Finite Group Databasebeta

← #6616254

Subgroups — #6616254

Order 512.

1705 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36116C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic, complemented
3#33109C211yesmaximal, characteristic, complemented
4#33098C212yesmaximal, complemented
5#27020C214yesmaximal, complemented
6#26708C214yesmaximal, complemented
7#26445C212yesmaximal, complemented
8#25902C211yesmaximal, characteristic, complemented
9#25273C212yesmaximal, complemented
10#24606C212yesmaximal, complemented
11#24299C211yesmaximal, characteristic
12#24067C212yesmaximal
13#23221C212yesmaximal, complemented
14#22969C212yesmaximal, complemented
15#22603C211yesmaximal, characteristic, complemented
16C4×#481C211yesmaximal, characteristic
17#21391C212yesmaximal
18C22×C4⋊Q8C2211yescharacteristic
19C2×#529C2212yes—
20C2×#487C2212yes—
21C2×#484C2212yes—
22C2×#481C2211yescharacteristic
23C2×#476C2212yes—
24C2×#432C2212yes—
25#2857C2211yescharacteristic
26#2855C2212yes—
27#2552C2211yescharacteristic
28#2547C2211yescharacteristic
29#2496—21no—
30#2486—21no—
31#2478—22no—
32#2453C2212yes—
33#2447C2212yes—
34#2419—22no—
35#2369—21no—
36#2369—22no—
37#2369—21no—
38#2369C2212yes—
39#2369C2214yes—
40#2369C2212yes—
41#2365C2211yescharacteristic
42D4×C4⋊C4—21no—
43D4×C4⋊C4C2211yescharacteristic
44#2344—21no—
45C4×#121C2211yescharacteristic
46C2×#391—22no—
47C2×#389—21no—
48C2×#389C2212yes—
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2086C2214yes—
54#2066C2214yes—
55#2059C2214yes—
56#2058C2212yes—
57#2042C2214yes—
58#2024C2214yes—
59#2023C2212yes—
60#1990C2212yes—
61#1989C2214yes—
62#1987C2212yes—
63#1983C2214yes—
64#1981C2212yes—
65#1947C2212yes—
66#1944C2214yes—
67#1939C2212yes—
68#1935C2214yes—
69#1928C2214yes—
70#1927C2214yes—
71#1927C2214yes—
72#1926C2212yes—
73#1920C2214yes—
74#1920C2214yes—
75#1919C2212yes—
76#1917C2214yes—
77#1874C2212yes—
78#1874C2211yescharacteristic
79#1872C2212yes—
80#1872C2212yes—
81#1872C2212yes—
82#1865C2214yes—
83#1862C2214yes—
84#1853C2212yes—
85#1853C2211yescharacteristic
86#1824C2211yescharacteristic
87#1817C2212yes—
88#1797C2211yescharacteristic
89C4×#106C2212yes—
90#1770C2212yes—
91#1763C2214yes—
92#1762C2214yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102#529—28no—
103#529—44no—
104C2×#126—22no—
105C2×#125C2312yes—
106C2×#124C2312yes—
107C2×#123—22no—
108C2×C22⋊Q8C2312yes—
109C2×#121C2311yescharacteristic
110C2×C4×Q8C2312yes—
111C2×C4×D4—22no—
112C2×C4×D4—21no—
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2312yes—
115C2×#117—22no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C42—21no—
125C22×C42—22no—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128#487—28no—
129#484—44no—
130#481—24no—
131#476—28no—
132#432—44no—
133C2×#107C2314yes—
134C2×#106C2314yes—
135C2×#106C2312yes—
136C2×#106C2312yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#102C2312yes—
144C2×#102C2314yes—
145C2×#102C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2311yescharacteristic
148C2×C22⋊C8C2311yescharacteristic
149C2×#97C2314yes—
150C2×C4×C8C2312yes—
151#400—22no—
152#398—22no—
153#398—22no—
154#397—22no—
155#397—22no—
156#397—22no—
157#391—24no—
158#391—24no—
159#391—24no—
160#391—24no—
161#391C2312yes—
162#391C2312yes—
163#391C2312yes—
164#390C2311yescharacteristic
165#389—21no—
166#389—22no—
167#389—21no—
168#389—22no—
169#389—22no—
170#389—24no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389C2311yescharacteristic
176#389C2314yes—
177#389C2312yes—
178#389C2312yes—
179#389C2314yes—
180#388—21no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#386—21no—
186#385—22no—
187#385—22no—
188#385—22no—
189#384—21no—
190#384C2311yescharacteristic
191#381—22no—
192#381—22no—
193#380—21no—
194#380—21no—
195#380—21no—
196#380—21no—
197#380—21no—
198#380C2311yescharacteristic
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—24no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D412yes—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—21no—
219C2×#95—22no—
220C2×#95—22no—
221C2×#95—21no—
222C2×#95D412yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#336C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—28no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—82no—
240C22×Q8—82no—
241C22×Q8C2412yes—
242C22×D4—21no—
243C22×D4C2411yescharacteristic
244C23×C4—22no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4C2×D412yes—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D412yes—
253C23×C4C2411yescharacteristic
