Finite Group Databasebeta

← #6528490

Subgroups — #6528490

Order 512.

1968 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic
3#29103C214yesmaximal
4#27960C214yesmaximal, complemented
5#27813C218yesmaximal, complemented
6#27727C218yesmaximal, complemented
7#21833C212yesmaximal, complemented
8#20732C212yesmaximal
9#20533C211yesmaximal, characteristic, complemented
10C2×#544—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C22×C22⋊Q8—21no—
13C22×C22⋊Q8—21no—
14C22×C4×D4C2211yescharacteristic
15C2×#431C2212yes—
16#2857C2214yes—
17#2659—22no—
18#2552C2214yescomplemented
19#2551C2214yescomplemented
20#2550—24no—
21#2542—24no—
22#2542C2218yes—
23C2×#399—22no—
24#2316—21no—
25C2×#390C2212yes—
26#2285C2212yes—
27C2×#384C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29#2099C22116yescomplemented
30#2071C2218yes—
31#2023C2218yes—
32#1990C2218yescomplemented
33#1990C2214yescomplemented
34#1990C2214yescomplemented
35#1988C2218yescomplemented
36#1934C22116yes—
37#1834C2218yescomplemented
38#1823C2214yes—
39#1823C2218yes—
40#1802C2214yes—
41#1802C2214yes—
42#1786C2214yes—
43C8×C4⋊C4C2218yescomplemented
44C4×#102C2218yes—
45C4×C22⋊C8C2214yes—
46C23×Q8—21no—
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49#544—48no—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—28no—
53C2×C4⋊Q8C2314yes—
54C2×C4⋊Q8C2314yes—
55C2×#125—24no—
56C2×C22⋊Q8—24no—
57C2×C22⋊Q8—28no—
58C2×C22⋊Q8—28no—
59C2×C22⋊Q8—48no—
60C2×C22⋊Q8—48no—
61C2×C4×D4—22no—
62C2×C4×D4—28no—
63C2×C4×D4C2314yes—
64C2×#117—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C42C2312yes—
76C22×C42C2311yescharacteristic
77#431—28no—
78C2×C4⋊C8C2314yes—
79C2×#102C2318yes—
80C2×#102C2318yes—
81C2×C22⋊C8C2314yes—
82C2×C4×C8C2314yes—
83C2×C4×C8C2318yes—
84#399—216no—
85#398—28no—
86#395—28no—
87#391—24no—
88#390—24no—
89#390C2318yes—
90#389—24no—
91#389—24no—
92#389—24no—
93#389C2318yes—
94#384—24no—
95#384C2312yes—
96#384C2312yes—
97#384C2314yes—
98#384C2318yes—
99#382—22no—
100#380—24no—
101#380—22no—
102#380C2314yes—
103C4×C4⋊C4—22no—
104C4×C4⋊C4D414yes—
105C4×C4⋊C4C2314yes—
106C4×C4⋊C4C2314yes—
107C2×#95D412yes—
108#340C2318yes—
109#340C2318yes—
110#340C23116yes—
111#340C23116yes—
112#336C2318yes—
113C25—21no—
114C22×Q8—24no—
115C22×Q8—24no—
116C22×Q8—44no—
117C22×Q8—44no—
118C22×Q8—44no—
119C22×D4—24no—
120C22×D4—21no—
121C22×D4—24no—
122C22×D4—42no—
123C22×D4C2412yes—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—24no—
127C23×C4—44no—
128C23×C4—44no—
129C23×C4—44no—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2411yescharacteristic
135C22×C8C2418yes—
136C4⋊Q8—28no—
137C4⋊Q8—28no—
138C4⋊Q8—48no—
139C4⋊Q8—416no—
140C4⋊Q8—48no—
141#125—416no—
142C22⋊Q8—416no—
143C22⋊Q8—432no—
144C22⋊Q8—432no—
145C4×D4—28no—
146C4×D4—28no—
147C4×D4—48no—
148#117—48no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—28no—
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—48no—
167C2×C4⋊C4D818yes—
168C2×C4⋊C4SD1618yes—
169C2×C4⋊C4C2×D414yes—
170C2×C4⋊C4C2×D414yes—
171C2×C4⋊C4C2412yes—
172C2×C4⋊C4C2414yes—
173C2×C4⋊C4C2414yes—
174C2×C4⋊C4C2414yes—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—48no—
181C2×C22⋊C4—48no—
182C2×C42—24no—
183C2×C42—44no—
184C2×C42C2×D414yes—
185C2×C42C2×D412yes—
186C2×C42#4118yes—
187C2×C42C2414yes—
188C2×C42C2412yes—
189C4⋊C8—28no—
190#102—216no—
191#102—48no—
192C22⋊C8—28no—
193C4×C8—28no—
194C4×C8—48no—
195#95—24no—
196#95—24no—
197#95—28no—
198#95C2×D418yes—
