Finite Group Databasebeta

← #6458399

Subgroups — #6458399

Order 512.

1782 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
1C2×#2628C211yesmaximal, characteristic
2#34052C211yesmaximal, characteristic
3#27971C2116yesmaximal, complemented
4#23747C214yesmaximal, complemented
5#21324C212yesmaximal, complemented
6#20765C212yesmaximal, complemented
7#20702C214yesmaximal
8#20377C211yesmaximal, characteristic
9C22×C4⋊Q8C2211yescharacteristic
10C22×#117C2212yes—
11C22×#117C2211yescharacteristic
12C2×#407C2212yes—
13C2×C4×M16C2212yes—
14#2628—24no—
15#2628C2218yes—
16C2×#400—21no—
17C2×#399—21no—
18C2×#395—21no—
19C2×#393—21no—
20#2317C2218yes—
21#2316—21no—
22#2316—21no—
23#2315—21no—
24#2315—21no—
25#2296—24no—
26C2×#389C2212yes—
27#2285C2212yes—
28#2285C2212yes—
29C2×#382C2211yescharacteristic
30C2×#380C2212yes—
31#2101C22116yescomplemented
32#2100C2218yescomplemented
33#2099C2218yescomplemented
34#1993C2218yescomplemented
35#1990C2218yescomplemented
36#1836C2218yescomplemented
37#1835C2218yescomplemented
38#1825C22116yes—
39#1824C2218yes—
40#1823C2218yes—
41#1750C2218yes—
42#1736C2218yes—
43#1736C2218yes—
44#1728C2212yes—
45C23×Q8—21no—
46C24×C4C2311yescharacteristic
47C22×M16C2314yes—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—28no—
50C2×C4⋊Q8C2314yes—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—28no—
54C2×#117—216no—
55C2×#117D414yes—
56C2×#117C2314yes—
57C2×#117C2314yes—
58C2×#117C2314yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2314yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4D411yescharacteristic
69C22×C22⋊C4D411yescharacteristic
70C22×C42—22no—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73#407—28no—
74C2×C22⋊C8C2318yes—
75C4×M16—28no—
76C2×#97C2314yes—
77C2×C4×C8C2314yes—
78#400—28no—
79#399—28no—
80#395—28no—
81#393—28no—
82#389—28no—
83#389—28no—
84#389—24no—
85#389C2318yes—
86#385—28no—
87#385—28no—
88#384C2314yes—
89#384C2314yes—
90#382—22no—
91#382C2314yes—
92#382C2314yes—
93#382C2314yes—
94#380—22no—
95#380—22no—
96#380—22no—
97#380—22no—
98#380—24no—
99#380C2318yes—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108#340C23116yes—
109#340C23116yes—
110#340C23116yes—
111#340C23116yes—
112#336C2318yes—
113C25C2×D411yescharacteristic
114C22×Q8—24no—
115C22×Q8—44no—
116C22×Q8—84no—
117C23×C4—24no—
118C23×C4—24no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—24no—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4#4118yes—
129C23×C4C2412yes—
130C23×C4C2411yescharacteristic
131C2×M16—28no—
132C2×M16—28no—
133C2×M16—28no—
134C22×C8C2418yes—
135C4⋊Q8—28no—
136C4⋊Q8—28no—
137C4⋊Q8—48no—
138#117—216no—
139#117—28no—
140#117—28no—
141#117—216no—
142#117—216no—
143#117—48no—
144#117—48no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—216no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—84no—
161C2×C4⋊C4C2×D414yes—
162C2×C4⋊C4C2×D414yes—
163C2×C4⋊C4C2418yes—
164C2×C4⋊C4C2414yes—
165C2×C4⋊C4C2414yes—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—48no—
172C2×C22⋊C4—84no—
173C2×C22⋊C4C2×D414yes—
174C2×C22⋊C4C2×D414yes—
175C2×C42—24no—
176C2×C42—24no—
177C2×C42—24no—
178C2×C42—48no—
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181C2×C42#4118yes—
