Finite Group Databasebeta

← #6488765

Subgroups — #6488765

Order 512.

1780 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35295C211yesmaximal, characteristic, complemented
2#34207C211yesmaximal, characteristic, complemented
3#33013C214yesmaximal, complemented
4#30129C212yesmaximal, complemented
5#30126C212yesmaximal, complemented
6#29368C212yesmaximal, complemented
7#29365C212yesmaximal, complemented
8#29199C212yesmaximal, complemented
9#28630C212yesmaximal, complemented
10#28127C214yesmaximal, complemented
11#27966C211yesmaximal, characteristic, complemented
12#27906C211yesmaximal, characteristic, complemented
13#25858C211yesmaximal, characteristic, complemented
14#25729C211yesmaximal, characteristic
15#22279C212yesmaximal, complemented
16#22227C212yesmaximal, complemented
17#20784C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×#520C2212yes—
20C2×#496C2212yes—
21C2×#488C2212yes—
22C2×#481C2214yes—
23C2×#449C2214yes—
24#2860C2212yes—
25#2856C2211yescharacteristic
26#2851C2211yescharacteristic
27#2647C2211yescharacteristic
28#2641C2211yescharacteristic
29#2624C2212yes—
30#2597—22no—
31#2559—22no—
32#2479—22no—
33#2476—22no—
34#2361—22no—
35#2361—22no—
36#2354—22no—
37D4×C4⋊C4—22no—
38#2330C2211yescharacteristic
39#2328C2211yescharacteristic
40#2326C2212yes—
41#2326C2212yes—
42#2324C2212yes—
43#2311C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C4×#127C2211yescharacteristic
48#2293C2212yes—
49C2×#384C2211yescharacteristic
50C4×#117C2211yescharacteristic
51#2100C2212yes—
52#2100C2212yes—
53#2094C2212yes—
54#2094C2212yes—
55#2066C2214yes—
56#2065C2214yes—
57#2055C2214yes—
58#2054C2214yes—
59#2043C2212yes—
60#2041C2212yes—
61#2013C2212yes—
62#2011C2212yes—
63#1992C2212yes—
64#1959C2212yes—
65#1958C2212yes—
66#1957C2212yes—
67#1927C2214yes—
68#1926C2214yes—
69#1880C2214yes—
70#1880C2214yes—
71#1879C2214yes—
72#1879C2214yes—
73#1879C2214yes—
74#1877C2214yes—
75#1865C2214yes—
76#1865C2214yes—
77#1864C2214yes—
78#1864C2214yes—
79#1844C2212yes—
80#1839C2212yes—
81#1836C2212yes—
82#1831C2212yes—
83#1823C2214yes—
84#1818C2212yes—
85#1798C2212yes—
86#1788C2214yes—
87#1768C2212yes—
88#1749C2211yescharacteristic
89C4×C22⋊C8C2211yescharacteristic
90#1746C2212yes—
91C22×#41C2312yes—
92C22×SD16C2314yes—
93C22×D8C2314yes—
94#532—28no—
95C2×C4⋊Q8C2311yescharacteristic
96C2×#127C2311yescharacteristic
97C2×#127C2311yescharacteristic
98C2×#126—22no—
99C2×#125C2311yescharacteristic
100C2×#124C2312yes—
101C2×#121—22no—
102C2×#121—22no—
103C2×#121C2312yes—
104#520—48no—
105C2×C4×Q8C2312yes—
106C2×C4×Q8C2312yes—
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—22no—
111C2×C4×D4—22no—
112C2×C4×D4C2312yes—
113C2×C4×D4C2312yes—
114C2×C4×D4C2312yes—
115C2×#117—22no—
116C2×#117—22no—
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C22×C4⋊C4—22no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4C2312yes—
122C22×C42C2311yescharacteristic
123#496—28no—
124#488—28no—
125#481—48no—
126#449—48no—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2312yes—
140C2×#97C2312yes—
141C2×C4×C8C2312yes—
142#400—22no—
143#399—22no—
144#398—22no—
145#397—22no—
146#394—22no—
147#394—22no—
148#394—22no—
149#394—22no—
150#391C2312yes—
151#391C2312yes—
152#390—22no—
153#390—22no—
154#390—22no—
155#390—22no—
156#390C2312yes—
157#390C2312yes—
158#390C2312yes—
159#390C2312yes—
160#390C2312yes—
161#390C2312yes—
162#389—22no—
163#389—22no—
164#389—22no—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—24no—
176#388—24no—
177#387—22no—
178#387—22no—
179#386—24no—
180#386—22no—
181#386—22no—
182#386—22no—
183#385—22no—
184#385—22no—
185#385—24no—
186#385—24no—
187#384—22no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2311yescharacteristic
192#382C2311yescharacteristic
193#380—24no—
194#380C2312yes—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4D412yes—
198C4×C4⋊C4D412yes—
199C4×C4⋊C4C2312yes—
200C4×C22⋊C4—22no—
201C4×C22⋊C4D412yes—
202C4×C22⋊C4D412yes—
203#376C2311yescharacteristic
204#376C2311yescharacteristic
205C43C2311yescharacteristic
206C43C2311yescharacteristic
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#336C2314yes—
212#336C2314yes—
213#336C2314yes—
214#336C2312yes—
215#336C2312yes—
216C2×#41—24no—
217C2×#41—24no—
218C2×#41—28no—
219C2×#41—44no—
220C22×Q8C2412yes—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4C2412yes—
226C22×D4C2412yes—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4C2×D412yes—
233C23×C4C2411yescharacteristic
234C2×SD16—28no—
235C2×SD16—44no—
236C2×SD16—44no—
237C2×D8—28no—
238C2×D8—44no—
239C2×D8—44no—
240C22×C8C2414yes—
241C22×C8C2414yes—
242C4⋊Q8—24no—
243#127—24no—
244#127—24no—
245#126—84no—
246#125—24no—
247#124—24no—
248#124#4118yes—
