Finite Group Databasebeta

← #410815

Subgroups — #410815

Order 512.

1544 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1544 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, complemented
2#20912C212yesmaximal
3#14136C211yesmaximal, characteristic
4#11296C214yesmaximal, complemented
5#11054C214yesmaximal, complemented
6#10733C212yesmaximal, complemented
7#10728C211yesmaximal, characteristic, complemented
8C22×#121C2211yescharacteristic
9C22×#117C2212yes—
10C22×#117C2211yescharacteristic
11C2×#418C2214yes—
12C2×#407C2212yes—
13#2317—24no—
14#2316C2214yes—
15#2315—22no—
16#2296—24no—
17C2×#390C2212yes—
18#2285—22no—
19C2×C4×C22⋊C4C2211yescharacteristic
20#1875—24no—
21#1874—24no—
22C2×#343C2214yes—
23#1736C2214yes—
24C2×#339C2218yes—
25#1438—22no—
26#1437—22no—
27C4⋊#95C2212yes—
28C23×D4—21no—
29C23×D4C2311yescharacteristic
30C24×C4C2311yescharacteristic
31C22×M16C2314yes—
32C2×#121—22no—
33C2×#121—22no—
34C2×#121—22no—
35C2×#121—24no—
36C2×#121Q814yes—
37C2×#117—22no—
38C2×#117—22no—
39C2×#117—22no—
40C2×#117—22no—
41C2×#117—28no—
42C2×#117—48no—
43C2×#117D414yes—
44C2×#117C2314yes—
45C22×C4⋊C4D414yes—
46C22×C4⋊C4C2311yescharacteristic
47C22×C4⋊C4C2312yes—
48C22×C22⋊C4—22no—
49C22×C22⋊C4—21no—
50C22×C22⋊C4—21no—
51C22×C22⋊C4—24no—
52C22×C22⋊C4D412yes—
53C22×C42—22no—
54C22×C42D412yes—
55C22×C42C2312yes—
56#418—48no—
57C2×#102—28no—
58#407—24no—
59#407—24no—
60C2×C22⋊C8—22no—
61C2×C22⋊C8—22no—
62#390—28no—
63#389—28no—
64#389—28no—
65#385—216no—
66#384—24no—
67#382—24no—
68#380—28no—
69C4×C4⋊C4—28no—
70C4×C22⋊C4—24no—
71C2×#95—21no—
72C2×#95—24no—
73#343—28no—
74#340—28no—
75#339—48no—
76C25—21no—
77C25C2×D411yescharacteristic
78C22×D4—24no—
79C22×D4—22no—
80C22×D4—22no—
81C22×D4—24no—
82C22×D4—21no—
83C22×D4—24no—
84C22×D4—42no—
85C22×D4—42no—
86C22×D4—84no—
87C22×D4C2×Q812yes—
88C23×C4—24no—
89C23×C4—24no—
90C23×C4—22no—
91C23×C4—21no—
92C23×C4—24no—
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D412yes—
95C23×C4C2×D412yes—
96C23×C4C2×D414yes—
97C23×C4C2×Q812yes—
98C23×C4#4114yes—
99C23×C4#4112yes—
100C23×C4C2411yescharacteristic
101C2×M16—28no—
102C2×M16—24no—
103C2×M16—24no—
104C2×M16—28no—
105C22×C8—24no—
106#121—28no—
107#121—28no—
108#121—44no—
109#121—44no—
110#117—216no—
111#117—216no—
112#117—28no—
113#117—28no—
114#117—48no—
115#117—48no—
116#117—48no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—28no—
124C2×C4⋊C4—28no—
125C2×C4⋊C4—48no—
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—41no—
138C2×C22⋊C4—41no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—48no—
141C2×C22⋊C4—48no—
142C2×C22⋊C4—84no—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42—28no—
146C2×C42—24no—
147C2×C42—24no—
148C2×C42—24no—
149C2×C42—24no—
150C2×C42—24no—
151C2×C42—48no—
152#102—416no—
153C22⋊C8—44no—
154C22⋊C8—44no—
155#95—216no—
156#95—42no—
157#95—84no—
158C24—22no—
159C24—22no—
160C24—22no—
161C24—42no—
162C24—42no—
163C24—44no—
164C24—44no—
165C24—41no—
166C24—42no—
167C24—41no—
168C24—42no—
169C24#12012yes—
170C24C22⋊Q812yes—
171C24#12412yes—
172C24C22×D411yescharacteristic
173C2×D4—24no—
174C2×D4—28no—
175C2×D4—48no—
176C2×D4—44no—
177C2×D4—42no—
178C2×D4—44no—
179C2×D4—48no—
180C2×D4—44no—
181C2×D4—88no—
182C2×D4—88no—
