Finite Group Databasebeta

← #193255

Subgroups — #193255

Order 512.

1122 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 1122 conjugacy classes of subgroups — too many to draw. Only its 282 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×#2369C211yesmaximal, characteristic
2#21365C212yesmaximal
3#21202C211yesmaximal, characteristic
4C2×#1558C211yesmaximal, characteristic
5#8929C214yesmaximal, complemented
6#8928C214yesmaximal, complemented
7#7904C212yesmaximal
8C22×C4⋊Q8C2211yescharacteristic
9C2×#432C2212yes—
10C2×#419—22no—
11C2×C4×M16C2211yescharacteristic
12#2369—24no—
13#2369—22no—
14#2369C2214yes—
15C2×#391—22no—
16C2×#389—22no—
17#2285C2211yescharacteristic
18C2×C4×C4⋊C4—22no—
19C2×C4×C4⋊C4C2212yes—
20#1960—22no—
21#1956C2212yes—
22#1956C2212yes—
23#1948C2214yes—
24#1947C2214yes—
25#1853—22no—
26#1852—22no—
27#1776—24no—
28#1558C418yescomplemented
29C2×#333C2214yes—
30#1279C2218yes—
31C23×Q8D412yes—
32C22×M16—21no—
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8—24no—
35C2×C4⋊Q8C2×C414yes—
36C2×C4⋊Q8D412yes—
37C2×C4⋊Q8Q814yes—
38C2×C4⋊Q8C2312yes—
39C2×#117—21no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4D412yes—
43C22×C4⋊C4D412yes—
44C22×C42—22no—
45C22×C42—22no—
46C22×C42C2311yescharacteristic
47#432—24no—
48#432C2×C418yes—
49C2×#107—24no—
50C2×#106—24no—
51C2×C4⋊C8C2314yes—
52#419—48no—
53C2×#103—24no—
54C2×#103—24no—
55C2×C22⋊C8—22no—
56C4×M16—24no—
57C2×#97—22no—
58C2×#97—21no—
59C2×C4×C8C2312yes—
60#391—28no—
61#391—28no—
62#389—28no—
63#389—28no—
64#384C2311yescharacteristic
65#384C2311yescharacteristic
66#382—21no—
67#380—22no—
68C4×C4⋊C4—28no—
69C4×C4⋊C4—28no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4D414yes—
74C4×C4⋊C4C2314yes—
75C2×#95—24no—
76C2×#95—22no—
77#340—24no—
78#340—24no—
79#333C2×C4116yescomplemented
80C22×Q8—22no—
81C22×Q8—24no—
82C22×Q8—44no—
83C22×Q8—44no—
84C22×Q8C4⋊C418yes—
85C22×Q8C2×D414yes—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D412yes—
91C2×M16—24no—
92C2×M16—42no—
93C2×M16—42no—
94C22×C8—22no—
95C22×C8—24no—
96C4⋊Q8—28no—
97C4⋊Q8—28no—
98C4⋊Q8—24no—
99C4⋊Q8—24no—
100C4⋊Q8C4⋊C418yes—
101C4⋊Q8C22×C414yes—
102C4⋊Q8C2×Q814yes—
103#117—44no—
104C2×C4⋊C4—28no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—28no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—28no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—28no—
116C2×C4⋊C4—44no—
117C2×C4⋊C4—84no—
118C2×C4⋊C4C2×D414yes—
119C2×C4⋊C4C2×D414yes—
120C2×C22⋊C4—41no—
121C2×C42—24no—
122C2×C42—24no—
123C2×C42—28no—
124C2×C42—48no—
125C2×C42—41no—
126C2×C42—84no—
127C2×C42C22×C412yes—
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D412yes—
130C2×C42C2×Q812yes—
131C2×C42#4114yes—
132C2×C42C2411yescharacteristic
133#107—48no—
134#106—48no—
135C4⋊C8—24no—
136C4⋊C8C22×C418yes—
137#103—48no—
138#103—48no—
139C22⋊C8—28no—
140#97—28no—
141#97—42no—
142C4×C8—24no—
143#95—216no—
144#95—24no—
145#95—24no—
146#95—42no—
147C24#12011yescharacteristic
148C2×Q8—24no—
149C2×Q8—28no—
150C2×Q8—44no—
151C2×Q8—416no—
152C2×Q8—44no—
153C2×Q8—88no—
154C2×Q8—82no—
155C2×Q8—82no—
156C2×Q8C2×C4⋊C418yes—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—28no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—28no—
165C22×C4—24no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—44no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4C2×C4⋊C412yes—
176C22×C4C4×D414yes—
177C22×C4#12011yescharacteristic
178C22×C4#12012yes—
179C22×C4C22⋊Q814yes—
180C22×C4#12312yes—
181C22×C4C22×D411yescharacteristic
182C22×C4C22×D412yes—
183M16—48no—
184C2×C8—24no—
185C2×C8—28no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—44no—
189C2×C8—44no—
190C4⋊C4—28no—
191C4⋊C4—28no—
192C4⋊C4—28no—
193C4⋊C4—24no—
194C4⋊C4—216no—
195C4⋊C4—24no—
196C4⋊C4—24no—
197C4⋊C4—44no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—44no—
201C4⋊C4—44no—
202C4⋊C4—88no—
203C4⋊C4#13718yes—
204C22⋊C4—82no—
205C42—48no—
206C42—48no—
207C42—82no—
208C42—88no—
209C42C2×C4⋊C412yes—
210C42C4×D414yes—
211C42C22⋊Q814yes—
212C42#12312yes—
213C42C23×C411yescharacteristic
214C42C22×Q811yescharacteristic
215C42C2×#4112yes—
216C23—22no—
217C23—22no—
218C23#38512yes—
219C23#45312yes—
220C23#45512yes—
221C23C2×#12011yescharacteristic
222Q8—48no—
223Q8—84no—
224Q8—88no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—28no—
234C2×C4—28no—
235C2×C4—28no—
236C2×C4—28no—
237C2×C4—416no—
238C2×C4—416no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—88no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4#38514yes—
252C2×C4#38512yes—
253C2×C4C22×C4⋊C411yescharacteristic
254C2×C4C2×C4×D412yes—
255C2×C4C2×C22⋊Q812yes—
256C2×C4C2×#12311yescharacteristic
257C2×C4C2×#13714yes—
258C2×C4#57514yes—
259C8—44no—
260C8—48no—
261C22—24no—
262C22—22no—
263C22—22no—
264C22—24no—
265C22#185614yes—
266C22C2×#38511yescharacteristic
267C22#300412yes—
268C22#300512yes—
269C22C2×#45311yescharacteristic
270C22C2×#45511yescharacteristic
271C4—21no—
272C4—21no—
273C4—22no—
274C4—24no—
275C4—48no—
276C4—48no—
277C4—84no—
278C4—161no—
279C4—161no—
280C4C4×#13714yes—
281C4#307814yes—
282C2—24no—
283C2#1853012yes—
284C2#1854611yescharacteristic
285C2#1854711yescharacteristic
286C2C2×#185611yescharacteristic
287C2#2087412yes—
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #8929C2semidirect
6. #8928C2semidirect
28. #1558C4semidirect
79. #333C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23692
#213652
#212022
C2×#15582
#89292
#89282
#79042