Finite Group Databasebeta

← #416743

Subgroups — #416743

Order 512.

1404 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2#14149C211yesmaximal, characteristic
3#13966C212yesmaximal
4#11390C214yesmaximal
5#11222C214yesmaximal
6#10854C212yesmaximal
7C4×#342C211yesmaximal, characteristic
8C2×#543C2212yes—
9#2857—22no—
10#2659—22no—
11#2549—24no—
12#2440—24no—
13#2315C2212yes—
14C2×#385C2212yes—
15C2×#380C2211yescharacteristic
16C2×#380C2212yes—
17C2×C4×C22⋊C4C2212yes—
18C2×C4×C22⋊C4C2211yescharacteristic
19C22×#95C2211yescharacteristic
20C2×#342C2212yes—
21C2×#342C2212yes—
22C2×#342C2218yes—
23#1442—24no—
24#1441—22no—
25#1436—22no—
26#1434—24no—
27C22⋊#95C2212yes—
28C22⋊#95C2218yes—
29C24×C4C2312yes—
30C24×C4C2311yescharacteristic
31#543—28no—
32#543—44no—
33C2×C4⋊Q8—22no—
34C2×C22⋊Q8—24no—
35C2×C22⋊Q8—28no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4D414yes—
39C22×C22⋊C4C2314yes—
40C22×C22⋊C4C2314yes—
41C22×C22⋊C4C2312yes—
42C22×C22⋊C4C2311yescharacteristic
43C22×C22⋊C4C2311yescharacteristic
44C22×C42—22no—
45C22×C42D412yes—
46#398—28no—
47#395—28no—
48#391—24no—
49#389—24no—
50#389—24no—
51#385—28no—
52#385—28no—
53#384—22no—
54#380—24no—
55#380—22no—
56#380—22no—
57#380—22no—
58#380—22no—
59#380—22no—
60#380Q814yescomplemented
61C4×C22⋊C4—22no—
62C4×C22⋊C4—22no—
63C4×C22⋊C4—24no—
64C4×C22⋊C4Q818yescomplemented
65C2×#95—21no—
66C2×#95—22no—
67C2×#95—28no—
68C2×#95—22no—
69C2×#95—22no—
70C2×#95—21no—
71C2×#95—22no—
72C2×#95—21no—
73C2×#95—22no—
74C2×#95—28no—
75C2×#95—22no—
76C2×#95D412yes—
77C2×#95D414yes—
78#342—24no—
79#342—24no—
80#342—48no—
81C25C2411yescharacteristic
82C22×Q8—24no—
83C23×C4—22no—
84C23×C4—24no—
85C23×C4—22no—
86C23×C4—24no—
87C23×C4—24no—
88C23×C4—24no—
89C23×C4—44no—
90C23×C4—82no—
91C23×C4—82no—
92C23×C4C2×D418yes—
93C23×C4C2×D412yes—
94C23×C4C2×D412yes—
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D412yes—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D412yes—
99C23×C4C2×Q814yes—
100C23×C4C2×Q812yes—
101C23×C4#4112yes—
102C4⋊Q8—84no—
103C22⋊Q8—432no—
104C22⋊Q8—48no—
105C22⋊Q8—48no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—28no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—44no—
114C2×C4⋊C4—44no—
115C2×C4⋊C4—84no—
116C2×C4⋊C4—82no—
117C2×C4⋊C4—82no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—28no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—28no—
134C2×C22⋊C4C2×Q818yes—
135C2×C22⋊C4C2×Q814yes—
136C2×C22⋊C4#4114yes—
137C2×C22⋊C4#4114yes—
138C2×C42—22no—
139C2×C42—22no—
140C2×C42—28no—
141C2×C42—82no—
142C2×C42—84no—
143#95—28no—
144#95—24no—
145#95—24no—
146#95—24no—
147#95—28no—
148#95—24no—
149#95—416no—
150#95—44no—
151#95—41no—
152#95—42no—
153#95—41no—
154#95—44no—
155#95—82no—
156#95—88no—
157C24—42no—
158C24—42no—
159C24C22×D411yescharacteristic
160C24C22×D412yes—
161C24C22×Q812yes—
162C24C22×Q811yescharacteristic
163C24C2×#4111yescharacteristic
164C24C2×#4112yes—
165C24C2×#4112yes—
166C24H(5,𝔽2)12yes—
167C242−1+412yes—
168C2×Q8—48no—
169C2×Q8—84no—
170C2×Q8—84no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—28no—
176C22×C4—24no—
177C22×C4—28no—
178C22×C4—28no—
179C22×C4—24no—
180C22×C4—216no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—48no—