199C24—24no—
200C24—22no—
201C24—42no—
202C24—42no—
203C24—42no—
204C24—42no—
205C24—42no—
206C24—42no—
207C24C22×D411yescharacteristic
208C2×Q8—48no—
209C2×Q8—48no—
210C2×Q8—48no—
211C2×Q8—48no—
212C2×Q8—84no—
213C2×Q8—84no—
214C2×Q8—816no—
215C2×D4—24no—
216C2×D4—24no—
217C2×D4—22no—
218C2×D4—24no—
219C2×D4—416no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D42−1+414yes—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—48no—
242C22×C4—48no—
243C22×C4—48no—
244C22×C4—816no—
245C22×C4#12414yes—
246C22×C4#12414yes—
247C22×C4C2×D818yes—
248C22×C4C2×D814yes—
249C22×C4C2×SD1618yes—
250C22×C4C2×SD1614yes—
251C22×C4C22×D411yescharacteristic
252C22×C4C22×D412yes—
253C22×C4C22×D414yes—
254C22×C4C22×D414yes—
255C22×C4C22×D412yes—
256C22×C4C2×#4114yes—
257C22×C4C2511yescharacteristic
258C2×C8—28no—
259C2×C8—28no—
260C2×C8—28no—
261C4⋊C4—24no—
262C4⋊C4—28no—
263C4⋊C4—28no—
264C4⋊C4—28no—
265C4⋊C4—42no—
266C4⋊C4—44no—
267C4⋊C4—48no—
268C4⋊C4—416no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—48no—
272C4⋊C4—416no—
273C4⋊C4—416no—
274C4⋊C4—82no—
275C4⋊C4—84no—
276C4⋊C4—82no—
277C4⋊C4—84no—
278C4⋊C4C2×D8116yes—
279C4⋊C4C2×SD16116yes—
280C22⋊C4—48no—
281C22⋊C4—48no—
282C22⋊C4—416no—
283C22⋊C4—416no—
284C42—28no—
285C42—44no—
286C42—48no—
287C42#12418yes—
288C42C2×#4114yes—
289C42C2×#4114yes—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—24no—
295C23—48no—
296C23—48no—
297C23—48no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—84no—
305C23—84no—
306C23—84no—
307C23—84no—
308C23C2×#12412yes—
309C23C22×D812yes—
310C23C22×SD1612yes—
311C23C23×D411yescharacteristic
312Q8—816no—
313Q8—168no—
314D4—28no—
315D4—48no—
316D4—44no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—28no—
326C2×C4—44no—
327C2×C4—48no—
328C2×C4—48no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—84no—
342C2×C4—88no—
343C2×C4—816no—
344C2×C4#486116yes—
345C2×C4C2×#12412yes—
346C2×C4C2×#12414yes—
347C2×C4C2×#12414yes—
348C2×C4C22×D818yes—
349C2×C4C22×D814yes—
350C2×C4C22×SD1618yes—
351C2×C4C22×SD1614yes—
352C2×C4C22×#4112yes—
353C2×C4C2×2−1+412yes—
354C2×C4#58512yes—
355C8—48no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—44no—
363C22—44no—
364C22—44no—
365C22—816no—
366C22—82no—
367C22—82no—
368C22—82no—
369C22—82no—
370C22C2×#48614yes—
371C22C22×#12411yescharacteristic
372C22#348511yescharacteristic
373C22#348611yescharacteristic
374C22C23×D811yescharacteristic
375C22C23×SD1611yescharacteristic
376C22#358611yescharacteristic
377C22#358611yescharacteristic
378C4—22no—
379C4—24no—
380C4—28no—
381C4—28no—
382C4—84no—
383C4—82no—
384C4—84no—
385C4—84no—
386C4C2×#48618yes—
387C4#349514yes—
388C2—24no—
389C2—84no—
390C2#3564411yescharacteristic
391C2C22×#48611yescharacteristic
392C2#6062011yescharacteristic
393C2#6169411yescharacteristic
394C2#6169511yescharacteristic
395C2#6181711yescharacteristic
396C2#6181811yescharacteristic
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. #27960C2semidirect
5. #27813C2semidirect
6. #27727C2semidirect
7. #21833C2semidirect
9. #20533C2semidirect
18. #2552C22semidirect
19. #2551C22semidirect
29. #2099C22semidirect
32. #1990C22semidirect
33. #1990C22semidirect
34. #1990C22semidirect
35. #1988C22semidirect
37. #1834C22semidirect
43. C8×C4⋊C4C22semidirect

Maximal subgroups

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

subgroupindex
#351672
C2×#25422
#291032
#279602
#278132
#277272
#218332
#207322
#205332