Finite Group Databasebeta

← #6993716

Subgroups — #6993716

Order 512.

1751 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36288C211yesmaximal, characteristic, complemented
2#35317C211yesmaximal, characteristic
3#32807C212yesmaximal, complemented
4#32050C214yesmaximal, complemented
5#31836C214yesmaximal, complemented
6#30205C212yesmaximal, complemented
7#29983C212yesmaximal, complemented
8#29663C211yesmaximal, characteristic
9#29555C212yesmaximal, complemented
10#28400C212yesmaximal, complemented
11#27813C212yesmaximal, complemented
12#25902C211yesmaximal, characteristic, complemented
13#25320C212yesmaximal, complemented
14#24678C212yesmaximal, complemented
15#24301C211yesmaximal, characteristic, complemented
16#23570C211yesmaximal, characteristic
17#22632C211yesmaximal, characteristic, complemented
18C2×#536—21no—
19C2×#536—21no—
20C2×#532C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C2×#487C2212yes—
23C2×#475C2212yes—
24C2×#461C2212yes—
25C2×C4×SD16C2212yes—
26#2868C2212yes—
27#2862C2212yes—
28#2858C2212yes—
29#2857C2211yescharacteristic
30#2658—22no—
31#2612—22no—
32#2574—22no—
33#2551C2211yescharacteristic
34#2546C2211yescharacteristic
35#2542—21no—
36#2542—21no—
37#2542C2212yes—
38#2542C2212yes—
39#2508—21no—
40#2501—22no—
41#2501C2214yes—
42#2492C2212yes—
43#2439—21no—
44#2429C2212yes—
45#2369—21no—
46#2365C2211yescharacteristic
47#2355C2211yescharacteristic
48#2346—21no—
49C4×C22⋊Q8C2211yescharacteristic
50#2295C2212yes—
51C4×#117C2211yescharacteristic
52#2099C2214yes—
53#2096C2214yes—
54#2072C2214yes—
55#2072C2214yes—
56#2058C2211yescharacteristic
57#2058C2212yes—
58#2055C2214yes—
59#2055C2214yes—
60#2041C2211yescharacteristic
61#2034C2212yes—
62#2024C2211yescharacteristic
63#2024C2212yes—
64#2024C2212yes—
65#2024C2212yes—
66#2024C2212yes—
67#2023C2214yes—
68#2014C2211yescharacteristic
69#2014C2212yes—
70#2013C2214yes—
71#2000C2212yes—
72#2000C2214yes—
73#1990C2212yes—
74#1990C2214yes—
75#1988C2212yes—
76#1968C2214yes—
77#1956C2214yes—
78#1948C2214yes—
79#1939C2212yes—
80#1934C2214yes—
81#1926C2214yes—
82#1923C2214yes—
83#1880C2214yes—
84#1872C2214yes—
85#1872C2212yes—
86#1853C2212yes—
87#1831C2212yes—
88#1803C2214yes—
89#1802C2212yes—
90#1797C2211yescharacteristic
91C8×C4⋊C4C2212yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1744C2212yes—
95C22×#41—21no—
96C23×Q8—21no—
97C22×SD16C2312yes—
98C22×SD16C2314yes—
99#536—48no—
100#536—48no—
101#532—44no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—44no—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2312yes—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2312yes—
114C2×#127C2311yescharacteristic
115C2×#124—22no—
116C2×#124—22no—
117C2×#124—21no—
118C2×#123—24no—
119C2×#123—24no—
120C2×C22⋊Q8—22no—
121C2×C22⋊Q8—22no—
122C2×C22⋊Q8—22no—
123C2×C22⋊Q8—22no—
124C2×C22⋊Q8C2311yescharacteristic
125C2×C22⋊Q8C2312yes—
126C2×#121—21no—
127C2×#121C2312yes—
128C2×C4×Q8C2311yescharacteristic
129C2×C4×Q8C2312yes—
130C2×C4×D4—21no—
131C2×C4×D4C2312yes—
132C2×#117D411yescharacteristic
133C2×#117D411yescharacteristic
134C22×C4⋊C4—21no—
135C22×C4⋊C4C2312yes—
136C22×C42C2311yescharacteristic
137#487—44no—
138#475—44no—
139#461—44no—
140C4×SD16—44no—
141C2×#106C2314yes—
142C2×#106C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×#102C2314yes—
153C2×#102C2312yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C4×C8C2312yes—
157C2×C4×C8C2314yes—
158#400—22no—
159#399—24no—
160#399—24no—
161#399—24no—
162#399—24no—
163#398—22no—
164#398—22no—
165#396—22no—
166#395—22no—
167#393—22no—
168#391—22no—
169#391—22no—
170#391—22no—
171#391C2312yes—
172#391C2312yes—
173#391C2312yes—
174#391C2311yescharacteristic
175#391C2312yes—
176#390—22no—
177#390—22no—
178#390C2314yes—
179#390C2314yes—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—22no—
187#389C2311yescharacteristic
188#389C2312yes—
189#388—22no—
190#388—22no—
191#387—22no—
192#387—21no—
193#387—22no—
194#387—22no—
195#387—22no—
196#385—21no—
197#384—22no—
198#384—22no—
199#384C2311yescharacteristic
200#384C2314yes—
201#384C2314yes—
202#382—21no—
203#382—22no—
204#382—22no—
205#380—21no—
206#380—21no—
207#380C2311yescharacteristic
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—21no—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4C2312yes—
221#376—22no—
222#376D412yes—
223C43C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2314yes—
234#336C2314yes—
235#336C2311yescharacteristic
236C2×#41—44no—
237C2×#41—44no—
238C2×#41—44no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8—44no—
246C22×Q8C2411yescharacteristic
247C22×Q8C2412yes—
248C22×D4—22no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4C2412yes—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2411yescharacteristic
257C2×SD16—42no—
258C2×SD16—42no—
259C2×SD16—42no—
260C2×SD16—44no—
