Finite Group Databasebeta

← #6854497

Subgroups — #6854497

Order 512.

1544 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36559C211yesmaximal, characteristic, complemented
2#36555C211yesmaximal, characteristic, complemented
3#33294C211yesmaximal, characteristic, complemented
4#31537C214yesmaximal, complemented
5#31535C214yesmaximal, complemented
6#30311C212yesmaximal, complemented
7#29195C212yesmaximal, complemented
8#29149C211yesmaximal, characteristic, complemented
9#29146C212yesmaximal, complemented
10#28519C211yesmaximal, characteristic, complemented
11#24814C212yesmaximal, complemented
12#24812C212yesmaximal, complemented
13#24331C212yesmaximal, complemented
14#24327C212yesmaximal, complemented
15#22272C212yesmaximal
16C4×#496C211yesmaximal, characteristic
17#21345C211yesmaximal, characteristic, complemented
18C2×#554C2212yes—
19C2×#496C2211yescharacteristic
20C2×#477C2212yes—
21C2×#476C2212yes—
22C2×C4×SD16C2212yes—
23C2×C4×D8C2212yes—
24#2868C2211yescharacteristic
25#2864C2212yes—
26#2859C2211yescharacteristic
27#2859C2212yes—
28#2587—22no—
29#2582C2211yescharacteristic
30#2566C2214yes—
31#2563C2212yes—
32#2538—21no—
33#2538—21no—
34#2538C2211yescharacteristic
35#2522—22no—
36#2497—21no—
37#2454C2212yes—
38#2453C2212yes—
39#2452—22no—
40#2452—22no—
41#2423—21no—
42#2360—21no—
43Q8×C4⋊C4C2211yescharacteristic
44D4×C4⋊C4—21no—
45D4×C4⋊C4—21no—
46D4×C4⋊C4C2211yescharacteristic
47#2345—21no—
48C4×#127C2212yes—
49C4×#127C2211yescharacteristic
50C4×#124C2212yes—
51C4×#124C2211yescharacteristic
52C42×Q8C2211yescharacteristic
53#2096C2212yes—
54#2093C2212yes—
55#2085C2214yes—
56#2085C2214yes—
57#2067C2214yes—
58#2065C2214yes—
59#2059C2214yes—
60#2059C2214yes—
61#2026C2214yes—
62#2025C2214yes—
63#2025C2214yes—
64#1992C2214yes—
65#1990C2212yes—
66#1990C2212yes—
67#1984C2212yes—
68#1973C2212yes—
69#1972C2212yes—
70#1961C2212yes—
71#1961C2211yescharacteristic
72#1959C2212yes—
73#1959C2211yescharacteristic
74#1939C2212yes—
75#1917C2214yes—
76#1880C2214yes—
77#1879C2214yes—
78#1879C2214yes—
79#1877C2214yes—
80#1864C2214yes—
81#1862C2214yes—
82#1847C2212yes—
83#1847C2211yescharacteristic
84#1836C2212yes—
85#1836C2212yes—
86#1836C2212yes—
87#1805C2214yes—
88#1804C2214yes—
89#1802C2212yes—
90#1802C2212yes—
91#1767C2212yes—
92#1767C2211yescharacteristic
93#1745C2212yes—
94C4×#97C2211yescharacteristic
95C22×SD16C2312yes—
96C22×D8C2312yes—
97#554—44no—
98#554—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2312yes—
102C2×#127C2311yescharacteristic
103C2×#125C2314yes—
104C2×#125C2312yes—
105C2×#124C2311yescharacteristic
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—24no—
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4C2312yes—
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2314yes—
121C2×C4×D4C2312yes—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—22no—
124#496Q818yescomplemented
125#477—44no—
126#476—44no—
127C4×SD16—44no—
128C4×D8—44no—
129C2×#107C2314yes—
130C2×#107C2312yes—
131C2×#106C2312yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2312yes—
140C2×#102C2314yes—
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×#97C2311yescharacteristic
144C2×#97C2312yes—
145C2×#97C2312yes—
146C2×#97C2311yescharacteristic
147C2×C4×C8C2312yes—
148#400—22no—
149#399—22no—
150#398—22no—
151#398—22no—
152#396—24no—
153#396—24no—
154#396—22no—
155#396—22no—
156#393—24no—
157#391—21no—
158#391C2312yes—
159#391C2311yescharacteristic
160#390—21no—
161#390—21no—
162#390—22no—
163#390—22no—
164#390C2314yes—
165#390C2312yes—
166#390C2311yescharacteristic
167#390C2312yes—
168#389—21no—
169#389—22no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389—22no—
174#389—21no—
175#389—22no—
176#389—22no—
177#389—21no—
178#389—22no—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#388—21no—
184#388—22no—
185#388—22no—
186#388—21no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#385—22no—
193#385—24no—
194#385—24no—
195#385—22no—
196#384—21no—
197#384—21no—
198#384—21no—
199#384C2311yescharacteristic
200#384C2312yes—
201#384C2312yes—
202#384C2311yescharacteristic
203#384C2311yescharacteristic
204#382C2314yes—
205#382C2312yes—
206#382C2311yescharacteristic
207#380—22no—
208#380—22no—
209#380—21no—
210#380—21no—
211#380—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4D412yes—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D412yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—22no—
223C43C2312yes—
224C43C2311yescharacteristic
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234#336C2314yes—
235C22×Q8—22no—
236C22×Q8C2×D411yescharacteristic
237C22×Q8C2411yescharacteristic
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C2411yescharacteristic
241C22×D4C2412yes—
242C23×C4—22no—
