Finite Group Databasebeta

← #9463

Subgroups — #9463

Order 256.

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

normal subgroups only

This group has 517 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#2433C211yesmaximal, characteristic
2#2308C211yesmaximal, characteristic, complemented
3#1783C211yesmaximal, characteristic, complemented
4#1510C212yesmaximal, complemented
5#1284C212yesmaximal, complemented
6C23×D4C2211yescharacteristic
7C2×#127C412yescomplemented
8C2×#121C2211yescharacteristic
9C2×#120—21no—
10C2×#120—21no—
11C22×C22⋊C4—21no—
12C2×#100—22no—
13#407C2212yes—
14#392—22no—
15#390C412yescomplemented
16#386—24no—
17#381C2212yes—
18#379—21no—
19C4×C22⋊C4C2211yescharacteristic
20#327—24no—
21C25D411yescharacteristic
22C25D411yescharacteristic
23C22×D4—21no—
24C22×D4—22no—
25C22×D4—22no—
26C22×D4—22no—
27C22×D4—22no—
28C22×D4—44no—
29C22×D4C2×C412yes—
30C22×D4C2×C411yescharacteristic, complemented
31C22×D4D412yes—
32C23×C4—22no—
33C23×C4C2311yescharacteristic
34C2×M16—22no—
35#127—44no—
36#121—22no—
37#121—22no—
38#120—44no—
39#120—44no—
40C2×C4⋊C4C2×C411yescharacteristic, complemented
41C2×C22⋊C4—21no—
42C2×C22⋊C4—22no—
43C2×C22⋊C4—22no—
44C2×C22⋊C4—22no—
45C2×C22⋊C4—22no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—22no—
48C2×C22⋊C4—22no—
49C2×C22⋊C4—44no—
50C2×C42C2×C412yescomplemented
51#100—44no—
52C22⋊C8—22no—
53C22⋊C8—22no—
54#95—21no—
55#95—22no—
56#95D414yescomplemented
57C24—21no—
58C24—24no—
59C24—21no—
60C24—21no—
61C24—21no—
62C24—21no—
63C24—42no—
64C24—42no—
65C24—42no—
66C24—81no—
67C24—81no—
68C24C22⋊C412yes—
69C24C22⋊C412yes—
70C24C2×D411yescharacteristic
71C2×D4—24no—
72C2×D4—24no—
73C2×D4—24no—
74C2×D4—22no—
75C2×D4—24no—
76C2×D4—42no—
77C2×D4—42no—
78C2×D4—42no—
79C2×D4—44no—
80C2×D4—44no—
81C2×D4—44no—
82C2×D4—44no—
83C2×D4—48no—
84C2×D4—42no—
85C2×D4—42no—
86C2×D4—44no—
87C2×D4—44no—
88C2×D4—41no—
89C2×D4—41no—
90C2×D4—82no—
91C2×D4—82no—
92C22×C4—22no—
93C22×C4—21no—
94C22×C4—24no—
95C22×C4—42no—
96C22×C4—44no—
97C22×C4—82no—
98C22×C4—82no—
99C22×C4C22×C411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D412yes—
102C22×C4#4112yes—
103M16—44no—
104C2×C8—42no—
105C4⋊C4—22no—
106C22⋊C4—24no—
107C22⋊C4—44no—
108C22⋊C4—44no—
109C22⋊C4—44no—
110C22⋊C4—44no—
111C22⋊C4—42no—
112C22⋊C4—82no—
113C22⋊C4—82no—
114C42—42no—
115C23—22no—
116C23—22no—
117C23—24no—
118C23—22no—
119C23—22no—
120C23—22no—
121C23—22no—
122C23—22no—
123C23—22no—
124C23—22no—
125C23—41no—
126C23—42no—
127C23—42no—
128C23—42no—
129C23—42no—
130C23—42no—
131C23—42no—
132C23—42no—
133C23—42no—
134C23—41no—
135C23—41no—
136C23—42no—
137C23—42no—
138C23—42no—
139C23—41no—
140C23—42no—
141C23—81no—
142C23—81no—
143C23—81no—
144C23—81no—
145C23—88no—
146C23—88no—
147C23#9914yes—
148C23C2×C22⋊C411yescharacteristic
149C23#12011yescharacteristic
150C23#12311yescharacteristic
151D4—48no—
152D4—48no—
153D4—82no—
154D4—88no—
155D4—88no—
156D4—82no—
157D4—82no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—24no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—44no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—44no—
169C2×C4—41no—
170C2×C4—44no—
171C2×C4—88no—
172C2×C4C4×D412yes—
173C2×C4#12112yes—
174C8—82no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—21no—
179C22—21no—
180C22—21no—
181C22—24no—
182C22—21no—
183C22—44no—
184C22—44no—
185C22—44no—
186C22—44no—
187C22—44no—
188C22—42no—
189C22—42no—
190C22—42no—
191C22—42no—
192C22—42no—
193C22—82no—
194C22—82no—
195C22—84no—
196C22—82no—
197C22—82no—
198C22—84no—
199C22—84no—
200C22—81no—
201C22—81no—
202C22—81no—
203C22#38611yescharacteristic
204C22C2×#9912yes—
205C22#45312yes—
206C4—22no—
207C4—22no—
208C4—44no—
209C4—84no—
210C4—81no—
211C2—22no—
212C2—22no—
213C2—41no—
214C2—44no—
215C2—82no—
216C2—161no—
217C2#188711yescharacteristic
218C2#189811yescharacteristic
219C2#191111yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radical#2433C2
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
2. #2308C2semidirect
3. #1783C2semidirect
4. #1510C2semidirect
5. #1284C2semidirect
7. C2×#127C4semidirect
15. #390C4semidirect
30. C22×D4C2×C4semidirect
40. C2×C4⋊C4C2×C4semidirect
50. C2×C42C2×C4semidirect
56. #95D4semidirect

Maximal subgroups

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

subgroupindex
#24332
#23082
#17832
#15102
#12842