Finite Group Databasebeta

← #6800727

Subgroups — #6800727

Order 512.

1507 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C4⋊Q8C211yesmaximal, characteristic, complemented
2#34945C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic
4#32444C214yesmaximal, complemented
5#31650C214yesmaximal
6#29983C212yesmaximal, complemented
7#29148C212yesmaximal
8#28717C211yesmaximal, characteristic, complemented
9#28419C212yesmaximal
10#28255C211yesmaximal, characteristic, complemented
11#28159C212yesmaximal
12#27738C212yesmaximal, complemented
13#25936C212yesmaximal, complemented
14#24329C212yesmaximal
15#22825C211yesmaximal, characteristic, complemented
16#22154C212yesmaximal, complemented
17C4×#461C211yesmaximal, characteristic
18C2×Q82C2212yes—
19C2×#487C2212yes—
20C2×#476C2212yes—
21C2×#462C2212yes—
22C2×#462C2212yes—
23C2×#461C2211yescharacteristic
24C2×C4×SD16C2212yes—
25#2868C2212yes—
26#2868C2211yescharacteristic
27#2573—22no—
28#2571C2211yescharacteristic
29#2568C2211yescharacteristic
30#2567C2212yes—
31#2567C2214yes—
32#2551—22no—
33#2551—22no—
34#2551—21no—
35#2551—21no—
36#2547C2211yescharacteristic
37#2538C2211yescharacteristic
38#2514C2212yes—
39#2494—21no—
40#2447C2212yes—
41#2421—21no—
42#2401—22no—
43Q8×C4⋊C4—21no—
44Q8×C4⋊C4—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2212yes—
48C4×#124C2212yes—
49C4×C22⋊Q8—21no—
50#2295C2212yes—
51#2283—21no—
52C42×Q8C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2099C2214yes—
55#2096C2214yes—
56#2074C2214yes—
57#2072C2214yes—
58#2058C2214yes—
59#2056C2214yes—
60#2024C2214yes—
61#2024C2214yes—
62#2023C2214yes—
63#2015C2214yes—
64#1990C2212yes—
65#1987C2212yes—
66#1984C2212yes—
67#1981C2211yescharacteristic
68#1971C2214yes—
69#1959C2214yes—
70#1948C2211yescharacteristic
71#1948C2212yes—
72#1948C2212yes—
73#1935C2214yes—
74#1927C2212yes—
75#1917C2214yes—
76#1880C2214yes—
77#1878C2214yes—
78#1878C2214yes—
79#1877C2212yes—
80#1865C2214yes—
81#1842C2211yescharacteristic
82#1836C2214yes—
83#1831C2212yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1802C2212yes—
89#1802C2212yes—
90#1802C2211yescharacteristic
91#1770C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×Q16C2314yes—
96C22×SD16C2312yes—
97Q82—44no—
98Q82—44no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2314yes—
109C2×#124C2312yes—
110C2×C22⋊Q8—21no—
111C2×C4×Q8—22no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2314yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C22×C42—21no—
122#487—44no—
123#476—44no—
124#462—28no—
125#462—44no—
126#461—24no—
127C4×SD16—44no—
128C2×#106C2312yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2312yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×#97C2314yes—
145C2×C4×C8C2312yes—
146#401—22no—
147#400—22no—
148#400—22no—
149#400—22no—
150#397—22no—
151#397—22no—
152#397—22no—
153#396—22no—
154#395—24no—
155#395—24no—
156#395—24no—
157#391—21no—
158#391C2312yes—
159#391C2314yes—
160#391C2312yes—
161#391C2312yes—
162#391C2312yes—
163#391C2312yes—
164#390C2312yes—
165#389—22no—
166#389—22no—
167#389—24no—
168#389—24no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389—21no—
174#389—21no—
175#389—22no—
176#389C2312yes—
177#389C2312yes—
178#388—22no—
179#388—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#385—22no—
184#384—22no—
185#384—21no—
186#384—22no—
187#384—21no—
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2311yescharacteristic
191#384C2312yes—
192#384C2314yes—
193#383—22no—
194#382—21no—
195#382—21no—
196#382C2312yes—
197#381—22no—
198#380—21no—
199#380—21no—
200#380—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4D412yes—
209C4×C4⋊C4D412yes—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222#376—22no—
223C43C2311yescharacteristic
224C43C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#336C2314yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236C22×Q8—24no—
237C22×Q8—22no—
238C22×Q8C2×D411yescharacteristic
239C22×Q8C2412yes—
240C22×Q8C2412yes—
241C22×D4C2411yescharacteristic
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C2×Q16—28no—
