Finite Group Databasebeta

← #6793522

Subgroups — #6793522

Order 512.

1493 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic, complemented
2#34943C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic, complemented
4#31433C214yesmaximal, complemented
5#31361C214yesmaximal, complemented
6#29942C212yesmaximal, complemented
7#29287C211yesmaximal, characteristic, complemented
8#29218C211yesmaximal, characteristic, complemented
9#28790C212yesmaximal, complemented
10#27814C212yesmaximal, complemented
11#26297C212yesmaximal, complemented
12#24966C212yesmaximal
13#24701C212yesmaximal, complemented
14#24300C212yesmaximal, complemented
15#22795C211yesmaximal, characteristic, complemented
16#21837C212yesmaximal, complemented
17C4×#474C211yesmaximal, characteristic, complemented
18C2×#533C2212yes—
19C2×#476C2212yes—
20C2×#474C2211yescharacteristic
21C2×#431C2212yes—
22#2868C2211yescharacteristic
23#2857C2212yes—
24#2572—22no—
25#2571C2211yescharacteristic
26#2563C2211yescharacteristic
27#2563C2211yescharacteristic
28#2552C2212yes—
29#2542—21no—
30#2538—21no—
31#2537C2211yescharacteristic
32#2495—21no—
33#2495C2212yes—
34#2441C2214yes—
35#2429—22no—
36#2423—22no—
37#2420—21no—
38#2420C2212yes—
39#2398—22no—
40#2365—21no—
41#2354C2212yes—
42Q8×C4⋊C4—21no—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#121—21no—
46C4×#121C2212yes—
47#2283—21no—
48#2283C2212yes—
49C42×D4C2211yescharacteristic
50C42×D4C2211yescharacteristic
51#2101C2214yes—
52#2100C2214yes—
53#2088C2214yes—
54#2085C2214yes—
55#2077C2214yes—
56#2077C2214yes—
57#2075C2214yes—
58#2059C2214yes—
59#2034C2214yes—
60#2025C2214yes—
61#1991C2214yes—
62#1991C2214yes—
63#1990C2212yes—
64#1988C2211yescharacteristic
65#1988C2212yes—
66#1987C2212yes—
67#1987C2212yes—
68#1987C2212yes—
69#1987C2212yes—
70#1954C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1948C2212yes—
74#1948C2212yes—
75#1947C2211yescharacteristic
76#1941C2214yes—
77#1934C2214yes—
78#1923C2214yes—
79#1918C2212yes—
80#1875C2214yes—
81#1862C2212yes—
82#1854C2214yes—
83#1842C2211yescharacteristic
84#1834C2212yes—
85#1825C2214yes—
86#1825C2214yes—
87#1802C2211yescharacteristic
88#1802C2212yes—
89#1802C2212yes—
90#1797C2212yes—
91C4×#107C2212yes—
92C8×C4⋊C4C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95#533—28no—
96#533—44no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#125C2314yes—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106C2×#121—21no—
107C2×#121C2312yes—
108C2×C4×Q8—21no—
109C2×C4×Q8—22no—
110C2×C4×Q8—22no—
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×#117C2314yes—
120C22×C42—21no—
121C22×C42C2312yes—
122#476—44no—
123#474—24no—
124#431—44no—
125C2×#107C2312yes—
126C2×#107C2312yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2312yes—
134C2×#102C2312yes—
135C2×#102C2312yes—
136C2×#102C2314yes—
137C2×C22⋊C8C2314yes—
138C2×C22⋊C8C2314yes—
139C2×C4×C8C2312yes—
140C2×C4×C8C2314yes—
141#400—22no—
142#400—22no—
143#399—24no—
144#398—24no—
145#397—22no—
146#397—22no—
147#396—24no—
148#394—22no—
149#394—22no—
150#393—22no—
151#393—22no—
152#391—21no—
153#391—21no—
154#391—22no—
155#390—21no—
156#390—22no—
157#390C2312yes—
158#389—21no—
159#389—22no—
160#389—21no—
161#389—21no—
162#389—22no—
163#389—24no—
164#389C2312yes—
165#389C2312yes—
166#389C2312yes—
167#389C2314yes—
168#388—22no—
169#388—24no—
170#388—22no—
171#387—24no—
172#386—22no—
173#386—22no—
174#384—22no—
175#384—21no—
176#384—21no—
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#384C2312yes—
180#384C2312yes—
181#383—22no—
182#383—22no—
183#382—21no—
184#382—21no—
185#382C2312yes—
186#382C2312yes—
187#381—22no—
188#381—22no—
189#380—22no—
190#380—21no—
191#380—21no—
192#380—21no—
193#380C2312yes—
194#380C2312yes—
195#380C2314yes—
196#380C2312yes—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4D411yescharacteristic
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C22⋊C4—22no—
209C4×C22⋊C4—21no—
210C4×C22⋊C4—21no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—21no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4C2312yes—
216#376—22no—
217C43C2311yescharacteristic
218#340C2314yes—
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#336C2314yes—
232#336C2312yes—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×Q8—21no—
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×C8C2412yes—
247C22×C8C2414yes—
248C4⋊Q8—24no—
249C4⋊Q8—22no—
250C4⋊Q8—24no—
