Finite Group Databasebeta

← #6434115

Subgroups — #6434115

Order 512.

1795 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35255C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic, complemented
3#29918C214yesmaximal, complemented
4#29917C214yesmaximal, complemented
5#29911C212yesmaximal, complemented
6#29910C212yesmaximal, complemented
7#29909C212yesmaximal, complemented
8#29906C212yesmaximal, complemented
9#23748C212yesmaximal, complemented
10#23737C212yesmaximal, complemented
11#23657C211yesmaximal, characteristic, complemented
12C2×#2083C211yesmaximal, characteristic, complemented
13#23630C211yesmaximal, characteristic, complemented
14#23627C211yesmaximal, characteristic, complemented
15#20915C212yesmaximal
16#20847C212yesmaximal, complemented
17#20684C211yesmaximal, characteristic, complemented
18C2×#537C2211yescharacteristic
19C2×#520C2212yes—
20C2×C4×#41C2211yescharacteristic
21C2×#428C2212yes—
22C22×#107C2212yes—
23C2×#417C2212yes—
24C2×#416C2212yes—
25C2×#408C2212yes—
26#2855C2212yes—
27#2854C2212yes—
28#2625C2212yes—
29#2624C2212yes—
30#2444—22no—
31#2443—22no—
32#2423—21no—
33#2422—21no—
34#2421—21no—
35#2420—21no—
36Q8×C4⋊C4—21no—
37Q8×C4⋊C4—21no—
38D4×C4⋊C4—21no—
39D4×C4⋊C4—21no—
40D4×C4⋊C4—22no—
41D4×C4⋊C4—22no—
42#2317C2211yescharacteristic
43#2317C2212yes—
44#2317C2214yes—
45#2311C2211yescharacteristic
46#2311C2211yescharacteristic
47#2297C2211yescharacteristic
48C2×#389C2211yescharacteristic
49#2285C2211yescharacteristic
50#2285C2212yes—
51C2×#384C2211yescharacteristic
52C2×#384C2212yes—
53#2101C2214yes—
54#2101C2214yes—
55#2100C2214yes—
56#2099C2214yes—
57#2097C2214yes—
58#2095C2214yes—
59#2093C2214yes—
60#2092C2214yes—
61#2085C2214yes—
62#2084C2212yes—
63#2083C2212yes—
64#2083C2212yes—
65#2083C2214yes—
66#2083C2212yes—
67#2074C2212yes—
68#2073C2212yes—
69#2072C2211yescharacteristic
70#2072C2211yescharacteristic
71#2072C2212yes—
72#2072C2212yes—
73#2071C2211yescharacteristic
74#2071C2211yescharacteristic
75#2071C2212yes—
76#2071C2212yes—
77#1875C2214yes—
78#1874C2214yes—
79#1873C2214yes—
80#1872C2214yes—
81#1854C2214yes—
82#1854C2214yes—
83#1852C2214yes—
84#1852C2214yes—
85#1818C2214yes—
86#1817C2212yes—
87#1817C2212yes—
88#1736C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C22×#41C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#537—28no—
100C2×#123—22no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8C2312yes—
103C2×#121—21no—
104C2×#121—22no—
105C2×#121C2312yes—
106#520—44no—
107#520—44no—
108C4×#41—28no—
109C2×C4×Q8—22no—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8D411yescharacteristic
112C2×C4×D4—24no—
113C2×C4×D4—22no—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4D411yescharacteristic
116C2×C4×D4D412yes—
117C2×C4×D4D412yes—
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⋊C4—21no—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2314yes—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134#428—44no—
135C2×#107—44no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#106C2314yes—
140#417—44no—
141#416—44no—
142C2×#103C2314yes—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×#102C2314yes—
146C2×#102C2312yes—
147C2×#102C2312yes—
148#408—28no—
149C2×C22⋊C8C2314yes—
150C2×C22⋊C8C2314yes—
151#400—22no—
152#399—22no—
153#398—22no—
154#397—22no—
155#397—22no—
156#397—22no—
157#394—22no—
158#394—22no—
159#391—21no—
160#391C2312yes—
161#391C2311yescharacteristic
162#391C2311yescharacteristic
163#390—21no—
164#390C2312yes—
165#390C2311yescharacteristic
166#390C2311yescharacteristic
167#389—22no—
168#389—22no—
169#389—24no—
170#389—24no—
171#389—21no—
172#389—21no—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#388—22no—
178#388—22no—
179#387—22no—
180#387—22no—
181#386—22no—
182#386—22no—
183#386—21no—
184#386—21no—
185#386—22no—
186#385—22no—
187#385—24no—
188#385—24no—
189#385—24no—
190#385—22no—
191#384—21no—
192#384—21no—
193#384—21no—
194#384—22no—
195#384—22no—
196#384C2312yes—
197#384C2314yes—
198#384C2311yescharacteristic
199#384C2312yes—
200#384C2311yescharacteristic
201#382—21no—
202#382C2312yes—
203#382C2314yes—
204#381—22no—
205#380—21no—
206#380—22no—
207#380—21no—
208#380—22no—
209#380—21no—
210#380—21no—
211#380C2312yes—
212#380C2312yes—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219C2×#95—21no—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2312yes—
237C2×#41—22no—
238C2×#41—22no—
239C2×#41—24no—
240C2×#41—24no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—24no—
244C2×#41#4118yes—
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2411yescharacteristic
247C22×D4—21no—
248C22×D4—22no—
249C22×D4C2×D412yes—
250C22×D4C2×D411yescharacteristic
251C22×D4C2411yescharacteristic
252C23×C4—24no—
253C23×C4—22no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D412yes—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D412yes—