249#121—216no—
250#121—28no—
251#121—84no—
252C4×Q8—44no—
253C4×Q8—44no—
254C4×D4—216no—
255C4×D4—42no—
256C4×D4—42no—
257C4×D4—44no—
258C4×D4—44no—
259C4×D4—48no—
260C4×D4—48no—
261C4×D4—84no—
262C4×D4—84no—
263#117—24no—
264#117—42no—
265#117—48no—
266#117—84no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—48no—
282C2×C4⋊C4—84no—
283C2×C4⋊C4C2×D411yescharacteristic
284C2×C4⋊C4C2×D411yescharacteristic
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4C2×D411yescharacteristic
308C2×C22⋊C4C2×D411yescharacteristic
309C2×C22⋊C4C2×D412yes—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2411yescharacteristic
320C2×C42C2411yescharacteristic
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323C2×C42C2411yescharacteristic
324C2×C42C2411yescharacteristic
325#106—44no—
326#106—44no—
327C4⋊C8—44no—
328#103—28no—
329#103—44no—
330#103—44no—
331#102—28no—
332#102—44no—
333#102—44no—
334#102—44no—
335#102—44no—
336C22⋊C8—24no—
337C22⋊C8—24no—
338#97—24no—
339C4×C8—24no—
340#95—22no—
341#95—22no—
342#95—24no—
343#95—24no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95C2×D412yes—
350#95C2×D412yes—
351#95C2×D412yes—
352#95C2×D412yes—
353C24—22no—
354C24—22no—
355C24C22×D411yescharacteristic
356#41—416no—
357#41—416no—
358C2×Q8—22no—
359C2×Q8—24no—
360C2×Q8—24no—
361C2×Q8C2×#4114yes—
362C2×D4—28no—
363C2×D4—28no—
364C2×D4—22no—
365C2×D4—24no—
366C2×D4—24no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—82no—
378C2×D4—82no—
379C2×D4—82no—
380C2×D4—82no—
381C2×D4C2×#4114yes—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—28no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—42no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—82no—
401C22×C4—82no—
402C22×C4—84no—
403C22×C4—84no—
404C22×C4—84no—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D412yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D412yes—
414C22×C4H(5,𝔽2)14yes—
415C22×C4C2511yescharacteristic
416SD16—88no—
417D8—88no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C4⋊C4—28no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—42no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—84no—
442C4⋊C4—82no—
443C4⋊C4—82no—
444C4⋊C4—82no—
445C4⋊C4—82no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—216no—
449C22⋊C4—42no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—82no—
457C22⋊C4—82no—
458C22⋊C4—82no—
459C22⋊C4—82no—
460C22⋊C4—84no—
461C22⋊C4—84no—
462C22⋊C4—82no—
463C22⋊C4—82no—
464C22⋊C4—82no—
465C42—24no—
466C42—24no—
467C42—24no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—42no—
474C42—42no—
475C42—44no—
476C42—44no—
477C42—82no—
478C42C2×#4114yes—
479C23—22no—
480C23—21no—
481C23—21no—
482C23—22no—
483C23—24no—
484C23—44no—
485C23—42no—
486C23—42no—
487C23—44no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23#53512yes—
496C23C23×D411yescharacteristic
497Q8—28no—
498Q8—44no—
499D4—28no—
500D4—416no—
501D4—416no—
502D4—44no—
503D4—44no—
504D4—44no—
505D4—164no—
506D4—164no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—28no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—48no—
530C2×C4—416no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—88no—
547C2×C4—164no—
548C2×C4—164no—
549C2×C4—164no—
550C2×C4#52514yes—
551C2×C4#53512yes—
552C2×C4#57714yes—
553C2×C4C22×#4112yes—
554C2×C4C2×H(5,𝔽2)12yes—
555C2×C4#58512yes—
556C8—44no—
557C8—44no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—42no—
562C22—42no—
563C22—44no—
564C22—48no—
565C22—84no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22C2×#52511yescharacteristic
576C22C2×#53511yescharacteristic
577C22#344911yescharacteristic
578C22C2×#57712yes—
579C22#358512yes—
580C4—22no—
581C4—24no—
582C4—24no—
583C4—42no—
584C4—42no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—44no—
590C4—82no—
591C4—82no—
592C4—82no—
593C4—82no—
594C4—82no—
595C4#351314yes—
596C2—42no—
597C2—84no—
598C2—162no—
599C2#3523311yescharacteristic
600C2#6058812yes—
601C2#6082011yescharacteristic
602C2#6083011yescharacteristic
603C2#6098912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#25729C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35295C2semidirect
2. #34207C2semidirect
3. #33013C2semidirect
4. #30129C2semidirect
5. #30126C2semidirect
6. #29368C2semidirect
7. #29365C2semidirect
8. #29199C2semidirect
9. #28630C2semidirect
10. #28127C2semidirect
11. #27966C2semidirect
12. #27906C2semidirect
13. #25858C2semidirect
15. #22279C2semidirect
16. #22227C2semidirect
17. #20784C2semidirect

Maximal subgroups

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

subgroupindex
#352952
#342072
#330132
#301292
#301262
#293682
#293652
#291992
#286302
#281272
#279662
#279062
#258582
#257292
#222792
#222272
#207842