Finite Group Databasebeta

← #6571300

Subgroups — #6571300

Order 512.

1624 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36504C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic, complemented
3#29928C212yesmaximal, complemented
4#29923C211yesmaximal, characteristic, complemented
5#27976C212yesmaximal, complemented
6#27972C211yesmaximal, characteristic, complemented
7#27652C212yesmaximal, complemented
8#27644C211yesmaximal, characteristic, complemented
9#26241C214yesmaximal, complemented
10#26233C212yesmaximal, complemented
11#24976C214yesmaximal, complemented
12#24968C212yesmaximal, complemented
13#23757C212yesmaximal, complemented
14#23755C211yesmaximal, characteristic, complemented
15#22739C212yesmaximal, complemented
16#22541C212yesmaximal, complemented
17#22530C211yesmaximal, characteristic, complemented
18C2×#556C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#485C2214yes—
21C2×#480C2212yes—
22C2×C8×Q8C2212yes—
23#2857C2211yescharacteristic
24#2850C2212yes—
25#2542C2211yescharacteristic
26#2540C2212yes—
27#2537C2211yescharacteristic
28#2536C2212yes—
29#2479—22no—
30#2467—22no—
31#2423—21no—
32#2407—21no—
33#2363—21no—
34#2355—21no—
35#2355C2212yes—
36#2355C2211yescharacteristic
37#2354—21no—
38#2354—21no—
39#2354C2212yes—
40#2352—22no—
41#2352C2214yes—
42#2314—22no—
43#2311—21no—
44C4×C22⋊Q8C2212yes—
45C4×#121C2211yescharacteristic
46#2285C2211yescharacteristic
47#2285C2212yes—
48#2285C2212yes—
49C2×#384C2211yescharacteristic
50#2274—21no—
51#2100C2211yescharacteristic
52#2095C2212yes—
53#2094C2211yescharacteristic
54#2084C2211yescharacteristic
55#2083C2212yes—
56#2077C2212yes—
57#2076C2214yes—
58#2072C2212yes—
59#2072C2212yes—
60#2072C2212yes—
61#2071C2211yescharacteristic
62#2071C2212yes—
63#2036C2214yes—
64#2036C2214yes—
65#2035C2214yes—
66#2034C2212yes—
67#2033C2214yes—
68#1983C2214yes—
69#1981C2212yes—
70#1958C2214yes—
71#1956C2212yes—
72#1954C2212yes—
73#1954C2214yes—
74#1949C2214yes—
75#1949C2214yes—
76#1947C2212yes—
77#1947C2212yes—
78#1875C2214yes—
79#1873C2214yes—
80#1872C2214yes—
81#1835C2214yes—
82#1834C2212yes—
83#1825C2214yes—
84#1823C2212yes—
85#1805C2214yes—
86#1802C2212yes—
87#1797C2212yes—
88#1797C2211yescharacteristic
89C4×#107C2214yes—
90C4×#106C2212yes—
91C4×#103C2214yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C22×#41D411yescharacteristic
96#556—28no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×#126—22no—
99C2×#124C2312yes—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8C2312yes—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2312yes—
104C2×#121—21no—
105C2×#121C2311yescharacteristic
106C2×#121C2312yes—
107C4×#41—28no—
108C2×C4×Q8D411yescharacteristic
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×D4—21no—
112C2×C4×D4D412yes—
113C2×C4×D4D411yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2312yes—
116C2×C4×D4C2311yescharacteristic
117C2×#117—21no—
118C2×#117—22no—
119C2×#117—21no—
120C2×#117D412yes—
121C2×#117D411yescharacteristic
122C2×#117C2314yes—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128#485—48no—
129#480—44no—
130C8×Q8—28no—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#106C2312yes—
134C2×#106C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#102C2314yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2314yes—
147C2×C22⋊C8C2312yes—
148C2×C4×C8C2312yes—
149C2×C4×C8C2314yes—
150#399—22no—
151#399—22no—
152#398—22no—
153#394—22no—
154#394—22no—
155#392—22no—
156#391—22no—
157#391—21no—
158#391C2312yes—
159#390—22no—
160#390—21no—
161#390—22no—
162#390—22no—
163#390C2311yescharacteristic
164#389—21no—
165#389—21no—
166#389C2311yescharacteristic
167#388—24no—
168#388—24no—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—21no—
174#387—24no—
175#387—24no—
176#387—22no—
177#387—22no—
178#386—24no—
179#386—22no—
180#386—22no—
181#386—21no—
182#385—22no—
183#384—21no—
184#384—21no—
185#384—21no—
186#384C2312yes—
187#384C2312yes—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#382—21no—
192#382—22no—
193#382C2314yes—
194#382C2312yes—
195#380—22no—
196#380—22no—
197#380—21no—
198#380—21no—
199#380—21no—
200#380—22no—
201#380—21no—
202#380C2312yes—
203#380C2314yes—
204#380C2311yescharacteristic
205#380C2312yes—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—24no—
220C4×C22⋊C4—21no—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233C2×#41—24no—
234C2×#41—24no—
235C2×#41—42no—
236C22×Q8C2×D411yescharacteristic
237C22×Q8C2412yes—
238C22×D4—21no—
239C22×D4—22no—
240C22×D4C2×D411yescharacteristic
241C22×D4C2×D412yes—
242C22×D4C2411yescharacteristic
243C23×C4—22no—
244C23×C4—21no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D412yes—
247C23×C4C2×D412yes—
248C23×C4C2×D411yescharacteristic
