Finite Group Databasebeta

← #375668

Subgroups — #375668

Order 512.

1295 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1295 conjugacy classes of subgroups — too many to draw. Only its 250 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#35540C211yesmaximal, characteristic
2#20398C212yesmaximal, complemented
3#11167C212yesmaximal, complemented
4#10967C212yesmaximal
5#10937C214yesmaximal, complemented
6#10916C214yesmaximal, complemented
7C22×#121C2211yescharacteristic
8C2×#518C2211yescharacteristic
9C22×#100C2214yes—
10C2×#407C2214yes—
11#2873C2214yes—
12#2483C414yes—
13#2422—24no—
14#2420—24no—
15#2405C414yes—
16#2317C2212yes—
17#2309C2211yescharacteristic
18#2280C2212yes—
19C4×#100—24no—
20C4×#100—24no—
21C2×#344C2218yes—
22#1742C2214yes—
23C2×#343C2214yes—
24C23×D4C2×C412yes—
25C23×D4C2311yescharacteristic
26C24×C4C2311yescharacteristic
27C22×M16C2318yes—
28C2×#123—24no—
29C2×#121—24no—
30C2×#121—44no—
31#518—44no—
32C2×C4×D4D414yes—
33C2×C4×D4D414yes—
34C2×#117C2312yes—
35C2×#117C2312yes—
36C22×C4⋊C4C2311yescharacteristic
37C22×C22⋊C4C2×C412yes—
38C22×C22⋊C4C2×C414yes—
39C2×#100—28no—
40C2×#100—216no—
41#407—216no—
42C2×C22⋊C8—44no—
43#405—48no—
44C4×M16—28no—
45#397—28no—
46#394—28no—
47#390—24no—
48#389—24no—
49#386—24no—
50#385—24no—
51#385C2×C418yes—
52#384—22no—
53#382—22no—
54#381—24no—
55#381C2×C414yes—
56#380—22no—
57#380—22no—
58#379C2×C414yes—
59#379C2×C414yes—
60C4×C22⋊C4—24no—
61#344—216no—
62#343—24no—
63#343—24no—
64#343—44no—
65C25C22×C411yescharacteristic
66C25C22×C412yes—
67C22×D4—24no—
68C22×D4—28no—
69C22×D4—24no—
70C22×D4—44no—
71C22×D4—84no—
72C22×D4C2×D414yes—
73C23×C4—22no—
74C23×C4—28no—
75C23×C4—24no—
76C23×C4C22⋊C414yes—
77C23×C4C22⋊C414yes—
78C23×C4C22×C411yescharacteristic
79C23×C4C22×C412yes—
80C23×C4C2411yescharacteristic
81C2×M16—28no—
82C2×M16—28no—
83C2×M16—28no—
84C2×M16—28no—
85C2×M16—28no—
86C2×M16#41116yes—
87C22×C8—28no—
88#123—88no—
89#121—416no—
90C4×D4—48no—
91C4×D4—48no—
92#117—28no—
93#117—28no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—28no—
98C2×C4⋊C4C2×D412yes—
99C2×C4⋊C4C2×D412yes—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—24no—
102C2×C22⋊C4—24no—
103C2×C22⋊C4—24no—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—48no—
106C2×C22⋊C4—44no—
107C2×C22⋊C4C22⋊C418yes—
108C2×C22⋊C4C22×C414yes—
109C2×C22⋊C4C22×C414yes—
110C2×C42—22no—
111C2×C42C2×D414yes—
112#100—232no—
113C22⋊C8—416no—
114#97—48no—
115C4×C8—48no—
116#95—24no—
117#95—24no—
118#95—28no—
119C24—21no—
120C24—22no—
121C24—24no—
122C24—28no—
123C24—24no—
124C24—42no—
125C24—81no—
126C24—82no—
127C24—82no—
128C24C2×C22⋊C414yes—
129C24C23×C411yescharacteristic
130C2×D4—28no—
131C2×D4—28no—
132C2×D4—28no—
133C2×D4—48no—
134C2×D4—48no—
135C2×D4—44no—
136C2×D4—48no—
137C2×D4—44no—
138C2×D4—84no—
139C2×D4—88no—
140C2×D4—88no—
141C22×C4—21no—
142C22×C4—24no—
143C22×C4—22no—
144C22×C4—24no—
145C22×C4—22no—
146C22×C4—24no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4—44no—
150C22×C4—44no—
151C22×C4—44no—
152C22×C4—44no—
153C22×C4—48no—
154C22×C4—44no—
155C22×C4—82no—
156C22×C4—84no—
157C22×C4—84no—
158C22×C4C2×C22⋊C414yes—
159C22×C4C2×C22⋊C414yes—
160C22×C4C22×D412yes—
161C22×C4C2×#4118yes—
162C22×C4H(5,𝔽2)14yes—
163M16—216no—
164M16—216no—
165M16—48no—
166M16—48no—
167C2×C8—216no—
168C2×C8—48no—
169C2×C8—48no—
170C4⋊C4—24no—
171C4⋊C4—24no—
172C4⋊C4—44no—
173C4⋊C4—44no—
174C4⋊C4—88no—
175C22⋊C4—28no—
176C22⋊C4—28no—
177C22⋊C4—48no—
178C22⋊C4—48no—
179C22⋊C4—48no—
180C22⋊C4—84no—
181C22⋊C4—84no—
182C22⋊C4—84no—
183C42—28no—
184C42—44no—
185C23—22no—
186C23—24no—
187C23—24no—
188C23—28no—
189C23—41no—
190C23—41no—
191C23—42no—
192C23—41no—
193C23—42no—
194C23—44no—
195C23—44no—
196C23—44no—
197C23—41no—
198C23—44no—
199C23—42no—
200C23—48no—
201C23—84no—
202C23—84no—
203C23—88no—
204C23—168no—
205C23C22×C22⋊C411yescharacteristic
206C23C4×#4114yes—
207C23#51812yes—
208D4—416no—
209D4—816no—
210C2×C4—24no—
211C2×C4—24no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—42no—
222C2×C4—88no—
223C2×C4—84no—
224C2×C4—164no—
225C2×C4—164no—
226C2×C4#52514yes—
227C2×C4#53214yes—
228C2×C4#53414yes—
229C2×C4#55114yes—
230C2×C4#55318yes—
231C2×C4#55914yes—
232C8—416no—
233C22—21no—
234C22—22no—
235C22—21no—
236C22—22no—
237C22—44no—
238C22—44no—
239C22—48no—
240C22—44no—
241C22—42no—
242C22—44no—
243C22—44no—
244C22—88no—
245C22—82no—
246C22—84no—
247C22—81no—
248C22—81no—
249C22—168no—
250C22#229712yes—
251C22#230811yescharacteristic
252C22#237414yes—
253C4—44no—
254C4—44no—
255C4—84no—
256C4—82no—
257C4—81no—
258C4#260218yes—
259C2—22no—
260C2—24no—
261C2—82no—
262C2—84no—
263C2#1416711yescharacteristic
264C2#2177811yescharacteristic
265C2#2177911yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20398C2semidirect
3. #11167C2semidirect
5. #10937C2semidirect
6. #10916C2semidirect

Maximal subgroups

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

subgroupindex
#355402
#203982
#111672
#109672
#109372
#109162