Finite Group Databasebeta

← #6795702

Subgroups — #6795702

Order 512.

1541 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#36180C211yesmaximal, characteristic, complemented
2#34943C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic, complemented
4#31702C214yesmaximal, complemented
5#31104C214yesmaximal, complemented
6#29942C212yesmaximal, complemented
7#29290C211yesmaximal, characteristic, complemented
8#29208C211yesmaximal, characteristic, complemented
9#28835C212yesmaximal, complemented
10#27735C212yesmaximal, complemented
11#27644C212yesmaximal, complemented
12#26300C212yesmaximal, complemented
13#25010C212yesmaximal
14#24708C212yesmaximal, complemented
15#22799C211yesmaximal, characteristic, complemented
16#21987C212yesmaximal, complemented
17C4×#476C211yesmaximal, characteristic, complemented
18C2×#554—21no—
19C2×#480C2214yes—
20C2×#476C2211yescharacteristic
21C2×#476C2212yes—
22C2×#432C2212yes—
23#2868C2211yescharacteristic
24#2857C2212yes—
25#2572—22no—
26#2571C2211yescharacteristic
27#2567—21no—
28#2563C2211yescharacteristic
29#2563C2211yescharacteristic
30#2552—21no—
31#2552C2212yes—
32#2537C2211yescharacteristic
33#2502—22no—
34#2495—21no—
35#2495C2212yes—
36#2440—22no—
37#2440C2214yes—
38#2420—21no—
39#2420C2212yes—
40#2398—22no—
41D4×C4⋊C4—21no—
42D4×C4⋊C4C2212yes—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#121—21no—
46C4×#121C2212yes—
47C2×#384C2212yes—
48#2283—21no—
49#2283C2212yes—
50C42×D4C2211yescharacteristic
51C42×D4C2211yescharacteristic
52#2101C2214yes—
53#2087C2214yes—
54#2085C2214yes—
55#2083C2214yes—
56#2077C2214yes—
57#2077C2214yes—
58#2075C2214yes—
59#2071C2214yes—
60#2057C2214yes—
61#2033C2214yes—
62#1990C2212yes—
63#1987C2212yes—
64#1987C2211yescharacteristic
65#1987C2212yes—
66#1987C2212yes—
67#1954C2212yes—
68#1948C2211yescharacteristic
69#1948C2212yes—
70#1948C2212yes—
71#1947C2212yes—
72#1947C2212yes—
73#1947C2212yes—
74#1947C2212yes—
75#1943C2214yes—
76#1936C2214yes—
77#1924C2214yes—
78#1920C2212yes—
79#1864C2212yes—
80#1853C2214yes—
81#1852C2214yes—
82#1847C2214yes—
83#1845C2212yes—
84#1842C2211yescharacteristic
85#1836C2214yes—
86#1823C2214yes—
87#1802C2212yes—
88#1802C2211yescharacteristic
89#1802C2212yes—
90#1797C2212yes—
91C4×#106C2212yes—
92#1770C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95#554—48no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×#125—22no—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8—24no—
104C2×#121—21no—
105C2×#121C2312yes—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4D411yescharacteristic
111C2×C4×D4D411yescharacteristic
112C2×C4×D4C2311yescharacteristic
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C22×C4⋊C4—22no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4C2314yes—
119C22×C42—21no—
120C22×C42C2312yes—
121#480—48no—
122#476—24no—
123#476—44no—
124#432—28no—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×#107C2314yes—
128C2×#106C2312yes—
129C2×#106C2314yes—
130C2×#106C2312yes—
131C2×C4⋊C8C2311yescharacteristic
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2311yescharacteristic
136C2×#102C2314yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×C22⋊C8C2314yes—
140C2×C22⋊C8C2314yes—
141C2×#97C2314yes—
142C2×C4×C8C2312yes—
143#400—22no—
144#400—22no—
145#398—24no—
146#398—24no—
147#397—22no—
148#397—22no—
149#395—24no—
150#394—22no—
151#394—22no—
152#393—22no—
153#393—22no—
154#391—21no—
155#391—21no—
156#391—22no—
157#390—21no—
158#390C2312yes—
159#389—22no—
160#389—22no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—22no—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#389C2314yes—
169#388—22no—
170#388—22no—
171#386—22no—
172#386—22no—
173#385—24no—
174#385—24no—
175#384—22no—
176#384—21no—
177#384—21no—
178#384—22no—
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#384C2312yes—
182#384C2312yes—
183#384C2314yes—
184#383—22no—
185#383—22no—
186#382—21no—
187#382—21no—
188#382C2312yes—
189#382C2312yes—
190#381—22no—
191#381—22no—
192#380—22no—
193#380—21no—
194#380—21no—
195#380—21no—
196#380C2312yes—
197#380C2312yes—
198#380C2312yes—
199#380C2314yes—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C22⋊C4—22no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—21no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218#376—22no—
219C43C2311yescharacteristic
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#336C2314yes—
232#336C2312yes—
233C22×Q8—21no—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×D4—22no—
237C22×D4C2×D411yescharacteristic
238C22×D4C2411yescharacteristic
239C23×C4—24no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2412yes—
