Finite Group Databasebeta

← #6986814

Subgroups — #6986814

Order 512.

1597 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36456C211yesmaximal, characteristic
2#35279C211yesmaximal, characteristic
3#32790C212yesmaximal, complemented
4#32261C214yesmaximal, complemented
5#32000C214yesmaximal, complemented
6#30291C212yesmaximal, complemented
7#30169C212yesmaximal, complemented
8#29939C211yesmaximal, characteristic, complemented
9#29160C212yesmaximal, complemented
10#28705C212yesmaximal, complemented
11#26282C212yesmaximal, complemented
12#25841C212yesmaximal, complemented
13#25813C211yesmaximal, characteristic, complemented
14#25340C212yesmaximal, complemented
15#24302C211yesmaximal, characteristic, complemented
16#23757C211yesmaximal, characteristic, complemented
17#22469C211yesmaximal, characteristic
18C2×Q82C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#500C2212yes—
22C2×#477C2212yes—
23C2×C8×D4C2212yes—
24#2857C2212yes—
25#2857C2212yes—
26#2857C2212yes—
27#2857C2211yescharacteristic
28#2628—22no—
29#2568—21no—
30#2567C2212yes—
31#2563C2212yes—
32#2563C2211yescharacteristic
33#2563C2212yes—
34#2542—21no—
35#2538C2212yes—
36#2538C2211yescharacteristic
37#2487C2214yes—
38#2452—22no—
39#2452—21no—
40#2439—21no—
41#2439—22no—
42#2409—22no—
43#2351—21no—
44Q8×C4⋊C4C2211yescharacteristic
45#2341—21no—
46#2317—21no—
47#2311—21no—
48C4×C22⋊Q8C2211yescharacteristic
49C4×C22⋊Q8C2212yes—
50#2285C2211yescharacteristic
51#2100C2214yes—
52#2100C2211yescharacteristic
53#2100C2212yes—
54#2095C2211yescharacteristic
55#2083C2214yes—
56#2083C2211yescharacteristic
57#2083C2212yes—
58#2076C2214yes—
59#2076C2214yes—
60#2076C2212yes—
61#2076C2214yes—
62#2076C2214yes—
63#2072C2211yescharacteristic
64#2057C2214yes—
65#2057C2214yes—
66#2057C2212yes—
67#2057C2212yes—
68#2057C2212yes—
69#2036C2212yes—
70#2036C2214yes—
71#2024C2212yes—
72#1990C2212yes—
73#1987C2212yes—
74#1987C2212yes—
75#1984C2212yes—
76#1947C2212yes—
77#1921C2214yes—
78#1921C2214yes—
79#1892C2214yes—
80#1872C2212yes—
81#1869C2214yes—
82#1863C2214yes—
83#1863C2214yes—
84#1852C2214yes—
85#1852C2212yes—
86#1817C2212yes—
87#1808C2214yes—
88#1808C2214yes—
89#1808C2214yes—
90#1797C2211yescharacteristic
91C4×#107C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C2×#336C2212yes—
95C22×#41D411yescharacteristic
96C23×Q8—21no—
97C23×C8C2314yes—
98Q82—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8C2311yescharacteristic
113C2×C22⋊Q8C2312yes—
114C4×#41—44no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×D4D412yes—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2312yes—
125C2×#117—21no—
126C2×#117D411yescharacteristic
127C2×#117D412yes—
128C22×C4⋊C4—21no—
129C22×C4⋊C4C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#500—44no—
133#477—44no—
134C8×D4—44no—
135C2×#107C2312yes—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#107C2312yes—
140C2×#107C2314yes—
141C2×C4⋊C8C2312yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2314yes—
148C2×C4×C8C2312yes—
149#400—24no—
150#399—24no—
151#396—22no—
152#396—22no—
153#396—22no—
154#395—24no—
155#393—24no—
156#393—22no—
157#393—22no—
158#393—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391—21no—
163#391—22no—
164#391C2311yescharacteristic
165#390—22no—
166#390—21no—
167#390—22no—
168#390—21no—
169#390—22no—
170#389—21no—
171#389—22no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—21no—
177#389—22no—
178#389C2314yes—
179#389C2314yes—
180#389C2311yescharacteristic
181#389C2312yes—
182#387—21no—
183#387—22no—
184#386—24no—
185#385—21no—
186#385—22no—
187#385—24no—
188#384—22no—
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#382—22no—
194#382—21no—
195#382—22no—
196#382—21no—
197#380—22no—
198#380—22no—
199#380—21no—
200#380—22no—
201#380—21no—
202#380C2312yes—
203#380C2311yescharacteristic
204#380C2312yes—
205#380C2312yes—
206#380C2314yes—
207#380C2314yes—
208#380C2312yes—
209#380C2312yes—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—21no—
217C4×C22⋊C4—22no—
218#340C2312yes—
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2314yes—
233#336C2314yes—
234C2×#41—42no—
235C2×#41—42no—
236C2×#41Q1618yes—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—44no—
242C22×Q8—82no—
243C22×Q8—82no—
244C22×Q8C2×D411yescharacteristic
245C22×Q8C2411yescharacteristic
246C22×D4C2×D411yescharacteristic
247C22×D4C2×D412yes—
248C23×C4—22no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2411yescharacteristic
254C23×C4C2412yes—
255C22×C8—28no—
256C22×C8—44no—