261C2×SD16—44no—
262C2×SD16—44no—
263C22×C8C2412yes—
264C22×C8C2414yes—
265C22×C8C2412yes—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—24no—
269C4⋊Q8—44no—
270C4⋊Q8—42no—
271C4⋊Q8—44no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274C4⋊Q8—48no—
275C4⋊Q8—44no—
276C4⋊Q8—44no—
277C4⋊Q8—48no—
278#127—42no—
279#124—44no—
280#124—48no—
281#124—48no—
282#123—416no—
283#123—416no—
284C22⋊Q8—22no—
285C22⋊Q8—41no—
286C22⋊Q8—48no—
287C22⋊Q8—48no—
288C22⋊Q8—48no—
289C22⋊Q8—48no—
290C22⋊Q8—44no—
291#121—42no—
292#121—42no—
293#121—44no—
294C4×Q8—24no—
295C4×Q8—24no—
296C4×Q8—42no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—44no—
300#117—24no—
301#117SD1618yescomplemented
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—22no—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361#106—42no—
362#106—44no—
363C4⋊C8—24no—
364C4⋊C8—44no—
365#103—42no—
366#103—44no—
367#103—44no—
368#103—44no—
369#103—44no—
370#102—24no—
371#102—44no—
372#102—42no—
373#102—42no—
374C22⋊C8—42no—
375C22⋊C8—42no—
376C4×C8—24no—
377C4×C8—44no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—22no—
384#95—21no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—21no—
391#95—22no—
392#95—22no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24—22no—
398C24C22×D411yescharacteristic
399#41—816no—
400C2×Q8—22no—
401C2×Q8—24no—
402C2×Q8—22no—
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8—24no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—44no—
409C2×Q8—42no—
410C2×Q8—44no—
411C2×Q8—42no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—42no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—82no—
419C2×Q8—82no—
420C2×Q8—816no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—42no—
435C2×D4—42no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—41no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4C2×SD1614yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C2511yescharacteristic
470SD16—48no—
471SD16—88no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—48no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—81no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4C2×SD1614yes—
512C4⋊C4C2×SD1614yes—
513C4⋊C4AGL(1,ℤ/8)14yes—
514C4⋊C42−1+412yes—
515C22⋊C4—24no—
516C22⋊C4—42no—
517C22⋊C4—44no—
518C22⋊C4—48no—
519C22⋊C4—44no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—81no—
524C22⋊C4—82no—
525C22⋊C4C2×SD1618yes—
526C42—22no—
527C42—22no—
528C42—24no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—24no—
533C42—22no—
534C42—42no—
535C42—44no—
536C42—42no—
537C42—42no—
538C42—82no—
539C42C2×SD1618yes—
540C42H(5,𝔽2)14yes—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23C22×SD1612yes—
555C23C23×D411yescharacteristic
556Q8—42no—
557Q8—44no—
558Q8—42no—
559Q8—44no—
560Q8—88no—
561Q8—84no—
562Q8—88no—
563Q8—164no—
564D4—42no—
565D4—42no—
566D4—42no—
567D4—42no—
568D4—84no—
569D4—84no—
570D4—88no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—28no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—88no—
618C2×C4—88no—
619C2×C4#53512yes—
620C2×C4#53512yes—
621C2×C4C22×SD1612yes—
622C2×C4C22×SD1614yes—
623C2×C4C22×SD1614yes—
624C2×C4C22×SD1612yes—
625C2×C4C2×AGL(1,ℤ/8)14yes—
626C2×C4C2×AGL(1,ℤ/8)12yes—
627C2×C4#57812yes—
628C2×C4C2×H(5,𝔽2)12yes—
629C2×C4C2×2−1+411yescharacteristic
630C2×C4#58511yescharacteristic
631C2×C4#58512yes—
632C8—42no—
633C8—44no—
634C8—42no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22C2×#53511yescharacteristic
650C22#348911yescharacteristic
651C22#350011yescharacteristic
652C22C23×SD1611yescharacteristic
653C22C22×AGL(1,ℤ/8)11yescharacteristic
654C22C2×#57811yescharacteristic
655C22#358611yescharacteristic
656C4—24no—
657C4—22no—
658C4—24no—
659C4—22no—
660C4—22no—
661C4—24no—
662C4—42no—
663C4—42no—
664C4—41no—
665C4—82no—
666C4—84no—
667C4—82no—
668C4—81no—
669C4—81no—
670C4—81no—
671C4—82no—
672C4#320814yes—
673C4#354012yes—
674C4#354712yes—
675C2—42no—
676C2—162no—
677C2#4299511yescharacteristic
678C2C2×#320811yescharacteristic
679C2#6081411yescharacteristic
680C2#6209911yescharacteristic
681C2#6214811yescharacteristic
682C2#6252611yescharacteristic
683C2#6258311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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
1. #36288C2semidirect
3. #32807C2semidirect
4. #32050C2semidirect
5. #31836C2semidirect
6. #30205C2semidirect
7. #29983C2semidirect
9. #29555C2semidirect
10. #28400C2semidirect
11. #27813C2semidirect
12. #25902C2semidirect
13. #25320C2semidirect
14. #24678C2semidirect
15. #24301C2semidirect
17. #22632C2semidirect
301. #117SD16semidirect

Maximal subgroups

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

subgroupindex
#362882
#353172
#328072
#320502
#318362
#302052
#299832
#296632
#295552
#284002
#278132
#259022
#253202
#246782
#243012
#235702
#226322