Finite Group Databasebeta

← #6541989

Subgroups — #6541989

Order 512.

1759 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic, complemented
2#34313C211yesmaximal, characteristic
3#29987C212yesmaximal, complemented
4#29980C211yesmaximal, characteristic, complemented
5#29258C211yesmaximal, characteristic, complemented
6#29254C212yesmaximal, complemented
7#28516C212yesmaximal, complemented
8#28508C211yesmaximal, characteristic, complemented
9#26475C214yesmaximal, complemented
10#26463C212yesmaximal, complemented
11#26052C214yesmaximal, complemented
12#26041C212yesmaximal, complemented
13#23781C212yesmaximal, complemented
14#23775C211yesmaximal, characteristic, complemented
15#22736C211yesmaximal, characteristic, complemented
16#22728C212yesmaximal, complemented
17#22545C212yesmaximal
18C2×#550C2211yescharacteristic
19C2×#520—21no—
20C2×#478C2214yes—
21C2×#478C2214yes—
22C2×#474C2214yes—
23C2×#446C2212yes—
24#2868C2211yescharacteristic
25#2859C2212yes—
26#2582—21no—
27#2567—21no—
28#2552C2212yes—
29#2550C2214yes—
30#2507—22no—
31#2451—22no—
32#2369—21no—
33#2366C2211yescharacteristic
34#2366C2212yes—
35#2361—21no—
36#2361—22no—
37#2355C2212yes—
38Q8×C4⋊C4—21no—
39Q8×C4⋊C4C2211yescharacteristic
40D4×C4⋊C4C2211yescharacteristic
41D4×C4⋊C4C2212yes—
42#2341—21no—
43#2324—21no—
44#2324—22no—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127C2212yes—
48C4×#127C2211yescharacteristic
49#2293C2211yescharacteristic
50#2293C2212yes—
51C42×Q8C2211yescharacteristic
52#2100C2211yescharacteristic
53#2099C2212yes—
54#2094C2212yes—
55#2094C2211yescharacteristic
56#2093C2212yes—
57#2093C2212yes—
58#2093C2212yes—
59#2088C2212yes—
60#2087C2214yes—
61#2065C2214yes—
62#2065C2212yes—
63#2059C2214yes—
64#2058C2211yescharacteristic
65#2057C2212yes—
66#2025C2214yes—
67#2023C2212yes—
68#2023C2211yescharacteristic
69#2023C2212yes—
70#1982C2214yes—
71#1982C2212yes—
72#1961C2214yes—
73#1957C2212yes—
74#1949C2214yes—
75#1948C2212yes—
76#1947C2212yes—
77#1947C2214yes—
78#1940C2214yes—
79#1939C2212yes—
80#1864C2214yes—
81#1862C2214yes—
82#1862C2214yes—
83#1847C2214yes—
84#1845C2212yes—
85#1831C2211yescharacteristic
86#1831C2212yes—
87#1804C2214yes—
88#1802C2212yes—
89#1776C2214yes—
90C4×#106C2212yes—
91#1767C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93#1762C2214yes—
94C4×#102C2212yes—
95C22×#41—21no—
96#550—24no—
97#550Q818yescomplemented
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#127C2311yescharacteristic
103C2×#127C2312yes—
104C2×#125C2314yes—
105C2×#121—21no—
106C2×#121—22no—
107#520—48no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8—22no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×D4—21no—
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4—21no—
118C2×C4×D4—22no—
119C2×C4×D4—21no—
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2312yes—
122C2×C4×D4C2312yes—
123C2×C4×D4C2312yes—
124C2×C4×D4C2312yes—
125C2×#117—22no—
126C2×#117—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—22no—
129#478—48no—
130#478—48no—
131#474—48no—
132#446—28no—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2312yes—
137C2×#106C2312yes—
138C2×#106C2314yes—
139C2×C4⋊C8C2314yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2314yes—
142C2×C4⋊C8C2312yes—
143C2×#102C2312yes—
144C2×#102C2314yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#97C2314yes—
