Finite Group Databasebeta

← #6986143

Subgroups — #6986143

Order 512.

1561 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36505C211yesmaximal, characteristic
2#35280C211yesmaximal, characteristic
3#32789C212yesmaximal, complemented
4#31390C214yesmaximal, complemented
5#31326C214yesmaximal, complemented
6#30190C212yesmaximal, complemented
7#30116C212yesmaximal, complemented
8#29688C211yesmaximal, characteristic, complemented
9#29159C212yesmaximal, complemented
10#28142C212yesmaximal, complemented
11#27960C211yesmaximal, characteristic, complemented
12#27680C211yesmaximal, characteristic, complemented
13#25341C212yesmaximal, complemented
14#24997C212yesmaximal, complemented
15#24332C212yesmaximal, complemented
16#23571C211yesmaximal, characteristic, complemented
17#22469C211yesmaximal, characteristic
18C2×#557C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#475C2212yes—
22C2×C4×Q16C2212yes—
23C2×C8×D4C2212yes—
24#2857C2212yes—
25#2857C2211yescharacteristic
26#2857C2212yes—
27#2851C2212yes—
28#2629—22no—
29#2582—21no—
30#2582C2212yes—
31#2562C2212yes—
32#2547C2211yescharacteristic
33#2538—21no—
34#2538C2211yescharacteristic
35#2538C2212yes—
36#2536C2212yes—
37#2475—22no—
38#2460—22no—
39#2446—21no—
40#2432—21no—
41#2417—22no—
42#2417C2214yes—
43#2367—21no—
44#2362—21no—
45#2355C2211yescharacteristic
46#2316—21no—
47#2311—21no—
48C4×C22⋊Q8C2211yescharacteristic
49C4×C22⋊Q8C2212yes—
50C2×#384C2211yescharacteristic
51#2101C2214yes—
52#2099C2212yes—
53#2099C2212yes—
54#2099C2212yes—
55#2096C2212yes—
56#2077C2214yes—
57#2076C2214yes—
58#2074C2214yes—
59#2059C2214yes—
60#2058C2212yes—
61#2057C2211yescharacteristic
62#2043C2211yescharacteristic
63#2043C2212yes—
64#2036C2212yes—
65#2035C2214yes—
66#2035C2214yes—
67#2026C2214yes—
68#2024C2211yescharacteristic
69#2016C2211yescharacteristic
70#2001C2212yes—
71#2001C2214yes—
72#1990C2212yes—
73#1984C2212yes—
74#1948C2212yes—
75#1927C2214yes—
76#1919C2214yes—
77#1892C2214yes—
78#1892C2214yes—
79#1892C2214yes—
80#1878C2214yes—
81#1872C2212yes—
82#1869C2214yes—
83#1865C2214yes—
84#1853C2212yes—
85#1834C2212yes—
86#1823C2214yes—
87#1817C2212yes—
88#1807C2214yes—
89#1803C2212yes—
90#1803C2212yes—
91#1797C2211yescharacteristic
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C2×#336C2212yes—
95C22×#41D411yescharacteristic
96C22×Q16C2312yes—
97C23×C8C2314yes—
98#557—44no—
99#533—44no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#125—22no—
105C2×#125—21no—
106C2×#125C2312yes—
107C2×#124—21no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8C2311yescharacteristic
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2314yes—
114C2×C22⋊Q8C2312yes—
115C4×#41—44no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D412yes—
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2312yes—
127C2×#117—21no—
128C2×#117D411yescharacteristic
129C2×#117D412yes—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135#475—44no—
136C4×Q16—44no—
137C8×D4—44no—
138C2×#107C2312yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2312yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2312yes—
152C2×C4×C8C2312yes—
153#400—22no—
154#398—24no—
155#398—24no—
156#398—22no—
157#397—22no—
158#396—24no—
159#396—24no—
160#396—22no—
161#394—22no—
162#394—22no—
163#391—22no—
164#391—21no—
165#391—21no—
166#391—22no—
167#391—21no—
168#391C2314yes—
169#391C2311yescharacteristic
170#390—22no—
171#390—21no—
172#389—22no—
173#389—22no—
174#389—21no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—21no—
179#389C2311yescharacteristic
180#389C2312yes—
181#389C2314yes—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—21no—
191#386—24no—
192#385—22no—
193#385—22no—
194#385—22no—
195#385—21no—
196#384—21no—
197#384—21no—
198#384C2311yescharacteristic
199#384C2312yes—
200#384C2312yes—
201#382C2312yes—
202#380—22no—
203#380—22no—
204#380—22no—
205#380—21no—
206#380—24no—
207#380C2311yescharacteristic
208#380C2312yes—
209#380C2312yes—
210#380C2312yes—
211#380C2314yes—
212#380C2312yes—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2314yes—
233#336C2311yescharacteristic
234C2×#41—24no—
235C2×#41—24no—
236C2×#41—42no—
237C22×Q8—21no—
238C22×Q8—21no—
239C22×Q8C2×D411yescharacteristic
240C22×Q8C2412yes—
241C22×Q8C2411yescharacteristic
242C22×D4C2×D412yes—
243C22×D4C2×D411yescharacteristic
244C23×C4—22no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D412yes—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D412yes—
249C23×C4C2411yescharacteristic
