Finite Group Databasebeta

← #6798542

Subgroups — #6798542

Order 512.

1677 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36164C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic, complemented
3#33236C211yesmaximal, characteristic, complemented
4#31857C214yesmaximal, complemented
5#31853C214yesmaximal, complemented
6#29815C212yesmaximal, complemented
7#29224C211yesmaximal, characteristic, complemented
8#28449C211yesmaximal, characteristic, complemented
9#27983C212yesmaximal, complemented
10#25603C212yesmaximal, complemented
11#25573C212yesmaximal, complemented
12#25571C212yesmaximal
13#25323C212yesmaximal, complemented
14#25319C212yesmaximal, complemented
15#21866C212yesmaximal, complemented
16C4×#489C211yesmaximal, characteristic, complemented
17#21270C211yesmaximal, characteristic, complemented
18C2×#554C2211yescharacteristic
19C2×#547C2211yescharacteristic
20C2×#533C2212yes—
21C2×#497C2212yes—
22C2×#489C2211yescharacteristic
23C2×#487C2212yes—
24C2×#487C2212yes—
25C2×#462C2212yes—
26C2×#461C2212yes—
27C2×#432C2212yes—
28#2868C2211yescharacteristic
29#2862C2211yescharacteristic
30#2857C2212yes—
31#2850C2212yes—
32#2568C2211yescharacteristic
33#2563C2211yescharacteristic
34#2552—21no—
35#2542C2212yes—
36#2538—21no—
37#2504C2212yes—
38#2503C2212yes—
39#2489—21no—
40#2489—21no—
41#2488—22no—
42#2488—22no—
43#2441—22no—
44#2429C2214yes—
45#2423—22no—
46#2365C2212yes—
47#2354—21no—
48Q8×C4⋊C4—21no—
49#2339—21no—
50#2339—21no—
51C4×C4⋊Q8C2211yescharacteristic
52C4×#124C2211yescharacteristic
53C2×#382C2212yes—
54C42×D4C2211yescharacteristic
55#2101C2214yes—
56#2074C2214yes—
57#2074C2214yes—
58#2056C2214yes—
59#2055C2214yes—
60#2016C2212yes—
61#2013C2212yes—
62#2001C2214yes—
63#2001C2214yes—
64#2000C2214yes—
65#2000C2214yes—
66#1993C2212yes—
67#1993C2211yescharacteristic
68#1988C2212yes—
69#1987C2212yes—
70#1960C2212yes—
71#1960C2212yes—
72#1960C2212yes—
73#1960C2214yes—
74#1959C2211yescharacteristic
75#1954C2212yes—
76#1939C2212yes—
77#1936C2212yes—
78#1935C2212yes—
79#1924C2214yes—
80#1880C2214yes—
81#1878C2214yes—
82#1872C2214yes—
83#1872C2214yes—
84#1872C2214yes—
85#1872C2214yes—
86#1836C2212yes—
87#1836C2211yescharacteristic
88#1805C2214yes—
89#1805C2214yes—
90#1767C2211yescharacteristic
91C4×#103C2212yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×#97C2211yescharacteristic
95C22×#41—21no—
96C22×Q16C2314yes—
97C22×SD16C2314yes—
98#554—44no—
99#547—44no—
100#533—28no—
101#533—44no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×#125—21no—
109C2×#125—22no—
110C2×#124C2311yescharacteristic
111C2×#123—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8C2314yes—
116C2×C22⋊Q8C2312yes—
117C2×C22⋊Q8C2314yes—
118C2×#121C2312yes—
119C2×C4×Q8—21no—
120C2×C4×Q8—22no—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2314yes—
123C2×C4×D4—21no—
124C2×C4×D4D411yescharacteristic
125C2×C4×D4D411yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×#117—22no—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C42C2312yes—
132#497—44no—
133#489—24no—
134#487—44no—
135#487—44no—
136#462—44no—
137#461—44no—
138#432—44no—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141C2×#103C2312yes—
142C2×#103C2314yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×C22⋊C8C2314yes—
152C2×C22⋊C8C2314yes—
153C2×#97C2311yescharacteristic
154C2×#97C2312yes—
155C2×#97C2311yescharacteristic
156C2×#97C2312yes—
157C2×C4×C8C2312yes—
158#400—22no—
159#400—22no—
160#400—22no—
161#399—24no—
162#398—24no—
163#397—22no—
164#397—22no—
165#396—24no—
166#393—22no—
167#391—21no—
168#391—21no—
169#391—21no—
170#391—21no—
171#391—21no—
172#391—21no—
173#391—21no—
174#391C2314yes—
175#391C2312yes—
176#391C2311yescharacteristic
177#390—22no—
178#390C2312yes—
179#390C2311yescharacteristic
180#389—22no—
181#389—21no—
182#389—22no—
183#389—22no—
184#389—24no—
185#389C2312yes—
186#388—24no—
187#388—22no—
188#388—22no—
189#388—21no—
190#388—21no—
191#387—24no—
192#386—22no—
193#386—22no—
194#386—22no—
195#386—22no—
196#384—22no—
197#384—21no—
198#384—21no—
199#384—21no—
200#384—21no—
201#384C2311yescharacteristic
202#382—22no—
203#382—22no—
204#382C2312yes—
205#382C2312yes—
206#382C2312yes—
207#382C2311yescharacteristic
208#382C2312yes—
209#381—22no—
210#381—22no—
211#380—22no—
212#380—22no—
213#380—21no—
214#380—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C22⋊C4—21no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4C2312yes—
226C2×#95—22no—
227C2×#95—22no—
228C43C2311yescharacteristic
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#340C2312yes—
236#340C2312yes—
237#336C2314yes—
238#336C2312yes—
239C2×#41—44no—
240C2×#41—44no—