243C23×C4—21no—
244C23×C4—24no—
245C23×C4—22no—
246C2×SD16—42no—
247C2×SD16—42no—
248C2×SD16C2×Q818yes—
249C2×D8—42no—
250C2×D8—42no—
251C2×D8C2×Q818yes—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—82no—
258C4⋊Q8—82no—
259#127—22no—
260#127C2×Q814yes—
261#125—44no—
262#125—44no—
263#125—44no—
264#124—22no—
265#124C2×Q814yes—
266C22⋊Q8—416no—
267C22⋊Q8—44no—
268C22⋊Q8—48no—
269C4×Q8—24no—
270C4×Q8—24no—
271C4×Q8—42no—
272C4×Q8—42no—
273C4×Q8—44no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—44no—
278C4×D4—44no—
279C4×D4—44no—
280C4×D4—44no—
281C4×D4—44no—
282C4×D4—84no—
283C4×D4—84no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—48no—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C42—21no—
334C2×C42—22no—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348#107—42no—
349#107—44no—
350#106—42no—
351#106—44no—
352C4⋊C8—24no—
353C4⋊C8—44no—
354#103—42no—
355#103—44no—
356#102—44no—
357#102—44no—
358#102—42no—
359#102—42no—
360#102—42no—
361#102—44no—
362#97—22no—
363#97—24no—
364#97—42no—
365#97C2×Q814yes—
366C4×C8—24no—
367#95—21no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—21no—
372#95—22no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384C24—21no—
385C24—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—21no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—44no—
393C2×Q8—82no—
394C2×Q8—82no—
395C2×Q8C22×Q812yes—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4C22×Q812yes—
411C2×D4C22×Q814yes—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—48no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—42no—
427C22×C4—82no—
428C22×C4—84no—
429C22×C4—84no—
430C22×C4—82no—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C2511yescharacteristic
442SD16—48no—
443D8—48no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8C22×Q814yes—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—48no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—81no—
480C4⋊C4—81no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—48no—
493C22⋊C4—44no—
494C22⋊C4—42no—
495C22⋊C4—44no—
496C22⋊C4—81no—
497C22⋊C4—81no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C22⋊C4—84no—
501C22⋊C4—84no—
502C42—22no—
503C42—22no—
504C42—24no—
505C42—22no—
506C42—24no—
507C42—42no—
508C42—42no—
509C42—42no—
510C42—42no—
511C42—41no—
512C42—41no—
513C42—42no—
514C42—82no—
515C42C22×Q812yes—
516C42H(5,𝔽2)14yes—
517C422−1+412yes—
518C23—22no—
519C23—24no—
520C23—41no—
521C23—42no—
522C23—42no—
523C23—41no—
524C23—42no—
525C23—41no—
526C23—41no—
527C23—42no—
528C23—41no—
529C23—42no—
530C23—42no—
531C23—41no—
532C23C23×D411yescharacteristic
533Q8—24no—
534Q8—44no—
535Q8—42no—
536Q8—164no—
537D4—24no—
538D4—24no—
539D4—24no—
540D4—48no—
541D4—48no—
542D4—42no—
543D4—42no—
544D4—42no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—88no—
591C2×C4—84no—
592C2×C4—164no—
593C2×C4—168no—
594C2×C4#53512yes—
595C2×C4#53512yes—
596C2×C4D4×Q812yes—
597C2×C4D4×Q812yes—
598C2×C4#57512yes—
599C2×C4#57512yes—
600C2×C4C23×Q811yescharacteristic
601C2×C4C2×H(5,𝔽2)12yes—
602C2×C4C2×2−1+411yescharacteristic
603C2×C4#58512yes—
604C8—24no—
605C8—42no—
606C8—44no—
607C22—44no—
608C22—44no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22—82no—
614C22—81no—
615C22—81no—
616C22—82no—
617C22—81no—
618C22—82no—
619C22—82no—
620C22—81no—
621C22C2×#53511yescharacteristic
622C22C2×D4×Q811yescharacteristic
623C22#350011yescharacteristic
624C22C2×#57511yescharacteristic
625C22C2×#57511yescharacteristic
626C22#358511yescharacteristic
627C22#358511yescharacteristic
628C4—22no—
629C4—22no—
630C4—22no—
631C4—42no—
632C4—42no—
633C4—41no—
634C4—41no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—82no—
639C4—81no—
640C4—82no—
641C4—82no—
642C4—81no—
643C4—84no—
644C4#321914yes—
645C4#350112yes—
646C4#354712yes—
647C2—84no—
648C2—82no—
649C2#3961811yescharacteristic
650C2C2×#321911yescharacteristic
651C2#6083011yescharacteristic
652C2#6136711yescharacteristic
653C2#6136711yescharacteristic
654C2#6258611yescharacteristic
655C2#6259211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#496C22
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. #36559C2semidirect
2. #36555C2semidirect
3. #33294C2semidirect
4. #31537C2semidirect
5. #31535C2semidirect
6. #30311C2semidirect
7. #29195C2semidirect
8. #29149C2semidirect
9. #29146C2semidirect
10. #28519C2semidirect
11. #24814C2semidirect
12. #24812C2semidirect
13. #24331C2semidirect
14. #24327C2semidirect
17. #21345C2semidirect
124. #496Q8semidirect

Maximal subgroups

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

subgroupindex
#365592
#365552
#332942
#315372
#315352
#303112
#291952
#291492
#291462
#285192
#248142
#248122
#243312
#243272
#222722
C4×#4962
#213452