Finite Group Databasebeta

← #6434111

Subgroups — #6434111

Order 512.

1754 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35307C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic
3#29919C214yesmaximal, complemented
4#29918C214yesmaximal, complemented
5#29911C212yesmaximal, complemented
6#29911C212yesmaximal, complemented
7#29909C212yesmaximal, complemented
8#29908C212yesmaximal, complemented
9#23749C212yesmaximal, complemented
10#23736C212yesmaximal, complemented
11#23658C211yesmaximal, characteristic, complemented
12#23656C211yesmaximal, characteristic, complemented
13#23629C211yesmaximal, characteristic, complemented
14C2×#2072C211yesmaximal, characteristic, complemented
15#20913C212yesmaximal
16#20849C212yesmaximal
17#20684C211yesmaximal, characteristic
18C2×#536C2211yescharacteristic
19C2×#520C2212yes—
20C2×C4×#41C2211yescharacteristic
21C2×#428C2212yes—
22C2×#427C2212yes—
23C2×#419C2212yes—
24C22×#103C2212yes—
25C2×#408C2212yes—
26#2855C2212yes—
27#2855C2212yes—
28#2625C2212yes—
29#2625C2212yes—
30#2447—21no—
31#2447—21no—
32#2446—21no—
33#2446—21no—
34#2419—22no—
35#2419—22no—
36Q8×C4⋊C4—21no—
37Q8×C4⋊C4—21no—
38D4×C4⋊C4—22no—
39D4×C4⋊C4—22no—
40D4×C4⋊C4—21no—
41D4×C4⋊C4—21no—
42#2317C2212yes—
43#2317C2214yes—
44#2316C2211yescharacteristic
45#2311C2211yescharacteristic
46#2310C2211yescharacteristic
47#2296C2211yescharacteristic
48C2×#391C2211yescharacteristic
49#2285C2211yescharacteristic
50#2285C2212yes—
51C2×#384C2211yescharacteristic
52C2×#384C2212yes—
53#2101C2214yes—
54#2101C2214yes—
55#2100C2214yes—
56#2099C2214yes—
57#2097C2214yes—
58#2097C2214yes—
59#2096C2214yes—
60#2095C2214yes—
61#2085C2212yes—
62#2085C2212yes—
63#2084C2211yescharacteristic
64#2084C2211yescharacteristic
65#2084C2212yes—
66#2084C2212yes—
67#2083C2211yescharacteristic
68#2083C2211yescharacteristic
69#2083C2212yes—
70#2083C2212yes—
71#2074C2214yes—
72#2072C2212yes—
73#2072C2212yes—
74#2072C2214yes—
75#2072C2212yes—
76#2072C2212yes—
77#1873C2214yes—
78#1873C2214yes—
79#1872C2214yes—
80#1872C2214yes—
81#1854C2214yes—
82#1854C2214yes—
83#1853C2214yes—
84#1852C2214yes—
85#1818C2214yes—
86#1817C2212yes—
87#1817C2212yes—
88#1736C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×Q8C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#536—28no—
100C2×C4⋊Q8—21no—
101C2×#124—21no—
102C2×#123—22no—
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2312yes—
105#520—44no—
106#520—44no—
107C4×#41—28no—
108C2×C4×Q8—22no—
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8D411yescharacteristic
111C2×C4×D4—24no—
112C2×C4×D4—22no—
113C2×C4×D4D412yes—
114C2×C4×D4D412yes—
115C2×C4×D4D411yescharacteristic
116C2×C4×D4D411yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C2×#117C2312yes—
120C2×#117C2314yes—
121C2×#117C2311yescharacteristic
122C2×#117C2312yes—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2314yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132#428—44no—
133#427—44no—
134C2×#107C2314yes—
135C2×#107C2312yes—
136C2×#107C2312yes—
137C2×#106C2314yes—
138C2×#106C2312yes—
139C2×#106C2312yes—
140#419—44no—
141C2×#103—44no—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146#408—28no—
147C2×C22⋊C8C2314yes—
148C2×C22⋊C8C2314yes—
149#400—22no—
150#400—22no—
151#398—22no—
152#398—22no—
153#397—22no—
154#397—22no—
155#397—22no—
156#397—22no—
157#391—21no—
158#391—21no—
159#391—21no—
160#391—21no—
161#391—21no—
162#391C2312yes—
163#391C2312yes—
164#391C2312yes—
165#391C2312yes—
166#390—21no—
167#389—21no—
168#389—21no—
169#389—22no—
170#389—22no—
171#389—24no—
172#389—24no—
173#389C2312yes—
174#389C2312yes—
175#389C2311yescharacteristic
176#389C2311yescharacteristic
177#389C2311yescharacteristic
178#389C2311yescharacteristic
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#386—22no—
184#385—22no—
185#385—22no—
186#385—21no—
187#385—21no—
188#385—24no—
189#385—24no—
190#385—24no—
191#384—21no—
192#384—21no—
193#384—22no—
194#384—22no—
195#384C2312yes—
196#384C2314yes—
197#384C2312yes—
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#382C2312yes—
201#382C2314yes—
202#381—22no—
203#381—22no—
204#381—22no—
205#380—21no—
206#380—21no—
207#380—22no—
208#380—22no—
209#380—22no—
210#380C2312yes—
211#380C2312yes—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C2×#95—21no—
219C2×#95—21no—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2312yes—
237C2×#41—22no—
238C2×#41—22no—
239C2×#41—24no—
240C2×#41#4118yes—
241C22×Q8—21no—
242C22×Q8—24no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—24no—
246C22×Q8C2×D411yescharacteristic
247C22×Q8C2412yes—
248C22×D4C2×D412yes—
249C22×D4C2×D411yescharacteristic
250C23×C4—21no—
251C23×C4—24no—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D412yes—
257C23×C4C2×D412yes—
