Finite Group Databasebeta

← #6508965

Subgroups — #6508965

Order 512.

1956 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic
3#27611C214yesmaximal, complemented
4#27601C214yesmaximal, complemented
5#24935C2116yesmaximal, complemented
6#21938C212yesmaximal, complemented
7#21738C211yesmaximal, characteristic
8C2×#1824C212yesmaximal, complemented
9C2×#518—21no—
10C22×C4×D4C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C22×C22⋊C8C2212yes—
13#2487—24no—
14#2487C2218yes—
15C2×#400—21no—
16C2×#399—21no—
17#2349—24no—
18D4×C4⋊C4C2218yes—
19#2316—21no—
20#2315—21no—
21#2315—21no—
22C2×#389C2212yes—
23C2×#386—21no—
24C2×#385—21no—
25#2285C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#380C2211yescharacteristic
29C2×#380C2212yes—
30#2099C2218yescomplemented
31#2084C2218yescomplemented
32#2077C22116yescomplemented
33#1920C22116yescomplemented
34#1869C2218yes—
35#1853C2218yes—
36#1853C2218yescomplemented
37#1853C2218yescomplemented
38#1824C2218yes—
39#1824C2218yescomplemented
40#1823C2218yescomplemented
41#1817C2212yes—
42#1807C2218yescomplemented
43#1779C2212yes—
44C2×#340C2218yescomplemented
45C23×D4D411yescharacteristic
46C24×C4—21no—
47C24×C4C2311yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8C2314yes—
50#518—48no—
51C2×C4×D4—28no—
52C2×C4×D4—28no—
53C2×C4×D4D414yes—
54C2×C4×D4C2314yes—
55C2×#117—22no—
56C22×C4⋊C4—28no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2314yes—
63C22×C4⋊C4C2314yes—
64C22×C4⋊C4C2314yes—
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4D411yescharacteristic
71C22×C22⋊C4D411yescharacteristic
72C22×C42C2311yescharacteristic
73C2×#106C23116yes—
74C2×C22⋊C8—28no—
75C2×C22⋊C8C2314yes—
76C2×C22⋊C8C2314yes—
77C2×C22⋊C8C2318yes—
78#400—28no—
79#399—28no—
80#389—28no—
81#389—24no—
82#389C2318yes—
83#386—28no—
84#385—28no—
85#385—28no—
86#385—28no—
87#384C2314yes—
88#382—22no—
89#380—22no—
90#380—22no—
91#380—22no—
92#380—22no—
93#380—22no—
94#380—24no—
95#380—22no—
96#380C2314yes—
97#380C2314yes—
98#380C2314yes—
99#380C2318yes—
100#380C2314yes—
101C4×C4⋊C4—22no—
102C4×C4⋊C4C2314yes—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108#340C23116yes—
109#340C23116yes—
110#340C23116yes—
111#336C2318yes—
112C25C2×D411yescharacteristic
113C25C2×D411yescharacteristic
114C22×D4—24no—
115C22×D4—24no—
116C22×D4SD1618yes—
117C22×D4C2×D414yes—
118C23×C4—24no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—24no—
122C23×C4—24no—
123C23×C4—24no—
124C23×C4—44no—
125C23×C4—44no—
126C23×C4—44no—
127C23×C4—84no—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4#4118yes—
135C23×C4C2412yes—
136C23×C4C2412yes—
137C23×C4C2411yescharacteristic
138C22×C8—28no—
139C22×C8—48no—
140C22×C8C2418yes—
141C4×D4—216no—
142C4×D4—28no—
143C4×D4—28no—
144C4×D4—432no—
145#117—48no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—48no—
160C2×C4⋊C4—48no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—48no—
163C2×C4⋊C4—84no—
164C2×C4⋊C4C2×D414yes—
165C2×C4⋊C4C2414yes—
166C2×C4⋊C4C2418yes—
167C2×C4⋊C4C2414yes—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—48no—
176C2×C22⋊C4—48no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—84no—
179C2×C22⋊C4C2×D414yes—
180C2×C22⋊C4C2×D414yes—
181C2×C42—24no—
182C2×C42—24no—
183C2×C42C2×D412yes—
184C2×C42C2412yes—
185#106—416no—
186C22⋊C8—28no—
187C22⋊C8—28no—
188C22⋊C8—48no—
189#95—24no—
190#95—24no—
191#95—24no—
192#95—28no—
193#95—24no—
194#95—28no—
195#95—24no—
196C24—22no—
197C24—22no—
198C24—42no—
199C24—42no—
200C24—42no—
201C24—42no—
