Finite Group Databasebeta

← #6660797

Subgroups — #6660797

Order 512.

1727 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35316C211yesmaximal, characteristic
2#35309C211yesmaximal, characteristic
3#32725C212yesmaximal, complemented
4#32724C212yesmaximal, complemented
5#30250C211yesmaximal, characteristic, complemented
6#30242C211yesmaximal, characteristic, complemented
7#30202C211yesmaximal, characteristic, complemented
8#30190C211yesmaximal, characteristic, complemented
9#29963C212yesmaximal, complemented
10#29934C212yesmaximal, complemented
11#29724C212yesmaximal, complemented
12#29674C212yesmaximal, complemented
13#27685C212yesmaximal, complemented
14#24764C214yesmaximal, complemented
15#24392C214yesmaximal
16#24212C212yesmaximal, complemented
17#20630C211yesmaximal, characteristic
18C2×#541C2211yescharacteristic
19C2×#536C2212yes—
20C22×C4⋊Q8C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×#470C2214yes—
23C2×#467C2214yes—
24C2×#462C2214yes—
25C2×#451C2214yes—
26C2×#424C2212yes—
27C2×#408C2212yes—
28#2857C2211yescharacteristic
29#2857C2211yescharacteristic
30#2857C2211yescharacteristic
31#2857C2211yescharacteristic
32#2857C2211yescharacteristic
33#2855C2212yes—
34#2851C2211yescharacteristic
35#2629C2211yescharacteristic
36#2621C2211yescharacteristic
37#2621C2212yes—
38#2621C2212yes—
39#2616C2212yes—
40#2547—22no—
41#2546—22no—
42#2486—22no—
43#2486—22no—
44#2468—22no—
45#2466—22no—
46#2462—22no—
47#2460—22no—
48#2323C2212yes—
49#2314C2211yescharacteristic
50#2310C2212yes—
51C4×C22⋊Q8C2212yes—
52#2285C2211yescharacteristic
53#2100C2212yes—
54#2099C2212yes—
55#2096C2212yes—
56#2096C2212yes—
57#2095C2212yes—
58#2095C2212yes—
59#2074C2212yes—
60#2074C2212yes—
61#2068C2214yes—
62#2066C2214yes—
63#2056C2214yes—
64#2056C2214yes—
65#2043C2212yes—
66#2043C2212yes—
67#2036C2214yes—
68#2035C2214yes—
69#2026C2212yes—
70#2026C2212yes—
71#2016C2212yes—
72#2016C2212yes—
73#2016C2212yes—
74#2016C2212yes—
75#2001C2214yes—
76#2001C2214yes—
77#1981C2214yes—
78#1949C2214yes—
79#1894C2214yes—
80#1892C2214yes—
81#1873C2214yes—
82#1854C2214yes—
83#1805C2214yes—
84#1798C2212yes—
85#1798C2212yes—
86#1797C2212yes—
87#1797C2212yes—
88#1787C2214yes—
89#1728C2211yescharacteristic
90C2×#336C2211yescharacteristic
91C22×#41C2311yescharacteristic
92C23×Q8C2312yes—
93C22×Q16C2314yes—
94C22×Q16C2314yes—
95C22×M16C2312yes—
96C23×C8C2312yes—
97#541—44no—
98#536—48no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×#126—22no—
107C2×#125—22no—
108C2×#125C2311yescharacteristic
109C2×#124—22no—
110C2×#123—22no—
111C2×#123—22no—
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2312yes—
114C2×C22⋊Q8C2312yes—
115C2×C22⋊Q8C2312yes—
116C2×C22⋊Q8C2312yes—
117C2×C22⋊Q8C2312yes—
118C2×C22⋊Q8C2312yes—
119C4×#41—44no—
120C2×C4×Q8D412yes—
121C2×C4×Q8C2312yes—
122C2×C4×D4D412yes—
123C2×C4×D4D412yes—
124C2×C4×D4D412yes—
125C2×#117C2312yes—
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C22×C4⋊C4—22no—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#470—48no—
134#467—48no—
135#462—48no—
136#451—48no—
137C2×#107C2314yes—
138C2×#106C2314yes—
139#424—44no—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150#408—44no—
151C2×C22⋊C8C2314yes—
152C2×C22⋊C8C2314yes—
153#398—22no—
154#398—24no—
155#398—24no—
156#398—22no—
157#397—22no—
158#397—24no—
159#397—24no—
160#396—22no—
161#395—22no—
162#394—22no—
163#394—22no—
164#394—22no—
165#391—24no—
166#391C2312yes—
167#391C2312yes—
168#391C2312yes—
169#391C2312yes—
170#391C2312yes—
171#391C2312yes—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389C2312yes—
179#389C2312yes—
180#388—22no—
181#388—22no—
182#388—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#385—22no—
190#385—22no—
191#385—22no—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#382C2311yescharacteristic
196#382C2312yes—
197#381—22no—
198#381—24no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—22no—
204#380—22no—
205#380C2312yes—
206#380C2312yes—
207#380C2312yes—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4C2312yes—
211C4×C22⋊C4—22no—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#336C2312yes—
217#336C2312yes—
218#336C2312yes—
219#336C2312yes—
220C2×#41—24no—
221C2×#41—22no—
222C2×#41—22no—
223C2×#41—42no—
224C22×Q8—22no—
225C22×Q8—24no—
226C22×Q8—24no—
227C22×Q8—44no—
228C22×Q8—44no—
229C22×Q8C2×D411yescharacteristic
230C22×Q8C2412yes—