246C2×Q16—44no—
247C2×Q16—44no—
248C2×SD16—24no—
249C2×SD16—24no—
250C2×SD16—42no—
251C22×C8C2412yes—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259C4⋊Q8—44no—
260C4⋊Q8—44no—
261C4⋊Q8—416no—
262C4⋊Q8—48no—
263C4⋊Q8—82no—
264C4⋊Q8—84no—
265C4⋊Q8—82no—
266C4⋊Q8—84no—
267C4⋊Q8#4114yes—
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—24no—
278C4×Q8—24no—
279C4×Q8—44no—
280C4×Q8—44no—
281C4×Q8—48no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—41no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—44no—
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2412yes—
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348#106—42no—
349#106—44no—
350C4⋊C8—22no—
351C4⋊C8—24no—
352C4⋊C8—24no—
353C4⋊C8—22no—
354C4⋊C8—42no—
355#103—24no—
356#103—28no—
357#103—42no—
358#103—42no—
359#103—44no—
360#103—44no—
361#103—44no—
362#102—42no—
363#102—44no—
364#97—44no—
365C4×C8—24no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388C24—21no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—48no—
403C2×Q8—44no—
404C2×Q8—82no—
405C2×Q8—82no—
406C2×Q8—84no—
407C2×Q8—84no—
408C2×Q8#13714yes—
409C2×Q8C2×#4114yes—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D42−1+412yes—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
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×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C2511yescharacteristic
444Q16—88no—
445SD16—48no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—81no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4—84no—
492C4⋊C4—82no—
493C4⋊C4—84no—
494C22⋊C4—42no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—44no—
500C22⋊C4—82no—
501C42—24no—
502C42—22no—
503C42—24no—
504C42—22no—
505C42—24no—
506C42—22no—
507C42—22no—
508C42—21no—
509C42—22no—
510C42—22no—
511C42—21no—
512C42—41no—
513C42—42no—
514C42—48no—
515C42—44no—
516C42—44no—
517C42—42no—
518C42—42no—
519C42—42no—
520C42#13714yes—
521C42C2×#4112yes—
522C42H(5,𝔽2)12yes—
523C42H(5,𝔽2)12yes—
524C422−1+412yes—
525C23—22no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23C23×D411yescharacteristic
533Q8—28no—
534Q8—24no—
535Q8—24no—
536Q8—42no—
537Q8—42no—
538Q8—42no—
539Q8—42no—
540Q8—42no—
541Q8—42no—
542Q8—164no—
543Q8—168no—
544D4—22no—
545D4—22no—
546D4—41no—
547D4—41no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4#53512yes—
596C2×C4#53512yes—
597C2×C4#54812yes—
598C2×C4#54812yes—
599C2×C4C2×#13714yes—
600C2×C4C2×#13712yes—
601C2×C4C2×#13712yes—
602C2×C4C2×#13712yes—
603C2×C4C2×#13712yes—
604C2×C4#57712yes—
605C2×C4C22×#4111yescharacteristic
606C2×C4C2×H(5,𝔽2)11yescharacteristic
607C2×C4C2×H(5,𝔽2)11yescharacteristic
608C2×C4C2×2−1+411yescharacteristic
609C2×C4C2×2−1+411yescharacteristic
610C2×C4#58511yescharacteristic
611C8—44no—
612C8—42no—
613C8—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22C2×#53511yescharacteristic
622C22C2×#54811yescharacteristic
623C22#348911yescharacteristic
624C22C22×#13711yescharacteristic
625C22C22×#13711yescharacteristic
626C22C2×#57711yescharacteristic
627C22#358511yescharacteristic
628C4—24no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—42no—
633C4—42no—
634C4—41no—
635C4—42no—
636C4—44no—
637C4—81no—
638C4—81no—
639C4—82no—
640C4—81no—
641C4—84no—
642C4—82no—
643C4—82no—
644C4#321814yes—
645C4#336914yes—
646C4#351411yescharacteristic
647C4#352011yescharacteristic
648C4#354711yescharacteristic
649C4#355211yescharacteristic
650C2—82no—
651C2#3912411yescharacteristic
652C2C2×#321811yescharacteristic
653C2#6083311yescharacteristic
654C2C2×#336911yescharacteristic
655C2#6131911yescharacteristic
656C2#6210811yescharacteristic
657C2#6214711yescharacteristic
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. Q8×C4⋊Q8C2semidirect
4. #32444C2semidirect
6. #29983C2semidirect
8. #28717C2semidirect
10. #28255C2semidirect
12. #27738C2semidirect
13. #25936C2semidirect
15. #22825C2semidirect
16. #22154C2semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊Q82
#349452
#331972
#324442
#316502
#299832
#291482
#287172
#284192
#282552
#281592
#277382
#259362
#243292
#228252
#221542
C4×#4612