202C24#12314yes—
203C24C2×SD1614yes—
204C24C2×SD1614yes—
205C24#13714yes—
206C24C22×D412yes—
207C24C22×D412yes—
208C24C22×D411yescharacteristic
209C2×D4—28no—
210C2×D4—28no—
211C2×D4—232no—
212C2×D4C2×SD16116yes—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—28no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—28no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—28no—
224C22×C4—24no—
225C22×C4—416no—
226C22×C4—44no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—48no—
237C22×C4—44no—
238C22×C4—48no—
239C22×C4—48no—
240C22×C4—88no—
241C22×C4—88no—
242C22×C4—84no—
243C22×C4—84no—
244C22×C4—84no—
245C22×C4—84no—
246C22×C4—816no—
247C22×C4#12314yes—
248C22×C4#12318yes—
249C22×C4C2×SD1614yes—
250C22×C4C22×D412yes—
251C22×C4C22×D412yes—
252C22×C4C22×D411yescharacteristic
253C22×C4C22×D412yes—
254C22×C4C22×D412yes—
255C22×C4C2×#4114yes—
256C22×C4C2×#4114yes—
257C22×C4C2×#4114yes—
258C22×C4C2511yescharacteristic
259C2×C8—28no—
260C2×C8—216no—
261C2×C8—88no—
262C4⋊C4—28no—
263C4⋊C4—28no—
264C4⋊C4—28no—
265C4⋊C4—216no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—48no—
269C4⋊C4—48no—
270C4⋊C4—84no—
271C4⋊C4—84no—
272C4⋊C4—84no—
273C4⋊C4—88no—
274C22⋊C4—28no—
275C22⋊C4—28no—
276C22⋊C4—48no—
277C22⋊C4—48no—
278C22⋊C4—48no—
279C22⋊C4—48no—
280C22⋊C4—88no—
281C42—24no—
282C42—24no—
283C42—48no—
284C23—22no—
285C23—24no—
286C23—22no—
287C23—24no—
288C23—24no—
289C23—22no—
290C23—24no—
291C23—22no—
292C23—28no—
293C23—44no—
294C23—44no—
295C23—48no—
296C23—48no—
297C23—48no—
298C23—82no—
299C23—84no—
300C23—82no—
301C23#483116yes—
302C23C2×#12314yes—
303C23C2×#12312yes—
304C23C22×SD1614yes—
305C23C22×SD1612yes—
306C23C22×SD1614yes—
307C23C2×#13714yes—
308C23C2×#13712yes—
309C23C23×D411yescharacteristic
310D4—232no—
311C2×C4—28no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—28no—
317C2×C4—28no—
318C2×C4—28no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—48no—
324C2×C4—44no—
325C2×C4—48no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—48no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—48no—
333C2×C4—48no—
334C2×C4—44no—
335C2×C4—84no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—88no—
339C2×C4—88no—
340C2×C4—816no—
341C2×C4—168no—
342C2×C4—168no—
343C2×C4C2×#12312yes—
344C2×C4C2×#12314yes—
345C2×C4C22×SD1614yes—
346C2×C4C22×#4112yes—
347C2×C4#58514yes—
348C8—48no—
349C22—28no—
350C22—24no—
351C22—24no—
352C22—28no—
353C22—22no—
354C22—24no—
355C22—22no—
356C22—24no—
357C22—28no—
358C22—24no—
359C22—416no—
360C22—44no—
361C22—42no—
362C22—42no—
363C22—88no—
364C22C2×#48314yes—
365C22C2×#48318yes—
366C22C22×#12311yescharacteristic
367C22#348512yes—
368C22C23×SD1611yescharacteristic
369C22C22×#13711yescharacteristic
370C22#358512yes—
371C4—24no—
372C4—22no—
373C4—24no—
374C4—42no—
375C4—42no—
376C4—88no—
377C4—84no—
378C4—84no—
379C4#349114yes—
380C2—24no—
381C2—28no—
382C2—44no—
383C2#3544811yescharacteristic
384C2C22×#48311yescharacteristic
385C2#6051811yescharacteristic
386C2#6169712yes—
387C2#6170812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27611C2semidirect
4. #27601C2semidirect
5. #24935C2semidirect
6. #21938C2semidirect
8. C2×#1824C2semidirect
30. #2099C22semidirect
31. #2084C22semidirect
32. #2077C22semidirect
33. #1920C22semidirect
36. #1853C22semidirect
37. #1853C22semidirect
39. #1824C22semidirect
40. #1823C22semidirect
42. #1807C22semidirect
44. C2×#340C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
#341812
#276112
#276012
#249352
#219382
#217382
C2×#18242