Finite Group Databasebeta

← #6795704

Subgroups — #6795704

Order 512.

1501 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2#34932C211yesmaximal, characteristic
3#33195C211yesmaximal, characteristic, complemented
4#31711C214yesmaximal, complemented
5#31100C214yesmaximal, complemented
6#29939C212yesmaximal, complemented
7#29287C211yesmaximal, characteristic, complemented
8#29208C211yesmaximal, characteristic, complemented
9#28835C212yesmaximal, complemented
10#27736C212yesmaximal, complemented
11#27646C212yesmaximal, complemented
12#26301C212yesmaximal, complemented
13#25011C212yesmaximal
14#24708C212yesmaximal, complemented
15#22799C211yesmaximal, characteristic, complemented
16#21988C212yesmaximal, complemented
17C4×#476C211yesmaximal, characteristic, complemented
18C2×#529—21no—
19C2×#482C2214yes—
20C2×#476C2211yescharacteristic
21C2×#474C2212yes—
22C2×#432C2212yes—
23#2868C2211yescharacteristic
24#2857C2212yes—
25#2571C2212yes—
26#2563—21no—
27#2553—22no—
28#2552C2211yescharacteristic
29#2547C2211yescharacteristic
30#2542C2211yescharacteristic
31#2538—21no—
32#2537C2211yescharacteristic
33#2497—22no—
34#2496C2212yes—
35#2492—22no—
36#2489—21no—
37#2439C2214yes—
38#2427—21no—
39#2422C2212yes—
40#2395—22no—
41#2369—21no—
42#2355—21no—
43#2354C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C22⋊Q8—21no—
47C4×#121C2212yes—
48#2285C2212yes—
49#2285C2212yes—
50C42×D4C2211yescharacteristic
51C42×D4C2211yescharacteristic
52#2100C2214yes—
53#2088C2214yes—
54#2085C2214yes—
55#2083C2214yes—
56#2077C2214yes—
57#2076C2214yes—
58#2076C2214yes—
59#2073C2214yes—
60#2057C2214yes—
61#2033C2214yes—
62#1990C2212yes—
63#1988C2212yes—
64#1988C2212yes—
65#1987C2212yes—
66#1987C2211yescharacteristic
67#1954C2212yes—
68#1948C2212yes—
69#1948C2212yes—
70#1948C2211yescharacteristic
71#1947C2212yes—
72#1947C2212yes—
73#1947C2212yes—
74#1947C2212yes—
75#1943C2214yes—
76#1936C2214yes—
77#1924C2214yes—
78#1920C2212yes—
79#1864C2212yes—
80#1854C2214yes—
81#1854C2214yes—
82#1847C2214yes—
83#1845C2212yes—
84#1842C2211yescharacteristic
85#1836C2214yes—
86#1825C2214yes—
87#1802C2212yes—
88#1802C2211yescharacteristic
89#1802C2212yes—
90#1797C2212yes—
91C4×#106C2212yes—
92#1770C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100#529—48no—
101C2×#126—22no—
102C2×#125—21no—
103C2×#124—21no—
104C2×#123—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—21no—
108C2×#121—21no—
109C2×#121C2312yes—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×D4—22no—
113C2×C4×D4D411yescharacteristic
114C2×C4×D4D411yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×#117—22no—
120C2×#117C2314yes—
121C22×C42—21no—
122C22×C42C2312yes—
123#482—48no—
124#476—24no—
125#474—44no—
126#432—28no—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#106C2312yes—
131C2×#106C2314yes—
132C2×#106C2312yes—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2311yescharacteristic
138C2×#102C2314yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2314yes—
143C2×#97C2314yes—
144C2×C4×C8C2312yes—
145#400—24no—
146#400—22no—
147#399—22no—
148#398—22no—
149#397—22no—
150#397—22no—
151#397—22no—
152#396—24no—
153#394—22no—
154#394—22no—
155#393—24no—
156#391—21no—
157#391—21no—
158#391—21no—
159#391—22no—
160#391—24no—
161#390—22no—
162#390C2312yes—
163#389—22no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—22no—
168#389C2312yes—
169#389C2312yes—
170#389C2314yes—
171#389C2312yes—
172#388—24no—
173#388—22no—
174#387—24no—
175#387—22no—
176#386—22no—
177#385—22no—
178#384—21no—
179#384—21no—
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#384C2312yes—
183#384C2312yes—
184#384C2312yes—
185#384C2312yes—
186#382—22no—
187#382—22no—
188#382C2314yes—
189#380—22no—
190#380—22no—
191#380—22no—
192#380—22no—
193#380—21no—
194#380C2314yes—
195#380C2312yes—
196#380C2312yes—
197#380C2312yes—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—21no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4C2312yes—
219C43C2311yescharacteristic
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
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—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2412yes—
245C22×C8C2412yes—
246C22×C8C2414yes—
247C22×C8C2412yes—
248C4⋊Q8—22no—
249C4⋊Q8—22no—
250C4⋊Q8—22no—
251C4⋊Q8—22no—