250C23×C4C2412yes—
251C2×Q16—24no—
252C2×Q16—42no—
253C2×Q16—42no—
254C22×C8—44no—
255C22×C8—44no—
256C22×C8—44no—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C4⋊Q8—24no—
261C4⋊Q8—42no—
262C4⋊Q8—42no—
263C4⋊Q8—44no—
264#125—44no—
265#125—44no—
266#125—48no—
267#124—44no—
268C22⋊Q8—22no—
269C22⋊Q8—24no—
270C22⋊Q8—41no—
271C22⋊Q8—42no—
272C22⋊Q8—42no—
273C22⋊Q8—42no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276C22⋊Q8—44no—
277C22⋊Q8—44no—
278C22⋊Q8—84no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—42no—
284C4×Q8—44no—
285C4×Q8—44no—
286C4×Q8—44no—
287C4×D4—24no—
288C4×D4—42no—
289C4×D4—44no—
290C4×D4—44no—
291#117—44no—
292#117—42no—
293#117—44no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—82no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C42—24no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—22no—
350C2×C42—44no—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#107—42no—
359#106—44no—
360#106—44no—
361C4⋊C8—24no—
362#103—44no—
363#103—42no—
364#103—42no—
365#103—44no—
366#103—42no—
367#103—42no—
368#103—44no—
369C22⋊C8—42no—
370C22⋊C8—42no—
371C22⋊C8—44no—
372C4×C8—24no—
373#95—21no—
374#95—21no—
375#95—22no—
376#95—21no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390C24C22×D411yescharacteristic
391C24C22×D412yes—
392#41—416no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q82−1+412yes—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—42no—
417C2×D4#13714yes—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—42no—
443C22×C4—41no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4#13514yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C2511yescharacteristic
459Q16—48no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—88no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—81no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C22⋊C4—24no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—81no—
518C22⋊C4—82no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—24no—
525C42—42no—
526C42—42no—
527C42—44no—
528C42—44no—
529C42—42no—
530C42—42no—
531C42—82no—
532C42H(5,𝔽2)14yes—
533C23—22no—
534C23—21no—
535C23—22no—
536C23—21no—
537C23—21no—
538C23—22no—
539C23—22no—
540C23—21no—
541C23—22no—
542C23—21no—
543C23—22no—
544C23—21no—
545C23C2×#13512yes—
546C23C2×#13714yes—
547C23C23×D411yescharacteristic
548Q8—24no—
549Q8—44no—
550Q8—42no—
551Q8—42no—
552Q8—42no—
553Q8—42no—
554Q8—42no—
555Q8—84no—
556Q8—84no—
557D4—28no—
558D4—48no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—28no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—44no—
580C2×C4—48no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—84no—
612C2×C4—164no—
613C2×C4#53412yes—
614C2×C4#53412yes—
615C2×C4C2×#13512yes—
616C2×C4C2×#13512yes—
617C2×C4C2×#13712yes—
618C2×C4#57712yes—
619C2×C4C2×H(5,𝔽2)12yes—
620C2×C4C2×2−1+411yescharacteristic
621C2×C4#58512yes—
622C2×C4#58511yescharacteristic
623C8—42no—
624C8—44no—
625C8—42no—
626C22—22no—
627C22—24no—
628C22—22no—
629C22—24no—
630C22—22no—
631C22—24no—
632C22—42no—
633C22—41no—
634C22—42no—
635C22—41no—
636C22—42no—
637C22—42no—
638C22—41no—
639C22—41no—
640C22C2×#53411yescharacteristic
641C22#348911yescharacteristic
642C22#350011yescharacteristic
643C22C22×#13511yescharacteristic
644C22C22×#13711yescharacteristic
645C22C2×#57711yescharacteristic
646C22#358611yescharacteristic
647C4—22no—
648C4—22no—
649C4—22no—
650C4—42no—
651C4—42no—
652C4—41no—
653C4—42no—
654C4—42no—
655C4—82no—
656C4—84no—
657C4—81no—
658C4—82no—
659C4—81no—
660C4—82no—
661C4—82no—
662C4—81no—
663C4#319014yes—
664C4#354112yes—
665C4#355112yes—
666C2—42no—
667C2—44no—
668C2#4290811yescharacteristic
669C2C2×#319011yescharacteristic
670C2#6076311yescharacteristic
671C2#6210111yescharacteristic
672C2#6213111yescharacteristic
673C2#6253111yescharacteristic
674C2#6257711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
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
3. #32789C2semidirect
4. #31390C2semidirect
5. #31326C2semidirect
6. #30190C2semidirect
7. #30116C2semidirect
8. #29688C2semidirect
9. #29159C2semidirect
10. #28142C2semidirect
11. #27960C2semidirect
12. #27680C2semidirect
13. #25341C2semidirect
14. #24997C2semidirect
15. #24332C2semidirect
16. #23571C2semidirect

Maximal subgroups

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

subgroupindex
#365052
#352802
#327892
#313902
#313262
#301902
#301162
#296882
#291592
#281422
#279602
#276802
#253412
#249972
#243322
#235712
#224692