Finite Group Databasebeta

← #6481935

Subgroups — #6481935

Order 512.

1606 conjugacy classes of subgroups.

The subgroup lattice has 361 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic
3#33147C214yesmaximal
4#29951C214yesmaximal
5#26088C218yesmaximal, complemented
6#26022C218yesmaximal, complemented
7#21981C212yesmaximal
8#21951C211yesmaximal, characteristic
9#20734C212yesmaximal
10C2×#544—21no—
11C22×C4×Q8C2211yescharacteristic
12C2×#431C2212yes—
13#2857C2214yes—
14#2659—22no—
15#2552C2214yes—
16#2551C2214yes—
17#2550—24no—
18#2355—24no—
19#2355C2218yes—
20#2316—21no—
21C2×#391C2212yes—
22C2×#387—22no—
23#2285C2212yes—
24C2×#384C2211yescharacteristic
25C2×#382—21no—
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29#2096C2218yes—
30#2066C22116yes—
31#2024C2218yes—
32#1988C2218yes—
33#1948C2218yes—
34#1948C2218yes—
35#1939C2218yes—
36#1919C22116yes—
37#1853C2218yes—
38#1844C2214yes—
39#1834C2214yes—
40#1824C2214yes—
41#1803C2214yes—
42#1803C2214yes—
43#1797C2214yes—
44#1787C2214yes—
45C4×#103C2218yes—
46C23×Q8C2311yescharacteristic
47#544—48no—
48C2×C4⋊Q8—24no—
49C2×C4⋊Q8C2314yes—
50C2×#125—24no—
51C2×C22⋊Q8—24no—
52C2×C4×Q8—22no—
53C2×C4×Q8—24no—
54C2×C4×Q8Q818yescomplemented
55C2×C4×Q8C2314yes—
56C2×#117—22no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C42—21no—
65C22×C42—22no—
66C22×C42C2312yes—
67C22×C42C2311yescharacteristic
68#431—28no—
69C2×#106C23116yes—
70C2×C4⋊C8C2314yes—
71C2×C4⋊C8C2318yes—
72C2×#103C2318yes—
73C2×#103C2318yes—
74C2×C22⋊C8C2314yes—
75C2×C4×C8C2314yes—
76#398—28no—
77#395—28no—
78#391—24no—
79#391—24no—
80#391C2318yes—
81#389—24no—
82#389—24no—
83#389—24no—
84#389C2318yes—
85#387—216no—
86#384—22no—
87#384C2312yes—
88#384C2312yes—
89#384C2314yes—
90#384C2314yes—
91#382—28no—
92#382—22no—
93#380—24no—
94#380—22no—
95#380C2314yes—
96C4×C4⋊C4—28no—
97C4×C4⋊C4—24no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4D414yes—
100C4×C4⋊C4C2318yes—
101C4×C4⋊C4C2314yes—
102C4×C4⋊C4C2314yes—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95D412yes—
107#340C2318yes—
108#340C2318yes—
109#340C23116yes—
110#336C2318yes—
111C22×Q8—21no—
112C22×Q8—24no—
113C22×Q8—22no—
114C22×Q8—42no—
115C22×Q8—42no—
116C22×Q8C2×Q814yes—
117C22×Q8C2412yes—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4C2×D412yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2411yescharacteristic
126C22×C8C2418yes—
127C4⋊Q8—48no—
128C4⋊Q8—416no—
129#125—416no—
130C22⋊Q8—416no—
131C4×Q8—216no—
132C4×Q8—28no—
133C4×Q8C2×Q8116yes—
134#117—48no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—82no—
154C2×C4⋊C4SD1618yes—
155C2×C4⋊C4C2×D414yes—
156C2×C4⋊C4C2×D414yes—
157C2×C4⋊C4C2×Q814yes—
158C2×C4⋊C4C2414yes—
159C2×C4⋊C4C2414yes—
160C2×C4⋊C4C2414yes—
161C2×C4⋊C4C2412yes—
162C2×C22⋊C4—24no—
163C2×C42—28no—
164C2×C42—24no—
165C2×C42—24no—
166C2×C42—28no—
167C2×C42—44no—
168C2×C42—84no—
169C2×C42C2×D414yes—
170C2×C42C2×D412yes—
171C2×C42C2×Q814yes—
