Finite Group Databasebeta

← #6531994

Subgroups — #6531994

Order 512.

1922 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2#34182C211yesmaximal, characteristic, complemented
3#27613C214yesmaximal, complemented
4#27585C214yesmaximal
5#25058C218yesmaximal, complemented
6#24992C218yesmaximal, complemented
7#21981C212yesmaximal, complemented
8#21818C211yesmaximal, characteristic, complemented
9#20732C212yesmaximal
10C22×C22⋊Q8—21no—
11C2×#519—21no—
12C22×C4×Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C2×#431C2212yes—
15#2538—24no—
16#2538C2218yes—
17C2×#396—22no—
18#2351—24no—
19Q8×C4⋊C4C2214yescomplemented
20Q8×C4⋊C4C2214yescomplemented
21#2316—22no—
22#2316—21no—
23C2×#390C2212yes—
24C2×#389C2212yes—
25#2285C2214yes—
26#2285C2212yes—
27C2×#384C2211yescharacteristic
28C2×#380—21no—
29#2100C2218yescomplemented
30#2094C2218yes—
31#2084C2218yescomplemented
32#2071C2218yes—
33#1984C2218yescomplemented
34#1948C2218yescomplemented
35#1926C22116yes—
36#1919C22116yescomplemented
37#1853C2218yes—
38#1844C2214yescomplemented
39#1834C2214yescomplemented
40#1823C2214yes—
41#1817C2214yes—
42#1802C2214yes—
43#1802C2214yes—
44#1786C2214yes—
45C4×#102C2218yes—
46C23×Q8D411yescharacteristic
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C2×C22⋊Q8—28no—
50C2×C22⋊Q8—48no—
51#519—48no—
52C2×C4×Q8—24no—
53C2×C4×Q8—24no—
54C2×C4×Q8—28no—
55C2×C4×Q8D414yes—
56C2×C4×Q8C2314yes—
57C2×C4×D4—22no—
58C2×C4×D4—28no—
59C2×C4×D4C2314yes—
60C2×#117—24no—
61C2×#117—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D412yes—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74#431—28no—
75C2×#106C23116yes—
76C2×C4⋊C8C2314yes—
77C2×#102C2318yes—
78C2×#102C2318yes—
79C2×C22⋊C8C2314yes—
80C2×C4×C8C2314yes—
81#396—216no—
82#390—24no—
83#390C2318yes—
84#389—24no—
85#389—24no—
86#389—28no—
87#389—28no—
88#389—24no—
89#389C2318yes—
90#384—22no—
91#384C2312yes—
92#384C2312yes—
93#384C2314yes—
94#384C2314yes—
95#384C2314yes—
96#382—24no—
97#382—22no—
98#380—24no—
99#380—22no—
100#380—28no—
101#380C2314yes—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4C2318yes—
105C4×C4⋊C4C2314yes—
106C2×#95—22no—
107#340C2318yes—
108#340C2318yes—
109#340C23116yes—
110#336C2318yes—
111#336C2318yes—
112C25—21no—
113C22×Q8—24no—
114C22×Q8—24no—
115C22×Q8SD1618yes—
116C22×Q8C2×D414yes—
117C22×D4—24no—
118C22×D4—21no—
119C22×D4—24no—
120C22×D4—42no—
121C22×D4C2412yes—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—24no—
125C23×C4—44no—
126C23×C4—44no—
127C23×C4—44no—
128C23×C4C2×D412yes—
129C23×C4C2×D412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2411yescharacteristic
133C22×C8C2418yes—
134C22⋊Q8—432no—
135C4×Q8—216no—
136C4×Q8—28no—
137C4×Q8—28no—
138C4×Q8—416no—
139C4×Q8—416no—
140C4×D4—28no—
141C4×D4—28no—
142C4×D4—48no—
143#117—416no—
144#117—48no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—48no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4C2×D414yes—
166C2×C4⋊C4C2×D418yes—
167C2×C4⋊C4C2412yes—
168C2×C4⋊C4C2414yes—
169C2×C4⋊C4C2414yes—
170C2×C4⋊C4C2414yes—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—48no—
177C2×C22⋊C4—84no—
178C2×C42—28no—
179C2×C42—28no—
180C2×C42—24no—
181C2×C42C2×D414yes—
182C2×C42C2×D412yes—
183C2×C42#4118yes—
184C2×C42C2414yes—
185C2×C42C2412yes—
186#106—416no—
187C4⋊C8—28no—
188#102—216no—
189#102—48no—
190C22⋊C8—28no—
191C4×C8—28no—
192#95—28no—
193#95—24no—
194#95—28no—
