Finite Group Databasebeta

← #6528565

Subgroups — #6528565

Order 512.

1602 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35173C211yesmaximal, characteristic
2C2×#2552C211yesmaximal, characteristic
3#29329C214yesmaximal
4#27972C214yesmaximal
5#27749C218yesmaximal, complemented
6#27729C218yesmaximal, complemented
7#21860C212yesmaximal
8#21385C212yesmaximal
9#20533C211yesmaximal, characteristic
10C2×#533C2212yes—
11C22×#125—21no—
12C2×#431C2212yes—
13#2857C2214yes—
14#2659—22no—
15#2584—24no—
16#2567C2214yes—
17#2552—24no—
18#2552C2218yes—
19#2546C2214yes—
20C2×#398—22no—
21#2316—21no—
22#2316—21no—
23C2×#389—21no—
24C2×#389C2212yes—
25C2×#384C2212yes—
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#2100C2218yes—
30#2072C22116yes—
31#2043C2218yes—
32#1990C2214yes—
33#1987C2214yes—
34#1981C2214yes—
35#1956C2218yes—
36#1956C2214yes—
37#1933C22116yes—
38#1872C2214yes—
39#1872C2218yes—
40#1872C2214yes—
41#1834C2218yes—
42#1803C2218yes—
43C8×C4⋊C4C2218yes—
44C4×#103C2218yes—
45C4×C22⋊C8C2214yes—
46#533—28no—
47#533—28no—
48C2×C4⋊Q8C2312yes—
49C2×C4⋊Q8C2314yes—
50C2×C4⋊Q8C2312yes—
51C2×#125—28no—
52C2×#125—22no—
53C2×#125—48no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8C2314yes—
56C2×C4×Q8—22no—
57C2×C4×Q8C2314yes—
58C2×#117—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C42—21no—
70C22×C42C2312yes—
71C22×C42C2311yescharacteristic
72#431—28no—
73C2×C4⋊C8C2314yes—
74C2×#103C23116yes—
75C2×#103C2318yes—
76C2×#103C2318yes—
77C2×C22⋊C8C2314yes—
78C2×C4×C8C2314yes—
79C2×C4×C8C2318yes—
80#398—28no—
81#398—216no—
82#395—28no—
83#391C2318yes—
84#389—24no—
85#389—24no—
86#389—24no—
87#389—24no—
88#389—28no—
89#389—24no—
90#389—24no—
91#389—24no—
92#389C2318yes—
93#384—24no—
94#384—22no—
95#384C2314yes—
96#384C2314yes—
97#384C2318yes—
98#380—24no—
99#380—24no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4D414yes—
102C4×C4⋊C4C2314yes—
103C4×C4⋊C4C2314yes—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95D412yes—
107#340C23116yes—
108#340C2318yes—
109#340C2318yes—
110#336C2318yes—
111C22×Q8—22no—
112C22×Q8C2414yes—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D412yes—
118C23×C4C2×D412yes—
119C23×C4C2×D411yescharacteristic
120C23×C4C2411yescharacteristic
121C22×C8C2418yes—
122C4⋊Q8—24no—
123C4⋊Q8—24no—
124C4⋊Q8—24no—
125C4⋊Q8—24no—
126C4⋊Q8—48no—
127#125—432no—
128#125—48no—
129C22⋊Q8—28no—
130C22⋊Q8—28no—
131C22⋊Q8—48no—
132C4×Q8—28no—
133C4×Q8—28no—
134C4×Q8—48no—
135#117—48no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—48no—
157C2×C4⋊C4—48no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4—84no—
161C2×C4⋊C4SD1618yes—
162C2×C4⋊C4Q1618yes—
163C2×C4⋊C4C2×D414yes—
164C2×C4⋊C4C2×D414yes—
165C2×C4⋊C4C2412yes—
166C2×C4⋊C4C2412yes—
167C2×C4⋊C4C2414yes—
168C2×C4⋊C4C2414yes—
169C2×C22⋊C4—24no—
170C2×C42—24no—
171C2×C42—24no—
172C2×C42—44no—
173C2×C42—44no—
174C2×C42C2×D414yes—
175C2×C42C2×D412yes—
176C2×C42#4118yes—
177C2×C42C2414yes—
178C2×C42C2412yes—
179C4⋊C8—28no—
180#103—48no—