172C2×C42#4114yes—
173C2×C42C2414yes—
174C2×C42C2412yes—
175#106—416no—
176C4⋊C8—28no—
177C4⋊C8—48no—
178#103—48no—
179#103—48no—
180C22⋊C8—28no—
181C4×C8—28no—
182#95—28no—
183#95—28no—
184#95—28no—
185#95—24no—
186#95—24no—
187#95—28no—
188#95C2×D418yes—
189C24C22×D411yescharacteristic
190C2×Q8—24no—
191C2×Q8—22no—
192C2×Q8—24no—
193C2×Q8—24no—
194C2×Q8—48no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8—48no—
198C2×Q8—48no—
199C2×Q8—84no—
200C2×Q8—84no—
201C2×Q8C22×Q814yes—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4C22⋊Q814yes—
226C22×C4C22⋊Q814yes—
227C22×C4C2×SD1618yes—
228C22×C4C2×SD1614yes—
229C22×C4#13714yes—
230C22×C4C22×D414yes—
231C22×C4C22×D411yescharacteristic
232C22×C4C22×D412yes—
233C22×C4C22×D414yes—
234C22×C4C22×D412yes—
235C22×C4C22×Q812yes—
236C22×C4C2×#4112yes—
237C22×C4C2511yescharacteristic
238C2×C8—28no—
239C2×C8—28no—
240C2×C8—28no—
241C4⋊C4—216no—
242C4⋊C4—24no—
243C4⋊C4—28no—
244C4⋊C4—22no—
245C4⋊C4—28no—
246C4⋊C4—28no—
247C4⋊C4—28no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—48no—
252C4⋊C4—416no—
253C4⋊C4—48no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C4⋊C4—84no—
257C4⋊C4—84no—
258C4⋊C4—84no—
259C4⋊C4C2×SD16116yes—
260C4⋊C4C22×Q814yes—
261C22⋊C4—48no—
262C42—216no—
263C42—28no—
264C42—44no—
265C42—48no—
266C42—88no—
267C42C22⋊Q818yes—
268C42C22×Q814yes—
269C42C2×#4114yes—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23C2×C22⋊Q812yes—
275C23C22×SD1612yes—
276C23C2×#13712yes—
277C23C23×D411yescharacteristic
278Q8—28no—
279Q8—48no—
280Q8—44no—
281Q8—168no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—44no—
296C2×C4—48no—
297C2×C4—48no—
298C2×C4—48no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—84no—
313C2×C4—88no—
314C2×C4#475116yes—
315C2×C4C2×C22⋊Q812yes—
316C2×C4C2×C22⋊Q814yes—
317C2×C4C2×C22⋊Q814yes—
318C2×C4C22×SD1618yes—
319C2×C4C22×SD1614yes—
320C2×C4C2×#13714yes—
321C2×C4C23×Q811yescharacteristic
322C2×C4C22×#4111yescharacteristic
323C2×C4#58512yes—
324C2×C4#58512yes—
325C8—48no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22C2×#47514yes—
333C22C22×C22⋊Q811yescharacteristic
334C22#348511yescharacteristic
335C22#348711yescharacteristic
336C22C23×SD1611yescharacteristic
337C22C22×#13711yescharacteristic
338C22#358511yescharacteristic
339C22#358611yescharacteristic
340C4—28no—
341C4—24no—
342C4—22no—
343C4—44no—
344C4—44no—
345C4—44no—
346C4—84no—
347C4—84no—
348C4—84no—
349C4C2×#47518yes—
350C4#345812yes—
351C4#349312yes—
352C2—24no—
353C2#3515111yescharacteristic
354C2C22×#47511yescharacteristic
355C2#6049211yescharacteristic
356C2#6169711yescharacteristic
357C2#6172311yescharacteristic
358C2#6188311yescharacteristic
359C2#6189211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #26088C2semidirect
6. #26022C2semidirect
54. C2×C4×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
C2×#23552
#331472
#299512
#260882
#260222
#219812
#219512
#207342