Finite Group Databasebeta

← #6803346

Subgroups — #6803346

Order 512.

1477 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2#34945C211yesmaximal, characteristic
3#33162C211yesmaximal, characteristic
4#31907C214yesmaximal, complemented
5#31215C214yesmaximal
6#29831C212yesmaximal, complemented
7#28737C212yesmaximal, complemented
8#28440C211yesmaximal, characteristic, complemented
9#28372C212yesmaximal
10#28283C211yesmaximal, characteristic, complemented
11#28254C212yesmaximal
12#26527C212yesmaximal, complemented
13#25365C212yesmaximal
14#25293C212yesmaximal
15Q8×C4⋊C8C211yesmaximal, characteristic, complemented
16#22148C212yesmaximal, complemented
17C4×#474C211yesmaximal, characteristic
18C2×#494C2212yes—
19C2×#475C2212yes—
20C2×#474C2211yescharacteristic
21C2×#464C2214yes—
22C2×#462C2212yes—
23C2×#462C2212yes—
24C2×#461C2212yes—
25C2×#461C2212yes—
26C2×#461C2212yes—
27C2×C8×Q8C2212yes—
28#2868C2211yescharacteristic
29#2862C2212yes—
30#2582—21no—
31#2573—22no—
32#2571—21no—
33#2571C2211yescharacteristic
34#2570—22no—
35#2568C2211yescharacteristic
36#2568C2212yes—
37#2547C2211yescharacteristic
38#2538C2211yescharacteristic
39#2514C2212yes—
40#2511—22no—
41#2494—21no—
42#2447C2212yes—
43#2446C2214yes—
44#2421—21no—
45#2401—22no—
46#2365C2212yes—
47#2355—21no—
48#2355—21no—
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2212yes—
52C4×#124C2212yes—
53C4×C22⋊Q8—21no—
54#2295C2212yes—
55#2283—21no—
56C42×Q8C2211yescharacteristic
57C42×D4C2211yescharacteristic
58#2097C2214yes—
59#2066C2214yes—
60#2066C2214yes—
61#2066C2214yes—
62#2056C2214yes—
63#2056C2214yes—
64#2056C2214yes—
65#2055C2214yes—
66#2016C2214yes—
67#2015C2214yes—
68#1988C2211yescharacteristic
69#1981C2212yes—
70#1981C2212yes—
71#1981C2212yes—
72#1981C2212yes—
73#1969C2214yes—
74#1969C2214yes—
75#1968C2214yes—
76#1956C2212yes—
77#1948C2212yes—
78#1947C2211yescharacteristic
79#1939C2212yes—
80#1939C2214yes—
81#1933C2214yes—
82#1918C2212yes—
83#1878C2214yes—
84#1863C2214yes—
85#1862C2212yes—
86#1842C2212yes—
87#1842C2211yescharacteristic
88#1805C2214yes—
89#1803C2212yes—
90#1802C2211yescharacteristic
91C4×#107C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#102C2212yes—
95C22×Q16C2314yes—
96C22×Q16C2314yes—
97C22×SD16C2314yes—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×#124C2312yes—
104C2×C22⋊Q8—21no—
105C2×C4×Q8—21no—
106C2×C4×Q8D411yescharacteristic
107C2×C4×Q8D411yescharacteristic
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2314yes—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×D4C2311yescharacteristic
112C2×C4×D4C2311yescharacteristic
113C2×C4×D4C2311yescharacteristic
114C22×C42—21no—
115#494—44no—
116#475—44no—
117#474—24no—
118#464—48no—
119#462—28no—
120#462—44no—
121#461—44no—
122#461—44no—
123#461—44no—
124C8×Q8—44no—
125C2×#107C2312yes—
126C2×#107C2312yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2314yes—
132C2×C4⋊C8C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2311yescharacteristic
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144C2×C4×C8C2312yes—
145C2×C4×C8C2314yes—
146#401—22no—
147#400—24no—
148#400—22no—
149#400—22no—
150#400—24no—
151#400—22no—
152#398—24no—
153#397—22no—
154#397—22no—
155#397—22no—
156#396—22no—
157#391—22no—
158#391—21no—
159#391—21no—
160#391—22no—
161#391—22no—
162#391C2314yes—
163#391C2312yes—
164#391C2312yes—
165#391C2312yes—
166#391C2312yes—
167#391C2314yes—
168#391C2314yes—
169#390C2312yes—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—21no—
175#389—22no—
176#389—21no—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#388—22no—
181#388—22no—
182#387—22no—
183#387—24no—
184#387—22no—
185#387—22no—
186#387—24no—
187#387—24no—
188#385—22no—
189#384—21no—
190#384—21no—
191#384—21no—
192#384—21no—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#383—22no—
196#382—22no—
197#382—21no—
198#382—21no—
199#382—22no—
200#382—22no—
201#382C2312yes—
202#381—22no—
203#380—21no—
204#380—21no—
205#380—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4D412yes—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D412yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—21no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—21no—
228#376—22no—
229C43C2311yescharacteristic
230C43C2312yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#340C2312yes—
236#336C2314yes—
237#336C2312yes—
238C22×Q8—21no—
239C22×Q8C2×D411yescharacteristic
240C22×Q8C2412yes—
