Finite Group Databasebeta

← #241190

Subgroups — #241190

Order 512.

1212 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2#21938C212yesmaximal
3#13908C211yesmaximal, characteristic
4#10570C214yesmaximal, complemented
5#10568C214yesmaximal, complemented
6C2×#1294C212yesmaximal
7#7907C211yesmaximal, characteristic
8C2×#543—21no—
9C22×C22⋊C8C2212yes—
10#2857—22no—
11#2659C2214yes—
12#2549—24no—
13#2440—24no—
14#2315—21no—
15C2×#380—21no—
16C2×#380C2212yes—
17C2×#380C2211yescharacteristic
18C2×C4×C22⋊C4C2211yescharacteristic
19C22×#95C2211yescharacteristic
20#1853—24no—
21#1853C2218yes—
22#1807—24no—
23#1444C2212yes—
24#1442C2212yes—
25#1436—22no—
26C22⋊#95—21no—
27#1294C2214yes—
28#1294C2214yes—
29#1280C2214yes—
30C24×C4—21no—
31C24×C4C2311yescharacteristic
32C23×C8—22no—
33#543—44no—
34#543—44no—
35C2×C4⋊Q8—22no—
36C2×C22⋊Q8—28no—
37C2×C22⋊Q8—24no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4D414yes—
40C22×C4⋊C4C2312yes—
41C22×C22⋊C4—22no—
42C22×C22⋊C4—21no—
43C22×C22⋊C4D411yescharacteristic
44C22×C22⋊C4D411yescharacteristic
45C22×C42D412yes—
46C2×#106—28no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—24no—
49C2×C22⋊C8Q818yescomplemented
50C2×C22⋊C8C2314yes—
51#398—28no—
52#395—28no—
53#391—24no—
54#389—24no—
55#389—24no—
56#385—28no—
57#384—22no—
58#380—24no—
59#380—22no—
60#380—22no—
61#380—22no—
62#380—22no—
63#380—44no—
64#380D414yes—
65#380C2314yes—
66C4×C22⋊C4—22no—
67C4×C22⋊C4—22no—
68C2×#95—21no—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95—22no—
72C2×#95—21no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95Q814yescomplemented
76C2×#95C2312yes—
77C2×#95C2311yescharacteristic
78C2×#95C2311yescharacteristic
79#340—28no—
80C25C2×D411yescharacteristic
81C22×Q8—24no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—24no—
85C23×C4—24no—
86C23×C4—22no—
87C23×C4—44no—
88C23×C4—42no—
89C23×C4—42no—
90C23×C4—82no—
91C23×C4—82no—
92C23×C4C2×D412yes—
93C23×C4C2×D412yes—
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D412yes—
96C23×C4C2×D412yes—
97C23×C4C2×Q814yes—
98C23×C4C2×Q812yes—
99C23×C4#4112yes—
100C23×C4C2411yescharacteristic
101C22×C8—24no—
102C22×C8—28no—
103C22×C8—84no—
104C4⋊Q8—84no—
105C22⋊Q8—432no—
106C22⋊Q8—48no—
107C22⋊Q8—48no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—28no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—44no—
115C2×C4⋊C4—44no—
116C2×C4⋊C4—82no—
117C2×C4⋊C4—84no—
118C2×C4⋊C4—82no—
119C2×C4⋊C4C2×D418yes—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—44no—
127C2×C22⋊C4—44no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4—42no—
131C2×C22⋊C4SD1614yes—
132C2×C22⋊C4SD1614yes—
133C2×C42—22no—
134C2×C42—22no—
135C2×C42—82no—
136#106—88no—
137C22⋊C8—28no—
138C22⋊C8—24no—
139C22⋊C8C2×Q818yes—
140#95—28no—
141#95—24no—
142#95—24no—
143#95—24no—
144#95—24no—
145#95—21no—
146#95—21no—
147#95—44no—
148#95—41no—
149#95—41no—
150#95—44no—
151#95—82no—
152#95C2×Q812yes—
153#95C2×Q812yes—
154C24—42no—
155C24—42no—
156C24#12012yes—
157C24C22⋊Q812yes—
158C24C4⋊Q812yes—
159C24C2×SD1612yes—
160C24C2×SD1612yes—
161C24AGL(1,ℤ/8)12yes—
162C24#13712yes—
