Finite Group Databasebeta

← #410817

Subgroups — #410817

Order 512.

1414 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1414 conjugacy classes of subgroups — too many to draw. Only its 242 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#34052C211yesmaximal, characteristic
2#20911C212yesmaximal
3#14138C211yesmaximal, characteristic
4#11296C214yesmaximal, complemented
5#11055C214yesmaximal, complemented
6#10733C212yesmaximal
7#10728C211yesmaximal, characteristic
8C22×C22⋊Q8C2211yescharacteristic
9C22×#117C2212yes—
10C22×#117C2211yescharacteristic
11C2×#419C2214yes—
12C2×#407C2212yes—
13#2317—24no—
14#2316C2214yes—
15#2315—22no—
16#2296—24no—
17C2×#391C2212yes—
18#2285—22no—
19C2×C4×C22⋊C4C2211yescharacteristic
20#1873—24no—
21#1872—24no—
22C2×#343C2214yes—
23#1736C2214yes—
24C2×#339C2218yes—
25#1440—22no—
26#1436—22no—
27C4⋊#95C2212yes—
28C23×Q8C2311yescharacteristic
29C24×C4C2311yescharacteristic
30C22×M16C2314yes—
31C2×C22⋊Q8—22no—
32C2×C22⋊Q8—22no—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8—24no—
35C2×C22⋊Q8Q814yes—
36C2×#117—22no—
37C2×#117—22no—
38C2×#117—22no—
39C2×#117—22no—
40C2×#117—28no—
41C2×#117—48no—
42C2×#117D414yes—
43C2×#117C2314yes—
44C22×C4⋊C4—21no—
45C22×C4⋊C4D414yes—
46C22×C4⋊C4C2311yescharacteristic
47C22×C4⋊C4C2312yes—
48C22×C22⋊C4—22no—
49C22×C22⋊C4—21no—
50C22×C22⋊C4—21no—
51C22×C22⋊C4D412yes—
52C22×C42—22no—
53C22×C42D412yes—
54C22×C42C2312yes—
55#419—48no—
56C2×#103—28no—
57#407—24no—
58#407—24no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8—22no—
61#391—28no—
62#389—28no—
63#389—28no—
64#385—216no—
65#384—24no—
66#382—24no—
67#380—28no—
68C4×C4⋊C4—28no—
69C4×C22⋊C4—24no—
70C2×#95—24no—
71C2×#95—21no—
72C2×#95—24no—
73#343—28no—
74#340—28no—
75#339—48no—
76C25C2×D411yescharacteristic
77C22×Q8—21no—
78C22×Q8—22no—
79C22×Q8—22no—
80C22×Q8—24no—
81C22×Q8—24no—
82C22×Q8C2×Q812yes—
83C23×C4—24no—
84C23×C4—24no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—22no—
88C23×C4—24no—
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D412yes—
91C23×C4C2×D412yes—
92C23×C4C2×D414yes—
93C23×C4C2×Q812yes—
94C23×C4#4114yes—
95C23×C4#4112yes—
96C23×C4C2411yescharacteristic
97C2×M16—28no—
98C2×M16—24no—
99C2×M16—24no—
100C2×M16—28no—
101C22×C8—24no—
102C22⋊Q8—28no—
103C22⋊Q8—28no—
104C22⋊Q8—44no—
105C22⋊Q8—44no—
106#117—216no—
107#117—216no—
108#117—28no—
109#117—28no—
110#117—48no—
111#117—48no—
112#117—48no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—28no—
121C2×C4⋊C4—28no—
122C2×C4⋊C4—48no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4C2×D418yes—
127C2×C4⋊C4C2×Q812yes—
128C2×C4⋊C4#4118yes—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—28no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—44no—
135C2×C22⋊C4—48no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—42no—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42—28no—
142C2×C42—24no—
143C2×C42—24no—
144C2×C42—24no—
145C2×C42—24no—
146C2×C42—24no—
147C2×C42—48no—
148#103—416no—
149C22⋊C8—44no—
150C22⋊C8—44no—
151#95—216no—
152#95—48no—
153#95—48no—
154#95—42no—
155C24—22no—
