Finite Group Databasebeta

← #6541824

Subgroups — #6541824

Order 512.

1777 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic
2#34334C211yesmaximal, characteristic
3#30116C212yesmaximal, complemented
4#29799C211yesmaximal, characteristic, complemented
5#29255C211yesmaximal, characteristic, complemented
6#28440C211yesmaximal, characteristic, complemented
7#28313C212yesmaximal, complemented
8#26949C214yesmaximal, complemented
9#26234C212yesmaximal
10#25760C212yesmaximal
11#25173C214yesmaximal, complemented
12#23934C212yesmaximal, complemented
13#23776C211yesmaximal, characteristic, complemented
14#23577C212yesmaximal, complemented
15#23187C212yesmaximal, complemented
16#23102C212yesmaximal, complemented
17#22839C211yesmaximal, characteristic
18C2×D4×Q8C2211yescharacteristic
19C2×#533C2212yes—
20C2×#494C2212yes—
21C2×#483C2214yes—
22C2×#462C2212yes—
23C2×#462C2214yes—
24C2×#451C2214yes—
25C2×#451C2214yes—
26C2×#441C2214yes—
27#2868C2211yescharacteristic
28#2857C2212yes—
29#2552—21no—
30#2542C2212yes—
31#2538—21no—
32#2441—22no—
33#2429C2214yes—
34#2423—22no—
35#2369—21no—
36#2368—22no—
37#2365C2211yescharacteristic
38#2365C2212yes—
39#2360C2212yes—
40#2355C2211yescharacteristic
41#2354—21no—
42Q8×C4⋊C4—21no—
43#2349—21no—
44D4×C4⋊C4C2211yescharacteristic
45Q8×C22⋊C4C2212yes—
46#2331—21no—
47#2329—22no—
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C4×C22⋊Q8C2212yes—
51#2295C2211yescharacteristic
52C2×#391C2212yes—
53#2275—21no—
54C42×D4C2211yescharacteristic
55#2099C2211yescharacteristic
56#2096C2212yes—
57#2096C2211yescharacteristic
58#2088C2212yes—
59#2066C2212yes—
60#2058C2211yescharacteristic
61#2043C2212yes—
62#2043C2212yes—
63#2035C2214yes—
64#2035C2214yes—
65#2024C2212yes—
66#2024C2211yescharacteristic
67#2016C2212yes—
68#2016C2212yes—
69#2016C2212yes—
70#2016C2212yes—
71#2001C2214yes—
72#2001C2214yes—
73#1981C2212yes—
74#1981C2212yes—
75#1971C2214yes—
76#1971C2214yes—
77#1956C2212yes—
78#1954C2212yes—
79#1948C2212yes—
80#1927C2214yes—
81#1924C2214yes—
82#1878C2214yes—
83#1878C2214yes—
84#1878C2214yes—
85#1872C2214yes—
86#1865C2214yes—
87#1831C2211yescharacteristic
88#1817C2212yes—
89C4×#106C2212yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1763C2214yes—
93C4×#103C2212yes—
94#1749C2212yes—
95C23×Q8C2312yes—
96C22×Q16C2314yes—
97C22×Q16C2314yes—
98C22×Q16C2314yes—
99D4×Q8—24no—
100D4×Q8—24no—
101#533—48no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×#125—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8C2312yes—
112C2×C22⋊Q8C2314yes—
113C2×C22⋊Q8C2314yes—
114C2×C4×Q8—21no—
115C2×C4×Q8—22no—
116C2×C4×Q8C2314yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2312yes—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D411yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×#117—22no—
126C22×C4⋊C4C2312yes—
127C22×C42C2312yes—
128#494—28no—
129#483—48no—
130#462—28no—
131#462—48no—
132#451—48no—
133#451—48no—
134#441—48no—
135C2×#106C2312yes—
136C2×#106C2312yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×C22⋊C8C2314yes—
150C2×C22⋊C8C2314yes—
151C2×#97C2314yes—
152C2×C4×C8C2312yes—
153#399—24no—
154#398—24no—
155#396—24no—
156#391—21no—
157#391—22no—
158#391—22no—
159#391—22no—
160#391—22no—
161#391C2312yes—
162#391C2311yescharacteristic
163#391C2312yes—
164#391C2314yes—
165#391C2311yescharacteristic
166#391C2312yes—
167#390—22no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—24no—
178#389C2312yes—
179#389C2311yescharacteristic
180#389C2311yescharacteristic
181#388—24no—
182#387—21no—
183#387—22no—
184#387—21no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—24no—
189#385—22no—
190#385—22no—
191#385—22no—
192#384—22no—
193#384—21no—
194#384—21no—
195#384—21no—
196#384—21no—
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#382—21no—
200#380—22no—
201#380—22no—
202#380—22no—
203#380C2312yes—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4C2312yes—
224#376—21no—
225C2×#95—22no—
226C2×#95—22no—
227C43C2311yescharacteristic
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235#336C2312yes—
236#336C2314yes—
237#336C2312yes—
238C22×Q8—24no—
239C22×Q8—24no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8C2414yes—
245C22×Q8C2412yes—
246C22×Q8C2411yescharacteristic
247C22×D4C2×D411yescharacteristic
248C23×C4—22no—
249C23×C4—22no—
250C23×C4C2×D412yes—
251C23×C4C2×D412yes—
252C23×C4C2412yes—
253C2×Q16—28no—
254C2×Q16—28no—
255C2×Q16—28no—
256C2×Q16—28no—
257C2×Q16—44no—
258C2×Q16—44no—
259C2×Q16—44no—
