Finite Group Databasebeta

← #6481933

Subgroups — #6481933

Order 512.

1866 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic
3#33148C214yesmaximal, complemented
4#29952C214yesmaximal
5#26090C218yesmaximal, complemented
6#26024C218yesmaximal, complemented
7#21981C212yesmaximal, complemented
8#21951C211yesmaximal, characteristic, complemented
9#20732C212yesmaximal
10C2×#544—21no—
11C22×C4×D4C2211yescharacteristic
12C22×#117—21no—
13C2×#431C2212yes—
14#2857C2214yes—
15#2659—22no—
16#2552C2214yescomplemented
17#2551C2214yescomplemented
18#2550—24no—
19#2354—24no—
20#2354C2218yes—
21#2316—21no—
22C2×#390C2212yes—
23C2×#388—22no—
24#2285C2212yes—
25C2×#384C2211yescharacteristic
26C2×#380—21no—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29#2094C2218yes—
30#2065C22116yes—
31#2023C2218yes—
32#1987C2218yescomplemented
33#1948C2218yescomplemented
34#1948C2218yescomplemented
35#1939C2218yescomplemented
36#1920C22116yescomplemented
37#1853C2218yes—
38#1844C2214yescomplemented
39#1834C2214yescomplemented
40#1823C2214yes—
41#1802C2214yes—
42#1802C2214yes—
43#1797C2214yes—
44#1786C2214yes—
45C4×#102C2218yes—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48#544—48no—
49C2×C4⋊Q8—24no—
50C2×C4⋊Q8C2314yes—
51C2×#125—24no—
52C2×C22⋊Q8—24no—
53C2×C4×D4—22no—
54C2×C4×D4—24no—
55C2×C4×D4Q818yescomplemented
56C2×C4×D4C2314yes—
57C2×#117—22no—
58C2×#117—28no—
59C2×#117—48no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C42—22no—
70C22×C42C2312yes—
71C22×C42C2311yescharacteristic
72#431—28no—
73C2×#106C23116yes—
74C2×C4⋊C8C2314yes—
75C2×C4⋊C8C2318yes—
76C2×#102C2318yes—
77C2×#102C2318yes—
78C2×C22⋊C8C2314yes—
79C2×C4×C8C2314yes—
80#398—28no—
81#395—28no—
82#391—24no—
83#390—24no—
84#390C2318yes—
85#389—24no—
86#389—24no—
87#389—24no—
88#389C2318yes—
89#388—216no—
90#384—22no—
91#384C2312yes—
92#384C2312yes—
93#384C2314yes—
94#384C2314yes—
95#382—22no—
96#380—24no—
97#380—22no—
98#380—28no—
99#380C2314yes—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4D414yes—
103C4×C4⋊C4C2318yes—
104C4×C4⋊C4C2314yes—
105C4×C4⋊C4C2314yes—
106C2×#95—22no—
107C2×#95D412yes—
108#340C2318yes—
109#340C2318yes—
110#340C23116yes—
111#336C2318yes—
112C25—21no—
113C22×Q8—24no—
114C22×D4—21no—
115C22×D4—22no—
116C22×D4—42no—
117C22×D4—42no—
118C22×D4C2×Q814yes—
119C22×D4C2412yes—
120C23×C4—21no—
121C23×C4—28no—
122C23×C4—24no—
123C23×C4—22no—
124C23×C4—44no—
125C23×C4—44no—
126C23×C4C2×D412yes—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2411yescharacteristic
131C22×C8C2418yes—
132C4⋊Q8—48no—
133C4⋊Q8—416no—
134#125—416no—
135C22⋊Q8—416no—
136C4×D4—216no—
137C4×D4—28no—
138C4×D4C2×Q8116yes—
139#117—48no—
140#117—432no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—82no—
158C2×C4⋊C4SD1618yes—
159C2×C4⋊C4C2×D414yes—
160C2×C4⋊C4C2×D414yes—
161C2×C4⋊C4C2×Q814yes—
162C2×C4⋊C4C2414yes—
163C2×C4⋊C4C2414yes—
164C2×C4⋊C4C2414yes—
165C2×C4⋊C4C2412yes—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—48no—
172C2×C22⋊C4—84no—
173C2×C42—28no—
174C2×C42—24no—
175C2×C42—44no—
176C2×C42—48no—
177C2×C42C2×D414yes—
178C2×C42C2×D412yes—
179C2×C42C2×Q814yes—
180C2×C42#4114yes—
181C2×C42C2414yes—
182C2×C42C2412yes—
183#106—416no—
184C4⋊C8—28no—
185C4⋊C8—48no—
186#102—48no—
187#102—48no—
188C22⋊C8—28no—
189C4×C8—28no—
190#95—28no—
191#95—24no—
192#95—24no—
193#95—28no—
194#95C2×D418yes—
195C24—24no—
196C24—22no—
197C24—42no—
198C24—42no—
199C24—42no—
200C24—42no—
201C24—42no—
202C24—42no—
203C24C22×D411yescharacteristic
