Finite Group Databasebeta

← #6856699

Subgroups — #6856699

Order 512.

1551 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic
2#34951C211yesmaximal, characteristic, complemented
3#33163C211yesmaximal, characteristic, complemented
4#32092C214yesmaximal, complemented
5#30888C214yesmaximal, complemented
6#29797C212yesmaximal
7#29257C212yesmaximal, complemented
8#28752C212yesmaximal
9#28748C211yesmaximal, characteristic, complemented
10#28403C212yesmaximal, complemented
11#28279C211yesmaximal, characteristic
12#27768C212yesmaximal, complemented
13#25650C212yesmaximal, complemented
14#24189C212yesmaximal, complemented
15#22821C211yesmaximal, characteristic, complemented
16#22148C212yesmaximal, complemented
17C4×#460C211yesmaximal, characteristic
18C2×#556C2212yes—
19C2×#488C2214yes—
20C2×#487C2212yes—
21C2×#474C2212yes—
22C2×#462C2212yes—
23C2×#462C2212yes—
24C2×#462C2212yes—
25C2×#460C2211yescharacteristic
26C2×C4×SD16C2212yes—
27#2868C2211yescharacteristic
28#2858C2212yes—
29#2585—22no—
30#2582C2211yescharacteristic
31#2568C2211yescharacteristic
32#2552C2211yescharacteristic
33#2547C2212yes—
34#2546C2211yescharacteristic
35#2545C2214yes—
36#2538—21no—
37#2538—21no—
38#2512—22no—
39#2512—22no—
40#2511—21no—
41#2509C2212yes—
42#2453C2212yes—
43#2446—21no—
44#2402—22no—
45#2354—21no—
46#2354—21no—
47C4×C4⋊Q8—21no—
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2212yes—
50C4×#127C2211yescharacteristic
51C4×#125C2212yes—
52#2295—21no—
53#2295C2212yes—
54C42×Q8C2211yescharacteristic
55C42×Q8C2211yescharacteristic
56#2097C2214yes—
57#2067C2214yes—
58#2043C2214yes—
59#2042C2214yes—
60#2042C2214yes—
61#2042C2214yes—
62#2042C2214yes—
63#2026C2214yes—
64#2015C2214yes—
65#2015C2214yes—
66#1984C2212yes—
67#1982C2211yescharacteristic
68#1982C2212yes—
69#1981C2212yes—
70#1981C2212yes—
71#1981C2212yes—
72#1981C2212yes—
73#1981C2212yes—
74#1969C2214yes—
75#1958C2214yes—
76#1956C2212yes—
77#1948C2211yescharacteristic
78#1947C2212yes—
79#1933C2214yes—
80#1926C2212yes—
81#1916C2214yes—
82#1880C2212yes—
83#1878C2214yes—
84#1865C2214yes—
85#1863C2214yes—
86#1863C2214yes—
87#1842C2211yescharacteristic
88#1831C2212yes—
89#1803C2211yescharacteristic
90#1802C2212yes—
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#102C2212yes—
95C22×Q16C2314yes—
96C22×SD16C2312yes—
97#556—44no—
98#556—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127C2311yescharacteristic
105C2×#126—24no—
106C2×#125C2312yes—
107C2×#124C2314yes—
108C2×#124C2314yes—
109C2×C22⋊Q8—22no—
110C2×C4×Q8—21no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2314yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4C2312yes—
121C2×#117—22no—
122#488—48no—
123#487—44no—
124#474—44no—
125#462—28no—
126#462—44no—
127#462—44no—
128#460—24no—
129C4×SD16—44no—
130C2×#107C2314yes—
131C2×#106C2312yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2312yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×C4×C8C2312yes—
151C2×C4×C8C2314yes—
152#400—22no—
153#400—22no—
154#400—22no—
155#400—22no—
156#398—22no—
157#398—22no—
158#398—22no—
159#398—22no—
160#397—24no—
161#396—24no—
162#396—24no—
163#391—21no—
164#391—21no—
165#391—21no—
166#391—21no—
167#391—21no—
168#391C2314yes—
169#391C2312yes—
170#391C2312yes—
171#390—21no—
172#390—21no—
173#390—22no—
174#390—22no—
175#390—22no—
176#390—22no—
177#390C2312yes—
178#389—22no—
179#389—22no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—22no—
184#389—22no—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2312yes—
190#388—24no—
191#388—24no—
192#387—22no—
193#387—22no—
194#387—22no—
195#387—22no—
196#387—22no—
197#387—22no—
198#384—21no—
199#384—21no—
200#384—21no—
201#384—21no—
202#384—21no—
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#382C2312yes—
206#380—22no—
207#380—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4D412yes—
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4D412yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2312yes—
227#376—22no—
228C43—21no—
229C43C2312yes—
230C43C2311yescharacteristic
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2314yes—
237#336C2312yes—
238C22×Q8—21no—
239C22×Q8C2×D411yescharacteristic
240C22×Q8C2414yes—
241C22×Q8C2411yescharacteristic
242C22×D4—21no—
243C22×D4—21no—
244C22×D4C2412yes—
245C23×C4—22no—
246C2×Q16—28no—
247C2×Q16—44no—
248C2×Q16—44no—
249C2×SD16—24no—
250C2×SD16—42no—
