Finite Group Databasebeta

← #6487847

Subgroups — #6487847

Order 512.

2166 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic, complemented
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3#32956C214yesmaximal
4#29968C214yesmaximal, complemented
5#26110C218yesmaximal, complemented
6#26044C218yesmaximal, complemented
7#21981C212yesmaximal, complemented
8#21951C211yesmaximal, characteristic, complemented
9#20732C212yesmaximal
10C2×#544—21no—
11C22×C4×D4C2211yescharacteristic
12C22×C4×D4C2212yes—
13C23×C4⋊C4—21no—
14C2×#476C2218yes—
15C2×#431C2212yes—
16#2857C2214yes—
17#2659—22no—
18#2552C2214yescomplemented
19#2551C2214yescomplemented
20#2550—24no—
21D4×C4⋊C4—24no—
22D4×C4⋊C4C2218yes—
23#2316—21no—
24C2×#389C2212yes—
25C2×#385—22no—
26#2285C2212yes—
27C2×#384C2211yescharacteristic
28C2×#380—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30#2100C2218yescomplemented
31#2088C22116yescomplemented
32#2058C2218yescomplemented
33#1948C2218yescomplemented
34#1939C2218yescomplemented
35#1864C22116yes—
36#1853C2218yes—
37#1844C2214yescomplemented
38#1834C2214yescomplemented
39#1823C2214yes—
40#1802C2214yes—
41#1802C2218yes—
42#1802C2214yes—
43#1797C2214yes—
44#1786C2214yes—
45C4×#102C2218yes—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C24×C4—22no—
49#544—48no—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8C2314yes—
52C2×#125—24no—
53C2×C22⋊Q8—24no—
54C2×C4×D4—22no—
55C2×C4×D4—24no—
56C2×C4×D4—24no—
57C2×C4×D4—48no—
58C2×C4×D4Q818yescomplemented
59C2×C4×D4C2318yes—
60C2×C4×D4C2314yes—
61C2×#117—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—28no—
66C22×C4⋊C4—48no—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—22no—
73C22×C42C2312yes—
74C22×C42C2311yescharacteristic
75#476—416no—
76#431—28no—
77C2×#106C23116yes—
78C2×C4⋊C8C2314yes—
79C2×C4⋊C8C2318yes—
80C2×#102C2318yes—
81C2×#102C2318yes—
82C2×C22⋊C8C2314yes—
83C2×C4×C8C2314yes—
84#398—28no—
85#395—28no—
86#391—24no—
87#389—24no—
88#389—24no—
89#389—24no—
90#389—24no—
91#389C2318yes—
92#389C2318yes—
93#385—216no—
94#384—22no—
95#384C2314yes—
96#384C2312yes—
97#384C2312yes—
98#384C2314yes—
99#382—22no—
100#380—24no—
101#380—22no—
102#380—28no—
103#380C2314yes—
104C4×C4⋊C4—22no—
105C4×C4⋊C4D414yes—
106C4×C4⋊C4C2314yes—
107C4×C4⋊C4C2314yes—
108C2×#95—22no—
109C2×#95D412yes—
110#340C2318yes—
111#340C2318yes—
112#340C23116yes—
113#336C2318yes—
114C25—21no—
115C22×Q8—24no—
116C22×D4—24no—
117C22×D4—24no—
118C22×D4—21no—
119C22×D4—22no—
120C22×D4C2×Q814yes—
121C22×D4C2412yes—
122C23×C4—21no—
123C23×C4—28no—
124C23×C4—24no—
125C23×C4—22no—
126C23×C4—28no—
127C23×C4—44no—
128C23×C4—44no—
129C23×C4—48no—
130C23×C4—48no—
131C23×C4—84no—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2411yescharacteristic
137C22×C8C2418yes—
138C4⋊Q8—48no—
139C4⋊Q8—416no—
140#125—416no—
141C22⋊Q8—416no—
142C4×D4—216no—
143C4×D4—216no—
144C4×D4—28no—
145C4×D4—48no—
146C4×D4—416no—
147C4×D4C2×Q8116yes—
148#117—48no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—48no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—432no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4SD1618yes—
170C2×C4⋊C4C2×D414yes—
171C2×C4⋊C4C2×D414yes—
172C2×C4⋊C4C2×Q814yes—
173C2×C4⋊C4C2414yes—
174C2×C4⋊C4C2414yes—
175C2×C4⋊C4C2414yes—
176C2×C4⋊C4C2412yes—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—48no—
182C2×C42—24no—
183C2×C42—44no—
184C2×C42C2×D414yes—
185C2×C42C2×D412yes—
186C2×C42C2×Q814yes—
187C2×C42#4114yes—
188C2×C42C2414yes—
189C2×C42C2412yes—
190#106—416no—
191C4⋊C8—28no—
192C4⋊C8—48no—
193#102—216no—
194#102—216no—
195C22⋊C8—28no—
196C4×C8—28no—
197#95—28no—
198#95—24no—
199#95—24no—
200#95—28no—
201#95C2×D418yes—
202C24—24no—
203C24—24no—
204C24—24no—
205C24—22no—
206C24—42no—
207C24—42no—