241C22×Q8C2414yes—
242C22×D4C2411yescharacteristic
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C2×Q16—28no—
247C2×Q16—44no—
248C2×Q16—44no—
249C2×Q16—44no—
250C2×Q16—44no—
251C2×Q16—44no—
252C2×SD16—28no—
253C2×SD16—44no—
254C2×SD16—44no—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C4⋊Q8—22no—
259C4⋊Q8—24no—
260C4⋊Q8—22no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8—24no—
266C4⋊Q8—42no—
267C4⋊Q8—42no—
268#124—24no—
269#124—42no—
270C22⋊Q8—44no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—41no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—24no—
329C2×C42—44no—
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D412yes—
337C2×C42C2×D411yescharacteristic
338C2×C42C2411yescharacteristic
339C2×C42C2412yes—
340C2×C42C2412yes—
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343#107—24no—
344#107—42no—
345#106—44no—
346C4⋊C8—24no—
347C4⋊C8—24no—
348C4⋊C8—22no—
349C4⋊C8—44no—
350C4⋊C8—42no—
351C4⋊C8—44no—
352C4⋊C8#4114yes—
353#103—28no—
354#103—44no—
355#103—44no—
356#103—44no—
357#103—44no—
358#103—44no—
359#103—44no—
360#102—24no—
361#102—24no—
362C4×C8—24no—
363C4×C8—44no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95—24no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390C24—21no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—22no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8#13714yes—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4H(5,𝔽2)12yes—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
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—81no—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C2511yescharacteristic
443Q16—88no—
444Q16—88no—
445SD16—88no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8C2×#4114yes—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—81no—
487C4⋊C4—82no—
488C4⋊C4—84no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C22⋊C4—42no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—44no—
497C22⋊C4—82no—
498C42—24no—
499C42—24no—
500C42—22no—
501C42—22no—
502C42—24no—
503C42—22no—
504C42—21no—
505C42—22no—
506C42—22no—
507C42—21no—
508C42—22no—
509C42—41no—
510C42—42no—
511C42—44no—
512C42—42no—
513C42—42no—
514C42—42no—
515C42—44no—
516C42—84no—
517C42#13514yes—
518C42C2×#4112yes—
519C42H(5,𝔽2)12yes—
520C42H(5,𝔽2)12yes—
521C422−1+412yes—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23C23×D411yescharacteristic
530Q8—28no—
531Q8—42no—
532Q8—44no—
533Q8—44no—
534Q8—42no—
535Q8—44no—
536Q8—42no—
537Q8—42no—
538Q8—44no—
539Q8—84no—
540D4—22no—
541D4—22no—
542D4—41no—
543D4—41no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4#53712yes—
592C2×C4#53712yes—
593C2×C4#55012yes—
594C2×C4#55012yes—
595C2×C4C2×#13512yes—
596C2×C4C2×#13512yes—
597C2×C4C2×#13512yes—
598C2×C4C2×#13714yes—
599C2×C4C2×#13712yes—
600C2×C4#57812yes—
601C2×C4C22×#4111yescharacteristic
602C2×C4C2×H(5,𝔽2)11yescharacteristic
603C2×C4C2×H(5,𝔽2)11yescharacteristic
604C2×C4C2×H(5,𝔽2)11yescharacteristic
605C2×C4C2×2−1+411yescharacteristic
606C2×C4#58511yescharacteristic
607C8—24no—
608C8—44no—
609C8—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22C2×#53711yescharacteristic
618C22C2×#55011yescharacteristic
619C22#348811yescharacteristic
620C22C22×#13511yescharacteristic
621C22C22×#13711yescharacteristic
622C22C2×#57811yescharacteristic
623C22#358611yescharacteristic
624C4—22no—
625C4—24no—
626C4—22no—
627C4—22no—
628C4—42no—
629C4—41no—
630C4—42no—
631C4—42no—
632C4—81no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4—84no—
637C4—81no—
638C4—82no—
639C4—81no—
640C4#324814yes—
641C4#339314yes—
642C4#351411yescharacteristic
643C4#352211yescharacteristic
644C4#354311yescharacteristic
645C4#355411yescharacteristic
646C2—82no—
647C2#3913711yescharacteristic
648C2C2×#324811yescharacteristic
649C2#6092511yescharacteristic
650C2C2×#339311yescharacteristic
651C2#6140411yescharacteristic
652C2#6195011yescharacteristic
653C2#6200011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36598C2semidirect
4. #31907C2semidirect
6. #29831C2semidirect
7. #28737C2semidirect
8. #28440C2semidirect
10. #28283C2semidirect
12. #26527C2semidirect
15. Q8×C4⋊C8C2semidirect
16. #22148C2semidirect

Maximal subgroups

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

subgroupindex
#365982
#349452
#331622
#319072
#312152
#298312
#287372
#284402
#283722
#282832
#282542
#265272
#253652
#252932
Q8×C4⋊C82
#221482
C4×#4742