182C2×C42C2412yes—
183C2×C42C2412yes—
184C22⋊C8—216no—
185#97—28no—
186C4×C8—28no—
187#95—28no—
188#95—24no—
189#95—24no—
190#95—28no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—24no—
195C24—22no—
196C24—22no—
197C24—42no—
198C24—42no—
199C24#12414yes—
200C24C22×D412yes—
201C24C22×D411yescharacteristic
202C2×Q8—48no—
203C2×Q8—88no—
204C2×Q8—816no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—232no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—28no—
216C22×C4—24no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—48no—
227C22×C4—44no—
228C22×C4—84no—
229C22×C4—84no—
230C22×C4#12314yes—
231C22×C4#12314yes—
232C22×C4#12418yes—
233C22×C4#12414yes—
234C22×C4C22×D412yes—
235C22×C4C22×D412yes—
236C22×C4C22×D412yes—
237C22×C4C22×D412yes—
238C22×C4C22×D412yes—
239C22×C4C22×D411yescharacteristic
240C22×C4C2×#4118yes—
241C22×C4C2×#4114yes—
242C22×C4C2×#4114yes—
243C22×C42−1+414yes—
244C22×C4C2511yescharacteristic
245M16—416no—
246C2×C8—28no—
247C2×C8—28no—
248C2×C8—28no—
249C4⋊C4—28no—
250C4⋊C4—28no—
251C4⋊C4—216no—
252C4⋊C4—28no—
253C4⋊C4—28no—
254C4⋊C4—44no—
255C4⋊C4—48no—
256C4⋊C4—48no—
257C4⋊C4—44no—
258C4⋊C4—84no—
259C4⋊C4—84no—
260C4⋊C4—84no—
261C4⋊C4—84no—
262C4⋊C4—88no—
263C22⋊C4—28no—
264C22⋊C4—28no—
265C22⋊C4—48no—
266C22⋊C4—48no—
267C22⋊C4—88no—
268C42—24no—
269C42—24no—
270C42—48no—
271C42—48no—
272C42#12318yes—
273C42C2×#4114yes—
274C42C2×#4114yes—
275C23—22no—
276C23—22no—
277C23—24no—
278C23—24no—
279C23—22no—
280C23—22no—
281C23—28no—
282C23—48no—
283C23—82no—
284C23—82no—
285C23—84no—
286C23C2×#12312yes—
287C23C2×#12414yes—
288C23C2×#12412yes—
289C23#53612yes—
290C23C23×D411yescharacteristic
291Q8—816no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—28no—
297C2×C4—216no—
298C2×C4—216no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—48no—
309C2×C4—48no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—48no—
313C2×C4—44no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—88no—
317C2×C4—88no—
318C2×C4—816no—
319C2×C4C2×#12314yes—
320C2×C4C2×#12314yes—
321C2×C4C2×#12312yes—
322C2×C4C2×#12414yes—
323C2×C4C2×#12412yes—
324C2×C4#53612yes—
325C2×C4#552116yes—
326C2×C4C22×#4112yes—
327C2×C4C22×#4112yes—
328C2×C4C2×2−1+412yes—
329C8—48no—
330C22—22no—
331C22—22no—
332C22—24no—
333C22—24no—
334C22—28no—
335C22—24no—
336C22—42no—
337C22—42no—
338C22—44no—
339C22—88no—
340C22#250314yes—
341C22C22×#12311yescharacteristic
342C22C22×#12411yescharacteristic
343C22C2×#53611yescharacteristic
344C22#358512yes—
345C22#358612yes—
346C4—24no—
347C4—24no—
348C4—24no—
349C4—42no—
350C4—42no—
351C4—84no—
352C4—84no—
353C4—88no—
354C4#250314yes—
355C4C2×#55214yes—
356C2—24no—
357C2—44no—
358C2C2×#250311yescharacteristic
359C2#6051811yescharacteristic
360C2#6051911yescharacteristic
361C2#6062112yes—
362C2#6087012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27971C2semidirect
4. #23747C2semidirect
5. #21324C2semidirect
6. #20765C2semidirect
31. #2101C22semidirect
32. #2100C22semidirect
33. #2099C22semidirect
34. #1993C22semidirect
35. #1990C22semidirect
36. #1836C22semidirect
37. #1835C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
#340522
#279712
#237472
#213242
#207652
#207022
#203772