Finite Group Databasebeta

← #207002

Subgroups — #207002

Order 512.

1272 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2#20523C211yesmaximal, characteristic
3#13965C211yesmaximal, characteristic
4#9885C214yesmaximal, complemented
5#9884C214yesmaximal, complemented
6#8023C212yesmaximal
7C2×#1294C212yesmaximal
8C2×#543—21no—
9C22×C22⋊C8C2212yes—
10#2857C2214yes—
11#2659—22no—
12#2549—24no—
13#2440C418yescomplemented
14#2315—21no—
15C2×#385—21no—
16C2×#385—21no—
17C2×#380C2211yescharacteristic
18C2×#380C2212yes—
19C2×C4×C22⋊C4C2211yescharacteristic
20C22×#95C2211yescharacteristic
21C8×C22⋊C4—24no—
22C4×C22⋊C8C2214yes—
23#1444—22no—
24C4⋊#95C2212yes—
25C4⋊#95C2212yes—
26#1294C2218yes—
27#1294C2214yes—
28#1294C2214yes—
29C24×C4—21no—
30C24×C4C2311yescharacteristic
31C23×C8—22no—
32#543—44no—
33#543—44no—
34C2×C4⋊Q8C2×C414yescomplemented
35C2×C22⋊Q8—24no—
36C2×C22⋊Q8—28no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4C2×C414yes—
41C22×C22⋊C4—22no—
42C22×C22⋊C4—21no—
43C22×C22⋊C4D411yescharacteristic
44C22×C22⋊C4D411yescharacteristic
45C22×C42C2312yes—
46C2×C22⋊C8—22no—
47C2×C22⋊C8—24no—
48C2×C22⋊C8Q818yescomplemented
49C2×C22⋊C8C2314yes—
50C2×C4×C8—24no—
51#398—28no—
52#395—28no—
53#391—24no—
54#389—24no—
55#389C2×C418yescomplemented
56#385—28no—
57#385—44no—
58#385—44no—
59#384C2×C414yescomplemented
60#380—24no—
61#380—22no—
62#380—22no—
63#380—24no—
64#380C2×C414yes—
65#380C2314yes—
66C4×C22⋊C4C2×C414yes—
67C4×C22⋊C4D414yes—
68C2×#95—22no—
69C2×#95—24no—
70C2×#95—22no—
71C2×#95—22no—
72C2×#95—21no—
73C2×#95D412yes—
74C2×#95C2312yes—
75C2×#95C2311yescharacteristic
76C2×#95C2311yescharacteristic
77#336—24no—
78C25C2×D411yescharacteristic
79C22×Q8—24no—
80C23×C4—22no—
81C23×C4—24no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—44no—
85C23×C4—42no—
86C23×C4—42no—
87C23×C4—42no—
88C23×C4—82no—
89C23×C4—82no—
90C23×C4C22×C412yes—
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D412yes—
93C23×C4C2×D412yes—
94C23×C4C2×D412yes—
95C23×C4C2×Q814yes—
96C23×C4#4114yes—
97C23×C4C2411yescharacteristic
98C22×C8—24no—
99C22×C8—28no—
100C22×C8—84no—
101C4⋊Q8—84no—
102C22⋊Q8—48no—
103C22⋊Q8—48no—
104C22⋊Q8—432no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—28no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—44no—
111C2×C4⋊C4—44no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—44no—
114C2×C4⋊C4—44no—
115C2×C4⋊C4—84no—
116C2×C4⋊C4—82no—
117C2×C4⋊C4—82no—
118C2×C4⋊C4—82no—
119C2×C4⋊C4C22×C418yes—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—44no—
123C2×C22⋊C4—44no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4C22⋊C414yes—
128C2×C22⋊C4C22⋊C414yes—
129C2×C22⋊C4D814yes—
130C2×C22⋊C4SD1614yes—
131C2×C22⋊C4SD1614yes—
132C2×C22⋊C4Q1614yes—
133C2×C42—82no—
134C2×C42C22×C414yes—
135C2×C42C2×D414yes—
136C22⋊C8—28no—
137C22⋊C8—24no—
138C22⋊C8C2×Q818yes—
139C4×C8—84no—
140#95—24no—
141#95—28no—
142#95—24no—
143#95—24no—
144#95—22no—
145#95—22no—
146#95—42no—
147#95—44no—
148#95—44no—
149#95C22⋊C418yes—
150C24—42no—
151C24—42no—
152C24C2×C22⋊C412yes—
153C24#12412yes—
154C24C4⋊Q812yes—
155C24C2×D812yes—
156C24C2×SD1612yes—
157C24C2×SD1612yes—
158C24C2×Q1612yes—
159C24C22×D411yescharacteristic