258C23×C4C2412yes—
259C23×C4C2411yescharacteristic
260C2×M16—24no—
261C2×M16—24no—
262C2×M16—24no—
263C22×C8—24no—
264C22×C8—24no—
265C22×C8—24no—
266C22×C8C2412yes—
267C22×C8C2412yes—
268C22×C8C2414yes—
269C4⋊Q8—44no—
270#124—44no—
271#123—48no—
272C22⋊Q8—28no—
273C22⋊Q8—28no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—48no—
277C4×D4—28no—
278C4×D4—28no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—416no—
282C4×D4—48no—
283#117—28no—
284#117—24no—
285#117—24no—
286#117—24no—
287#117—24no—
288#117—22no—
289#117—22no—
290#117—44no—
291#117—44no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
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—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—82no—
319C2×C4⋊C4—84no—
320C2×C4⋊C4—82no—
321C2×C4⋊C4—84no—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2411yescharacteristic
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C42—28no—
346C2×C42—24no—
347C2×C42—24no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#107—42no—
359#107—42no—
360#107—44no—
361#106—42no—
362#106—42no—
363#106—44no—
364#103—44no—
365#103—44no—
366#103—44no—
367#103—44no—
368C22⋊C8—28no—
369C22⋊C8—28no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—24no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390C24C22×D412yes—
391C24C22×D411yescharacteristic
392#41—216no—
393#41—216no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—48no—
404C2×Q8—48no—
405C2×Q8#12314yes—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4#12318yes—
411C2×D4#12314yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—41no—
433C22×C4—42no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—48no—
440C22×C4—44no—
441C22×C4—42no—
442C22×C4—82no—
443C22×C4—84no—
444C22×C4—82no—
445C22×C4—84no—
446C22×C4—81no—
447C22×C4—81no—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C2×#4118yes—
457C22×C4C2×#4114yes—
458C22×C4H(5,𝔽2)12yes—
459C22×C4H(5,𝔽2)12yes—
460C22×C42−1+414yes—
461C22×C4C2511yescharacteristic
462M16—48no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—48no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—24no—
490C4⋊C4—42no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—81no—
504C4⋊C4—82no—
505C4⋊C4—81no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—84no—
509C4⋊C4—82no—
510C4⋊C4—84no—
511C22⋊C4—24no—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—48no—
515C22⋊C4—42no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C42—24no—
523C42—24no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—24no—
528C42—22no—
529C42—48no—
530C42—42no—
531C42—42no—
532C42—44no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23C2×#12314yes—
544C23C2×#12312yes—
545C23#53512yes—
546C23#53612yes—
547C23C23×D411yescharacteristic
548Q8—28no—
549Q8—44no—
550Q8—44no—
551Q8—84no—
552D4—216no—
553D4—28no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—44no—
575C2×C4—48no—
576C2×C4—44no—
577C2×C4—48no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—84no—
606C2×C4—164no—
607C2×C4—162no—
608C2×C4—168no—
609C2×C4C2×#12314yes—
610C2×C4C2×#12312yes—
611C2×C4#53512yes—
612C2×C4#53612yes—
613C2×C4#53918yes—
614C2×C4#53918yes—
615C2×C4#57712yes—
616C2×C4#57812yes—
617C2×C4C22×#4112yes—
618C2×C4C2×H(5,𝔽2)11yescharacteristic
619C2×C4C2×H(5,𝔽2)11yescharacteristic
620C2×C4C2×2−1+412yes—
621C8—42no—
622C8—42no—
623C8—44no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—24no—
631C22—24no—
632C22—24no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22#262314yes—
638C22C22×#12311yescharacteristic
639C22C2×#53511yescharacteristic
640C22C2×#53611yescharacteristic
641C22C2×#57711yescharacteristic
642C22C2×#57811yescharacteristic
643C22#358511yescharacteristic
644C22#358611yescharacteristic
645C4—24no—
646C4—22no—
647C4—22no—
648C4—41no—
649C4—42no—
650C4—41no—
651C4—42no—
652C4—82no—
653C4—81no—
654C4—81no—
655C4—81no—
656C4—84no—
657C4—82no—
658C4—81no—
659C4—82no—
660C4—82no—
661C4#262314yes—
662C4C2×#53912yes—
663C4C2×#53912yes—
664C2—24no—
665C2—44no—
666C2C2×#262311yescharacteristic
667C2#6052211yescharacteristic
668C2#6052611yescharacteristic
669C2#6083311yescharacteristic
670C2#6083711yescharacteristic
671C2#6087811yescharacteristic
672C2#6088511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29919C2semidirect
4. #29918C2semidirect
5. #29911C2semidirect
6. #29911C2semidirect
7. #29909C2semidirect
8. #29908C2semidirect
9. #23749C2semidirect
10. #23736C2semidirect
11. #23658C2semidirect
12. #23656C2semidirect
13. #23629C2semidirect
14. C2×#2072C2semidirect

Maximal subgroups

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

subgroupindex
#353072
#341832
#299192
#299182
#299112
#299092
#299082
#237492
#237362
#236582
#236562
#236292
C2×#20722
#209132
#208492
#206842