Finite Group Databasebeta

← #391896

Subgroups — #391896

Order 512.

1478 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1478 conjugacy classes of subgroups — too many to draw. Only its 258 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×#2857C211yesmaximal, characteristic
2#14683C212yesmaximal
3#11381C214yesmaximal
4#11353C218yesmaximal
5C22×C4⋊Q8C411yescharacteristic, complemented
6C22×C22⋊Q8C2212yes—
7C23×C42C2211yescharacteristic
8#2857—22no—
9#2857—24no—
10#2857—22no—
11C2×#391C412yescomplemented
12C2×#389C414yescomplemented
13C2×#384C411yescharacteristic, complemented
14C2×#383—22no—
15C2×#380C2214yes—
16C2×#342C22116yes—
17#1434—24no—
18#1434—22no—
19C22⋊#95C2218yes—
20C22⋊#95C2214yes—
21C23×Q8C2×C412yes—
22C24×C4C2311yescharacteristic
23C24×C4C2312yes—
24C2×C4⋊Q8—22no—
25C2×C4⋊Q8—21no—
26C2×C4⋊Q8—21no—
27C2×C4⋊Q8—44no—
28C2×C22⋊Q8—24no—
29C2×C22⋊Q8—24no—
30C2×C22⋊Q8—48no—
31C22×C4⋊C4C2×C412yes—
32C22×C4⋊C4C2×C414yes—
33C22×C4⋊C4C2×C414yes—
34C22×C4⋊C4C2×C414yes—
35C22×C22⋊C4C2318yes—
36C22×C22⋊C4C2314yes—
37C22×C42—44no—
38C22×C42C2×C411yescharacteristic
39C22×C42C2×C412yes—
40C22×C42C2×C411yescharacteristic
41C22×C42D412yes—
42C22×C42D414yes—
43C22×C42D412yes—
44#391—24no—
45#391—24no—
46#389—28no—
47#389—28no—
48#384—22no—
49#384—21no—
50#384—21no—
51#383—48no—
52#380—28no—
53#380—28no—
54C2×#95—24no—
55C2×#95—28no—
56C2×#95—24no—
57C2×#95—28no—
58C2×#95C2×C418yes—
59#342—416no—
60C25C2411yescharacteristic
61C22×Q8—22no—
62C22×Q8—22no—
63C22×Q8—44no—
64C22×Q8—44no—
65C22×Q8—44no—
66C23×C4—24no—
67C23×C4—28no—
68C23×C4—28no—
69C23×C4—44no—
70C23×C4—42no—
71C23×C4—44no—
72C23×C4—42no—
73C23×C4C22×C412yes—
74C23×C4C22×C411yescharacteristic
75C23×C4C22×C414yes—
76C23×C4C22×C414yes—
77C23×C4C22×C411yescharacteristic
78C23×C4C22×C412yes—
79C23×C4C2×D412yes—
80C23×C4C2×D412yes—
81C23×C4C2×D414yes—
82C23×C4C2×D414yes—
83C4⋊Q8—44no—
84C4⋊Q8—48no—
85C4⋊Q8—44no—
86C22⋊Q8—416no—
87C22⋊Q8—416no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—28no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—44no—
96C2×C4⋊C4—48no—
97C2×C4⋊C4—48no—
98C2×C4⋊C4—48no—
99C2×C22⋊C4—28no—
100C2×C22⋊C4—216no—
101C2×C22⋊C4—216no—
102C2×C22⋊C4—232no—
103C2×C42—24no—
104C2×C42—28no—
105C2×C42—44no—
106C2×C42—42no—
107C2×C42—48no—
108C2×C42—44no—
109C2×C42C22⋊C412yes—
110C2×C42C22⋊C411yescharacteristic
111C2×C42C22⋊C414yes—
112C2×C42C22⋊C412yes—
113C2×C42C22⋊C414yes—
114C2×C42C22⋊C412yes—
115C2×C42C22⋊C411yescharacteristic
116#95—216no—
117#95—416no—
118#95—48no—
119#95—416no—
120#95—48no—
121C24—41no—
122C24—41no—
123C24—42no—
124C24C23×C411yescharacteristic
125C24C22×D412yes—