160C2×Q8—48no—
161C2×Q8—84no—
162C2×Q8—84no—
163C22×C4—24no—
164C22×C4—28no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—44no—
171C22×C4—42no—
172C22×C4—44no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—48no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—48no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—84no—
192C22×C4—84no—
193C22×C4—84no—
194C22×C4—84no—
195C22×C4—88no—
196C22×C4—88no—
197C22×C4—84no—
198C22×C4—84no—
199C22×C4C2×C22⋊C412yes—
200C22×C4C2×C22⋊C414yes—
201C22×C4C2×C22⋊C414yes—
202C22×C4C4×Q818yes—
203C22×C4C22⋊Q814yes—
204C22×C4C22⋊Q814yes—
205C22×C4#12412yes—
206C22×C4C4⋊Q812yes—
207C22×C4C2×D814yes—
208C22×C4C2×SD1614yes—
209C22×C4C2×SD1614yes—
210C22×C4C2×Q1614yes—
211C22×C4C23×C411yescharacteristic
212C22×C4C22×D411yescharacteristic
213C22×C4C22×Q812yes—
214C22×C4C2×#4112yes—
215C2×C8—28no—
216C2×C8—48no—
217C2×C8—88no—
218C4⋊C4—48no—
219C4⋊C4—48no—
220C4⋊C4—416no—
221C4⋊C4—84no—
222C4⋊C4—84no—
223C4⋊C4—84no—
224C4⋊C4—88no—
225C4⋊C4—88no—
226C4⋊C4—84no—
227C22⋊C4—44no—
228C22⋊C4—44no—
229C22⋊C4—48no—
230C22⋊C4—84no—
231C22⋊C4#102116yes—
232C22⋊C4#103116yes—
233C42—44no—
234C42—44no—
235C23—22no—
236C23—24no—
237C23—22no—
238C23—24no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—44no—
243C23—44no—
244C23—42no—
245C23—44no—
246C23—44no—
247C23#39114yes—
248C23C2×#10214yes—
249C23C2×#10314yes—
250C23#48614yes—
251C23#48714yes—
252C23#49814yes—
253C23#50014yes—
254C23C22×C22⋊C411yescharacteristic
255C23C2×#12411yescharacteristic
256C23C2×C4⋊Q811yescharacteristic
257C23C22×D811yescharacteristic
258C23C22×SD1611yescharacteristic
259C23C22×SD1611yescharacteristic
260C23C22×Q1611yescharacteristic
261Q8—168no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—48no—
274C2×C4—44no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—48no—
284C2×C4—48no—
285C2×C4—48no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—816no—
290C2×C4—82no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—168no—
302C2×C4#39114yes—
303C2×C4C2×#10218yes—
304C2×C4C2×#10318yes—
305C2×C4C2×C4×Q812yes—
306C2×C4C2×C22⋊Q812yes—
307C8—48no—
308C22—21no—
309C22—21no—
310C22—24no—
311C22—21no—
312C22—21no—
313C22—24no—
314C22—22no—
315C22—22no—
316C22—28no—
317C22—22no—
318C22—22no—
319C22—28no—
320C22—44no—
321C22—44no—
322C22—44no—
323C22—44no—
324C22#199018yes—
325C22C2×#39111yescharacteristic
326C22#288312yes—
327C22C22×#10211yescharacteristic
328C22C22×#10311yescharacteristic
329C22#290112yes—
330C22C2×#48611yescharacteristic
331C22C2×#48711yescharacteristic
332C22C2×#49811yescharacteristic
333C22C2×#50011yescharacteristic
334C4—22no—
335C4—22no—
336C4—24no—
337C4—28no—
338C4—28no—
339C4—84no—
340C4—88no—
341C4—161no—
342C4—161no—
343C2—21no—
344C2—21no—
345C2—21no—
346C2—21no—
347C2—28no—
348C2#1761312yes—
349C2#1761412yes—
350C2#2130711yescharacteristic
351C2C2×#199011yescharacteristic
352C2#2133311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9885C2semidirect
5. #9884C2semidirect
13. #2440C4semidirect
34. C2×C4⋊Q8C2×C4semidirect
48. C2×C22⋊C8Q8semidirect
55. #389C2×C4semidirect
59. #384C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350022
#205232
#139652
#98852
#98842
#80232
C2×#12942