163C24C22×D411yescharacteristic
164C2×Q8—48no—
165C2×Q8—84no—
166C2×Q8—84no—
167C22×C4—24no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—28no—
174C22×C4—22no—
175C22×C4—42no—
176C22×C4—44no—
177C22×C4—42no—
178C22×C4—44no—
179C22×C4—48no—
180C22×C4—42no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—48no—
192C22×C4—48no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—84no—
201C22×C4—82no—
202C22×C4—82no—
203C22×C4—84no—
204C22×C4—84no—
205C22×C4—84no—
206C22×C4—88no—
207C22×C4—88no—
208C22×C4#12011yescharacteristic
209C22×C4#12011yescharacteristic
210C22×C4C22⋊Q814yes—
211C22×C4C22⋊Q814yes—
212C22×C4C22⋊Q812yes—
213C22×C4C4⋊Q814yes—
214C22×C4C4⋊Q814yes—
215C22×C4C4⋊Q812yes—
216C22×C4C2×SD1614yes—
217C22×C4C2×SD1614yes—
218C22×C4C22×D412yes—
219C22×C4C22×Q812yes—
220C22×C4C22×Q811yescharacteristic
221C22×C4C2×#4111yescharacteristic
222C2×C8—28no—
223C2×C8—48no—
224C2×C8—88no—
225C4⋊C4—48no—
226C4⋊C4—48no—
227C4⋊C4—48no—
228C4⋊C4—416no—
229C4⋊C4—48no—
230C4⋊C4—84no—
231C4⋊C4—84no—
232C4⋊C4—88no—
233C22⋊C4—28no—
234C22⋊C4—28no—
235C22⋊C4—44no—
236C22⋊C4—44no—
237C22⋊C4—48no—
238C22⋊C4—84no—
239C42—44no—
240C42—82no—
241C23—22no—
242C23—24no—
243C23—22no—
244C23—24no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—44no—
249C23—44no—
250C23—42no—
251C23—44no—
252C23—44no—
253C23#39514yes—
254C23#44814yes—
255C23#45014yes—
256C23#47514yes—
257C23#47614yes—
258C23#49814yes—
259C23#50114yes—
260C23C2×#12011yescharacteristic
261C23C2×C22⋊Q811yescharacteristic
262C23C2×C4⋊Q811yescharacteristic
263C23C22×SD1611yescharacteristic
264C23C22×SD1611yescharacteristic
265C23C2×AGL(1,ℤ/8)11yescharacteristic
266C23C2×#13711yescharacteristic
267Q8—168no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—48no—
296C2×C4—48no—
297C2×C4—48no—
298C2×C4—42no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—816no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—168no—
312C2×C4#39512yes—
313C2×C4#39512yes—
314C2×C4C2×C22⋊Q812yes—
315C2×C4C2×C4⋊Q812yes—
316C8—48no—
317C22—21no—
318C22—21no—
319C22—24no—
320C22—21no—
321C22—21no—
322C22—24no—
323C22—22no—
324C22—22no—
325C22—28no—
326C22—22no—
327C22—22no—
328C22—28no—
329C22—44no—
330C22—44no—
331C22—44no—
332C22—44no—
333C22#205818yes—
334C22C2×#39511yescharacteristic
335C22C2×#45011yescharacteristic
336C22C2×#44811yescharacteristic
337C22#301812yes—
338C22#302612yes—
339C22C2×#47611yescharacteristic
340C22C2×#47511yescharacteristic
341C22C2×#49811yescharacteristic
342C22C2×#50111yescharacteristic
343C4—22no—
344C4—22no—
345C4—22no—
346C4—22no—
347C4—44no—
348C4—44no—
349C4—44no—
350C4—82no—
351C4—88no—
352C4—162no—
353C2—21no—
354C2—21no—
355C2—28no—
356C2—21no—
357C2—21no—
358C2#1838112yes—
359C2#1838212yes—
360C2C2×#205811yescharacteristic
361C2#2356011yescharacteristic
362C2#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. #10570C2semidirect
5. #10568C2semidirect
49. C2×C22⋊C8Q8semidirect
75. C2×#95Q8semidirect

Maximal subgroups

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

subgroupindex
#350022
#219382
#139082
#105702
#105682
C2×#12942
#79072