241C2×#41—44no—
242C22×Q8—21no—
243C22×Q8—21no—
244C22×Q8—21no—
245C22×Q8C2411yescharacteristic
246C22×Q8C2414yes—
247C22×D4—22no—
248C22×D4C2×D411yescharacteristic
249C22×D4C2411yescharacteristic
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—21no—
253C23×C4C2×D412yes—
254C23×C4C2×D412yes—
255C23×C4C2412yes—
256C2×Q16—44no—
257C2×Q16—44no—
258C2×Q16—44no—
259C2×SD16—44no—
260C2×SD16—44no—
261C2×SD16—44no—
262C22×C8C2414yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C4⋊Q8—24no—
266C4⋊Q8—22no—
267C4⋊Q8—24no—
268C4⋊Q8—24no—
269C4⋊Q8—24no—
270C4⋊Q8—24no—
271C4⋊Q8—24no—
272C4⋊Q8—44no—
273C4⋊Q8#4114yescomplemented
274#125—48no—
275#125—44no—
276#124—22no—
277#124—22no—
278#123—48no—
279C22⋊Q8—28no—
280C22⋊Q8—48no—
281C22⋊Q8—44no—
282C22⋊Q8—44no—
283C22⋊Q8—44no—
284C22⋊Q8—48no—
285C22⋊Q8—44no—
286C22⋊Q8—44no—
287#121—44no—
288C4×Q8—28no—
289C4×Q8—24no—
290C4×Q8—44no—
291C4×Q8—48no—
292C4×Q8—44no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—42no—
297C4×D4—44no—
298#117—48no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358C4⋊C8—24no—
359C4⋊C8—44no—
360#103—24no—
361#103—42no—
362#103—44no—
363#103—44no—
364#103—44no—
365#103—44no—
366#103—44no—
367#103—44no—
368#102—24no—
369#102—42no—
370C22⋊C8—44no—
371C22⋊C8—44no—
372#97—22no—
373#97—22no—
374#97—24no—
375#97—42no—
376C4×C8—24no—
377#95—21no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—21no—
386#95—21no—
387#95—24no—
388#95—22no—
389#95—22no—
390#95—24no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24—21no—
399C24C22×D412yes—
400#41—816no—
401C2×Q8—24no—
402C2×Q8—21no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—22no—
407C2×Q8—24no—
408C2×Q8—21no—
409C2×Q8—22no—
410C2×Q8—24no—
411C2×Q8—24no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8—81no—
419C2×Q8—81no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4#13714yes—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C2511yescharacteristic
467Q16—88no—
468SD16—88no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—84no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4C2×#4114yes—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—42no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—81no—
518C22⋊C4—81no—
519C22⋊C4—82no—
520C22⋊C4—82no—
521C22⋊C4—84no—
522C42—24no—
523C42—24no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—41no—
530C42—41no—
531C42—48no—
532C42—42no—
533C42—44no—
534C42C2×#4112yes—
535C42H(5,𝔽2)14yes—
536C42H(5,𝔽2)12yes—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23C2×#13714yes—
551C23C23×D411yescharacteristic
552Q8—48no—
553Q8—42no—
554Q8—44no—
555Q8—44no—
556Q8—44no—
557Q8—42no—
558Q8—44no—
559Q8—84no—
560Q8—162no—
561D4—28no—
562D4—42no—
563D4—42no—
564D4—88no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—48no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—84no—
614C2×C4—88no—
615C2×C4—84no—
616C2×C4—164no—
617C2×C4#53512yes—
618C2×C4#53512yes—
619C2×C4#54612yes—
620C2×C4#54612yes—
621C2×C4C2×#13712yes—
622C2×C4#57512yes—
623C2×C4C22×#4111yescharacteristic
624C2×C4C2×H(5,𝔽2)12yes—
625C2×C4C2×H(5,𝔽2)11yescharacteristic
626C2×C4#58512yes—
627C8—42no—
628C8—42no—
629C8—44no—
630C22—24no—
631C22—24no—
632C22—24no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22C2×#53511yescharacteristic
645C22C2×#54611yescharacteristic
646C22#350011yescharacteristic
647C22C22×#13711yescharacteristic
648C22C2×#57511yescharacteristic
649C22#358611yescharacteristic
650C22#358611yescharacteristic
651C4—22no—
652C4—22no—
653C4—22no—
654C4—42no—
655C4—41no—
656C4—41no—
657C4—42no—
658C4—42no—
659C4—44no—
660C4—81no—
661C4—84no—
662C4—81no—
663C4—82no—
664C4—82no—
665C4—81no—
666C4—82no—
667C4#321514yes—
668C4#351712yes—
669C4#354612yes—
670C2—44no—
671C2—161no—
672C2#3895611yescharacteristic
673C2C2×#321511yescharacteristic
674C2#6082611yescharacteristic
675C2#6115711yescharacteristic
676C2#6115911yescharacteristic
677C2#6257711yescharacteristic
678C2#6259111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
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
2. #36134C2semidirect
3. #33236C2semidirect
4. #31857C2semidirect
5. #31853C2semidirect
6. #29815C2semidirect
7. #29224C2semidirect
8. #28449C2semidirect
9. #27983C2semidirect
10. #25603C2semidirect
11. #25573C2semidirect
13. #25323C2semidirect
14. #25319C2semidirect
15. #21866C2semidirect
16. C4×#489C2semidirect
17. #21270C2semidirect
273. C4⋊Q8#41semidirect

Maximal subgroups

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

subgroupindex
#361642
#361342
#332362
#318572
#318532
#298152
#292242
#284492
#279832
#256032
#255732
#255712
#253232
#253192
#218662
C4×#4892
#212702