181#103—416no—
182#103—48no—
183C22⋊C8—28no—
184C4×C8—28no—
185C4×C8—48no—
186#95—24no—
187#95—28no—
188#95—28no—
189#95—28no—
190#95C2×D418yes—
191C24C22×D411yescharacteristic
192C2×Q8—24no—
193C2×Q8—24no—
194C2×Q8—28no—
195C2×Q8—28no—
196C2×Q8—44no—
197C2×Q8—82no—
198C2×Q8—82no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—41no—
210C22×C4—42no—
211C22×C4—41no—
212C22×C4—42no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—44no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4#12414yes—
222C22×C4#12414yes—
223C22×C4C2×SD1618yes—
224C22×C4C2×SD1614yes—
225C22×C4C2×Q1618yes—
226C22×C4C2×Q1614yes—
227C22×C4C22×D412yes—
228C22×C4C22×D411yescharacteristic
229C22×C4C22×D414yes—
230C22×C4C22×D414yes—
231C22×C4C22×D412yes—
232C22×C4C2×#4114yes—
233C22×C4C2511yescharacteristic
234C2×C8—28no—
235C2×C8—28no—
236C2×C8—28no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—28no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—44no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—48no—
248C4⋊C4—48no—
249C4⋊C4—44no—
250C4⋊C4—416no—
251C4⋊C4—416no—
252C4⋊C4—48no—
253C4⋊C4—84no—
254C4⋊C4—84no—
255C4⋊C4—88no—
256C4⋊C4—84no—
257C4⋊C4—84no—
258C4⋊C4—84no—
259C4⋊C4C2×SD16116yes—
260C4⋊C4C2×Q16116yes—
261C22⋊C4—48no—
262C42—28no—
263C42—44no—
264C42—48no—
265C42—48no—
266C42#12418yes—
267C42C2×#4114yes—
268C42C2×#4114yes—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23C2×#12412yes—
274C23C22×SD1612yes—
275C23C22×Q1612yes—
276C23C23×D411yescharacteristic
277Q8—48no—
278Q8—48no—
279Q8—88no—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—28no—
289C2×C4—44no—
290C2×C4—48no—
291C2×C4—48no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—44no—
301C2×C4—42no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—84no—
308C2×C4—88no—
309C2×C4—84no—
310C2×C4#487116yes—
311C2×C4C2×#12412yes—
312C2×C4C2×#12414yes—
313C2×C4C2×#12414yes—
314C2×C4C22×SD1618yes—
315C2×C4C22×SD1614yes—
316C2×C4C22×Q1618yes—
317C2×C4C22×Q1614yes—
318C2×C4C22×#4112yes—
319C2×C4#58512yes—
320C2×C4#58512yes—
321C8—48no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22C2×#48714yes—
329C22C22×#12411yescharacteristic
330C22#348511yescharacteristic
331C22#348711yescharacteristic
332C22C23×SD1611yescharacteristic
333C22C23×Q1611yescharacteristic
334C22#358511yescharacteristic
335C22#358611yescharacteristic
336C4—28no—
337C4—22no—
338C4—24no—
339C4—28no—
340C4—84no—
341C4—82no—
342C4—84no—
343C4—84no—
344C4—82no—
345C4C2×#48718yes—
346C4#351214yes—
347C2—24no—
348C2#3564911yescharacteristic
349C2C22×#48711yescharacteristic
350C2#6062111yescharacteristic
351C2#6169511yescharacteristic
352C2#6169811yescharacteristic
353C2#6188111yescharacteristic
354C2#6188411yescharacteristic
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. #27749C2semidirect
6. #27729C2semidirect

Maximal subgroups

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

subgroupindex
#351732
C2×#25522
#293292
#279722
#277492
#277292
#218602
#213852
#205332