Finite Group Databasebeta

← #7534633

Subgroups — #7534633

Order 512.

1999 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59815C211yesmaximal, characteristic
2#36559C211yesmaximal, characteristic, complemented
3#31322C214yesmaximal, complemented
4#29206C211yesmaximal, characteristic, complemented
5#29139C211yesmaximal, characteristic, complemented
6#28223C212yesmaximal, complemented
7#26232C212yesmaximal, complemented
8#25543C214yesmaximal, complemented
9#25387C212yesmaximal, complemented
10#25012C212yesmaximal, complemented
11#24969C212yesmaximal, complemented
12#24120C212yesmaximal, complemented
13#23173C211yesmaximal, characteristic, complemented
14#22703C212yesmaximal, complemented
15#22485C212yesmaximal, complemented
16C4×#475C211yesmaximal, characteristic, complemented
17C4×#460C211yesmaximal, characteristic, complemented
18C2×#554C2212yes—
19C2×#550—21no—
20C2×#547C2212yes—
21C2×#481C2214yes—
22C2×#475C2211yescharacteristic
23C2×#465C2214yes—
24C2×#460C2211yescharacteristic
25C2×C4×SD16C2212yes—
26C2×C8×D4C2212yes—
27C2×C8×D4C2212yes—
28#2868C2211yescharacteristic
29#2859—21no—
30#2682—22no—
31#2566—22no—
32#2563C2212yes—
33#2541C2212yes—
34#2538—21no—
35#2538C2212yes—
36#2538C2212yes—
37#2537C2212yes—
38#2452—22no—
39#2452—22no—
40#2445—22no—
41#2416C2214yes—
42#2407—22no—
43D4×C4⋊C4C2212yes—
44D4×C4⋊C4C2212yes—
45D4×C4⋊C4C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127—21no—
48C4×#127C2211yescharacteristic
49C4×#127C2211yescharacteristic
50C42×Q8C2211yescharacteristic
51#2077C2214yes—
52#2058C2214yes—
53#2035C2214yes—
54#2034C2214yes—
55#2034C2214yes—
56#2034C2214yes—
57#2023C2214yes—
58#2000C2214yes—
59#1990C2212yes—
60#1988C2211yescharacteristic
61#1982C2211yescharacteristic
62#1957C2212yes—
63#1954C2212yes—
64#1954C2212yes—
65#1948C2212yes—
66#1948C2212yes—
67#1948C2211yescharacteristic
68#1948C2211yescharacteristic
69#1926C2212yes—
70#1919C2212yes—
71#1893C2214yes—
72#1880C2212yes—
73#1874C2214yes—
74#1872C2214yes—
75#1869C2214yes—
76#1865C2212yes—
77#1853C2214yes—
78#1853C2214yes—
79#1844C2212yes—
80#1834C2212yes—
81#1823C2214yes—
82#1807C2214yes—
83#1803C2211yescharacteristic
84#1803C2211yescharacteristic
85#1802C2212yes—
86#1802C2212yes—
87C4×#106C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90C4×#103C2212yes—
91C4×#102C2212yes—
92C2×#340C2214yes—
93C22×#41—22no—
94C22×SD16C2312yes—
95C23×C8C2314yes—
96#554—44no—
97#554—44no—
98#550—48no—
99#547—28no—
100#547—44no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127—22no—
106C2×#127D411yescharacteristic
107C2×#127C2311yescharacteristic
108C2×#125—22no—
109C2×#123—24no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8C2314yes—
113C2×#121—22no—
114C2×#121—24no—
115C2×#121C2314yes—
116C2×#121C2314yes—
117C2×C4×Q8—21no—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×D4—22no—
122C2×C4×D4—22no—
123C2×C4×D4—22no—
124C2×C4×D4—22no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4C2312yes—
128C2×C4×D4C2312yes—
129C2×C4×D4C2312yes—
130C2×C4×D4C2312yes—
131C22×C4⋊C4C2314yes—
132C22×C4⋊C4C2314yes—
133#481—48no—
134#475—24no—
135#465—48no—
136#460—24no—
137C4×SD16—44no—
138C8×D4—28no—
139C8×D4—44no—
140C2×#106C2312yes—
141C2×#106C2312yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×#106C2312yes—
145C2×C4⋊C8C2311yescharacteristic
146C2×C4⋊C8C2311yescharacteristic
