C4×D412yes—
157C2×C4#12112yes—
158C8—82no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22—21no—
164C22—21no—
165C22—24no—
166C22—21no—
167C22—44no—
168C22—44no—
169C22—44no—
170C22—44no—
171C22—44no—
172C22—42no—
173C22—42no—
174C22—42no—
175C22—42no—
176C22—42no—
177C22—82no—
178C22—82no—
179C22—84no—
180C22—84no—
181C22—84no—
182C22#38611yescharacteristic
183C22C2×#9912yes—
184C22#45512yes—
185C4—22no—
186C4—22no—
187C4—82no—
188C4—84no—
189C4—81no—
190C4—81no—
191C2—22no—
192C2—22no—
193C2—41no—
194C2—44no—
195C2—82no—
196C2#188511yescharacteristic
197C2#189811yescharacteristic
198C2#191111yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC23×D4C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1511C2semidirect
5. #1284C2semidirect
7. C2×#124C4semidirect
13. #390C4semidirect
23. C22×Q8C2×C4semidirect
36. C2×C4⋊C4C2×C4semidirect
48. C2×C42C2×C4semidirect

Maximal subgroups

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

subgroupindex
#24352
#23082
#17832
#15112
#12842