183C2×D4—88no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—28no—
191C22×C4—216no—
192C22×C4—28no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—28no—
197C22×C4—22no—
198C22×C4—44no—
199C22×C4—44no—
200C22×C4—42no—
201C22×C4—44no—
202C22×C4—48no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—84no—
211C22×C4—84no—
212C22×C4#12011yescharacteristic
213C22×C4#12011yescharacteristic
214C22×C4C22⋊Q814yes—
215C22×C4C22⋊Q814yes—
216C22×C4C22⋊Q812yes—
217C22×C4#12318yes—
218C22×C4#12414yes—
219C22×C4#12414yes—
220C22×C4#12412yes—
221C22×C4C22×D412yes—
222C22×C4C22×Q811yescharacteristic
223C22×C4C2×#4112yes—
224C22×C4C2×#4114yes—
225C22×C4C2×#4111yescharacteristic
226C22×C4H(5,𝔽2)14yes—
227M16—48no—
228M16—48no—
229C2×C8—44no—
230C2×C8—44no—
231C2×C8—44no—
232C4⋊C4—216no—
233C4⋊C4—24no—
234C4⋊C4—28no—
235C4⋊C4—28no—
236C4⋊C4—48no—
237C4⋊C4—42no—
238C4⋊C4—88no—
239C4⋊C4—88no—
240C22⋊C4—216no—
241C22⋊C4—48no—
242C22⋊C4—48no—
243C22⋊C4—48no—
244C22⋊C4—44no—
245C22⋊C4—44no—
246C22⋊C4—816no—
247C42—28no—
248C42—28no—
249C42—48no—
250C42—48no—
251C42—48no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—24no—
257C23—24no—
258C23—24no—
259C23—24no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—44no—
265C23—44no—
266C23—42no—
267C23—44no—
268C23—82no—
269C23—82no—
270C23—82no—
271C23—82no—
272C23—84no—
273C23—82no—
274C23—82no—
275C23—84no—
276C23—84no—
277C23—84no—
278C23—81no—
279C23—81no—
280C23—82no—
281C23—82no—
282C23—82no—
283C23#39314yes—
284C23C2×#12011yescharacteristic
285C23C2×C22⋊Q811yescharacteristic
286C23C2×#12312yes—
287C23C2×#12411yescharacteristic
288C23#53512yes—
289D4—48no—
290D4—816no—
291D4—84no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—28no—
299C2×C4—28no—
300C2×C4—28no—
301C2×C4—28no—
302C2×C4—28no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—48no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—48no—
311C2×C4—42no—
312C2×C4—48no—
313C2×C4—48no—
314C2×C4—84no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—816no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4#39312yes—
323C2×C4#39312yes—
324C2×C4C2×C22⋊Q812yes—
325C2×C4#52514yes—
326C2×C4C2×#12412yes—
327C2×C4#53414yes—
328C2×C4#55318yes—
329C2×C4#55314yes—
330C2×C4#55414yes—
331C8—84no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—24no—
341C22—24no—
342C22—24no—
343C22—44no—
344C22—44no—
345C22—84no—
346C22—84no—
347C22—88no—
348C22—82no—
349C22—82no—
350C22—82no—
351C22—164no—
352C22—162no—
353C22—162no—
354C22C2×#39311yescharacteristic
355C22#248712yes—
356C22#250412yes—
357C22#261311yescharacteristic
358C22#284911yescharacteristic
359C22#301812yes—
360C22#302112yes—
361C4—22no—
362C4—22no—
363C4—48no—
364C4—84no—
365C4—82no—
366C4—88no—
367C4#257914yes—
368C4#259114yes—
369C2—22no—
370C2—22no—
371C2—44no—
372C2—162no—
373C2#1509111yescharacteristic
374C2#2346111yescharacteristic
375C2#2346311yescharacteristic
376C2#2962012yes—
377C2#3299212yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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
1. #34052C2semidirect
4. #11296C2semidirect
5. #11054C2semidirect
6. #10733C2semidirect
7. #10728C2semidirect

Maximal subgroups

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

subgroupindex
#340522
#209122
#141362
#112962
#110542
#107332
#107282