147C2×C4⋊C8C2311yescharacteristic
148C2×C4⋊C8C2311yescharacteristic
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2312yes—
157C2×C22⋊C8C2314yes—
158C2×C22⋊C8C2314yes—
159C2×C22⋊C8C2314yes—
160C2×C4×C8C2312yes—
161C2×C4×C8C2312yes—
162#396—24no—
163#396—24no—
164#396—24no—
165#394—24no—
166#394—24no—
167#393—24no—
168#392—24no—
169#391—21no—
170#390—22no—
171#390—22no—
172#390—22no—
173#390—22no—
174#390—21no—
175#390—21no—
176#390D412yes—
177#390D412yes—
178#390D412yes—
179#390C2312yes—
180#389—22no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—21no—
186#389C2312yes—
187#386—24no—
188#385—24no—
189#385—24no—
190#385—24no—
191#384—21no—
192#384—21no—
193#384D411yescharacteristic
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#384C2312yes—
197#384C2312yes—
198#382—22no—
199#380—22no—
200#380C2314yes—
201#380C2314yes—
202#380C2314yes—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C43—21no—
212C43C2311yescharacteristic
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2312yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#340C2312yes—
223#336C2312yes—
224#336C2312yes—
225C2×#41—48no—
226C2×#41—48no—
227C2×#41—84no—
228C22×Q8—21no—
229C22×Q8—22no—
230C22×Q8C2411yescharacteristic
231C22×D4—24no—
232C22×D4—24no—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4C2×D412yes—
236C22×D4C2×D412yes—
237C22×D4C2×D412yes—
238C22×D4C2412yes—
239C23×C4—24no—
240C23×C4—24no—
241C23×C4—22no—
242C23×C4C2414yes—
243C2×SD16—24no—
244C2×SD16—24no—
245C2×SD16—42no—
246C22×C8—28no—
247C22×C8—44no—
248C22×C8—44no—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C4⋊Q8—22no—
254C4⋊Q8—42no—
255C4⋊Q8—42no—
256C4⋊Q8—82no—
257C4⋊Q8—82no—
258C4⋊Q8#4114yes—
259#127—22no—
260#127—22no—
261#127—48no—
262#127SD1618yescomplemented
263#125—48no—
264#123—416no—
265C22⋊Q8—28no—
266C22⋊Q8—44no—
267C22⋊Q8—48no—
268C22⋊Q8—416no—
269#121—28no—
270#121—44no—
271#121—48no—
272#121—416no—
273#121—44no—
274#121—44no—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—22no—
281C4×Q8—44no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—42no—
289C4×D4—42no—
290C4×D4—48no—
291C4×D4—48no—
292C4×D4—44no—
293C4×D4—82no—
294C4×D4—84no—
295C4×D4—84no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—28no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4C2×D414yes—
337C2×C22⋊C4C2×D414yes—
338C2×C22⋊C4C2×D414yes—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#106—24no—
354#106—24no—
355#106—42no—
356#106—44no—
357#106—44no—
358C4⋊C8—22no—
359C4⋊C8—22no—
360C4⋊C8—22no—
361C4⋊C8#4114yescomplemented
362#103—24no—
363#103—24no—
364#103—42no—
365#102—24no—
366#102—24no—
367#102—42no—
368#102—44no—
369#102—44no—
370C22⋊C8—28no—
371C22⋊C8—44no—
372C22⋊C8—44no—
373C4×C8—24no—
374C4×C8—24no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—24no—
382#95—24no—
383C24—22no—
384C24C22×D414yes—
385#41—832no—
386C2×Q8—24no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—44no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—82no—
396C2×Q8—82no—
397C2×Q8H(5,𝔽2)12yes—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—81no—
418C2×D4—81no—
419C2×D4—84no—
420C2×D4—84no—
421C2×D4—84no—
422C2×D4C2×SD1618yes—
423C2×D4C2×SD1618yes—
424C2×D4C2×SD1618yes—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
