Finite Group Databasebeta

← #6484523

Subgroups — #6484523

Order 512.

2268 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic, complemented
2C2×#2366C211yesmaximal, characteristic
3#32956C214yesmaximal, complemented
4#29952C214yesmaximal, complemented
5#26530C218yesmaximal, complemented
6#26464C218yesmaximal, complemented
7#21951C211yesmaximal, characteristic, complemented
8#21803C212yesmaximal
9#21359C212yesmaximal, complemented
10C2×#544—21no—
11C22×#127C2211yescharacteristic
12C22×C4×D4—21no—
13C22×C4×D4C2212yes—
14C2×#476C2218yescomplemented
15C2×#431C2212yes—
16#2857C2214yes—
17#2659—22no—
18#2552C2214yescomplemented
19#2551C2214yescomplemented
20#2550—24no—
21#2366—24no—
22#2366C2218yes—
23#2316—21no—
24C2×#390—21no—
25C2×#390C2212yes—
26C2×#386—22no—
27#2285C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#2094C2218yescomplemented
31#2065C22116yescomplemented
32#2023C2218yescomplemented
33#1982C2218yes—
34#1957C2214yes—
35#1957C2214yes—
36#1948C2214yescomplemented
37#1948C2214yescomplemented
38#1939C2218yescomplemented
39#1864C22116yescomplemented
40#1853C2214yes—
41#1853C2214yes—
42#1802C2218yescomplemented
43#1797C2214yes—
44#1786C2218yes—
45C4×#106C2218yescomplemented
46C23×D4—21no—
47C23×D4—21no—
48C23×D4C2312yes—
49C24×C4—22no—
50#544—48no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8C2314yes—
53C2×#127—22no—
54C2×#127—24no—
55C2×#127Q818yescomplemented
56C2×#127C2314yes—
57C2×#125—24no—
58C2×C22⋊Q8—24no—
59C2×C4×D4—28no—
60C2×C4×D4—24no—
61C2×C4×D4—48no—
62C2×C4×D4—48no—
63C2×C4×D4C2318yes—
64C2×#117—22no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C42—21no—
73C22×C42C2312yes—
74C22×C42C2311yescharacteristic
75#476—416no—
76#431—28no—
77C2×#106C2318yes—
78C2×#106C2318yes—
79C2×C4⋊C8C2314yes—
80C2×C4⋊C8C2318yes—
81C2×#102C23116yes—
82C2×#102C23116yes—
83C2×C22⋊C8C2314yes—
84C2×C4×C8C2314yes—
85#398—28no—
86#395—28no—
87#391—24no—
88#390—28no—
89#390—24no—
90#390C2318yes—
91#389—24no—
92#389—24no—
93#389—24no—
94#389C2318yes—
95#386—216no—
96#384C2314yes—
97#384C2312yes—
98#384C2312yes—
99#382—22no—
100#380—24no—
101#380—22no—
102#380C2314yes—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4D414yes—
106C4×C4⋊C4C2314yes—
107C4×C4⋊C4C2314yes—
108C4×C4⋊C4C2314yes—
109C2×#95—21no—
110C2×#95D412yes—
111#340C2318yes—
112#340C2318yes—
113#336C2318yes—
114C25—22no—
115C22×Q8—24no—
116C22×D4—28no—
117C22×D4—28no—
118C22×D4—22no—
119C22×D4—24no—
120C22×D4—24no—
121C22×D4—24no—
122C22×D4—44no—
123C22×D4—44no—
124C22×D4—44no—
125C22×D4—44no—
126C22×D4—82no—
127C22×D4—82no—
128C22×D4C2×Q818yes—
129C22×D4C2414yes—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—28no—
133C23×C4—48no—
134C23×C4—48no—
135C23×C4—84no—
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C22×C8C2418yes—
142C4⋊Q8—48no—
143C4⋊Q8—416no—
144#127—216no—
145#127—28no—
146#127C2×Q8116yes—
147#125—416no—
148C22⋊Q8—416no—
149C4×D4—216no—
150C4×D4—48no—
151C4×D4—416no—
152C4×D4—432no—
153#117—48no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4SD1618yes—
168C2×C4⋊C4C2×D414yes—
169C2×C4⋊C4C2×D414yes—
170C2×C4⋊C4C2412yes—
171C2×C4⋊C4C2414yes—
172C2×C4⋊C4C2414yes—
173C2×C4⋊C4C2412yes—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—48no—
179C2×C22⋊C4—48no—
180C2×C22⋊C4—84no—
181C2×C42—24no—
182C2×C42—24no—
183C2×C42—44no—
184C2×C42—44no—
185C2×C42C2×D414yes—
186C2×C42C2×D412yes—
187C2×C42C2×Q814yes—
188C2×C42#4114yes—
189C2×C42C2414yes—
190C2×C42C2412yes—
191#106—216no—
192#106—48no—
193C4⋊C8—28no—
194C4⋊C8—48no—
195#102—416no—
196#102—416no—
197C22⋊C8—28no—
198C4×C8—28no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—28no—
203#95C2×D418yes—
204C24—28no—
205C24—24no—
206C24—44no—
207C24—44no—
208C24—44no—
209C24—44no—
210C24—44no—
211C24—44no—
