Finite Group Databasebeta

← #6461575

Subgroups — #6461575

Order 512.

1855 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35267C211yesmaximal, characteristic
2#34209C211yesmaximal, characteristic, complemented
3#29509C214yesmaximal, complemented
4#29508C214yesmaximal, complemented
5#29501C212yesmaximal, complemented
6#29500C212yesmaximal, complemented
7#29495C212yesmaximal, complemented
8#29494C212yesmaximal, complemented
9#23657C211yesmaximal, characteristic, complemented
10#23627C211yesmaximal, characteristic
11C2×#2043C211yesmaximal, characteristic, complemented
12#23432C211yesmaximal, characteristic
13#21234C212yesmaximal
14#21088C212yesmaximal, complemented
15#20930C212yesmaximal
16#20877C212yesmaximal, complemented
17#20368C211yesmaximal, characteristic, complemented
18C22×#137C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C2×#519C2212yes—
21C2×#428C2212yes—
22C2×#419C2212yes—
23C2×#416C2212yes—
24C22×#103C2212yes—
25C22×#103C2212yes—
26C2×#405C2212yes—
27#2863C2212yes—
28#2860C2212yes—
29#2645C2212yes—
30#2644C2212yes—
31#2510—22no—
32#2497—21no—
33#2496—21no—
34#2495—21no—
35#2494—21no—
36#2481—22no—
37#2365—21no—
38#2365—21no—
39#2365—21no—
40#2365—21no—
41#2344—22no—
42#2344—22no—
43#2327C2214yes—
44#2317C2211yescharacteristic
45#2316C2211yescharacteristic
46#2311C2211yescharacteristic
47#2311C2211yescharacteristic
48#2310C2212yes—
49C4×C4⋊Q8C2212yes—
50C4×C4⋊Q8C2212yes—
51#2297C2211yescharacteristic
52C2×#389C2211yescharacteristic
53#2285C2211yescharacteristic
54C4×#117C2211yescharacteristic
55#2085C2214yes—
56#2084C2212yes—
57#2083C2212yes—
58#2074C2212yes—
59#2073C2212yes—
60#2072C2211yescharacteristic
61#2072C2211yescharacteristic
62#2071C2211yescharacteristic
63#2071C2211yescharacteristic
64#2043C2212yes—
65#2043C2212yes—
66#2043C2214yes—
67#2016C2212yes—
68#2016C2212yes—
69#2013C2212yes—
70#2013C2212yes—
71#1973C2214yes—
72#1971C2214yes—
73#1969C2214yes—
74#1968C2214yes—
75#1935C2214yes—
76#1935C2214yes—
77#1933C2214yes—
78#1933C2214yes—
79#1880C2214yes—
80#1880C2214yes—
81#1878C2214yes—
82#1878C2214yes—
83#1865C2214yes—
84#1865C2214yes—
85#1863C2214yes—
86#1863C2214yes—
87#1746C2214yes—
88#1745C2212yes—
89#1744C2212yes—
90#1736C2212yes—
91#1735C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×Q8C2312yes—
97C2×#137—28no—
98C2×#137—28no—
99C2×#137—44no—
100C22×Q16C2314yes—
101C22×SD16C2314yes—
102C22×M16C2312yes—
103C23×C8C2312yes—
104C2×C4⋊Q8—28no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×#126—22no—
110#519—44no—
111#519—44no—
112C2×C4×Q8—22no—
113C2×C4×Q8—22no—
114C2×C4×Q8C2314yes—
115C2×C4×Q8C2314yes—
116C2×#117—24no—
117C2×#117C2312yes—
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#428—44no—
128C2×#107C2314yes—
129C2×#106C2314yes—
130#419—28no—
131#416—44no—
132C2×#103—28no—
133C2×#103—44no—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144#405—28no—
145C2×#97C2314yes—
146C2×C4×C8C2314yes—
147#401—22no—
148#400—22no—
149#400—22no—
150#400—22no—
151#397—22no—
152#397—22no—
153#397—22no—
154#396—22no—
155#391—22no—
156#391—22no—
157#391C2314yes—
158#391C2314yes—
159#391C2312yes—
160#391C2312yes—
161#391C2311yescharacteristic
162#391C2311yescharacteristic
163#390C2312yes—
164#390C2312yes—
165#390C2311yescharacteristic
166#390C2311yescharacteristic
167#389—21no—
168#389—21no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—21no—
174#389—21no—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#388—22no—
182#388—22no—
183#388—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—24no—
188#387—24no—
189#387—24no—
190#387—24no—
191#386—21no—
192#386—21no—
193#386—22no—
194#386—22no—
195#385—22no—
196#384—21no—
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#383—22no—
200#382—21no—
201#382C2312yes—
202#381—24no—
203#380—21no—
204#380—21no—
205#380—21no—
206#380—21no—
207#380—21no—
208#380—21no—
209#380—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4D411yescharacteristic
220C4×C22⋊C4—21no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4D412yes—
224#376C2312yes—
225C2×#95—21no—
226C43C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235#336C2312yes—
236#336C2312yes—
237#336C2312yes—
238#336C2314yes—
239#336C2314yes—
240C2×#41—24no—
241C2×#41—24no—
242C2×#41—22no—
243C2×#41—22no—
244C22×Q8—22no—
245C22×Q8—24no—
246C22×Q8—24no—
247C22×Q8—24no—
248C22×Q8—24no—
249C22×Q8—24no—
250C22×Q8#4118yes—