249C23×C4C2411yescharacteristic
250C23×C4C2412yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C4⋊Q8—24no—
255#126—48no—
256#124—28no—
257C22⋊Q8—24no—
258C22⋊Q8—42no—
259C22⋊Q8—44no—
260C22⋊Q8—44no—
261C22⋊Q8—44no—
262#121—22no—
263#121—41no—
264#121—44no—
265#121—44no—
266C4×Q8—24no—
267C4×Q8—28no—
268C4×Q8#4118yes—
269C4×D4—24no—
270C4×D4—28no—
271C4×D4—24no—
272C4×D4—28no—
273C4×D4—24no—
274C4×D4—44no—
275#117—24no—
276#117—28no—
277#117—44no—
278#117—44no—
279#117—44no—
280#117—44no—
281#117—84no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
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—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2414yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—24no—
334C2×C42—28no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—21no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2412yes—
348#107—28no—
349#107—44no—
350#106—24no—
351#106—42no—
352C4⋊C8—24no—
353C4⋊C8—28no—
354#103—44no—
355#103—44no—
356#103—44no—
357#103—44no—
358#102—42no—
359#102—42no—
360#102—44no—
361C22⋊C8—42no—
362C22⋊C8—44no—
363C22⋊C8—42no—
364C22⋊C8—44no—
365C4×C8—24no—
366C4×C8—28no—
367#95—21no—
368#95—24no—
369#95—24no—
370#95—22no—
371#95—21no—
372#95—22no—
373#95—24no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—22no—
382C24—21no—
383C24C22×D412yes—
384C24C22×D411yescharacteristic
385#41—416no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—24no—
392C2×Q8#12314yes—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—22no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—44no—
406C2×D4—81no—
407C2×D4—82no—
408C2×D4—82no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—41no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—82no—
441C22×C4—82no—
442C22×C4#13514yes—
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C2511yescharacteristic
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—81no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4#12314yes—
492C4⋊C4#12318yes—
493C4⋊C4H(5,𝔽2)12yes—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—42no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—42no—
502C22⋊C4—44no—
503C22⋊C4—42no—
504C22⋊C4—44no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—82no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—82no—
511C22⋊C4—82no—
512C42—24no—
513C42—22no—
514C42—24no—
515C42—22no—
516C42—22no—
517C42—44no—
518C42—42no—
519C42—44no—
520C42—82no—
521C42—84no—
522C42—84no—
523C42C2×#4114yes—
524C42C2×#4118yes—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23C2×#13512yes—
546C23C23×D411yescharacteristic
547Q8—28no—
548Q8—44no—
549Q8—44no—
550D4—48no—
551D4—44no—
552D4—42no—
553D4—42no—
554D4—84no—
555D4—164no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—28no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—44no—
575C2×C4—48no—
576C2×C4—48no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—84no—
605C2×C4—164no—
606C2×C4C2×#12314yes—
607C2×C4C2×#12312yes—
608C2×C4C2×#12314yes—
609C2×C4C2×#12312yes—
610C2×C4C2×#13512yes—
611C2×C4C2×#13512yes—
612C2×C4#57512yes—
613C2×C4#57812yes—
614C2×C4C22×#4112yes—
615C2×C4C2×H(5,𝔽2)11yescharacteristic
616C2×C4#58512yes—
617C2×C4#58511yescharacteristic
618C8—44no—
619C8—42no—
620C8—42no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22C22×#12311yescharacteristic
643C22#348811yescharacteristic
644C22#350011yescharacteristic
645C22C22×#13511yescharacteristic
646C22C2×#57511yescharacteristic
647C22C2×#57811yescharacteristic
648C22#358511yescharacteristic
649C4—22no—
650C4—24no—
651C4—22no—
652C4—24no—
653C4—22no—
654C4—42no—
655C4—41no—
656C4—42no—
657C4—82no—
658C4—81no—
659C4—81no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—84no—
664C4#309114yes—
665C4#351712yes—
666C4#352412yes—
667C2—42no—
668C2—82no—
669C2—161no—
670C2#3607611yescharacteristic
671C2C2×#309111yescharacteristic
672C2#6052611yescharacteristic
673C2#6194911yescharacteristic
674C2#6199811yescharacteristic
675C2#6252811yescharacteristic
676C2#6259211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36504C2
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
2. #34188C2semidirect
3. #29928C2semidirect
4. #29923C2semidirect
5. #27976C2semidirect
6. #27972C2semidirect
7. #27652C2semidirect
8. #27644C2semidirect
9. #26241C2semidirect
10. #26233C2semidirect
11. #24976C2semidirect
12. #24968C2semidirect
13. #23757C2semidirect
14. #23755C2semidirect
15. #22739C2semidirect
16. #22541C2semidirect
17. #22530C2semidirect

Maximal subgroups

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

subgroupindex
#365042
#341882
#299282
#299232
#279762
#279722
#276522
#276442
#262412
#262332
#249762
#249682
#237572
#237552
#227392
#225412
#225302