246C22×C8C2412yes—
247C22×C8C2414yes—
248C22×C8C2412yes—
249C4⋊Q8—22no—
250C4⋊Q8—22no—
251C4⋊Q8—22no—
252C4⋊Q8—22no—
253C4⋊Q8—42no—
254C4⋊Q8—42no—
255C4⋊Q8—44no—
256C4⋊Q8—82no—
257#125—48no—
258C22⋊Q8—416no—
259C22⋊Q8—48no—
260#121—24no—
261#121—42no—
262#121—44no—
263C4×Q8—44no—
264C4×Q8—82no—
265C4×D4—24no—
266C4×D4—22no—
267C4×D4—24no—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—41no—
275C4×D4—48no—
276C4×D4—84no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—48no—
298C2×C4⋊C4—84no—
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2414yes—
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4C2×D412yes—
321C2×C22⋊C4C2×D412yes—
322C2×C22⋊C4C2×D412yes—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—44no—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2411yescharacteristic
338C2×C42C2412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341#107—44no—
342#107—44no—
343#107—44no—
344#106—24no—
345#106—42no—
346#106—44no—
347C4⋊C8—24no—
348C4⋊C8—24no—
349C4⋊C8—22no—
350C4⋊C8—42no—
351C4⋊C8#4114yes—
352#102—28no—
353#102—24no—
354#102—24no—
355C22⋊C8—28no—
356C22⋊C8—44no—
357#97—28no—
358C4×C8—24no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95—24no—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379C24—21no—
380C24C22×D412yes—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—44no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—81no—
387C2×Q8—81no—
388C2×Q8—82no—
389C2×Q8—82no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—82no—
400C2×D4AGL(1,ℤ/8)14yes—
401C2×D4H(5,𝔽2)12yes—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—48no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—81no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C2511yescharacteristic
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8C2×#4114yes—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—42no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4—82no—
486C4⋊C4—81no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—42no—
494C22⋊C4—44no—
495C22⋊C4—48no—
496C22⋊C4—44no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C42—24no—
504C42—24no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—21no—
509C42—22no—
510C42—21no—
511C42—41no—
512C42—44no—
513C42—42no—
514C42—42no—
515C42—42no—
516C42—44no—
517C42#13714yes—
518C42C2×#4112yes—
519C422−1+412yes—
520C422−1+412yes—
521C422−1+412yes—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23C2×AGL(1,ℤ/8)14yes—
536C23C23×D411yescharacteristic
537Q8—84no—
538Q8—84no—
539Q8—164no—
540D4—28no—
541D4—22no—
542D4—22no—
543D4—41no—
544D4—41no—
545D4—88no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—168no—
598C2×C4#53612yes—
599C2×C4#53612yes—
600C2×C4#54712yes—
601C2×C4#54712yes—
602C2×C4C2×AGL(1,ℤ/8)12yes—
603C2×C4C2×#13712yes—
604C2×C4C2×#13712yes—
605C2×C4C2×#13712yes—
606C2×C4#57612yes—
607C2×C4C22×#4111yescharacteristic
608C2×C4C2×H(5,𝔽2)11yescharacteristic
609C2×C4C2×2−1+411yescharacteristic
610C2×C4C2×2−1+411yescharacteristic
611C2×C4C2×2−1+411yescharacteristic
612C2×C4#58511yescharacteristic
613C8—24no—
614C8—44no—
615C8—42no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22C2×#53611yescharacteristic
631C22C2×#54711yescharacteristic
632C22#348811yescharacteristic
633C22C22×AGL(1,ℤ/8)11yescharacteristic
634C22C22×#13711yescharacteristic
635C22C2×#57611yescharacteristic
636C22#358611yescharacteristic
637C4—22no—
638C4—22no—
639C4—22no—
640C4—42no—
641C4—41no—
642C4—42no—
643C4—42no—
644C4—81no—
645C4—82no—
646C4—84no—
647C4—82no—
648C4—81no—
649C4—82no—
650C4—81no—
651C4—82no—
652C4#323814yes—
653C4#335714yes—
654C4#351511yescharacteristic
655C4#352211yescharacteristic
656C4#354411yescharacteristic
657C4#355511yescharacteristic
658C2—44no—
659C2—82no—
660C2#3892111yescharacteristic
661C2C2×#323811yescharacteristic
662C2#6088811yescharacteristic
663C2C2×#335711yescharacteristic
664C2#6125211yescharacteristic
665C2#6196011yescharacteristic
666C2#6197911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36180C2semidirect
3. #33197C2semidirect
4. #31702C2semidirect
5. #31104C2semidirect
6. #29942C2semidirect
7. #29290C2semidirect
8. #29208C2semidirect
9. #28835C2semidirect
10. #27735C2semidirect
11. #27644C2semidirect
12. #26300C2semidirect
14. #24708C2semidirect
15. #22799C2semidirect
16. #21987C2semidirect
17. C4×#476C2semidirect

Maximal subgroups

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

subgroupindex
#361802
#349432
#331972
#317022
#311042
#299422
#292902
#292082
#288352
#277352
#276442
#263002
#250102
#247082
#227992
#219872
C4×#4762