Finite Group Databasebeta

← #238220

Subgroups — #238220

Order 512.

1138 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic, complemented
2#21803C211yesmaximal, characteristic
3#21803C211yesmaximal, characteristic
4C2×#1982C211yesmaximal, characteristic
5#10354C214yesmaximal, complemented
6#10351C214yesmaximal, complemented
7#8039C211yesmaximal, characteristic, complemented
8#7893C211yesmaximal, characteristic, complemented
9#7892C211yesmaximal, characteristic, complemented
10C22×#127C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#102—21no—
15C22×#102—21no—
16#2851—22no—
17#2646—22no—
18#2571—21no—
19#2571C2212yescomplemented
20#2570—22no—
21#2570—22no—
22#2552—21no—
23#2552C2212yescomplemented
24#2317—21no—
25#2317—21no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1982—22no—
30#1982—22no—
31#1982C2214yes—
32#1982C2214yes—
33#1957—21no—
34#1957—21no—
35#1957C2212yes—
36#1957C2212yes—
37#1786—22no—
38#1786—22no—
39#1296C2212yescomplemented
40#1296C2212yescomplemented
41#1279C2214yescomplemented
42#1279C2214yescomplemented
43C23×D4—21no—
44C23×D4—21no—
45C23×D4—21no—
46C23×C8—21no—
47C2×#127—21no—
48C2×#127—21no—
49C2×#127—21no—
50C2×#127—22no—
51C2×#127—22no—
52C2×#127—22no—
53C2×#127Q814yes—
54C2×#127C2312yes—
55C2×#125—22no—
56C2×#117—21no—
57C2×#117—21no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C42—21no—
61C22×C42C2311yescharacteristic
62#431—22no—
63#431—22no—
64#431—22no—
65#431—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2312yes—
72C2×#102—24no—
73C2×#102—24no—
74C2×#102—24no—
75C2×#102—24no—
76C2×#102—24no—
77C2×#102—24no—
78C2×#102—44no—
79C2×#102—44no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8C2312yes—
84#400—24no—
85#400—24no—
86#398—24no—
87#398—24no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94#387—24no—
95#387—24no—
96#385—24no—
97#385—24no—
98#384—22no—
99#384—22no—
100#384D411yescharacteristic
101#384D411yescharacteristic
102#384C2311yescharacteristic
103#384C2311yescharacteristic
104#382—22no—
105#382—22no—
106#382—22no—
107#382—21no—
108#382—21no—
109#380—21no—
110#380—21no—
111#380D412yes—
112#380D412yes—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4D412yes—
118C4×C4⋊C4D412yes—
119C4×C4⋊C4C2312yes—
120C4×C4⋊C4C2312yes—
121C2×#95—21no—
122C25—41no—
123C22×D4—24no—
124C22×D4—24no—
125C22×D4—24no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—41no—
130C22×D4—41no—
131C22×D4—41no—
132C22×D4—42no—
133C22×D4—42no—
134C22×D4—42no—
135C22×D4—42no—
136C22×D4—42no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4—82no—
140C22×D4—82no—
141C22×D4—82no—
142C22×D4—82no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—42no—
152C22×C8—42no—
153C22×C8—42no—
154#127—24no—
155#127—24no—
156#127—24no—
157#127—22no—
158#127—22no—
159#127—22no—
160#127—42no—
161#127—42no—
162#127—42no—
163#127C2×Q814yes—
164#125—84no—
165#117—42no—
166#117—42no—
167#117—42no—
168#117—42no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D412yes—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—82no—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×Q812yes—
204C2×C42#4112yes—
205C2×C42#4112yes—
206C2×C42#4112yes—
207C2×C42C2411yescharacteristic
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—42no—
213C4⋊C8—42no—
214#102—48no—
215#102—48no—
216#102—84no—
217#102—84no—
218#102—84no—
219#102—84no—
220C22⋊C8—42no—
221C22⋊C8—42no—
222C4×C8—22no—
223C4×C8—22no—
224C4×C8—42no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235C24—42no—
236C24—44no—
237C24—82no—
238C24—82no—
239C24—82no—
240C24—82no—
241C24—82no—
242C24—82no—
243C24#12011yescharacteristic
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—24no—
247C2×D4—44no—
248C2×D4—44no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—44no—
