Finite Group Databasebeta

← #6849043

Subgroups — #6849043

Order 512.

1489 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2#34929C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic, complemented
4#31763C214yesmaximal, complemented
5#31141C214yesmaximal, complemented
6#29691C212yesmaximal, complemented
7#29141C211yesmaximal, characteristic, complemented
8#28797C212yesmaximal, complemented
9#28284C211yesmaximal, characteristic, complemented
10#27757C212yesmaximal, complemented
11#27649C212yesmaximal, complemented
12#26231C212yesmaximal
13#24970C212yesmaximal
14#24711C212yesmaximal, complemented
15#22799C211yesmaximal, characteristic
16#21870C212yesmaximal, complemented
17C4×#477C211yesmaximal, characteristic, complemented
18C2×#533C2212yes—
19C2×#485C2214yes—
20C2×#477C2211yescharacteristic
21C2×#477C2212yes—
22C2×#475C2212yes—
23C2×#432C2212yes—
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2571C2212yes—
27#2563—21no—
28#2556—22no—
29#2551C2211yescharacteristic
30#2546C2211yescharacteristic
31#2542C2211yescharacteristic
32#2538—21no—
33#2538C2211yescharacteristic
34#2506—21no—
35#2497—22no—
36#2492—22no—
37#2492C2212yes—
38#2452—21no—
39#2439C2214yes—
40#2429C2212yes—
41#2396—22no—
42#2369—21no—
43#2355—21no—
44#2354C2212yes—
45C4×C4⋊Q8—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×C22⋊Q8C2212yes—
49#2293—21no—
50#2285C2212yes—
51C42×Q8C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2085C2214yes—
54#2075C2214yes—
55#2074C2214yes—
56#2068C2214yes—
57#2057C2214yes—
58#2043C2214yes—
59#2036C2214yes—
60#2036C2214yes—
61#2035C2214yes—
62#2024C2214yes—
63#1993C2214yes—
64#1988C2212yes—
65#1987C2211yescharacteristic
66#1981C2212yes—
67#1960C2214yes—
68#1956C2212yes—
69#1954C2212yes—
70#1948C2212yes—
71#1948C2212yes—
72#1947C2211yescharacteristic
73#1947C2212yes—
74#1944C2214yes—
75#1935C2214yes—
76#1924C2214yes—
77#1921C2212yes—
78#1873C2214yes—
79#1873C2214yes—
80#1863C2212yes—
81#1842C2211yescharacteristic
82#1834C2212yes—
83#1825C2214yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2211yescharacteristic
87#1803C2212yes—
88#1803C2212yes—
89#1803C2212yes—
90#1797C2212yes—
91C4×#107C2212yes—
92#1770C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95#533—28no—
96#533—44no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×#125—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8C2312yes—
109C2×C22⋊Q8C2312yes—
110C2×C22⋊Q8C2312yes—
111C2×C4×Q8—22no—
112C2×C4×Q8—21no—
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×#117—22no—
121C2×#117C2314yes—
122C22×C42C2312yes—
123#485—48no—
124#477—24no—
125#477—44no—
126#475—44no—
127#432—44no—
128C2×#107C2312yes—
129C2×#107C2312yes—
130C2×#107C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2312yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146C2×#97C2314yes—
147C2×C4×C8C2312yes—
148#400—24no—
149#400—22no—
150#399—22no—
151#398—22no—
152#398—22no—
153#396—22no—
154#396—22no—
155#396—24no—
156#395—22no—
157#393—24no—
158#393—22no—
159#391—21no—
160#391—22no—
161#391—22no—
162#391—24no—
163#391C2312yes—
164#391C2312yes—
165#390—22no—
166#390—21no—
167#390—21no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—22no—
177#389—22no—
178#389C2312yes—
179#389C2314yes—
180#389C2312yes—
181#388—24no—
182#388—22no—
183#388—22no—
184#387—22no—
185#387—24no—
186#385—22no—
187#384—21no—
188#384—21no—
189#384—21no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#382—22no—
195#382C2314yes—
196#380—22no—
197#380—22no—
198#380—22no—
199#380C2312yes—
200#380C2314yes—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—22no—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4C2312yes—
222C43—21no—
223C43C2311yescharacteristic
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#336C2314yes—
234#336C2312yes—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8C2412yes—
239C22×Q8C2411yescharacteristic
240C22×D4C2×D411yescharacteristic
241C23×C4C2×D412yes—
242C23×C4C2×D412yes—
243C23×C4C2412yes—
244C22×C8C2412yes—
245C22×C8C2412yes—
246C22×C8C2414yes—
247C4⋊Q8—24no—
248C4⋊Q8—24no—
249C4⋊Q8—22no—
250C4⋊Q8—22no—
251C4⋊Q8—22no—
252C4⋊Q8—22no—
253C4⋊Q8—42no—