151C2×C4×C8C2312yes—
152#398—24no—
153#398—24no—
154#395—24no—
155#391—22no—
156#391—22no—
157#391—21no—
158#391—24no—
159#390—21no—
160#390—22no—
161#390—22no—
162#390—21no—
163#390—21no—
164#390—22no—
165#390C2311yescharacteristic
166#390C2312yes—
167#390C2311yescharacteristic
168#390C2312yes—
169#389—21no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—24no—
179#389C2314yes—
180#389C2312yes—
181#389C2311yescharacteristic
182#389C2311yescharacteristic
183#389C2312yes—
184#388—22no—
185#388—21no—
186#388—22no—
187#388—21no—
188#387—24no—
189#386—22no—
190#386—22no—
191#386—22no—
192#385—22no—
193#385—22no—
194#385—22no—
195#384—22no—
196#384—21no—
197#384—21no—
198#384C2311yescharacteristic
199#384C2312yes—
200#384C2311yescharacteristic
201#384C2312yes—
202#384C2312yes—
203#382C2314yes—
204#380—21no—
205#380—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D412yes—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D412yes—
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C43C2312yes—
222C43C2311yescharacteristic
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2314yes—
233#336C2311yescharacteristic
234#336C2312yes—
235C2×#41—28no—
236C2×#41—44no—
237C2×#41—44no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8C2×D411yescharacteristic
241C22×D4—22no—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4—21no—
245C22×D4C2411yescharacteristic
246C22×D4C2412yes—
247C23×C4—22no—
248C23×C4—21no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—22no—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C4⋊Q8—24no—
256C4⋊Q8—44no—
257C4⋊Q8—44no—
258C4⋊Q8—82no—
259#127—22no—
260#127—24no—
261#127C2×Q814yes—
262#127C2×Q818yes—
263#125—44no—
264#125—44no—
265#121—216no—
266#121—28no—
267C4×Q8—24no—
268C4×Q8—42no—
269C4×Q8—44no—
270C4×Q8—44no—
271C4×Q8—84no—
272C4×Q8C2×Q814yes—
273C4×Q8#4114yes—
274C4×D4—22no—
275C4×D4—24no—
276C4×D4—48no—
277C4×D4—44no—
278C4×D4—44no—
279C4×D4—44no—
280C4×D4—48no—
281C4×D4—44no—
282C4×D4—44no—
283C4×D4—84no—
284C4×D4—82no—
285C4×D4C2×Q814yes—
286C4×D4C2×Q818yes—
287#117—48no—
288#117—44no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—48no—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C42—22no—
337C2×C42—21no—
338C2×C42—22no—
339C2×C42—24no—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353#107—28no—
354#107—44no—
355#107—44no—
356#106—24no—
357#106—24no—
358#106—44no—
359C4⋊C8—24no—
360C4⋊C8—28no—
361C4⋊C8—24no—
362C4⋊C8—28no—
363#102—28no—
364#102—24no—
365#102—24no—
366#102—28no—
367#102—44no—
368#102—44no—
369#102—44no—
370#97—28no—
371C4×C8—24no—
372#95—21no—
373#95—22no—
374#95—21no—
375#95—24no—
376#95—24no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C24—22no—
387C24—21no—
388#41—432no—
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—42no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—82no—
396C2×Q8—82no—
397C2×Q8C22⋊Q814yes—
398C2×D4—28no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—28no—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—24no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—82no—
413C2×D4—81no—
414C2×D4—82no—
415C2×D4—81no—
416C2×D4C22×Q812yes—
417C2×D4C22×Q814yes—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C2511yescharacteristic
