Finite Group Databasebeta

← #6436594

Subgroups — #6436594

Order 512.

3268 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3508C211yesmaximal, characteristic
2C2×#3123C212yesmaximal, complemented
3#35309C211yesmaximal, characteristic, complemented
4#29700C212yesmaximal, complemented
5#29694C212yesmaximal, complemented
6#29673C212yesmaximal, complemented
7#29663C212yesmaximal, complemented
8#29645C212yesmaximal, complemented
9#29625C212yesmaximal, complemented
10#29604C214yesmaximal, complemented
11#23582C214yesmaximal, complemented
12#23576C212yesmaximal, complemented
13#23570C212yesmaximal, complemented
14#20744C212yesmaximal, complemented
15#20720C211yesmaximal, characteristic, complemented
16#3508—216no—
17C2×#559—22no—
18C2×#550—22no—
19C2×#546—22no—
20C2×#546—22no—
21C2×#541C2211yescharacteristic
22C2×#540C2212yes—
23C2×#540C2212yes—
24C2×#535C2212yes—
25C2×#533C2211yescharacteristic
26C2×#532C2212yes—
27C2×#532C2212yes—
28C22×#127C2211yescharacteristic
29C2×#518C2212yes—
30C2×C4×#41C2211yescharacteristic
31#3123—216no—
32C2×#485C2214yes—
33C2×#478C2214yescomplemented
34C2×#461C2214yescomplemented
35C2×#450C2214yes—
36C2×#440C2214yescomplemented
37C2×C4×SD16C2214yescomplemented
38C2×#433C2212yes—
39#2857C2211yescharacteristic, complemented
40#2855C2212yes—
41#2851C2211yescharacteristic, complemented
42#2629C2211yescharacteristic, complemented
43#2621C2211yescharacteristic, complemented
44#2616C2212yes—
45#2468—22no—
46#2466—22no—
47#2462—22no—
48#2460—22no—
49#2323C2212yescomplemented
50#2314C2211yescharacteristic
51C4×C22⋊Q8C2212yescomplemented
52#2059C2214yescomplemented
53#2058C2212yescomplemented
54#2057C2212yescomplemented
55#2041C2214yes—
56#2041C2212yes—
57#2041C2212yes—
58#2034C2214yescomplemented
59#2034C2214yescomplemented
60#2033C2214yescomplemented
61#2033C2214yescomplemented
62#2026C2214yescomplemented
63#2024C2212yescomplemented
64#2024C2212yescomplemented
65#2014C2212yes—
66#2014C2212yes—
67#2014C2214yes—
68#2000C2214yescomplemented
69#2000C2214yescomplemented
70#2000C2214yescomplemented
71#2000C2214yescomplemented
72#1825C2214yes—
73#1817C2212yes—
74#1804C2214yescomplemented
75#1802C2214yescomplemented
76#1798C2212yescomplemented
77#1797C2212yescomplemented
78#1786C2214yes—
79#1780C2212yes—
80#1762C2214yescomplemented
81C4×#102C2214yescomplemented
82#1750C2212yescomplemented
83#1749C2211yescharacteristic, complemented
84C4×C22⋊C8C2211yescharacteristic, complemented
85C22×#41—22no—
86C22×#41—22no—
87C22×#41C2311yescharacteristic
88C23×D4—22no—
89C23×D4C2312yes—
90C22×SD16C2314yes—
91C22×SD16C2314yes—
92#559—232no—
93#550—232no—
94#546—232no—
95#546—232no—
96#541—28no—
97#540—216no—
98#540—216no—
99#535—28no—
100#535—28no—
101#533—28no—
102#532—216no—
103#532—216no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#127—28no—
107C2×#127—22no—
108C2×#127—22no—
109C2×#127C2312yes—
110C2×#127C2312yes—
111C2×#126—22no—
112C2×#125—22no—
113C2×#125C2311yescharacteristic
114C2×#125C2312yes—
115C2×#124—22no—
116C2×#124—22no—
117C2×#124—22no—
118C2×#123—24no—
119C2×#123—24no—
120C2×C22⋊Q8—22no—
121C2×C22⋊Q8C2312yes—
122C2×C22⋊Q8C2312yes—
123C2×C22⋊Q8C2312yes—
124C2×#121—22no—
125C2×#121—22no—
126C2×#121—24no—
127C2×#121—24no—
128C2×#121C2314yes—
129C2×#121C2314yes—
130C2×#120—22no—
131C2×#120—22no—
132#518—28no—
133#518D4116yes—
134C4×#41—28no—
135C2×C4×Q8—22no—
136C2×C4×Q8D412yes—
137C2×C4×Q8C2312yes—
138C2×C4×D4—24no—
139C2×C4×D4—22no—
140C2×C4×D4—22no—
141C2×C4×D4D412yes—
142C2×C4×D4D412yes—
143C2×C4×D4D412yes—
144C2×C4×D4C2314yes—
145C2×C4×D4C2314yes—
146C2×#117C2312yes—
147C2×#117C2311yescharacteristic
148C2×#117C2311yescharacteristic
149C2×#117C2311yescharacteristic
150C22×C22⋊C4—24no—
151C22×C42C2311yescharacteristic
152C22×C42C2311yescharacteristic
153C22×C42C2311yescharacteristic
154#485—216no—
155#478—216no—
156#461—216no—
157#450—216no—
158#440—216no—
159C4×SD16—216no—
160#433—28no—
161C2×#107C2314yes—
162C2×#106C2314yes—