254C4⋊Q8—42no—
255C4⋊Q8—42no—
256C4⋊Q8—42no—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259C4⋊Q8—82no—
260#125—48no—
261C22⋊Q8—24no—
262C22⋊Q8—24no—
263C22⋊Q8—24no—
264C22⋊Q8—42no—
265C22⋊Q8—42no—
266C22⋊Q8—42no—
267C22⋊Q8—48no—
268C22⋊Q8—84no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×Q8—48no—
276C4×Q8—41no—
277C4×Q8—44no—
278C4×Q8#4114yes—
279C4×D4—24no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—42no—
283#117—48no—
284#117—44no—
285#117—44no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C42—21no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—24no—
340C2×C42—21no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#107—24no—
352#107—24no—
353#107—44no—
354#106—28no—
355C4⋊C8—22no—
356C4⋊C8—22no—
357C4⋊C8—24no—
358C4⋊C8—24no—
359C4⋊C8—42no—
360#103—24no—
361#103—42no—
362#103—44no—
363#103—44no—
364#103—44no—
365#103—44no—
366#103—44no—
367C22⋊C8—44no—
368C22⋊C8—44no—
369#97—44no—
370C4×C8—24no—
371#95—22no—
372#95—22no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—24no—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389C24C22×D412yes—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8—44no—
407C2×Q8—82no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×Q8C2×#4112yes—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C2511yescharacteristic
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—81no—
491C4⋊C4—84no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4C2×#4112yes—
495C4⋊C4H(5,𝔽2)12yes—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—42no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—82no—
503C22⋊C4—84no—
504C22⋊C4—84no—
505C42—24no—
506C42—24no—
507C42—24no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—21no—
513C42—21no—
514C42—41no—
515C42—42no—
516C42—42no—
517C42—44no—
518C42—44no—
519C42—42no—
520C42—42no—
521C42—81no—
522C42—84no—
523C42#13714yes—
524C42C2×#4112yes—
525C42H(5,𝔽2)12yes—
526C422−1+412yes—
527C422−1+412yes—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23C23×D411yescharacteristic
536Q8—22no—
537Q8—22no—
538Q8—41no—
539Q8—41no—
540Q8—42no—
541Q8—42no—
542Q8—42no—
543Q8—42no—
544Q8—84no—
545Q8—88no—
546Q8—164no—
547D4—44no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
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—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—48no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—84no—
599C2×C4#53612yes—
600C2×C4#53612yes—
601C2×C4#54612yes—
602C2×C4#54612yes—
603C2×C4C2×#13712yes—
604C2×C4C2×#13712yes—
605C2×C4C2×#13712yes—
606C2×C4#57512yes—
607C2×C4#57812yes—
608C2×C4C22×#4111yescharacteristic
609C2×C4C2×H(5,𝔽2)11yescharacteristic
610C2×C4C2×H(5,𝔽2)11yescharacteristic
611C2×C4C2×2−1+411yescharacteristic
612C2×C4C2×2−1+411yescharacteristic
613C2×C4#58511yescharacteristic
614C8—42no—
615C8—42no—
616C8—44no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22C2×#53611yescharacteristic
625C22C2×#54611yescharacteristic
626C22#348811yescharacteristic
627C22C22×#13711yescharacteristic
628C22C2×#57511yescharacteristic
629C22C2×#57811yescharacteristic
630C22#358511yescharacteristic
631C4—22no—
632C4—22no—
633C4—22no—
634C4—41no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—84no—
642C4—82no—
643C4—82no—
644C4—81no—
645C4—82no—
646C4—82no—
647C4#323714yes—
648C4#332814yes—
649C4#350711yescharacteristic
650C4#351111yescharacteristic
651C4#354711yescharacteristic
652C4#354811yescharacteristic
653C2—82no—
654C2#3947011yescharacteristic
655C2C2×#323711yescharacteristic
656C2#6089311yescharacteristic
657C2C2×#332811yescharacteristic
658C2#6116611yescharacteristic
659C2#6196511yescharacteristic
660C2#6198511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33196C2semidirect
4. #31763C2semidirect
5. #31141C2semidirect
6. #29691C2semidirect
7. #29141C2semidirect
8. #28797C2semidirect
9. #28284C2semidirect
10. #27757C2semidirect
11. #27649C2semidirect
14. #24711C2semidirect
16. #21870C2semidirect
17. C4×#477C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#349292
#331962
#317632
#311412
#296912
#291412
#287972
#282842
#277572
#276492
#262312
#249702
#247112
#227992
#218702
C4×#4772