251C4⋊Q8—22no—
252C4⋊Q8—22no—
253C4⋊Q8—44no—
254C4⋊Q8—82no—
255C4⋊Q8#4114yes—
256#125—28no—
257#125—44no—
258C22⋊Q8—48no—
259C22⋊Q8—48no—
260C22⋊Q8—84no—
261#121—24no—
262#121—42no—
263#121—44no—
264C4×Q8—48no—
265C4×Q8—48no—
266C4×Q8—44no—
267C4×Q8—82no—
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—84no—
279#117—28no—
280#117—44no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
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—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2414yes—
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
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—24no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—44no—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2411yescharacteristic
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#107—24no—
343#107—42no—
344#106—44no—
345#106—44no—
346C4⋊C8—22no—
347C4⋊C8—22no—
348C4⋊C8—24no—
349C4⋊C8—24no—
350C4⋊C8—42no—
351#102—24no—
352#102—24no—
353#102—44no—
354C22⋊C8—44no—
355C22⋊C8—44no—
356C4×C8—24no—
357C4×C8—44no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378C24—21no—
379C24C22×D412yes—
380C2×Q8—24no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—42no—
384C2×Q8—44no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—81no—
388C2×Q8—81no—
389C2×D4—22no—
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×D42−1+412yes—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—81no—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C2511yescharacteristic
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8—24no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—82no—
479C4⋊C4—81no—
480C4⋊C4—82no—
481C4⋊C4—84no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4C2×#4114yes—
485C22⋊C4—24no—
486C22⋊C4—24no—
487C22⋊C4—42no—
488C22⋊C4—48no—
489C22⋊C4—42no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C22⋊C4—82no—
497C22⋊C4—82no—
498C22⋊C4—84no—
499C42—24no—
500C42—24no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—21no—
506C42—21no—
507C42—41no—
508C42—48no—
509C42—42no—
510C42—42no—
511C42—42no—
512C42—44no—
513C42—44no—
514C42#13514yes—
515C42C2×#4112yes—
516C42H(5,𝔽2)12yes—
517C422−1+412yes—
518C422−1+412yes—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23C23×D411yescharacteristic
534Q8—48no—
535Q8—88no—
536Q8—84no—
537D4—22no—
538D4—22no—
539D4—44no—
540D4—41no—
541D4—41no—
542D4—88no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—48no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4#53612yes—
595C2×C4#53612yes—
596C2×C4#54612yes—
597C2×C4#54612yes—
598C2×C4C2×#13512yes—
599C2×C4C2×#13512yes—
600C2×C4C2×#13512yes—
601C2×C4#57512yes—
602C2×C4#57712yes—
603C2×C4C22×#4111yescharacteristic
604C2×C4C2×H(5,𝔽2)11yescharacteristic
605C2×C4C2×2−1+411yescharacteristic
606C2×C4C2×2−1+411yescharacteristic
607C2×C4C2×2−1+411yescharacteristic
608C2×C4#58511yescharacteristic
609C8—42no—
610C8—42no—
611C8—44no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22C2×#53611yescharacteristic
627C22C2×#54611yescharacteristic
628C22#348911yescharacteristic
629C22C22×#13511yescharacteristic
630C22C2×#57511yescharacteristic
631C22C2×#57711yescharacteristic
632C22#358511yescharacteristic
633C4—22no—
634C4—22no—
635C4—22no—
636C4—41no—
637C4—42no—
638C4—44no—
639C4—42no—
640C4—42no—
641C4—82no—
642C4—81no—
643C4—81no—
644C4—82no—
645C4—81no—
646C4—82no—
647C4—84no—
648C4#323314yes—
649C4#330314yes—
650C4#351811yescharacteristic
651C4#352311yescharacteristic
652C4#355211yescharacteristic
653C4#355411yescharacteristic
654C2—82no—
655C2—82no—
656C2#3891111yescharacteristic
657C2C2×#323311yescharacteristic
658C2#6089211yescharacteristic
659C2C2×#330311yescharacteristic
660C2#6116011yescharacteristic
661C2#6209811yescharacteristic
662C2#6214411yescharacteristic
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. #36134C2semidirect
3. #33197C2semidirect
4. #31433C2semidirect
5. #31361C2semidirect
6. #29942C2semidirect
7. #29287C2semidirect
8. #29218C2semidirect
9. #28790C2semidirect
10. #27814C2semidirect
11. #26297C2semidirect
13. #24701C2semidirect
14. #24300C2semidirect
15. #22795C2semidirect
16. #21837C2semidirect
17. C4×#474C2semidirect

Maximal subgroups

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

subgroupindex
#361342
#349432
#331972
#314332
#313612
#299422
#292872
#292182
#287902
#278142
#262972
#249662
#247012
#243002
#227952
#218372
C4×#4742