254C22×C8C2412yes—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C4⋊Q8—28no—
258C4⋊Q8—28no—
259C4⋊Q8—28no—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8#4118yes—
263#126—48no—
264#125—24no—
265#125—42no—
266#124—24no—
267#124—42no—
268#123—48no—
269C22⋊Q8—24no—
270C22⋊Q8—24no—
271#121—22no—
272#121#4114yes—
273C4×Q8—24no—
274C4×Q8—42no—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—42no—
278C4×D4—48no—
279C4×D4—44no—
280#117—48no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—84no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—21no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—44no—
340C2×C42—84no—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2412yes—
346C2×C42C2412yes—
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349#107—44no—
350#106—28no—
351#106—24no—
352#106—24no—
353C4⋊C8—24no—
354C4⋊C8—44no—
355#103—28no—
356#103—28no—
357#103—28no—
358#103—44no—
359#102—24no—
360#102—28no—
361#102—24no—
362C22⋊C8—22no—
363C22⋊C8—22no—
364C22⋊C8—24no—
365#97—44no—
366C4×C8—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—22no—
372#95—22no—
373#95—21no—
374#95—21no—
375#95—22no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388C24—21no—
389C24C22×D411yescharacteristic
390C2×Q8—28no—
391C2×Q8—28no—
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—24no—
396C2×Q8—82no—
397C2×Q8—82no—
398C2×Q8—82no—
399C2×Q8—82no—
400C2×Q8—816no—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—21no—
404C2×D4—22no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4C2×#4112yes—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—41no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—42no—
434C22×C4—44no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—82no—
444C22×C4#13714yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C2×#4112yes—
455C22×C4H(5,𝔽2)12yes—
456C22×C4C2511yescharacteristic
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—22no—
466C4⋊C4—24no—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—28no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—81no—
496C4⋊C4—82no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4C2×#4118yes—
500C4⋊C4C2×#4112yes—
501C4⋊C42−1+414yes—
502C22⋊C4—24no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—81no—
511C42—22no—
512C42—22no—
513C42—24no—
514C42—24no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—44no—
521C42—44no—
522C42—42no—
523C42—44no—
524C42—84no—
525C42—84no—
526C42C2×#4114yes—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23#54612yes—
539C23C2×#13712yes—
540C23#57712yes—
541C23C23×D411yescharacteristic
542Q8—44no—
543Q8—44no—
544Q8—88no—
545Q8—88no—
546D4—24no—
547D4—48no—
548D4—42no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—28no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—44no—
577C2×C4—48no—
578C2×C4—416no—
579C2×C4—44no—
580C2×C4—48no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—84no—
602C2×C4—88no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4#52512yes—
606C2×C4#52512yes—
607C2×C4#54612yes—
608C2×C4C2×#13712yes—
609C2×C4C2×#13712yes—
610C2×C4#57512yes—
611C2×C4C22×#4111yescharacteristic
612C2×C4C22×#4112yes—
613C2×C4C2×H(5,𝔽2)11yescharacteristic
614C2×C4C2×2−1+412yes—
615C8—42no—
616C8—42no—
617C8—44no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22#333114yes—
632C22C2×#52511yescharacteristic
633C22C2×#54611yescharacteristic
634C22#348611yescharacteristic
635C22C22×#13711yescharacteristic
636C22C2×#57511yescharacteristic
637C22C2×#57711yescharacteristic
638C22#358511yescharacteristic
639C4—22no—
640C4—22no—
641C4—22no—
642C4—44no—
643C4—41no—
644C4—48no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—82no—
651C4—81no—
652C4—81no—
653C4—82no—
654C4—82no—
655C4#311714yes—
656C4#348212yes—
657C4#352212yes—
658C2—24no—
659C2—82no—
660C2#3642211yescharacteristic
661C2C2×#311711yescharacteristic
662C2#6057911yescharacteristic
663C2C2×#333111yescharacteristic
664C2#6116611yescharacteristic
665C2#6183011yescharacteristic
666C2#6184111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2369C2semidirect
3. #33109C2semidirect
4. #33098C2semidirect
5. #27020C2semidirect
6. #26708C2semidirect
7. #26445C2semidirect
8. #25902C2semidirect
9. #25273C2semidirect
10. #24606C2semidirect
13. #23221C2semidirect
14. #22969C2semidirect
15. #22603C2semidirect

Maximal subgroups

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

subgroupindex
#361162
C2×#23692
#331092
#330982
#270202
#267082
#264452
#259022
#252732
#246062
#242992
#240672
#232212
#229692
#226032
C4×#4812
#213912