257C22×C8—44no—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—48no—
269C4⋊Q8—44no—
270C22⋊Q8—22no—
271C22⋊Q8—24no—
272C22⋊Q8—41no—
273C22⋊Q8—42no—
274C22⋊Q8—82no—
275C22⋊Q8—84no—
276C22⋊Q8—84no—
277C22⋊Q8—84no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—42no—
281C4×Q8—44no—
282C4×Q8—44no—
283C4×Q8—44no—
284C4×D4—24no—
285C4×D4—42no—
286C4×D4—44no—
287C4×D4—44no—
288#117—44no—
289#117—42no—
290#117—44no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—44no—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#107—24no—
354#107—42no—
355#107—44no—
356#107—44no—
357#107—44no—
358#107—44no—
359C4⋊C8—24no—
360#103—24no—
361#103—24no—
362#103—44no—
363C22⋊C8—24no—
364C22⋊C8—42no—
365C22⋊C8—44no—
366C4×C8—24no—
367#95—21no—
368#95—21no—
369#95—22no—
370#95—21no—
371#95—22no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—21no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—22no—
382#95—22no—
383#95—22no—
384C24C22×D412yes—
385C24C22×D411yescharacteristic
386#41—416no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—21no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—42no—
397C2×Q8—44no—
398C2×Q8—82no—
399C2×Q8—82no—
400C2×Q8—81no—
401C2×Q8—82no—
402C2×Q8—81no—
403C2×Q8—82no—
404C2×Q8—816no—
405C2×Q8C2×Q1614yes—
406C2×Q82−1+412yes—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4C2×Q1614yes—
412C2×D4C2×Q1618yes—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—42no—
436C22×C4—41no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4C2×Q1614yes—
444C22×C4C2×Q1618yes—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C2511yescharacteristic
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—88no—
463C2×C82−1+414yes—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—81no—
499C4⋊C4—84no—
500C4⋊C4—82no—
501C4⋊C4—84no—
502C22⋊C4—24no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—82no—
509C22⋊C4—81no—
510C22⋊C4—84no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C22⋊C4—82no—
514C22⋊C4—84no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—24no—
519C42—42no—
520C42—42no—
521C42—44no—
522C42—44no—
523C42—42no—
524C42—44no—
525C42—42no—
526C42H(5,𝔽2)14yes—
527C23—22no—
528C23—21no—
529C23—22no—
530C23—21no—
531C23—21no—
532C23—22no—
533C23—21no—
534C23—22no—
535C23—21no—
536C23—22no—
537C23—22no—
538C23—21no—
539C23C22×Q1612yes—
540C23C22×Q1614yes—
541C23C23×D411yescharacteristic
542Q8—28no—
543Q8—24no—
544Q8—42no—
545Q8—88no—
546Q8—88no—
547Q8—162no—
548Q8—164no—
549D4—28no—
550D4—48no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—28no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—48no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—44no—
602C2×C4—84no—
603C2×C4—88no—
604C2×C4#53412yes—
605C2×C4#53412yes—
606C2×C4C22×Q1612yes—
607C2×C4C22×Q1614yes—
608C2×C4C22×Q1612yes—
609C2×C4#57512yes—
610C2×C4#57612yes—
611C2×C4C2×H(5,𝔽2)12yes—
612C2×C4C2×2−1+411yescharacteristic
613C2×C4C2×2−1+412yes—
614C2×C4#58511yescharacteristic
615C8—24no—
616C8—42no—
617C8—44no—
618C22—24no—
619C22—22no—
620C22—22no—
621C22—24no—
622C22—24no—
623C22—22no—
624C22—42no—
625C22—41no—
626C22—41no—
627C22—42no—
628C22—42no—
629C22—41no—
630C22—41no—
631C22—42no—
632C22C2×#53411yescharacteristic
633C22#348911yescharacteristic
634C22#349911yescharacteristic
635C22C23×Q1611yescharacteristic
636C22C2×#57511yescharacteristic
637C22C2×#57611yescharacteristic
638C22#358511yescharacteristic
639C4—22no—
640C4—24no—
641C4—22no—
642C4—22no—
643C4—42no—
644C4—41no—
645C4—42no—
646C4—42no—
647C4—81no—
648C4—82no—
649C4—84no—
650C4—82no—
651C4—81no—
652C4—82no—
653C4—81no—
654C4—82no—
655C4#318914yes—
656C4#355212yes—
657C4#356112yes—
658C2—42no—
659C2—44no—
660C2#4291211yescharacteristic
661C2C2×#318911yescharacteristic
662C2#6075611yescharacteristic
663C2#6210011yescharacteristic
664C2#6213611yescharacteristic
665C2#6247211yescharacteristic
666C2#6250811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
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. #32790C2semidirect
4. #32261C2semidirect
5. #32000C2semidirect
6. #30291C2semidirect
7. #30169C2semidirect
8. #29939C2semidirect
9. #29160C2semidirect
10. #28705C2semidirect
11. #26282C2semidirect
12. #25841C2semidirect
13. #25813C2semidirect
14. #25340C2semidirect
15. #24302C2semidirect
16. #23757C2semidirect

Maximal subgroups

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

subgroupindex
#364562
#352792
#327902
#322612
#320002
#302912
#301692
#299392
#291602
#287052
#262822
#258412
#258132
#253402
#243022
#237572
#224692