191C22×C4—48no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—48no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—44no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—48no—
208C22×C4—48no—
209C22×C4—48no—
210C22×C4—84no—
211C22×C4—82no—
212C22×C4—82no—
213C22×C4—88no—
214C22×C4—88no—
215C22×C4#12011yescharacteristic
216C22×C4#12012yes—
217C22×C4#12011yescharacteristic
218C22×C4C22⋊Q812yes—
219C22×C4C22⋊Q814yes—
220C22×C4C22⋊Q814yes—
221C22×C4C22⋊Q812yes—
222C22×C4#12414yes—
223C22×C4#12414yes—
224C22×C4C4⋊Q812yes—
225C22×C4C4⋊Q812yes—
226C22×C4C4⋊Q814yes—
227C22×C4C4⋊Q814yes—
228C4⋊C4—48no—
229C4⋊C4—48no—
230C4⋊C4—416no—
231C4⋊C4—84no—
232C4⋊C4—84no—
233C4⋊C4—84no—
234C4⋊C4—84no—
235C4⋊C4—88no—
236C22⋊C4—28no—
237C22⋊C4—28no—
238C22⋊C4—28no—
239C22⋊C4—28no—
240C22⋊C4—28no—
241C22⋊C4—44no—
242C22⋊C4—44no—
243C22⋊C4—48no—
244C22⋊C4—44no—
245C22⋊C4—44no—
246C22⋊C4—44no—
247C42—44no—
248C42—82no—
249C42—88no—
250C23—22no—
251C23—24no—
252C23—22no—
253C23—24no—
254C23—42no—
255C23—41no—
256C23—41no—
257C23—44no—
258C23—44no—
259C23—42no—
260C23—41no—
261C23—41no—
262C23—44no—
263C23—44no—
264C23C2×#12011yescharacteristic
265C23C2×C22⋊Q811yescharacteristic
266C23C2×C22⋊Q812yes—
267C23C2×#12411yescharacteristic
268C23C2×#12411yescharacteristic
269C23C2×C4⋊Q811yescharacteristic
270C23C2×C4⋊Q812yes—
271C23#53314yes—
272C23#53511yescharacteristic
273C23#53611yescharacteristic
274C23#54414yes—
275C23#54614yes—
276C23#54814yes—
277C23#55412yes—
278C23#55512yes—
279C23#55612yes—
280C23#55712yes—
281Q8—168no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—48no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—48no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—416no—
308C2×C4—44no—
309C2×C4—48no—
310C2×C4—48no—
311C2×C4—48no—
312C2×C4—44no—
313C2×C4—48no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—88no—
320C2×C4—82no—
321C2×C4—88no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—88no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—816no—
328C2×C4#39512yes—
329C2×C4#39514yes—
330C2×C4#39512yes—
331C22—21no—
332C22—22no—
333C22—21no—
334C22—22no—
335C22—21no—
336C22—22no—
337C22—21no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—28no—
342C22—22no—
343C22—22no—
344C22—28no—
345C22—44no—
346C22—44no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22C2×#39511yescharacteristic
352C22#243511yescharacteristic
353C22#243611yescharacteristic
354C22#243914yes—
355C22#255014yes—
356C22#256311yescharacteristic
357C22#256811yescharacteristic
358C22#266011yescharacteristic
359C22#286211yescharacteristic
360C22#302512yes—
361C22#302612yes—
362C4—22no—
363C4—22no—
364C4—24no—
365C4—44no—
366C4—44no—
367C4—84no—
368C4—88no—
369C4—88no—
370C4—84no—
371C2—21no—
372C2—24no—
373C2—21no—
374C2—21no—
375C2—21no—
376C2—24no—
377C2#1523011yescharacteristic
378C2#2356411yescharacteristic
379C2#2356511yescharacteristic
380C2#2590112yes—
381C2#2594912yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, prime radical, cosocleC25C24
LCSG ▷ C25 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
60. #380Q8semidirect
64. C4×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#350022
#141492
#139662
#113902
#112222
#108542
C4×#3422