195#95—24no—
196C24—24no—
197C24—22no—
198C24—42no—
199C24—42no—
200C24—42no—
201C24—42no—
202C24—42no—
203C24—42no—
204C24C22×D411yescharacteristic
205C2×Q8—28no—
206C2×Q8—28no—
207C2×Q8—232no—
208C2×Q8C2×SD16116yes—
209C2×D4—24no—
210C2×D4—24no—
211C2×D4—22no—
212C2×D4—24no—
213C2×D4—416no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D42−1+414yes—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—48no—
237C22×C4—48no—
238C22×C4—48no—
239C22×C4—816no—
240C22×C4#12314yes—
241C22×C4#12314yes—
242C22×C4C2×SD1618yes—
243C22×C4C2×SD1614yes—
244C22×C4AGL(1,ℤ/8)14yes—
245C22×C4C22×D414yes—
246C22×C4C22×D411yescharacteristic
247C22×C4C22×D412yes—
248C22×C4C22×D414yes—
249C22×C4C22×D412yes—
250C22×C4C2×#4114yes—
251C22×C4C2511yescharacteristic
252C2×C8—28no—
253C2×C8—28no—
254C2×C8—28no—
255C4⋊C4—216no—
256C4⋊C4—28no—
257C4⋊C4—24no—
258C4⋊C4—28no—
259C4⋊C4—28no—
260C4⋊C4—28no—
261C4⋊C4—44no—
262C4⋊C4—48no—
263C4⋊C4—48no—
264C4⋊C4—42no—
265C4⋊C4—48no—
266C4⋊C4—416no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—82no—
270C4⋊C4—82no—
271C4⋊C4—84no—
272C4⋊C4—84no—
273C4⋊C4—84no—
274C22⋊C4—48no—
275C22⋊C4—48no—
276C22⋊C4—416no—
277C22⋊C4—88no—
278C42—28no—
279C42—28no—
280C42—416no—
281C42—48no—
282C42#12318yes—
283C42C2×#4114yes—
284C42C2×#4114yes—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—24no—
290C23—48no—
291C23—48no—
292C23—48no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—84no—
300C23—84no—
301C23—84no—
302C23—84no—
303C23C2×#12312yes—
304C23C22×SD1612yes—
305C23C2×AGL(1,ℤ/8)12yes—
306C23C23×D411yescharacteristic
307Q8—232no—
308D4—28no—
309D4—48no—
310D4—44no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—216no—
314C2×C4—24no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—24no—
318C2×C4—28no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—84no—
336C2×C4—88no—
337C2×C4—84no—
338C2×C4—816no—
339C2×C4#481116yes—
340C2×C4C2×#12312yes—
341C2×C4C2×#12314yes—
342C2×C4C2×#12314yes—
343C2×C4C22×SD1618yes—
344C2×C4C22×SD1614yes—
345C2×C4C2×AGL(1,ℤ/8)14yes—
346C2×C4C22×#4112yes—
347C2×C4C2×2−1+412yes—
348C2×C4#58512yes—
349C8—48no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—44no—
357C22—44no—
358C22—44no—
359C22—816no—
360C22—82no—
361C22—82no—
362C22—82no—
363C22—82no—
364C22C2×#48114yes—
365C22C22×#12311yescharacteristic
366C22#348611yescharacteristic
367C22#348711yescharacteristic
368C22C23×SD1611yescharacteristic
369C22C22×AGL(1,ℤ/8)11yescharacteristic
370C22#358511yescharacteristic
371C22#358611yescharacteristic
372C4—22no—
373C4—28no—
374C4—24no—
375C4—44no—
376C4—82no—
377C4—84no—
378C4—84no—
379C4—84no—
380C4C2×#48118yes—
381C4#349514yes—
382C2—24no—
383C2—84no—
384C2#3567211yescharacteristic
385C2C22×#48111yescharacteristic
386C2#6052011yescharacteristic
387C2#6181811yescharacteristic
388C2#6182511yescharacteristic
389C2#6188111yescharacteristic
390C2#6188811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34182C2semidirect
3. #27613C2semidirect
5. #25058C2semidirect
6. #24992C2semidirect
7. #21981C2semidirect
8. #21818C2semidirect
19. Q8×C4⋊C4C22semidirect
20. Q8×C4⋊C4C22semidirect
29. #2100C22semidirect
31. #2084C22semidirect
33. #1984C22semidirect
34. #1948C22semidirect
36. #1919C22semidirect
38. #1844C22semidirect
39. #1834C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
#341822
#276132
#275852
#250582
#249922
#219812
#218182
#207322