Finite Group Databasebeta

← #6800253

Subgroups — #6800253

Order 512.

1613 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
1Q8×C22⋊Q8C211yesmaximal, characteristic
2C2×#2582C211yesmaximal, characteristic
3#32957C211yesmaximal, characteristic
4#32595C214yesmaximal, complemented
5#32563C214yesmaximal, complemented
6#29966C212yesmaximal
7#29111C212yesmaximal
8#28196C212yesmaximal, complemented
9#28124C212yesmaximal
10#27960C211yesmaximal, characteristic
11#27647C211yesmaximal, characteristic
12#25480C212yesmaximal, complemented
13#24994C212yesmaximal, complemented
14#24333C212yesmaximal, complemented
15#22510C211yesmaximal, characteristic
16#22084C212yesmaximal
17C4×#451C211yesmaximal, characteristic
18C2×Q82C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#483C2212yes—
22C2×#477C2212yes—
23C2×#451C2211yescharacteristic
24C2×#446C2212yes—
25C2×C4×Q16C2212yes—
26#2857C2212yes—
27#2857C2211yescharacteristic
28#2582—21no—
29#2582—21no—
30#2582—22no—
31#2582C2212yes—
32#2582C2211yescharacteristic
33#2582C2212yes—
34#2582C2212yes—
35#2582C2214yes—
36#2567C2212yes—
37#2567C2211yescharacteristic
38#2451—21no—
39#2451—22no—
40#2440—22no—
41#2440—21no—
42C2×#398—22no—
43Q8×C4⋊C4—21no—
44Q8×C4⋊C4C2211yescharacteristic
45Q8×C22⋊C4—21no—
46C4×C22⋊Q8C2212yes—
47C4×C22⋊Q8C2211yescharacteristic
48C2×#391—21no—
49C2×#391C2212yes—
50C2×#389—21no—
51C2×#389C2212yes—
52C2×#384C2211yescharacteristic
53#2099C2214yes—
54#2099C2214yes—
55#2086C2214yes—
56#2086C2214yes—
57#2084C2212yes—
58#2072C2212yes—
59#2072C2214yes—
60#2066C2214yes—
61#2057C2214yes—
62#2056C2214yes—
63#2024C2214yes—
64#1990C2212yes—
65#1985C2214yes—
66#1984C2212yes—
67#1948C2212yes—
68#1944C2214yes—
69#1935C2214yes—
70#1935C2214yes—
71#1935C2214yes—
72#1927C2214yes—
73#1920C2214yes—
74#1878C2214yes—
75#1878C2212yes—
76#1872C2211yescharacteristic
77#1863C2214yes—
78#1863C2214yes—
79#1863C2212yes—
80#1852C2211yescharacteristic
81#1852C2212yes—
82#1834C2212yes—
83#1823C2212yes—
84#1823C2212yes—
85#1823C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1797C2212yes—
89#1797C2211yescharacteristic
90#1787C2211yescharacteristic
91#1787C2212yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8D411yescharacteristic
96C23×Q8C2311yescharacteristic
97C22×Q16C2312yes—
98Q82—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8C2311yescharacteristic
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8—44no—
111C2×C4×Q8—44no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D412yes—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2312yes—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4D412yes—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133#483—28no—
134#477—44no—
135#451—24no—
136#446—44no—
137C4×Q16—44no—
138C2×#107C2314yes—
139C2×#107C2312yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×#103C2312yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2311yescharacteristic
152C2×C22⋊C8C2311yescharacteristic
153C2×#97C2314yes—
154C2×C4×C8C2312yes—
155#398—24no—
156#398—24no—
157#398—24no—
158#398—24no—
159#398—22no—
160#398—22no—
161#398—22no—
162#395—22no—
163#395—22no—
164#395—22no—
165#391—21no—
166#391—22no—
167#391—22no—
168#391—24no—
169#391—22no—
170#391—21no—
171#391—22no—
172#391—22no—
173#391—22no—
174#391—21no—
175#391—22no—
176#391C2312yes—
177#391C2314yes—
178#391C2311yescharacteristic
179#391C2314yes—
180#389—22no—
181#389—22no—
182#389—24no—
183#389—21no—
184#389—22no—
185#389—21no—
186#389—22no—
187#389—22no—
188#389—22no—
189#389—21no—
190#389—22no—
191#389C2312yes—
192#389C2311yescharacteristic
193#389C2312yes—
194#389C2312yes—
195#389C2312yes—
196#389C2314yes—
197#389C2314yes—
198#387—21no—
199#387—22no—
200#385—21no—
201#385—22no—
202#384—21no—
203#384—22no—
204#384—21no—
205#384—21no—
206#384C2311yescharacteristic
207#384C2312yes—
208#384C2312yes—
209#384C2312yes—
210#380—22no—
211#380—21no—
212#380C2312yes—
213#380C2311yescharacteristic
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—22no—
219C4×C22⋊C4—21no—
220C2×#95—22no—
221C2×#95—22no—
222C2×#95—22no—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2314yes—
233#336C2312yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—24no—
237C22×Q8—24no—
238C22×Q8—24no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—42no—
245C22×Q8—42no—
246C22×Q8C2×D411yescharacteristic
247C22×Q8C2×D412yes—