251C2×SD16—42no—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C4⋊Q8—24no—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—42no—
262C4⋊Q8—42no—
263C4⋊Q8—44no—
264#127—22no—
265#127—22no—
266#126—416no—
267#125—24no—
268#125—42no—
269#124—44no—
270#124—44no—
271#124—44no—
272#124—44no—
273C22⋊Q8—48no—
274C4×Q8—24no—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—22no—
281C4×Q8—42no—
282C4×Q8—41no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×Q8—44no—
286C4×Q8#4114yes—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—44no—
290C4×D4—44no—
291#117—48no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C42—21no—
334C2×C42—24no—
335C2×C42—22no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354#107—44no—
355#106—42no—
356C4⋊C8—22no—
357C4⋊C8—24no—
358C4⋊C8—24no—
359C4⋊C8—22no—
360C4⋊C8—42no—
361#103—28no—
362#103—28no—
363#103—44no—
364#103—42no—
365#103—44no—
366#103—44no—
367#103—44no—
368#103—44no—
369#102—24no—
370#102—24no—
371#102—42no—
372#102—44no—
373#102—44no—
374C4×C8—24no—
375C4×C8—44no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—24no—
384#95—24no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396C24—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—21no—
401C2×Q8—24no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—21no—
410C2×Q8—24no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8C2×#4112yes—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C2511yescharacteristic
449Q16—88no—
450SD16—48no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—48no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—81no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4C2×#4112yes—
499C4⋊C4H(5,𝔽2)12yes—
500C22⋊C4—48no—
501C22⋊C4—44no—
502C22⋊C4—48no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—84no—
506C22⋊C4—84no—
507C42—24no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—24no—
512C42—22no—
513C42—21no—
514C42—22no—
515C42—22no—
516C42—21no—
517C42—22no—
518C42—41no—
519C42—42no—
520C42—42no—
521C42—42no—
522C42—48no—
523C42—42no—
524C42—44no—
525C42—42no—
526C42—42no—
527C42—81no—
528C42#13514yes—
529C42C2×#4112yes—
530C42H(5,𝔽2)12yes—
531C42H(5,𝔽2)12yes—
532C422−1+412yes—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23C23×D411yescharacteristic
541Q8—22no—
542Q8—22no—
543Q8—44no—
544Q8—41no—
545Q8—41no—
546Q8—44no—
547Q8—44no—
548Q8—44no—
549Q8—44no—
550Q8—84no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—84no—
556D4—84no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—88no—
603C2×C4#53512yes—
604C2×C4#53512yes—
605C2×C4#54812yes—
606C2×C4#54812yes—
607C2×C4C2×#13512yes—
608C2×C4C2×#13512yes—
609C2×C4C2×#13512yes—
610C2×C4#57512yes—
611C2×C4#57812yes—
612C2×C4C22×#4111yescharacteristic
613C2×C4C2×H(5,𝔽2)11yescharacteristic
614C2×C4C2×H(5,𝔽2)11yescharacteristic
615C2×C4C2×H(5,𝔽2)11yescharacteristic
616C2×C4C2×2−1+411yescharacteristic
617C2×C4#58511yescharacteristic
618C8—42no—
619C8—44no—
620C8—42no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22C2×#53511yescharacteristic
629C22C2×#54811yescharacteristic
630C22#348811yescharacteristic
631C22C22×#13511yescharacteristic
632C22C2×#57511yescharacteristic
633C22C2×#57811yescharacteristic
634C22#358611yescharacteristic
635C4—22no—
636C4—22no—
637C4—22no—
638C4—42no—
639C4—42no—
640C4—41no—
641C4—42no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—82no—
647C4—84no—
648C4—82no—
649C4—82no—
650C4—81no—
651C4#321214yes—
652C4#336414yes—
653C4#350611yescharacteristic
654C4#350811yescharacteristic
655C4#354811yescharacteristic
656C4#355411yescharacteristic
657C2—162no—
658C2#3972511yescharacteristic
659C2C2×#321211yescharacteristic
660C2#6083711yescharacteristic
661C2C2×#336411yescharacteristic
662C2#6132311yescharacteristic
663C2#6195411yescharacteristic
664C2#6198511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34951C2semidirect
3. #33163C2semidirect
4. #32092C2semidirect
5. #30888C2semidirect
7. #29257C2semidirect
9. #28748C2semidirect
10. #28403C2semidirect
12. #27768C2semidirect
13. #25650C2semidirect
14. #24189C2semidirect
15. #22821C2semidirect
16. #22148C2semidirect

Maximal subgroups

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

subgroupindex
#365942
#349512
#331632
#320922
#308882
#297972
#292572
#287522
#287482
#284032
#282792
#277682
#256502
#241892
#228212
#221482
C4×#4602