126C24H(5,𝔽2)14yes—
127C242−1+418yes—
128C2×Q8—44no—
129C2×Q8—44no—
130C2×Q8—44no—
131C2×Q8—44no—
132C2×Q8—44no—
133C2×Q8—44no—
134C2×Q8—816no—
135C22×C4—22no—
136C22×C4—24no—
137C22×C4—24no—
138C22×C4—22no—
139C22×C4—24no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—24no—
143C22×C4—24no—
144C22×C4—24no—
145C22×C4—28no—
146C22×C4—28no—
147C22×C4—28no—
148C22×C4—24no—
149C22×C4—28no—
150C22×C4—24no—
151C22×C4—24no—
152C22×C4—24no—
153C22×C4—28no—
154C22×C4—28no—
155C22×C4—41no—
156C22×C4—42no—
157C22×C4—44no—
158C22×C4—42no—
159C22×C4—41no—
160C22×C4—42no—
161C22×C4—44no—
162C22×C4—42no—
163C22×C4—432no—
164C22×C4—416no—
165C22×C4—44no—
166C22×C4—48no—
167C22×C4—44no—
168C22×C4—48no—
169C22×C4—48no—
170C22×C4—416no—
171C22×C4—48no—
172C22×C4—416no—
173C22×C4C2×C22⋊C414yes—
174C22×C4C2×C22⋊C412yes—
175C22×C4C2×C22⋊C412yes—
176C22×C4C2×C22⋊C411yescharacteristic
177C22×C4C2×C22⋊C412yes—
178C22×C4C2×C22⋊C411yescharacteristic
179C4⋊C4—48no—
180C4⋊C4—44no—
181C4⋊C4—44no—
182C4⋊C4—416no—
183C4⋊C4—48no—
184C4⋊C4—48no—
185C4⋊C4—48no—
186C22⋊C4—416no—
187C22⋊C4—432no—
188C42—28no—
189C42—24no—
190C42—22no—
191C42—24no—
192C42—44no—
193C42—42no—
194C42#9914yes—
195C23—21no—
196C23—22no—
197C23—21no—
198C23—22no—
199C23—22no—
200C23—24no—
201C23—44no—
202C23—48no—
203C23—44no—
204C23—48no—
205C23C22×C22⋊C411yescharacteristic
206C23#51812yes—
207C23#51914yes—
208C23#53514yes—
209C23#53618yes—
210C23#563116yes—
211Q8—88no—
212Q8—88no—
213C2×C4—216no—
214C2×C4—28no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—48no—
230C2×C4—48no—
231C2×C4—416no—
232C2×C4—416no—
233C2×C4—44no—
234C2×C4—48no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—48no—
239C2×C4—44no—
240C2×C4—88no—
241C2×C4—88no—
242C2×C4—88no—
243C2×C4—88no—
244C2×C4—816no—
245C2×C4—816no—
246C2×C4—88no—
247C2×C4C2×#9914yes—
248C2×C4C2×#9912yes—
249C2×C4#41014yes—
250C2×C4#41012yes—
251C22—22no—
252C22—24no—
253C22—22no—
254C22—24no—
255C22—24no—
256C22—28no—
257C22—28no—
258C22—24no—
259C22—416no—
260C22#230911yescharacteristic
261C22#231012yes—
262C22#238114yes—
263C22#282118yes—
264C22C22×#9911yescharacteristic
265C22C2×#41011yescharacteristic
266C22C2×#41012yes—
267C4—22no—
268C4—22no—
269C4—24no—
270C4—24no—
271C4—42no—
272C4—44no—
273C4—84no—
274C4—84no—
275C4—168no—
276C2—28no—
277C2—24no—
278C2#1466311yescharacteristic
279C2#2179311yescharacteristic
280C2#2179411yescharacteristic
281C2#2182512yes—
282C2#2182612yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C22×C4⋊Q8C4semidirect
11. C2×#391C4semidirect
12. C2×#389C4semidirect
13. C2×#384C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
#146832
#113812
#113532