252C4⋊Q8—42no—
253C4⋊Q8—42no—
254C4⋊Q8—44no—
255C4⋊Q8—82no—
256#126—48no—
257#125—44no—
258#124—44no—
259#123—48no—
260C22⋊Q8—44no—
261C22⋊Q8—44no—
262C22⋊Q8—84no—
263#121—24no—
264#121—42no—
265#121—44no—
266C4×Q8—44no—
267C4×Q8—44no—
268C4×D4—22no—
269C4×D4—24no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—42no—
277C4×D4—41no—
278C4×D4—48no—
279#117—48no—
280#117—44no—
281#117—44no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4C2×D412yes—
324C2×C22⋊C4C2×D412yes—
325C2×C22⋊C4C2×D412yes—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—21no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—24no—
334C2×C42—44no—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#107—44no—
346#107—44no—
347#107—44no—
348#106—24no—
349#106—42no—
350#106—44no—
351C4⋊C8—24no—
352C4⋊C8—24no—
353C4⋊C8—22no—
354C4⋊C8—42no—
355C4⋊C8#4114yes—
356#102—28no—
357#102—24no—
358#102—24no—
359C22⋊C8—28no—
360C22⋊C8—44no—
361#97—28no—
362C4×C8—24no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—24no—
371#95—22no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381C24—21no—
382C24C22×D412yes—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—44no—
389C2×Q8—42no—
390C2×Q8—82no—
391C2×Q8—82no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—82no—
403C2×D4H(5,𝔽2)12yes—
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—24no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—81no—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C2511yescharacteristic
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8C2×#4114yes—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—81no—
486C4⋊C4—82no—
487C4⋊C4—84no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C42—24no—
505C42—24no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—21no—
510C42—22no—
511C42—21no—
512C42—41no—
513C42—42no—
514C42—42no—
515C42—42no—
516C42—44no—
517C42—44no—
518C42—84no—
519C42AGL(1,ℤ/8)14yes—
520C42C2×#4112yes—
521C422−1+412yes—
522C422−1+412yes—
523C422−1+412yes—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23C23×D411yescharacteristic
539Q8—84no—
540Q8—84no—
541Q8—164no—
542D4—22no—
543D4—22no—
544D4—44no—
545D4—41no—
546D4—41no—
547D4—88no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—48no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
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—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4#53612yes—
600C2×C4#53612yes—
601C2×C4#54712yes—
602C2×C4#54712yes—
603C2×C4C2×AGL(1,ℤ/8)12yes—
604C2×C4C2×AGL(1,ℤ/8)12yes—
605C2×C4C2×AGL(1,ℤ/8)12yes—
606C2×C4#57512yes—
607C2×C4#57812yes—
608C2×C4C22×#4111yescharacteristic
609C2×C4C2×H(5,𝔽2)11yescharacteristic
610C2×C4C2×2−1+411yescharacteristic
611C2×C4C2×2−1+411yescharacteristic
612C2×C4C2×2−1+411yescharacteristic
613C2×C4#58511yescharacteristic
614C8—24no—
615C8—44no—
616C8—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22C2×#53611yescharacteristic
632C22C2×#54711yescharacteristic
633C22#348811yescharacteristic
634C22C22×AGL(1,ℤ/8)11yescharacteristic
635C22C2×#57511yescharacteristic
636C22C2×#57811yescharacteristic
637C22#358511yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—42no—
642C4—41no—
643C4—42no—
644C4—42no—
645C4—81no—
646C4—82no—
647C4—84no—
648C4—82no—
649C4—81no—
650C4—82no—
651C4—81no—
652C4—82no—
653C4#323514yes—
654C4#335314yes—
655C4#351511yescharacteristic
656C4#352211yescharacteristic
657C4#354411yescharacteristic
658C4#355511yescharacteristic
659C2—82no—
660C2—82no—
661C2#3892111yescharacteristic
662C2C2×#323511yescharacteristic
663C2#6089311yescharacteristic
664C2C2×#335311yescharacteristic
665C2#6125611yescharacteristic
666C2#6196111yescharacteristic
667C2#6198511yescharacteristic
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. #36194C2semidirect
3. #33195C2semidirect
4. #31711C2semidirect
5. #31100C2semidirect
6. #29939C2semidirect
7. #29287C2semidirect
8. #29208C2semidirect
9. #28835C2semidirect
10. #27736C2semidirect
11. #27646C2semidirect
12. #26301C2semidirect
14. #24708C2semidirect
15. #22799C2semidirect
16. #21988C2semidirect
17. C4×#476C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#349322
#331952
#317112
#311002
#299392
#292872
#292082
#288352
#277362
#276462
#263012
#250112
#247082
#227992
#219882
C4×#4762