231C22×Q8C2412yes—
232C22×Q8C2412yes—
233C22×D4C2×D411yescharacteristic
234C22×D4C2×D411yescharacteristic
235C22×D4C2×D411yescharacteristic
236C23×C4—22no—
237C23×C4C2×D412yes—
238C23×C4C2×D412yes—
239C23×C4C2×D412yes—
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C2×Q16—44no—
244C2×Q16—44no—
245C2×Q16—44no—
246C2×Q16—44no—
247C2×Q16—44no—
248C2×Q16—44no—
249C2×M16—24no—
250C2×M16—42no—
251C2×M16—42no—
252C22×C8—24no—
253C22×C8—42no—
254C22×C8—42no—
255C22×C8C2414yes—
256C22×C8C2414yes—
257C4⋊Q8—28no—
258C4⋊Q8—42no—
259C4⋊Q8—42no—
260C4⋊Q8—42no—
261C4⋊Q8—48no—
262C4⋊Q8—48no—
263#126—48no—
264#125—42no—
265#125—48no—
266#124—48no—
267#123—48no—
268#123—48no—
269C22⋊Q8—28no—
270C22⋊Q8—28no—
271C22⋊Q8—24no—
272C22⋊Q8—28no—
273C22⋊Q8—24no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276C22⋊Q8—44no—
277C4×Q8—28no—
278C4×Q8—44no—
279C4×D4—44no—
280C4×D4—44no—
281C4×D4—44no—
282#117—24no—
283#117—24no—
284#117—42no—
285#117—44no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—48no—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4C2×D412yes—
325C2×C22⋊C4C2×D412yes—
326C2×C22⋊C4C2×D412yes—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—42no—
332C2×C42—42no—
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340#107—44no—
341#106—44no—
342C4⋊C8—28no—
343C4⋊C8—44no—
344#103—44no—
345#103—44no—
346#103—44no—
347#103—44no—
348#103—44no—
349#103—44no—
350#103—44no—
351#103—44no—
352C22⋊C8—28no—
353C22⋊C8—44no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—24no—
358#95—22no—
359#95—22no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370C24C22×D411yescharacteristic
371C24C22×D411yescharacteristic
372C24C22×D411yescharacteristic
373#41—48no—
374#41—48no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—24no—
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—41no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—88no—
394C2×Q8—88no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4H(5,𝔽2)14yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C42−1+412yes—
443C22×C4C2511yescharacteristic
444Q16—88no—
445Q16—88no—
446M16—84no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—84no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—82no—
480C4⋊C4—82no—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C22⋊C4—42no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C42—24no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—42no—
500C42—42no—
501C42—44no—
502C42—44no—
503C42—82no—
504C42H(5,𝔽2)14yes—
505C23—22no—
506C23—22no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—22no—
518C23#53512yes—
519C23#53712yes—
520C23C23×D411yescharacteristic
521Q8—44no—
522Q8—44no—
523Q8—44no—
524Q8—44no—
525Q8—44no—
526Q8—44no—
527Q8—44no—
528Q8—88no—
529D4—44no—
530D4—44no—
531D4—44no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—48no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—88no—
581C2×C4#53414yes—
582C2×C4#53512yes—
583C2×C4#53712yes—
584C2×C4#57814yes—
585C2×C4C2×H(5,𝔽2)12yes—
586C2×C4C2×H(5,𝔽2)12yes—
587C2×C4C2×H(5,𝔽2)11yescharacteristic
588C2×C4C2×2−1+411yescharacteristic
589C8—44no—
590C8—44no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—24no—
596C22—42no—
597C22—42no—
598C22—41no—
599C22—41no—
600C22—42no—
601C22—41no—
602C22—42no—
603C22—41no—
604C22—41no—
605C22—42no—
606C22C2×#53411yescharacteristic
607C22C2×#53511yescharacteristic
608C22C2×#53711yescharacteristic
609C22C2×#57812yes—
610C22#358612yes—
611C4—22no—
612C4—22no—
613C4—24no—
614C4—24no—
615C4—41no—
616C4—42no—
617C4—42no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—82no—
625C4—82no—
626C4#353612yes—
627C4#353712yes—
628C2—42no—
629C2—42no—
630C2—81no—
631C2#3743911yescharacteristic
632C2#6075912yes—
633C2#6082111yescharacteristic
634C2#6083111yescharacteristic
635C2#6092512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32725C2semidirect
4. #32724C2semidirect
5. #30250C2semidirect
6. #30242C2semidirect
7. #30202C2semidirect
8. #30190C2semidirect
9. #29963C2semidirect
10. #29934C2semidirect
11. #29724C2semidirect
12. #29674C2semidirect
13. #27685C2semidirect
14. #24764C2semidirect
16. #24212C2semidirect

Maximal subgroups

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

subgroupindex
#353162
#353092
#327252
#327242
#302502
#302422
#302022
#301902
#299632
#299342
#297242
#296742
#276852
#247642
#243922
#242122
#206302