156C24—22no—
157C24—42no—
158C24—42no—
159C24#12012yes—
160C24C22⋊Q812yes—
161C24#12412yes—
162C24C22×D411yescharacteristic
163C2×Q8—24no—
164C2×Q8—28no—
165C2×Q8—42no—
166C2×Q8—44no—
167C2×Q8—44no—
168C2×Q8—44no—
169C2×Q8—48no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—28no—
177C22×C4—216no—
178C22×C4—28no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—28no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—48no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—84no—
202C22×C4—84no—
203C22×C4#12011yescharacteristic
204C22×C4#12011yescharacteristic
205C22×C4C22⋊Q814yes—
206C22×C4C22⋊Q814yes—
207C22×C4C22⋊Q812yes—
208C22×C4#12318yes—
209C22×C4#12414yes—
210C22×C4#12414yes—
211C22×C4#12412yes—
212C22×C4C22×D412yes—
213C22×C4C22×Q811yescharacteristic
214C22×C4C2×#4112yes—
215C22×C4C2×#4114yes—
216C22×C4C2×#4111yescharacteristic
217C22×C4H(5,𝔽2)14yes—
218M16—48no—
219M16—48no—
220C2×C8—44no—
221C2×C8—44no—
222C2×C8—44no—
223C4⋊C4—216no—
224C4⋊C4—24no—
225C4⋊C4—28no—
226C4⋊C4—28no—
227C4⋊C4—48no—
228C4⋊C4—42no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—88no—
232C4⋊C4—88no—
233C22⋊C4—216no—
234C22⋊C4—48no—
235C22⋊C4—48no—
236C22⋊C4—48no—
237C22⋊C4—48no—
238C42—28no—
239C42—28no—
240C42—48no—
241C42—48no—
242C42—48no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—24no—
248C23—24no—
249C23—24no—
250C23—24no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—82no—
256C23—82no—
257C23—82no—
258C23—82no—
259C23#39314yes—
260C23C2×#12011yescharacteristic
261C23C2×C22⋊Q811yescharacteristic
262C23C2×#12312yes—
263C23C2×#12411yescharacteristic
264C23#53512yes—
265Q8—48no—
266Q8—84no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—28no—
274C2×C4—28no—
275C2×C4—28no—
276C2×C4—28no—
277C2×C4—28no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—48no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—48no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—48no—
290C2×C4—48no—
291C2×C4—84no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—816no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4#39312yes—
303C2×C4#39312yes—
304C2×C4C2×C22⋊Q812yes—
305C2×C4#52514yes—
306C2×C4C2×#12412yes—
307C2×C4#53414yes—
308C2×C4#55318yes—
309C2×C4#55314yes—
310C2×C4#55414yes—
311C8—84no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—24no—
321C22—24no—
322C22—24no—
323C22—44no—
324C22—44no—
325C22—84no—
326C22—84no—
327C22C2×#39311yescharacteristic
328C22#248712yes—
329C22#250412yes—
330C22#261311yescharacteristic
331C22#284911yescharacteristic
332C22#301812yes—
333C22#302212yes—
334C4—22no—
335C4—22no—
336C4—48no—
337C4—84no—
338C4—82no—
339C4—88no—
340C4—82no—
341C4#257914yes—
342C4#259114yes—
343C2—22no—
344C2—22no—
345C2—44no—
346C2#1509111yescharacteristic
347C2#2346111yescharacteristic
348C2#2346411yescharacteristic
349C2#2962112yes—
350C2#3299512yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11296C2semidirect
5. #11055C2semidirect

Maximal subgroups

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

subgroupindex
#340522
#209112
#141382
#112962
#110552
#107332
#107282