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—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—84no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D414yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C2511yescharacteristic
456SD16—48no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—88no—
469C2×C8C2×#4114yes—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—42no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—81no—
499C4⋊C4—81no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4C2×#4114yes—
503C4⋊C4H(5,𝔽2)12yes—
504C22⋊C4—28no—
505C22⋊C4—28no—
506C22⋊C4—44no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—82no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C42—24no—
517C42—22no—
518C42—22no—
519C42—21no—
520C42—21no—
521C42—41no—
522C42—42no—
523C42—42no—
524C42—42no—
525C42—44no—
526C42—42no—
527C42—42no—
528C42C2×SD1614yes—
529C42C2×#4112yes—
530C42C2×#4112yes—
531C42H(5,𝔽2)12yes—
532C422−1+412yes—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23C22×SD1618yes—
547C23C23×D411yescharacteristic
548Q8—22no—
549Q8—22no—
550Q8—41no—
551Q8—41no—
552Q8—48no—
553Q8—164no—
554D4—216no—
555D4—216no—
556D4—48no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—84no—
562D4—816no—
563D4—84no—
564D4—168no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—44no—
605C2×C4—816no—
606C2×C4—88no—
607C2×C4—168no—
608C2×C4#54612yes—
609C2×C4#54612yes—
610C2×C4#54712yes—
611C2×C4#54712yes—
612C2×C4C22×SD1612yes—
613C2×C4C22×SD1612yes—
614C2×C4C22×SD1612yes—
615C2×C4#57612yes—
616C2×C4#57712yes—
617C2×C4C22×#4111yescharacteristic
618C2×C4C22×#4111yescharacteristic
619C2×C4C2×H(5,𝔽2)11yescharacteristic
620C2×C4C2×H(5,𝔽2)11yescharacteristic
621C2×C4C2×H(5,𝔽2)11yescharacteristic
622C2×C4C2×2−1+411yescharacteristic
623C8—24no—
624C8—42no—
625C8—42no—
626C8—42no—
627C22—28no—
628C22—28no—
629C22—28no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22C2×#54611yescharacteristic
642C22C2×#54711yescharacteristic
643C22#348411yescharacteristic
644C22C23×SD1611yescharacteristic
645C22C2×#57611yescharacteristic
646C22C2×#57711yescharacteristic
647C22#358611yescharacteristic
648C4—22no—
649C4—22no—
650C4—22no—
651C4—44no—
652C4—42no—
653C4—44no—
654C4—41no—
655C4—42no—
656C4—81no—
657C4—81no—
658C4—82no—
659C4—82no—
660C4—84no—
661C4#329514yes—
662C4#333514yes—
663C4#348311yescharacteristic
664C4#351911yescharacteristic
665C4#352211yescharacteristic
666C4#354811yescharacteristic
667C2—48no—
668C2—162no—
669C2#5903911yescharacteristic
670C2C2×#329511yescharacteristic
671C2#6120211yescharacteristic
672C2C2×#333511yescharacteristic
673C2#6127811yescharacteristic
674C2#6156211yescharacteristic
675C2#6166811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#59815C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36559C2semidirect
3. #31322C2semidirect
4. #29206C2semidirect
5. #29139C2semidirect
6. #28223C2semidirect
7. #26232C2semidirect
8. #25543C2semidirect
9. #25387C2semidirect
10. #25012C2semidirect
11. #24969C2semidirect
12. #24120C2semidirect
13. #23173C2semidirect
14. #22703C2semidirect
15. #22485C2semidirect
16. C4×#475C2semidirect
17. C4×#460C2semidirect
262. #127SD16semidirect
361. C4⋊C8#41semidirect

Maximal subgroups

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

subgroupindex
#598152
#365592
#313222
#292062
#291392
#282232
#262322
#255432
#253872
#250122
#249692
#241202
#231732
#227032
#224852
C4×#4752
C4×#4602