163C2×C4⋊C8C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#102C2314yes—
167C2×#102C2314yes—
168C2×#102C2314yes—
169C2×#102C2314yes—
170C2×#102C2314yes—
171C2×#102C2314yes—
172C2×C22⋊C8C2314yes—
173C2×C22⋊C8C2314yes—
174C2×C22⋊C8C2312yes—
175C2×C22⋊C8C2312yes—
176C2×#97C2312yes—
177C2×C4×C8C2312yes—
178#398—22no—
179#398—22no—
180#397—22no—
181#396—22no—
182#395—22no—
183#394—22no—
184#394—22no—
185#394—22no—
186#391C2312yes—
187#391C2312yes—
188#389—22no—
189#389—22no—
190#389C2312yes—
191#389C2312yes—
192#388—22no—
193#388—22no—
194#388—22no—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#385—22no—
202#385—22no—
203#385—22no—
204#384C2311yescharacteristic
205#382C2311yescharacteristic
206#381—22no—
207#380—22no—
208#380C2312yes—
209#380C2312yes—
210#380C2312yes—
211C4×C4⋊C4—22no—
212C4×C4⋊C4C2312yes—
213C4×C22⋊C4—22no—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2312yes—
219#336C2312yes—
220#336C2314yes—
221C25—22no—
222C2×#41—216no—
223C2×#41—216no—
224C2×#41—24no—
225C2×#41—42no—
226C2×#41—42no—
227C2×#41—84no—
228C2×#41—84no—
229C2×#41—84no—
230C2×#41—84no—
231C22×Q8—22no—
232C22×Q8C2×D411yescharacteristic
233C22×Q8C2412yes—
234C22×D4—216no—
235C22×D4—24no—
236C22×D4—24no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—24no—
241C22×D4—24no—
242C22×D4—44no—
243C22×D4—44no—
244C22×D4—84no—
245C22×D4—84no—
246C22×D4C2×D411yescharacteristic
247C22×D4C2×D411yescharacteristic
248C22×D4C2×D411yescharacteristic
249C22×D4C2×D418yes—
250C22×D4C2414yes—
251C23×C4—24no—
252C23×C4—24no—
253C23×C4C2×D412yes—
254C23×C4C2×D412yes—
255C23×C4C2×D412yes—
256C23×C4C2411yescharacteristic
257C23×C4C2411yescharacteristic
258C23×C4C2411yescharacteristic
259C2×SD16—28no—
260C2×SD16—28no—
261C2×SD16—44no—
262C2×SD16—44no—
263C2×SD16—44no—
264C2×SD16—44no—
265C22×C8C2414yes—
266C22×C8C2414yes—
267C4⋊Q8—28no—
268C4⋊Q8—24no—
269#127—216no—
270#127—216no—
271#127—28no—
272#127—28no—
273#126—48no—
274#125—28no—
275#125—24no—
276#125—48no—
277#124—216no—
278#124—216no—
279#124—216no—
280#123—232no—
281#123—232no—
282C22⋊Q8—216no—
283C22⋊Q8—28no—
284C22⋊Q8—44no—
285C22⋊Q8—42no—
286C22⋊Q8—42no—
287#121—216no—
288#121—216no—
289#121—232no—
290#121—232no—
291#121—28no—
292#121—28no—
293#121—28no—
294#121—28no—
295#120—216no—
296#120—216no—
297C4×Q8—216no—
298C4×Q8—28no—
299C4×Q8—28no—
300C4×D4—216no—
301C4×D4—216no—
302C4×D4—232no—
303C4×D4—28no—
304C4×D4—28no—
305C4×D4—28no—
306C4×D4—28no—
307C4×D4—28no—
308C4×D4C2×D4116yes—
309C4×D4C2×D4116yes—
310#117—24no—
311#117—24no—
312#117—24no—
313#117C2×D418yes—
314#117C2×D418yes—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2414yes—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—232no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C42—22no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—28no—
362#106—28no—
363C4⋊C8—28no—
364#103—28no—
365#103—28no—
366#102—28no—
367#102—28no—
368#102—44no—
369#102—44no—
370#102—44no—
371#102—44no—
372C22⋊C8—28no—
373C22⋊C8—44no—
374C22⋊C8—42no—
375C22⋊C8—42no—
376#97—24no—
377C4×C8—24no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389C24—24no—
390C24—28no—
391C24—44no—
392C24—44no—
393C24—84no—
394C24—84no—
395C24C22×D411yescharacteristic
396C24C22×D411yescharacteristic
397C24C22×D411yescharacteristic
398#41—416no—
399#41—832no—
400#41—832no—
401C2×Q8—28no—
402C2×Q8—22no—
403C2×Q8—24no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—41no—
408C2×Q8—41no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×D4—28no—
412C2×D4—28no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—216no—
416C2×D4—216no—
417C2×D4—216no—
418C2×D4—28no—
419C2×D4—22no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—416no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4—41no—
429C2×D4—41no—
430C2×D4—41no—
431C2×D4—41no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—416no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—84no—
440C2×D4—84no—
441C2×D4—832no—
442C2×D4—84no—
