Finite Group Databasebeta

← #213552

Subgroups — #213552

Order 512.

1202 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1202 conjugacy classes of subgroups — too many to draw. Only its 226 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
1C2×#2326C211yesmaximal, characteristic
2#21822C212yesmaximal, complemented
3#9084C218yesmaximal, complemented
4#8699C212yesmaximal, complemented
5#8302C212yesmaximal
6C22×#124C411yescharacteristic
7C22×C4×Q8C2211yescharacteristic
8C22×C4×D4C2211yescharacteristic
9C2×#432C2214yes—
10#2326—28no—
11C2×#391C411yescharacteristic
12C2×#390C411yescharacteristic
13C2×#388C2214yes—
14C2×#385C414yes—
15C2×#384C411yescharacteristic
16C2×#376C2211yescharacteristic
17#1985—28no—
18#1817C2214yes—
19#1817C2214yes—
20C2×#323C2218yes—
21#1280C2218yes—
22C23×Q8C2×C411yescharacteristic
23C23×D4C2×C411yescharacteristic
24C24×C4C2×C412yes—
25C2×#124—24no—
26C2×#124—24no—
27C2×#124—24no—
28C2×C4×Q8—44no—
29C2×C4×Q8D418yes—
30C2×C4×D4—24no—
31C2×C4×D4—44no—
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4C2×C414yes—
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4C2×C412yes—
36C22×C22⋊C4C2×C414yes—
37C22×C22⋊C4C2×C414yes—
38C22×C22⋊C4C2312yes—
39C22×C42C2×C411yescharacteristic
40C22×C42C2×C411yescharacteristic
41C22×C42C2311yescharacteristic
42C22×C42C2312yes—
43#432—48no—
44C2×C4⋊C8—24no—
45C2×C22⋊C8C2318yes—
46C2×#97—24no—
47#391—24no—
48#390—24no—
49#388—28no—
50#388—28no—
51#385—216no—
52#384—24no—
53#376—24no—
54C2×#95C2×C412yes—
55C2×#95C2×C414yes—
56C2×#95C2312yes—
57#336—28no—
58#336—28no—
59#323—48no—
60C25C22×C412yes—
61C22×Q8—22no—
62C22×Q8—22no—
63C22×Q8—42no—
64C22×Q8—42no—
65C22×Q8C22⋊C414yes—
66C22×D4—22no—
67C22×D4—22no—
68C22×D4—42no—
69C22×D4—42no—
70C22×D4M1614yes—
71C23×C4—24no—
72C23×C4—44no—
73C23×C4—84no—
74C23×C4C22×C412yes—
75C23×C4C22×C411yescharacteristic
76C23×C4C22×C412yes—
77C23×C4C22×C412yes—
78C23×C4C22×C412yes—
79C23×C4C22×C411yescharacteristic
80C23×C4C22×C412yes—
81C23×C4C2411yescharacteristic
82C22×C8—28no—
83#124—232no—
84C4×Q8—48no—
85C4×Q8—48no—
86C4×D4—416no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—28no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—44no—
92C2×C4⋊C4—84no—
93C2×C4⋊C4C22⋊C414yes—
94C2×C4⋊C4C22⋊C418yes—
95C2×C22⋊C4—28no—
96C2×C22⋊C4—28no—
97C2×C22⋊C4—28no—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—28no—
100C2×C22⋊C4—28no—
101C2×C22⋊C4—84no—
102C2×C42—22no—
103C2×C42—22no—
104C2×C42—24no—
105C2×C42—28no—
106C2×C42—22no—
107C2×C42—42no—
108C2×C42M1614yes—
109C2×C42C2×D414yes—
110C2×C42C2×D414yes—
111C2×C42C2×D414yes—
112C4⋊C8—48no—
113C22⋊C8—216no—
114#97—48no—
115#95—24no—
116#95—28no—
117#95—24no—
118C24—22no—
119C24—22no—
120C24—24no—
121C24—24no—
122C24—44no—
123C24—44no—
124C24C2×M1614yes—
125C24C23×C411yescharacteristic
126C2×Q8—28no—
127C2×Q8—44no—
128C2×Q8—44no—
129C2×Q8—48no—
130C2×Q8—48no—
131C2×D4—28no—
132C2×D4—48no—
133C2×D4—44no—
134C2×D4—44no—
135C2×D4—48no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—24no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—24no—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—24no—
149C22×C4—24no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—44no—
153C22×C4—48no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—44no—
158C22×C4—42no—
159C22×C4—88no—
160C22×C4—816no—
161C22×C4C2×C22⋊C414yes—
162C22×C4C2×C22⋊C412yes—
163C22×C4C2×C22⋊C414yes—
164C22×C4C2×C22⋊C412yes—
165C22×C4C2×M1612yes—
166C22×C4C2×M1612yes—
167C22×C4C2×M1614yes—
168C22×C4C22×D412yes—
169C22×C4H(5,𝔽2)14yes—
170C2×C8—416no—
171C2×C8—48no—
172C4⋊C4—44no—
173C4⋊C4—48no—
174C4⋊C4—44no—
175C4⋊C4—88no—
176C22⋊C4—232no—
177C22⋊C4—48no—
178C22⋊C4—88no—
179C42—24no—
180C42—24no—
181C42—28no—
182C42—28no—
183C42—28no—
184C42—44no—
185C23—22no—
186C23—22no—
187C23—24no—
188C23—28no—
189C23—48no—
190C23—48no—
191C23—42no—
192C23—42no—
193C23—44no—
194C23—44no—
195C23—44no—
196C23—84no—
197C23—88no—
198C23—84no—
199C23C22×C22⋊C411yescharacteristic
200C23#51812yes—
201C23C22×M1611yescharacteristic
202C23C22×M1611yescharacteristic
203C23#56812yes—
204Q8—416no—
205D4—416no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—24no—
211C2×C4—22no—
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—24no—
215C2×C4—24no—
216C2×C4—24no—
217C2×C4—28no—
218C2×C4—24no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—44no—
232C2×C4—84no—
233C2×C4—816no—
234C2×C4—84no—
235C2×C4—84no—
236C2×C4#40714yes—
237C2×C4#53414yes—
238C8—88no—
239C22—24no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—22no—
244C22—48no—
245C22—44no—
246C22—88no—
247C22—88no—
248C22—84no—
249C22—82no—
250C22—82no—
251C22#230811yescharacteristic
252C22C2×#40711yescharacteristic
253C22#287911yescharacteristic
254C22#288412yes—
255C22#288812yes—
256C22#297312yes—
257C22#297512yes—
258C4—24no—
259C4—24no—
260C4—44no—
261C4—44no—
262C4—42no—
263C4—44no—
264C4—82no—
265C4—82no—
266C2—22no—
267C2—22no—
268C2—84no—
269C2#1737212yes—
270C2#1740912yes—
271C2#2174211yescharacteristic
272C2#2175111yescharacteristic
273C2#2175511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21822C2semidirect
3. #9084C2semidirect
4. #8699C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23262
#218222
#90842
#86992
#83022