212C24C22×D411yescharacteristic
213C2×Q8—48no—
214C2×Q8—84no—
215C2×Q8—84no—
216C2×D4—28no—
217C2×D4—216no—
218C2×D4—216no—
219C2×D4—24no—
220C2×D4—28no—
221C2×D4—28no—
222C2×D4—416no—
223C2×D4—44no—
224C2×D4—44no—
225C2×D4—416no—
226C2×D4—48no—
227C2×D4—48no—
228C2×D4—432no—
229C2×D4—84no—
230C2×D4—84no—
231C2×D4—816no—
232C2×D4C22×Q818yes—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—416no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—416no—
254C22×C4—88no—
255C22×C4—832no—
256C22×C4C22⋊Q814yes—
257C22×C4C22⋊Q814yes—
258C22×C4C2×SD1618yes—
259C22×C4C2×SD1614yes—
260C22×C4AGL(1,ℤ/8)14yes—
261C22×C4C22×D414yes—
262C22×C4C22×D412yes—
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D414yes—
265C22×C4C22×D412yes—
266C22×C4C22×Q812yes—
267C22×C4C2×#4112yes—
268C22×C4C2511yescharacteristic
269C2×C8—28no—
270C2×C8—28no—
271C2×C8—28no—
272C4⋊C4—24no—
273C4⋊C4—28no—
274C4⋊C4—22no—
275C4⋊C4—28no—
276C4⋊C4—28no—
277C4⋊C4—24no—
278C4⋊C4—42no—
279C4⋊C4—48no—
280C4⋊C4—416no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—84no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C4⋊C4—84no—
287C4⋊C4C2×SD16116yes—
288C4⋊C4H(5,𝔽2)14yes—
289C22⋊C4—416no—
290C22⋊C4—48no—
291C22⋊C4—416no—
292C22⋊C4—88no—
293C42—28no—
294C42—44no—
295C42—48no—
296C42—48no—
297C42C22⋊Q818yes—
298C42C22×Q814yes—
299C42C2×#4114yes—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—28no—
305C23—416no—
306C23—416no—
307C23—44no—
308C23—44no—
309C23—44no—
310C23—44no—
311C23—44no—
312C23—44no—
313C23—416no—
314C23—88no—
315C23—88no—
316C23—88no—
317C23—88no—
318C23C2×C22⋊Q812yes—
319C23C22×SD1612yes—
320C23C2×AGL(1,ℤ/8)12yes—
321C23C23×D411yescharacteristic
322Q8—168no—
323D4—216no—
324D4—416no—
325D4—48no—
326D4—432no—
327D4—816no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—48no—
341C2×C4—44no—
342C2×C4—48no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—832no—
355C2×C4—84no—
356C2×C4—84no—
357C2×C4#476116yescomplemented
358C2×C4C2×C22⋊Q812yes—
359C2×C4C2×C22⋊Q814yes—
360C2×C4C2×C22⋊Q814yes—
361C2×C4C22×SD1618yes—
362C2×C4C22×SD1614yes—
363C2×C4C2×AGL(1,ℤ/8)14yes—
364C2×C4C23×Q811yescharacteristic
365C2×C4C22×#4111yescharacteristic
366C2×C4C2×H(5,𝔽2)12yes—
367C2×C4#58512yes—
368C8—48no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—48no—
376C22—48no—
377C22—48no—
378C22—832no—
379C22—84no—
380C22—84no—
381C22—84no—
382C22—84no—
383C22C2×#47614yes—
384C22C22×C22⋊Q811yescharacteristic
385C22#348411yescharacteristic
386C22#348711yescharacteristic
387C22C23×SD1611yescharacteristic
388C22C22×AGL(1,ℤ/8)11yescharacteristic
389C22#358511yescharacteristic
390C22#358511yescharacteristic
391C4—24no—
392C4—22no—
393C4—28no—
394C4—44no—
395C4—44no—
396C4—84no—
397C4—82no—
398C4—84no—
399C4C2×#47618yes—
400C4#346012yes—
401C4#350112yes—
402C2—24no—
403C2—88no—
404C2#3517411yescharacteristic
405C2C22×#47611yescharacteristic
406C2#6049011yescharacteristic
407C2#6156011yescharacteristic
408C2#6158711yescharacteristic
409C2#6188111yescharacteristic
410C2#6189111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
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. #32956C2semidirect
4. #29952C2semidirect
5. #26530C2semidirect
6. #26464C2semidirect
7. #21951C2semidirect
9. #21359C2semidirect
14. C2×#476C22semidirect
18. #2552C22semidirect
19. #2551C22semidirect
30. #2094C22semidirect
31. #2065C22semidirect
32. #2023C22semidirect
36. #1948C22semidirect
37. #1948C22semidirect
38. #1939C22semidirect
39. #1864C22semidirect
42. #1802C22semidirect
45. C4×#106C22semidirect
55. C2×#127Q8semidirect
357. C2×C4#476semidirect

Maximal subgroups

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

subgroupindex
#351672
C2×#23662
#329562
#299522
#265302
#264642
#219512
#218032
#213592