204C2×Q8—48no—
205C2×Q8—84no—
206C2×Q8—84no—
207C2×D4—24no—
208C2×D4—22no—
209C2×D4—24no—
210C2×D4—24no—
211C2×D4—48no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—48no—
215C2×D4C22×Q814yes—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—216no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—48no—
234C22×C4—48no—
235C22×C4—432no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4C22⋊Q814yes—
240C22×C4C22⋊Q814yes—
241C22×C4C2×SD1618yes—
242C22×C4C2×SD1614yes—
243C22×C4AGL(1,ℤ/8)14yes—
244C22×C4C22×D414yes—
245C22×C4C22×D411yescharacteristic
246C22×C4C22×D412yes—
247C22×C4C22×D414yes—
248C22×C4C22×D412yes—
249C22×C4C22×Q812yes—
250C22×C4C2×#4112yes—
251C22×C4C2511yescharacteristic
252C2×C8—28no—
253C2×C8—28no—
254C2×C8—28no—
255C4⋊C4—24no—
256C4⋊C4—28no—
257C4⋊C4—22no—
258C4⋊C4—28no—
259C4⋊C4—28no—
260C4⋊C4—28no—
261C4⋊C4—44no—
262C4⋊C4—48no—
263C4⋊C4—416no—
264C4⋊C4—48no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—82no—
268C4⋊C4—82no—
269C4⋊C4—84no—
270C4⋊C4—84no—
271C4⋊C4—84no—
272C4⋊C4C2×SD16116yes—
273C4⋊C4C22×Q814yes—
274C22⋊C4—216no—
275C22⋊C4—48no—
276C22⋊C4—416no—
277C22⋊C4—88no—
278C42—28no—
279C42—44no—
280C42—416no—
281C42—48no—
282C42C22⋊Q818yes—
283C42C22×Q814yes—
284C42C2×#4114yes—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—24no—
290C23—48no—
291C23—48no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—48no—
299C23—84no—
300C23—84no—
301C23—84no—
302C23—84no—
303C23C2×C22⋊Q812yes—
304C23C22×SD1612yes—
305C23C2×AGL(1,ℤ/8)12yes—
306C23C23×D411yescharacteristic
307Q8—168no—
308D4—28no—
309D4—48no—
310D4—44no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—44no—
324C2×C4—48no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—48no—
328C2×C4—432no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—84no—
339C2×C4—88no—
340C2×C4#476116yes—
341C2×C4C2×C22⋊Q812yes—
342C2×C4C2×C22⋊Q814yes—
343C2×C4C2×C22⋊Q814yes—
344C2×C4C22×SD1618yes—
345C2×C4C22×SD1614yes—
346C2×C4C2×AGL(1,ℤ/8)14yes—
347C2×C4C23×Q811yescharacteristic
348C2×C4C22×#4111yescharacteristic
349C2×C4#58512yes—
350C2×C4#58512yes—
351C8—48no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—44no—
359C22—44no—
360C22—44no—
361C22—816no—
362C22—82no—
363C22—82no—
364C22—82no—
365C22—82no—
366C22C2×#47614yes—
367C22C22×C22⋊Q811yescharacteristic
368C22#348511yescharacteristic
369C22#348711yescharacteristic
370C22C23×SD1611yescharacteristic
371C22C22×AGL(1,ℤ/8)11yescharacteristic
372C22#358511yescharacteristic
373C22#358611yescharacteristic
374C4—24no—
375C4—22no—
376C4—28no—
377C4—44no—
378C4—44no—
379C4—84no—
380C4—84no—
381C4—84no—
382C4C2×#47618yes—
383C4#345812yes—
384C4#349312yes—
385C2—24no—
386C2—84no—
387C2#3515111yescharacteristic
388C2C22×#47611yescharacteristic
389C2#6049211yescharacteristic
390C2#6169511yescharacteristic
391C2#6172011yescharacteristic
392C2#6188111yescharacteristic
393C2#6188811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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
1. #35167C2semidirect
3. #33148C2semidirect
5. #26090C2semidirect
6. #26024C2semidirect
7. #21981C2semidirect
8. #21951C2semidirect
16. #2552C22semidirect
17. #2551C22semidirect
32. #1987C22semidirect
33. #1948C22semidirect
34. #1948C22semidirect
35. #1939C22semidirect
36. #1920C22semidirect
38. #1844C22semidirect
39. #1834C22semidirect
55. C2×C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
C2×#23542
#331482
#299522
#260902
#260242
#219812
#219512
#207322