259C2×D4—44no—
260C2×D4—44no—
261C2×D4—48no—
262C2×D4—48no—
263C2×D4—88no—
264C2×D4—82no—
265C2×D4—82no—
266C2×D4—82no—
267C2×D4—82no—
268C2×D4—88no—
269C2×D4—88no—
270C2×D4—88no—
271C2×D4—82no—
272C2×D4—82no—
273C2×D4—84no—
274C2×D4—84no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—81no—
288C22×C4—81no—
289C22×C4—81no—
290C22×C4—81no—
291C22×C4#12011yescharacteristic
292C22×C4#12011yescharacteristic
293C22×C4#12011yescharacteristic
294C22×C4C22⋊Q812yes—
295C22×C4C22⋊Q812yes—
296C22×C4C22⋊Q812yes—
297C22×C4#12412yes—
298C22×C4#12412yes—
299C22×C4#12412yes—
300C22×C4#13512yes—
301C22×C4#13512yes—
302C22×C4AGL(1,ℤ/8)12yes—
303C22×C4AGL(1,ℤ/8)12yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—84no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—81no—
336C4⋊C4—81no—
337C4⋊C4—84no—
338C4⋊C4—84no—
339C4⋊C4—84no—
340C4⋊C4—84no—
341C22⋊C4—44no—
342C22⋊C4—44no—
343C42—42no—
344C42—42no—
345C42—44no—
346C42—44no—
347C42—82no—
348C42C22⋊Q812yes—
349C42C22⋊Q812yes—
350C42C22⋊Q812yes—
351C42#12412yes—
352C42#12412yes—
353C42#12412yes—
354C42C22×Q811yescharacteristic
355C42C2×#4111yescharacteristic
356C42C2×#4111yescharacteristic
357C42C2×#4111yescharacteristic
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—44no—
363C23—88no—
364C23—88no—
365C23—82no—
366C23—82no—
367C23—82no—
368C23—82no—
369C23—82no—
370C23—82no—
371C23—88no—
372C23—164no—
373C23—164no—
374C23—164no—
375C23—164no—
376C23#39312yes—
377C23#44912yes—
378C23#44912yes—
379C23C2×#12011yescharacteristic
380D4—48no—
381D4—44no—
382D4—44no—
383D4—48no—
384D4—84no—
385D4—88no—
386D4—88no—
387D4—82no—
388D4—82no—
389D4—84no—
390D4—88no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—84no—
420C2×C4—84no—
421C2×C4#39312yes—
422C2×C4#39312yes—
423C2×C4#39312yes—
424C2×C4#44912yes—
425C2×C4#44912yes—
426C2×C4#47814yes—
427C2×C4#47814yes—
428C2×C4#48814yes—
429C2×C4#49014yes—
430C2×C4C2×C22⋊Q811yescharacteristic
431C2×C4C2×C22⋊Q811yescharacteristic
432C2×C4C2×C22⋊Q811yescharacteristic
433C2×C4C2×#12411yescharacteristic
434C2×C4C2×#12411yescharacteristic
435C2×C4C2×#12411yescharacteristic
436C2×C4C2×#13511yescharacteristic
437C2×C4C2×#13511yescharacteristic
438C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
439C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
440C2×C4#57612yes—
441C2×C4#57712yes—
442C8—82no—
443C8—82no—
444C8—82no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—84no—
452C22—84no—
453C22—84no—
454C22—1616no—
455C22—162no—
456C22—162no—
457C22—162no—
458C22—162no—
459C22#202514yes—
460C22C2×#39311yescharacteristic
461C22C2×#44911yescharacteristic
462C22C2×#44911yescharacteristic
463C22#300811yescharacteristic
464C22#301011yescharacteristic
465C22#302111yescharacteristic
466C22#302111yescharacteristic
467C4—22no—
468C4—21no—
469C4—21no—
470C4—22no—
471C4—82no—
472C4—82no—
473C4—82no—
474C4—82no—
475C4—82no—
476C4—82no—
477C4#202514yes—
478C4C2×#47811yescharacteristic
479C4C2×#47811yescharacteristic
480C4#308112yes—
481C4C2×#48811yescharacteristic
482C4C2×#49011yescharacteristic
483C4#313612yes—
484C2—24no—
485C2—164no—
486C2#1889611yescharacteristic
487C2#1889811yescharacteristic
488C2#1891611yescharacteristic
489C2#1891811yescharacteristic
490C2C2×#202511yescharacteristic
491C2#2345611yescharacteristic
492C2#2346311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35181C2semidirect
5. #10354C2semidirect
6. #10351C2semidirect
7. #8039C2semidirect
8. #7893C2semidirect
9. #7892C2semidirect
19. #2571C22semidirect
23. #2552C22semidirect
39. #1296C22semidirect
40. #1296C22semidirect
41. #1279C22semidirect
42. #1279C22semidirect

Maximal subgroups

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

subgroupindex
#351812
#218032
C2×#19822
#103542
#103512
#80392
#78932
#78922