261C23×C4C2412yes—
262C23×C4C2411yescharacteristic
263C2×M16—24no—
264C2×M16—24no—
265C2×M16—24no—
266C22×C8—24no—
267C22×C8—24no—
268C22×C8—24no—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C22×C8C2414yes—
272#123—84no—
273C22⋊Q8—28no—
274C22⋊Q8—44no—
275#121—28no—
276#121—44no—
277#121—48no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—48no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—28no—
284C4×D4—28no—
285C4×D4—416no—
286C4×D4—48no—
287#117—28no—
288#117—24no—
289#117—24no—
290#117—24no—
291#117—24no—
292#117—44no—
293#117—44no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—82no—
321C2×C4⋊C4—84no—
322C2×C4⋊C4—82no—
323C2×C4⋊C4—82no—
324C2×C4⋊C4—84no—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C42—28no—
353C2×C42—24no—
354C2×C42—24no—
355C2×C42—22no—
356C2×C42—21no—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—44no—
365#107—44no—
366#107—44no—
367#106—44no—
368#103—42no—
369#103—42no—
370#103—44no—
371#102—42no—
372#102—42no—
373#102—44no—
374C22⋊C8—28no—
375C22⋊C8—28no—
376#95—22no—
377#95—22no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—22no—
383#95—24no—
384#95—22no—
385#95—24no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392C24—21no—
393C24C22×D412yes—
394C24C22×D411yescharacteristic
395#41—216no—
396#41—216no—
397#41—48no—
398#41—48no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8#12314yes—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—44no—
418C2×D4—82no—
419C2×D4—82no—
420C2×D4#12318yes—
421C2×D4#12314yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—41no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—48no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—81no—
453C22×C4—81no—
454C22×C4—82no—
455C22×C4—84no—
456C22×C4—82no—
457C22×C4—84no—
458C22×C4—81no—
459C22×C4—81no—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C2×#4118yes—
469C22×C4C2×#4114yes—
470C22×C4H(5,𝔽2)12yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C42−1+414yes—
473C22×C4C2511yescharacteristic
474M16—48no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—48no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—42no—
501C4⋊C4—48no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—48no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—42no—
526C22⋊C4—44no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—81no—
530C22⋊C4—82no—
531C22⋊C4—81no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C42—24no—
536C42—24no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—24no—
541C42—22no—
542C42—48no—
543C42—44no—
544C42—42no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23C2×#12314yes—
563C23C2×#12312yes—
564C23#53512yes—
565C23#53612yes—
566C23C23×D411yescharacteristic
567Q8—28no—
568Q8—42no—
569Q8—42no—
570D4—216no—
571D4—28no—
572D4—42no—
573D4—42no—
574D4—84no—
575D4—164no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—44no—
597C2×C4—48no—
598C2×C4—44no—
599C2×C4—48no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4—162no—
627C2×C4—164no—
628C2×C4—168no—
629C2×C4C2×#12314yes—
630C2×C4C2×#12312yes—
631C2×C4#53512yes—
632C2×C4#53612yes—
633C2×C4#53918yes—
634C2×C4#53918yes—
635C2×C4#57612yes—
636C2×C4#57812yes—
637C2×C4C22×#4112yes—
638C2×C4C2×H(5,𝔽2)11yescharacteristic
639C2×C4C2×H(5,𝔽2)11yescharacteristic
640C2×C4C2×2−1+412yes—
641C8—42no—
642C8—42no—
643C8—44no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22—81no—
664C22#262314yes—
665C22C22×#12311yescharacteristic
666C22C2×#53511yescharacteristic
667C22C2×#53611yescharacteristic
668C22C2×#57611yescharacteristic
669C22C2×#57811yescharacteristic
670C22#358511yescharacteristic
671C22#358611yescharacteristic
672C4—24no—
673C4—22no—
674C4—22no—
675C4—41no—
676C4—42no—
677C4—41no—
678C4—42no—
679C4—81no—
680C4—82no—
681C4—82no—
682C4—84no—
683C4—82no—
684C4—82no—
685C4—81no—
686C4—81no—
687C4#262314yes—
688C4C2×#53912yes—
689C4C2×#53912yes—
690C2—24no—
691C2—44no—
692C2—161no—
693C2C2×#262311yescharacteristic
694C2#6052111yescharacteristic
695C2#6052611yescharacteristic
696C2#6083211yescharacteristic
697C2#6083711yescharacteristic
698C2#6088011yescharacteristic
699C2#6088711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#537C22
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
2. #34183C2semidirect
3. #29918C2semidirect
4. #29917C2semidirect
5. #29911C2semidirect
6. #29910C2semidirect
7. #29909C2semidirect
8. #29906C2semidirect
9. #23748C2semidirect
10. #23737C2semidirect
11. #23657C2semidirect
12. C2×#2083C2semidirect
13. #23630C2semidirect
14. #23627C2semidirect
16. #20847C2semidirect
17. #20684C2semidirect

Maximal subgroups

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

subgroupindex
#352552
#341832
#299182
#299172
#299112
#299102
#299092
#299062
#237482
#237372
#236572
C2×#20832
#236302
#236272
#209152
#208472
#206842