260C2×Q16—44no—
261C2×Q16—44no—
262C22×C8C2412yes—
263C22×C8C2414yes—
264C22×C8C2412yes—
265C4⋊Q8—28no—
266C4⋊Q8—24no—
267C4⋊Q8—22no—
268C4⋊Q8—24no—
269C4⋊Q8—24no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272C4⋊Q8#4114yes—
273#125—48no—
274C22⋊Q8—216no—
275C22⋊Q8—24no—
276C22⋊Q8—24no—
277C22⋊Q8—28no—
278C22⋊Q8—44no—
279C22⋊Q8—48no—
280C22⋊Q8—44no—
281C22⋊Q8—44no—
282C4×Q8—22no—
283C4×Q8—28no—
284C4×Q8—44no—
285C4×Q8—44no—
286C4×Q8—44no—
287C4×Q8—48no—
288C4×Q8—44no—
289C4×Q8#4114yes—
290C4×D4—28no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—44no—
296#117—48no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C42—21no—
339C2×C42—24no—
340C2×C42—21no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—44no—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#106—24no—
357#106—24no—
358#106—44no—
359C4⋊C8—24no—
360C4⋊C8—24no—
361#103—24no—
362#103—28no—
363#103—28no—
364#103—28no—
365#103—24no—
366#103—44no—
367#103—44no—
368#103—44no—
369#103—44no—
370C22⋊C8—28no—
371C22⋊C8—44no—
372#97—44no—
373C4×C8—24no—
374#95—21no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95—24no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—21no—
391#95—22no—
392#95—24no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396C24C22×D412yes—
397C2×Q8—24no—
398C2×Q8—28no—
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—24no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—82no—
414C2×Q8—82no—
415C2×Q8—88no—
416C2×Q8—88no—
417C2×Q8C2×#4112yes—
418C2×Q8C2×#4114yes—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4#13714yes—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—28no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C2511yescharacteristic
457Q16—416no—
458Q16—88no—
459Q16—88no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8H(5,𝔽2)14yes—
469C4⋊C4—24no—
470C4⋊C4—28no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—44no—
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—81no—
496C4⋊C4—81no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—84no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4C2×#4112yes—
504C4⋊C4H(5,𝔽2)14yes—
505C22⋊C4—28no—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—48no—
510C22⋊C4—44no—
511C22⋊C4—84no—
512C42—24no—
513C42—22no—
514C42—22no—
515C42—24no—
516C42—22no—
517C42—22no—
518C42—42no—
519C42—42no—
520C42—41no—
521C42—48no—
522C42—42no—
523C42—44no—
524C42—82no—
525C42—82no—
526C42—82no—
527C42#13714yes—
528C42C2×#4112yes—
529C42C2×#4112yes—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23C2×#13714yes—
537C23C23×D411yescharacteristic
538Q8—24no—
539Q8—28no—
540Q8—48no—
541Q8—44no—
542Q8—42no—
543Q8—44no—
544Q8—44no—
545Q8—44no—
546Q8—44no—
547Q8—88no—
548D4—28no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—48no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—84no—
600C2×C4—88no—
601C2×C4—164no—
602C2×C4#52512yes—
603C2×C4#52512yes—
604C2×C4C2×#13712yes—
605C2×C4C2×#13712yes—
606C2×C4C2×#13712yes—
607C2×C4C2×#13712yes—
608C2×C4#57812yes—
609C2×C4C22×#4111yescharacteristic
610C2×C4C22×#4111yescharacteristic
611C2×C4C2×H(5,𝔽2)12yes—
612C2×C4C2×H(5,𝔽2)12yes—
613C8—24no—
614C8—44no—
615C8—42no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22C2×#52511yescharacteristic
624C22#348411yescharacteristic
625C22#349811yescharacteristic
626C22C22×#13711yescharacteristic
627C22C22×#13711yescharacteristic
628C22C2×#57811yescharacteristic
629C22#358611yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—42no—
635C4—44no—
636C4—42no—
637C4—42no—
638C4—44no—
639C4—42no—
640C4—42no—
641C4—41no—
642C4—84no—
643C4—81no—
644C4—82no—
645C4—82no—
646C4—81no—
647C4#311314yes—
648C4#350812yes—
649C4#351912yes—
650C2—44no—
651C2#3574111yescharacteristic
652C2C2×#311311yescharacteristic
653C2#6059411yescharacteristic
654C2#6158611yescharacteristic
655C2#6160211yescharacteristic
656C2#6242411yescharacteristic
657C2#6244811yescharacteristic
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. #30116C2semidirect
4. #29799C2semidirect
5. #29255C2semidirect
6. #28440C2semidirect
7. #28313C2semidirect
8. #26949C2semidirect
11. #25173C2semidirect
12. #23934C2semidirect
13. #23776C2semidirect
14. #23577C2semidirect
15. #23187C2semidirect
16. #23102C2semidirect

Maximal subgroups

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

subgroupindex
#361342
#343342
#301162
#297992
#292552
#284402
#283132
#269492
#262342
#257602
#251732
#239342
#237762
#235772
#231872
#231022
#228392