251C22×Q8C2411yescharacteristic
252C22×Q8C2414yes—
253C22×D4—21no—
254C22×D4C2411yescharacteristic
255C23×C4—22no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2411yescharacteristic
262#137—432no—
263C2×Q16—28no—
264C2×Q16—28no—
265C2×Q16—44no—
266C2×SD16—28no—
267C2×SD16—28no—
268C2×SD16—44no—
269C2×M16—24no—
270C2×M16—24no—
271C2×M16#4118yes—
272C22×C8—24no—
273C22×C8—24no—
274C22×C8—24no—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2414yes—
278C4⋊Q8—28no—
279C4⋊Q8—28no—
280C4⋊Q8—28no—
281C4⋊Q8—28no—
282#126—84no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×Q8—44no—
286C4×Q8—44no—
287C4×Q8—48no—
288C4×Q8—48no—
289#117—24no—
290#117—22no—
291#117—22no—
292#117—24no—
293#117—24no—
294#117—24no—
295#117—416no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—82no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2412yes—
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#107—44no—
358#106—44no—
359#103—28no—
360#103—28no—
361#103—28no—
362#103—28no—
363#103—44no—
364#103—44no—
365#103—42no—
366#103—42no—
367#102—42no—
368#102—42no—
369#97—28no—
370C4×C8—28no—
371#95—22no—
372#95—22no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395C24—21no—
396C24C22×D411yescharacteristic
397#41—48no—
398#41—48no—
399C2×Q8—28no—
400C2×Q8—24no—
401C2×Q8—216no—
402C2×Q8—216no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—48no—
414C2×Q8—48no—
415C2×Q8C2×#4118yes—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—28no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—81no—
446C22×C4—81no—
447C22×C4—81no—
448C22×C4—81no—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C2×#4114yes—
459C22×C4C2×#4114yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C4C2511yescharacteristic
463Q16—416no—
464SD16—416no—
465M16—216no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—48no—
475C2×C8C2×#4118yes—
476C4⋊C4—28no—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—28no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—42no—
510C22⋊C4—44no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C22⋊C4—81no—
516C22⋊C4—81no—
517C22⋊C4—82no—
518C22⋊C4—82no—
519C42—22no—
520C42—22no—
521C42—24no—
522C42—24no—
523C42—24no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—42no—
529C42—42no—
530C42—48no—
531C42—48no—
532C42—42no—
533C42—44no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23#52512yes—
546C23#53212yes—
547C23#53512yes—
548C23C23×D411yescharacteristic
549Q8—28no—
550Q8—28no—
551Q8—416no—
552Q8—44no—
553Q8—44no—
554Q8—42no—
555Q8—42no—
556D4—42no—
557D4—42no—
558D4—84no—
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—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—28no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—416no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4—162no—
611C2×C4#52512yes—
612C2×C4#53212yes—
613C2×C4#53512yes—
614C2×C4#55118yes—
615C2×C4#55318yes—
616C2×C4#57712yes—
617C2×C4#57812yes—
618C2×C4C22×#4112yes—
619C2×C4C22×#4112yes—
620C2×C4C2×H(5,𝔽2)11yescharacteristic
621C2×C4C2×H(5,𝔽2)11yescharacteristic
622C8—28no—
623C8—42no—
624C8—42no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22#260314yes—
639C22C2×#52511yescharacteristic
640C22C2×#53211yescharacteristic
641C22C2×#53511yescharacteristic
642C22C2×#57711yescharacteristic
643C22C2×#57811yescharacteristic
644C22#358611yescharacteristic
645C22#358611yescharacteristic
646C4—22no—
647C4—22no—
648C4—48no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—41no—
654C4—41no—
655C4—81no—
656C4—81no—
657C4—81no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4#260314yes—
663C4C2×#55112yes—
664C4C2×#55312yes—
665C2—24no—
666C2—161no—
667C2C2×#260311yescharacteristic
668C2#6058211yescharacteristic
669C2#6058311yescharacteristic
670C2#6072311yescharacteristic
671C2#6072611yescharacteristic
672C2#6083611yescharacteristic
673C2#6083711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34209C2semidirect
3. #29509C2semidirect
4. #29508C2semidirect
5. #29501C2semidirect
6. #29500C2semidirect
7. #29495C2semidirect
8. #29494C2semidirect
9. #23657C2semidirect
11. C2×#2043C2semidirect
14. #21088C2semidirect
16. #20877C2semidirect
17. #20368C2semidirect

Maximal subgroups

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

subgroupindex
#352672
#342092
#295092
#295082
#295012
#295002
#294952
#294942
#236572
#236272
C2×#20432
#234322
#212342
#210882
#209302
#208772
#203682