248C22×Q8C2×D411yescharacteristic
249C22×Q8C2411yescharacteristic
250C22×Q8C2412yes—
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D412yes—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D412yes—
258C23×C4C2411yescharacteristic
259C2×Q16—24no—
260C2×Q16—24no—
261C2×Q16—42no—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C22×C8C2414yes—
265C4⋊Q8—24no—
266C4⋊Q8—44no—
267C4⋊Q8—48no—
268C4⋊Q8—44no—
269C22⋊Q8—22no—
270C22⋊Q8—82no—
271C22⋊Q8—84no—
272C22⋊Q8#4114yes—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×Q8—42no—
281C4×Q8—42no—
282C4×Q8—42no—
283C4×Q8—42no—
284C4×Q8—44no—
285C4×Q8—44no—
286C4×Q8—44no—
287C4×Q8—44no—
288C4×Q8—44no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—42no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—82no—
321C2×C4⋊C4—84no—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2414yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—44no—
347C2×C42—44no—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2412yes—
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356#107—42no—
357#107—44no—
358#106—28no—
359C4⋊C8—24no—
360C4⋊C8—44no—
361#103—28no—
362#103—28no—
363#103—24no—
364#103—24no—
365#103—42no—
366#103—42no—
367#103—44no—
368C22⋊C8—22no—
369C22⋊C8—24no—
370C22⋊C8—22no—
371#97—44no—
372C4×C8—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—22no—
380#95—21no—
381#95—22no—
382#95—21no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—22no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—22no—
392C24C22×D411yescharacteristic
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—42no—
410C2×Q8—416no—
411C2×Q8—48no—
412C2×Q8—48no—
413C2×Q8—42no—
414C2×Q8—44no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—81no—
418C2×Q8—82no—
419C2×Q8—81no—
420C2×Q8—82no—
421C2×Q8#13714yes—
422C2×Q8C2×#4112yes—
423C2×Q82−1+414yes—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—42no—
444C22×C4—41no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4#13714yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C2×#4112yes—
466C22×C42−1+412yes—
467C22×C4C2511yescharacteristic
468Q16—48no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—81no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4—81no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4C2×#4112yes—
519C22⋊C4—24no—
520C22⋊C4—44no—
521C22⋊C4—42no—
522C42—22no—
523C42—22no—
524C42—24no—
525C42—22no—
526C42—24no—
527C42—42no—
528C42—42no—
529C42—44no—
530C42—44no—
531C42—44no—
532C42—42no—
533C42—44no—
534C42—42no—
535C42H(5,𝔽2)14yes—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23#54812yes—
541C23C2×#13712yes—
542C23#57612yes—
543C23C23×D411yescharacteristic
544Q8—28no—
545Q8—24no—
546Q8—24no—
547Q8—24no—
548Q8—48no—
549Q8—48no—
550Q8—44no—
551Q8—42no—
552Q8—42no—
553Q8—42no—
554Q8—162no—
555Q8—164no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—44no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4—88no—
612C2×C4#53412yes—
613C2×C4#53412yes—
614C2×C4#54812yes—
615C2×C4C2×#13714yes—
616C2×C4C2×#13712yes—
617C2×C4C2×#13712yes—
618C2×C4C2×#13712yes—
619C2×C4C22×#4111yescharacteristic
620C2×C4C2×H(5,𝔽2)12yes—
621C2×C4C2×2−1+412yes—
622C2×C4C2×2−1+411yescharacteristic
623C8—44no—
624C8—42no—
625C8—42no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22#337514yes—
633C22C2×#53411yescharacteristic
634C22C2×#54811yescharacteristic
635C22#348611yescharacteristic
636C22C22×#13711yescharacteristic
637C22C22×#13711yescharacteristic
638C22C2×#57611yescharacteristic
639C22#358611yescharacteristic
640C4—22no—
641C4—24no—
642C4—22no—
643C4—22no—
644C4—42no—
645C4—41no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—44no—
650C4—42no—
651C4—82no—
652C4—82no—
653C4—81no—
654C4—84no—
655C4—82no—
656C4—81no—
657C4#319514yes—
658C4#352012yes—
659C4#353812yes—
660C2—24no—
661C2#3912111yescharacteristic
662C2C2×#319511yescharacteristic
663C2#6076211yescharacteristic
664C2C2×#337511yescharacteristic
665C2#6132211yescharacteristic
666C2#6182911yescharacteristic
667C2#6184411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #32595C2semidirect
5. #32563C2semidirect
8. #28196C2semidirect
12. #25480C2semidirect
13. #24994C2semidirect
14. #24333C2semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊Q82
C2×#25822
#329572
#325952
#325632
#299662
#291112
#281962
#281242
#279602
#276472
#254802
#249942
#243332
#225102
#220842
C4×#4512