443C2×D4—84no—
444C2×D4—84no—
445C2×D4—84no—
446C2×D4C22×D418yes—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—216no—
463C22×C4—216no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—84no—
472C22×C4—84no—
473C22×C4—84no—
474C22×C4—84no—
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D414yes—
482C22×C4C22×D412yes—
483C22×C4C2511yescharacteristic
484SD16—416no—
485SD16—416no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C4⋊C4—28no—
493C4⋊C4—216no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4#12018yes—
511C4⋊C4C22×D418yes—
512C22⋊C4—216no—
513C22⋊C4—216no—
514C22⋊C4—28no—
515C22⋊C4—216no—
516C22⋊C4—216no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C22⋊C4#12018yes—
526C42—24no—
527C42—24no—
528C42—24no—
529C42—28no—
530C42—22no—
531C42—22no—
532C42—44no—
533C42—82no—
534C42C22×D414yes—
535C42C22×D414yes—
536C23—22no—
537C23—21no—
538C23—21no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—28no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—22no—
550C23—22no—
551C23—416no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—816no—
559C23—168no—
560C23—164no—
561C23—164no—
562C23C2×#12012yes—
563C23C23×D411yescharacteristic
564Q8—44no—
565Q8—416no—
566Q8—44no—
567Q8—44no—
568D4—28no—
569D4—28no—
570D4—416no—
571D4—416no—
572D4—416no—
573D4—44no—
574D4—44no—
575D4—44no—
576D4—44no—
577D4—44no—
578D4—816no—
579D4—816no—
580D4—816no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—28no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—28no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—48no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—816no—
620C2×C4—816no—
621C2×C4C2×#12012yes—
622C2×C4C2×#12014yes—
623C2×C4C2×#12014yes—
624C2×C4#57714yes—
625C2×C4C23×D412yes—
626C2×C4#58512yes—
627C2×C4#58512yes—
628C8—44no—
629C8—44no—
630C22—22no—
631C22—24no—
632C22—42no—
633C22—41no—
634C22—41no—
635C22—42no—
636C22—42no—
637C22—41no—
638C22—41no—
639C22—48no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—42no—
645C22—42no—
646C22—48no—
647C22—48no—
648C22—84no—
649C22—84no—
650C22—84no—
651C22—84no—
652C22—1616no—
653C22C22×#12011yescharacteristic
654C22#344911yescharacteristic
655C22#344911yescharacteristic
656C22C2×#57712yes—
657C22#358512yes—
658C4—22no—
659C4—24no—
660C4—24no—
661C4—24no—
662C4—41no—
663C4—41no—
664C4—42no—
665C4—48no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4#344914yes—
671C2—42no—
672C2—81no—
673C2—81no—
674C2—88no—
675C2#3463211yescharacteristic
676C2#6037312yes—
677C2#6099412yes—
678C2#6101211yescharacteristic
679C2#6101911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3508C2
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
2. C2×#3123C2semidirect
3. #35309C2semidirect
4. #29700C2semidirect
5. #29694C2semidirect
6. #29673C2semidirect
7. #29663C2semidirect
8. #29645C2semidirect
9. #29625C2semidirect
10. #29604C2semidirect
11. #23582C2semidirect
12. #23576C2semidirect
13. #23570C2semidirect
14. #20744C2semidirect
15. #20720C2semidirect
33. C2×#478C22semidirect
34. C2×#461C22semidirect
36. C2×#440C22semidirect
37. C2×C4×SD16C22semidirect
39. #2857C22semidirect
41. #2851C22semidirect
42. #2629C22semidirect
43. #2621C22semidirect
49. #2323C22semidirect
51. C4×C22⋊Q8C22semidirect
52. #2059C22semidirect
53. #2058C22semidirect
54. #2057C22semidirect
58. #2034C22semidirect
59. #2034C22semidirect
60. #2033C22semidirect
61. #2033C22semidirect
62. #2026C22semidirect
63. #2024C22semidirect
64. #2024C22semidirect
68. #2000C22semidirect
69. #2000C22semidirect
70. #2000C22semidirect
71. #2000C22semidirect
74. #1804C22semidirect
75. #1802C22semidirect
76. #1798C22semidirect
77. #1797C22semidirect
80. #1762C22semidirect
81. C4×#102C22semidirect
82. #1750C22semidirect
83. #1749C22semidirect
84. C4×C22⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
C2×#35082
C2×#31232
#353092
#297002
#296942
#296732
#296632
#296452
#296252
#296042
#235822
#235762
#235702
#207442
#207202