208C24—42no—
209C24—42no—
210C24C22×D411yescharacteristic
211C2×Q8—48no—
212C2×Q8—84no—
213C2×Q8—84no—
214C2×D4—24no—
215C2×D4—24no—
216C2×D4—216no—
217C2×D4—216no—
218C2×D4—24no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4C22×Q814yes—
223C2×D4H(5,𝔽2)14yes—
224C2×D42−1+414yes—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—216no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—416no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—48no—
244C22×C4—48no—
245C22×C4—432no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—416no—
250C22×C4—88no—
251C22×C4—832no—
252C22×C4C22⋊Q814yes—
253C22×C4C22⋊Q814yes—
254C22×C4C2×SD1618yes—
255C22×C4C2×SD1614yes—
256C22×C4AGL(1,ℤ/8)14yes—
257C22×C4C22×D414yes—
258C22×C4C22×D411yescharacteristic
259C22×C4C22×D412yes—
260C22×C4C22×D414yes—
261C22×C4C22×D412yes—
262C22×C4C22×Q812yes—
263C22×C4C2×#4112yes—
264C22×C4C2511yescharacteristic
265C2×C8—28no—
266C2×C8—28no—
267C2×C8—28no—
268C4⋊C4—24no—
269C4⋊C4—28no—
270C4⋊C4—22no—
271C4⋊C4—28no—
272C4⋊C4—28no—
273C4⋊C4—28no—
274C4⋊C4—44no—
275C4⋊C4—48no—
276C4⋊C4—416no—
277C4⋊C4—416no—
278C4⋊C4—48no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—82no—
282C4⋊C4—84no—
283C4⋊C4—82no—
284C4⋊C4—84no—
285C4⋊C4—84no—
286C4⋊C4C2×SD16116yes—
287C4⋊C4C22×Q814yes—
288C22⋊C4—216no—
289C22⋊C4—416no—
290C22⋊C4—48no—
291C42—28no—
292C42—44no—
293C42—48no—
294C42C22⋊Q818yes—
295C42C22×Q814yes—
296C42C2×#4114yes—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—216no—
302C23—24no—
303C23—24no—
304C23—24no—
305C23—48no—
306C23—48no—
307C23—48no—
308C23—48no—
309C23—48no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—48no—
315C23C2×C22⋊Q812yes—
316C23C22×SD1612yes—
317C23C2×AGL(1,ℤ/8)12yes—
318C23C23×D411yescharacteristic
319Q8—168no—
320D4—216no—
321D4—24no—
322D4—24no—
323D4—24no—
324D4#54418yescomplemented
325C2×C4—22no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—48no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—48no—
342C2×C4—432no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—832no—
353C2×C4—84no—
354C2×C4—88no—
355C2×C4#476116yescomplemented
356C2×C4C2×C22⋊Q812yes—
357C2×C4C2×C22⋊Q814yes—
358C2×C4C2×C22⋊Q814yes—
359C2×C4C22×SD1618yes—
360C2×C4C22×SD1614yes—
361C2×C4C2×AGL(1,ℤ/8)14yes—
362C2×C4C23×Q811yescharacteristic
363C2×C4C22×#4111yescharacteristic
364C2×C4C2×H(5,𝔽2)12yes—
365C2×C4C2×2−1+412yes—
366C8—48no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—28no—
374C22—432no—
375C22—44no—
376C22—44no—
377C22—44no—
378C22—44no—
379C22—44no—
380C22—44no—
381C22C2×#47614yes—
382C22C22×C22⋊Q811yescharacteristic
383C22#348411yescharacteristic
384C22#348611yescharacteristic
385C22C23×SD1611yescharacteristic
386C22C22×AGL(1,ℤ/8)11yescharacteristic
387C22#358511yescharacteristic
388C22#358611yescharacteristic
389C4—24no—
390C4—28no—
391C4—22no—
392C4—44no—
393C4—44no—
394C4—84no—
395C4—84no—
396C4—84no—
397C4C2×#47618yes—
398C4C2×#54412yes—
399C4#349212yes—
400C2—24no—
401C2—48no—
402C2#3522411yescharacteristic
403C2C22×#47611yescharacteristic
404C2#6049211yescharacteristic
405C2#6156011yescharacteristic
406C2#6158511yescharacteristic
407C2#6181811yescharacteristic
408C2#6182511yescharacteristic
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
4. #29968C2semidirect
5. #26110C2semidirect
6. #26044C2semidirect
7. #21981C2semidirect
8. #21951C2semidirect
18. #2552C22semidirect
19. #2551C22semidirect
30. #2100C22semidirect
31. #2088C22semidirect
32. #2058C22semidirect
33. #1948C22semidirect
34. #1939C22semidirect
37. #1844C22semidirect
38. #1834C22semidirect
58. C2×C4×D4Q8semidirect
324. D4#544semidirect
355. 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×D4×C4⋊C42
#329562
#299682
#261102
#260442
#219812
#219512
#207322