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8H(5,𝔽2)14yes—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—82no—
486C4⋊C4—81no—
487C4⋊C4—84no—
488C4⋊C4—84no—
489C4⋊C4—84no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4C22⋊Q818yes—
493C4⋊C4C22⋊Q814yes—
494C4⋊C4C22×Q812yes—
495C4⋊C42−1+414yes—
496C22⋊C4—24no—
497C22⋊C4—28no—
498C22⋊C4—44no—
499C22⋊C4—42no—
500C22⋊C4—44no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—44no—
504C22⋊C4—82no—
505C22⋊C4—81no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C42—24no—
510C42—22no—
511C42—24no—
512C42—22no—
513C42—24no—
514C42—22no—
515C42—22no—
516C42—42no—
517C42—41no—
518C42—42no—
519C42—44no—
520C42—42no—
521C42—44no—
522C42—84no—
523C42AGL(1,ℤ/8)14yes—
524C42C22×Q814yes—
525C42C22×Q812yes—
526C42C2×#4112yes—
527C42C2×#4114yes—
528C23—24no—
529C23—22no—
530C23—42no—
531C23—41no—
532C23—42no—
533C23—41no—
534C23—42no—
535C23—41no—
536C23—42no—
537C23—41no—
538C23—42no—
539C23—41no—
540C23—42no—
541C23—41no—
542C23C23×D411yescharacteristic
543Q8—28no—
544Q8—48no—
545Q8—164no—
546D4—28no—
547D4—24no—
548D4—416no—
549D4—48no—
550D4—44no—
551D4—42no—
552D4—88no—
553D4—84no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—416no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—48no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—164no—
599C2×C4—164no—
600C2×C4—164no—
601C2×C4C2×C22⋊Q812yes—
602C2×C4C2×C22⋊Q814yes—
603C2×C4C2×C22⋊Q814yes—
604C2×C4C2×C22⋊Q812yes—
605C2×C4C2×AGL(1,ℤ/8)12yes—
606C2×C4C2×AGL(1,ℤ/8)12yes—
607C2×C4C2×AGL(1,ℤ/8)12yes—
608C2×C4#57512yes—
609C2×C4#57612yes—
610C2×C4C23×Q811yescharacteristic
611C2×C4C22×#4111yescharacteristic
612C2×C4C2×H(5,𝔽2)12yes—
613C2×C4C2×2−1+412yes—
614C8—24no—
615C8—42no—
616C8—44no—
617C22—44no—
618C22—42no—
619C22—44no—
620C22—42no—
621C22—44no—
622C22—42no—
623C22—82no—
624C22—81no—
625C22—82no—
626C22—81no—
627C22—82no—
628C22—81no—
629C22—82no—
630C22—81no—
631C22C22×C22⋊Q811yescharacteristic
632C22#348611yescharacteristic
633C22#349811yescharacteristic
634C22C22×AGL(1,ℤ/8)11yescharacteristic
635C22C2×#57511yescharacteristic
636C22C2×#57611yescharacteristic
637C22#358611yescharacteristic
638C4—24no—
639C4—24no—
640C4—22no—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—42no—
645C4—44no—
646C4—42no—
647C4—41no—
648C4—42no—
649C4—81no—
650C4—82no—
651C4—81no—
652C4—84no—
653C4—82no—
654C4#307714yes—
655C4#350312yes—
656C4#352212yes—
657C2—84no—
658C2—82no—
659C2#3574811yescharacteristic
660C2C2×#307711yescharacteristic
661C2#6049411yescharacteristic
662C2#6182811yescharacteristic
663C2#6183511yescharacteristic
664C2#6242311yescharacteristic
665C2#6245211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#550C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36563C2semidirect
3. #29987C2semidirect
4. #29980C2semidirect
5. #29258C2semidirect
6. #29254C2semidirect
7. #28516C2semidirect
8. #28508C2semidirect
9. #26475C2semidirect
10. #26463C2semidirect
11. #26052C2semidirect
12. #26041C2semidirect
13. #23781C2semidirect
14. #23775C2semidirect
15. #22736C2semidirect
16. #22728C2semidirect
97. #550Q8semidirect

Maximal subgroups

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

subgroupindex
#365632
#343132
#299872
#299802
#292582
#292542
#285162
#285082
#264752
#264632
#260522
#260412
#237812
#237752
#227362
#227282
#225452