Finite Group Databasebeta

← #241155

Subgroups — #241155

Order 512.

1128 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1128 conjugacy classes of subgroups — too many to draw. Only its 210 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×#2646C211yesmaximal, characteristic
2#21942C212yesmaximal
3#13965C211yesmaximal, characteristic
4#10572C218yesmaximal, complemented
5#8019C212yesmaximal
6#7909C211yesmaximal, characteristic
7C2×#407C2212yes—
8#2646—24no—
9#2646—22no—
10#2646C2214yes—
11C2×#400—22no—
12C2×#398—22no—
13C2×#385—22no—
14C2×#385—22no—
15C2×#380C2212yes—
16C2×#380C2211yescharacteristic
17C2×C4×C22⋊C4C2211yescharacteristic
18C22×#95C2211yescharacteristic
19#1853—22no—
20#1853C2214yes—
21#1852—22no—
22#1852C2214yes—
23#1809—24no—
24#1444C2214yes—
25C4⋊#95—22no—
26#1294C2218yes—
27#1280C2214yes—
28C24×C4D412yes—
29C24×C4C2311yescharacteristic
30C22×M16—22no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—24no—
34C22×C4⋊C4D412yes—
35C22×C4⋊C4D412yes—
36C22×C4⋊C4C2312yes—
37C22×C22⋊C4—21no—
38C22×C22⋊C4—22no—
39C22×C22⋊C4—21no—
40C22×C42—21no—
41C2×#107—24no—
42C2×#106—24no—
43#407—24no—
44#407Q818yes—
45C2×C22⋊C8—22no—
46C2×C22⋊C8C2314yes—
47#400—28no—
48#400—28no—
49#398—28no—
50#398—28no—
51#385—28no—
52#385—28no—
53#385—48no—
54#380—22no—
55#380—22no—
56#380—22no—
57#380—22no—
58#380D414yes—
59#380C2314yes—
60C4×C22⋊C4—22no—
61C4×C22⋊C4Q814yes—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95—22no—
66C2×#95—22no—
67C2×#95—21no—
68C2×#95—24no—
69C2×#95D412yes—
70C2×#95C2312yes—
71C2×#95C2312yes—
72#340—24no—
73#340—24no—
74C25C2×D411yescharacteristic
75C23×C4—22no—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4—44no—
82C23×C4—44no—
83C23×C4—82no—
84C23×C4—82no—
85C23×C4C2×D414yes—
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D412yes—
88C23×C4C2×D412yes—
89C23×C4C2×Q814yes—
90C23×C4C2×Q812yes—
91C23×C4#4112yes—
92C23×C4C2411yescharacteristic
93C2×M16—28no—
94C2×M16—82no—
95C2×M16—82no—
96C22×C8—24no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—28no—
100C2×C4⋊C4—28no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—48no—
104C2×C4⋊C4—48no—
105C2×C4⋊C4—82no—
106C2×C4⋊C4—82no—
107C2×C4⋊C4—82no—
108C2×C4⋊C4—84no—
109C2×C4⋊C4—84no—
110C2×C4⋊C4—84no—
111C2×C4⋊C4C2×D414yes—
112C2×C4⋊C4C2×D414yes—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—24no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—48no—
119C2×C22⋊C4—44no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—41no—
122C2×C42—24no—
123C2×C42—82no—
124#107—84no—
125#106—84no—
126C22⋊C8—28no—
127C22⋊C8—24no—
128C22⋊C8C2×Q818yes—
129#95—28no—
130#95—28no—
131#95—24no—
132#95—24no—
133#95—24no—
134#95—24no—
135#95—22no—
136#95—48no—
137#95—42no—
138#95C2×Q814yes—
139C24—24no—
140C24—22no—
141C24—22no—
142C24—42no—
143C24#12011yescharacteristic
144C24#12011yescharacteristic
145C24C22⋊Q812yes—
146C24C4⋊Q812yes—
147C24C22×D411yescharacteristic
148C22×C4—24no—
149C22×C4—24no—
150C22×C4—24no—
151C22×C4—24no—
152C22×C4—24no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—24no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—44no—
163C22×C4—44no—
164C22×C4—44no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—44no—
173C22×C4—44no—
174C22×C4—44no—
175C22×C4—44no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—44no—
179C22×C4—42no—
180C22×C4—44no—
181C22×C4—82no—
182C22×C4—84no—
183C22×C4—82no—
184C22×C4—82no—
185C22×C4—82no—
186C22×C4—816no—
187C22×C4#12012yes—
188C22×C4C22⋊Q814yes—
189C22×C4C22⋊Q814yes—
190C22×C4C22⋊Q812yes—
191C22×C4C4⋊Q814yes—
192C22×C4C4⋊Q814yes—
193C22×C4C4⋊Q812yes—
194C22×C4C22×D412yes—
195C22×C4C22×Q812yes—
196C22×C4C22×Q811yescharacteristic
197C22×C4C2×#4111yescharacteristic
198M16—88no—
199C2×C8—28no—
200C2×C8—44no—
201C2×C8—44no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—82no—
207C4⋊C4—82no—
208C4⋊C4—88no—
209C4⋊C4—88no—
210C4⋊C4—816no—
211C22⋊C4—216no—
212C22⋊C4—48no—
213C22⋊C4—48no—
214C22⋊C4—82no—
215C22⋊C4—82no—
216C42—84no—
217C23—22no—
218C23—21no—
219C23—21no—
220C23—22no—
221C23—22no—
222C23—24no—
223C23—24no—
224C23—24no—
225C23—24no—
226C23—22no—
227C23—21no—
228C23—21no—
229C23—22no—
230C23—42no—
231C23—42no—
232C23—48no—
233C23—48no—
234C23#39512yes—
235C23#39512yes—
236C23#45312yes—
237C23#45612yes—
238C23C2×#12011yescharacteristic
239C23C2×C22⋊Q811yescharacteristic
240C23C2×C4⋊Q811yescharacteristic
241C23#57514yes—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—28no—
247C2×C4—42no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—44no—
257C2×C4—48no—
258C2×C4—48no—
259C2×C4—44no—
260C2×C4—48no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—88no—
271C2×C4—88no—
272C2×C4—88no—
273C2×C4—88no—
274C2×C4—84no—
275C2×C4—84no—
276C2×C4—84no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4#39514yes—
281C2×C4C2×C22⋊Q812yes—
282C2×C4C2×C4⋊Q812yes—
283C8—48no—
284C22—22no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—24no—
291C22—24no—
292C22—24no—
293C22—24no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—48no—
298C22—48no—
299C22—42no—
300C22—42no—
301C22#206014yes—
302C22C2×#39511yescharacteristic
303C22#300512yes—
304C22C2×#45311yescharacteristic
305C22C2×#45611yescharacteristic
306C22#302612yes—
307C22#308012yes—
308C22#316612yes—
309C4—22no—
310C4—22no—
311C4—24no—
312C4—48no—
313C4—84no—
314C4—84no—
315C4—84no—
316C4—162no—
317C2—22no—
318C2—24no—
319C2—22no—
320C2—42no—
321C2#1832112yes—
322C2#1832812yes—
323C2#2355311yescharacteristic
324C2C2×#206011yescharacteristic
325C2#2356511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10572C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26462
#219422
#139652
#105722
#80192
#79092