Finite Group Databasebeta

← #128170

Subgroups — #128170

Order 512.

574 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 130 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#20683C211yesmaximal, characteristic
2#17357C212yesmaximal
3#7327C214yesmaximal, complemented
4C22×#117C2211yescharacteristic
5#2920—22no—
6C2×#407C412yes—
7#1818—22no—
8#1818C412yes—
9#1818C2212yes—
10#1728—22no—
11#1464—22no—
12#1464C2214yes—
13#1463C414yes—
14#1462—22no—
15#1458—22no—
16#1458C414yes—
17C24×C4C2×C411yescharacteristic
18C22×M16—22no—
19C2×#117—42no—
20C2×#117—42no—
21C2×#117C2×C411yescharacteristic
22C2×#117C2×C412yes—
23C2×#117D411yescharacteristic
24C2×#117D412yes—
25C2×#117Q811yescharacteristic
26C2×#117C2311yescharacteristic
27C22×C4⋊C4C2×C412yes—
28C22×C22⋊C4D411yescharacteristic
29C22×C22⋊C4Q811yescharacteristic
30C22×C42D412yes—
31#432—24no—
32#432—42no—
33#431—24no—
34#431—42no—
35#423—42no—
36#407—24no—
37#407—42no—
38#407—42no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8C2×C414yes—
41C4×M16—42no—
42#336—24no—
43#336—24no—
44#324—24no—
45#324C2×C418yes—
46#323—24no—
47#323—24no—
48C25C4⋊C411yescharacteristic
49C23×C4—22no—
50C23×C4—22no—
51C23×C4—44no—
52C23×C4C4211yescharacteristic
53C23×C4C4211yescharacteristic
54C23×C4C22⋊C411yescharacteristic
55C23×C4C22⋊C412yes—
56C23×C4C22⋊C411yescharacteristic
57C23×C4C4⋊C411yescharacteristic
58C23×C4C4⋊C412yes—
59C23×C4C22×C411yescharacteristic
60C2×M16—48no—
61C2×M16—44no—
62C2×M16—42no—
63C2×M16—42no—
64C22×C8—24no—
65#117—42no—
66#117—42no—
67#117—42no—
68#117—42no—
69#117—42no—
70#117—42no—
71#117—42no—
72#117—42no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4C4212yes—
76C2×C4⋊C4C22⋊C412yes—
77C2×C4⋊C4C4⋊C412yes—
78C2×C4⋊C4C22×C412yes—
79C2×C22⋊C4—41no—
80C2×C22⋊C4—41no—
81C2×C22⋊C4—41no—
82C2×C22⋊C4—41no—
83C2×C22⋊C4C22⋊C412yes—
84C2×C22⋊C4C4⋊C411yescharacteristic
85C2×C22⋊C4C4⋊C411yescharacteristic
86C2×C22⋊C4C4⋊C412yes—
87C2×C22⋊C4C2×D411yescharacteristic
88C2×C22⋊C4C2×Q811yescharacteristic
89C2×C42—22no—
90C2×C42—22no—
91C2×C42—22no—
92C2×C42—22no—
93C2×C42C22⋊C412yes—
94C2×C42C22⋊C412yes—
95C2×C42C4⋊C412yes—
96C2×C42C2×D412yes—
97C4⋊C8—44no—
98C4⋊C8—44no—
99C22⋊C8—28no—
100C22⋊C8—44no—
101C22⋊C8—44no—
102C22⋊C8—44no—
103#97—44no—
104C4×C8—44no—
105C24—21no—
106C24—21no—
107C24—42no—
108C24—82no—
109C24#9511yescharacteristic
110C24#9511yescharacteristic
111C24C2×C4⋊C411yescharacteristic
112C22×C4—24no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—24no—
122C22×C4—24no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—48no—
129C22×C4—48no—
130C22×C4—82no—
131C22×C4—82no—
132C22×C4#9511yescharacteristic
133C22×C4#9512yes—
134C22×C4#9512yes—
135C22×C4#9511yescharacteristic
136C22×C4#9912yes—
137C22×C4#9912yes—
138C22×C4C2×C4211yescharacteristic
139C22×C4C2×C22⋊C411yescharacteristic
140C22×C4C2×C22⋊C412yes—
141C22×C4C2×C4⋊C412yes—
142M16—84no—
143M16—84no—
144M16—88no—
145C2×C8—48no—
146C2×C8—48no—
147C2×C8—44no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C22⋊C4—42no—
153C22⋊C4—41no—
154C22⋊C4—41no—
155C22⋊C4—42no—
156C22⋊C4—41no—
157C22⋊C4—41no—
158C42—22no—
159C42—22no—
160C42—22no—
161C42—22no—
162C42—22no—
163C42—22no—
164C42—22no—
165C42—22no—
166C23—22no—
167C23—21no—
168C23—21no—
169C23—22no—
170C23—22no—
171C23—42no—
172C23—42no—
173C23—42no—
174C23—42no—
175C23—42no—
176C23—82no—
177C23—82no—
178C23—88no—
179C23#34211yescharacteristic
180C23#34211yescharacteristic
181C23C2×#9511yescharacteristic
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—44no—
194C2×C4—44no—
195C2×C4—44no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4#34212yes—
208C2×C4#34414yes—
209C2×C4C2×#9912yes—
210C8—84no—
211C8—88no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—21no—
216C22—21no—
217C22—44no—
218C22—42no—
219C22—42no—
220C22—41no—
221C22—41no—
222C22—82no—
223C22—82no—
224C22—84no—
225C22—84no—
226C22C2×#34211yescharacteristic
227C22C2×#34412yes—
228C4—22no—
229C4—42no—
230C4—42no—
231C4—42no—
232C4—42no—
233C4—41no—
234C4—41no—
235C4#130614yes—
236C2—22no—
237C2—41no—
238C2—41no—
239C2—81no—
240C2—81no—
241C2C2×#130611yescharacteristic
242C2#1258912yes—
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×#117C23
prime radicalC25C4⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7327C2semidirect

Maximal subgroups

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

subgroupindex
#206832
#173572
#73272