Finite Group Databasebeta

← #6434110

Subgroups — #6434110

Order 512.

1852 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35254C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic, complemented
3#29917C214yesmaximal, complemented
4#29916C214yesmaximal, complemented
5#29910C212yesmaximal, complemented
6#29910C212yesmaximal, complemented
7#29907C212yesmaximal, complemented
8#29906C212yesmaximal, complemented
9#23749C212yesmaximal, complemented
10#23735C212yesmaximal, complemented
11#23658C211yesmaximal, characteristic, complemented
12#23656C211yesmaximal, characteristic, complemented
13#23628C211yesmaximal, characteristic, complemented
14C2×#2071C211yesmaximal, characteristic, complemented
15#20914C212yesmaximal
16#20849C212yesmaximal, complemented
17#20684C211yesmaximal, characteristic, complemented
18C2×#535C2211yescharacteristic
19C2×#520C2212yes—
20C2×C4×#41C2211yescharacteristic
21C2×#428C2212yes—
22C2×#427C2212yes—
23C2×#418C2212yes—
24C22×#102C2212yes—
25C2×#408C2212yes—
26#2854C2212yes—
27#2854C2212yes—
28#2624C2212yes—
29#2624C2212yes—
30#2448—21no—
31#2448—21no—
32#2445—21no—
33#2445—21no—
34#2418—22no—
35#2418—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#2317C2214yescomplemented
44#2316C2211yescharacteristic
45#2311C2211yescharacteristic
46#2309C2211yescharacteristic
47#2296C2211yescharacteristic
48C2×#390C2211yescharacteristic
49#2285C2211yescharacteristic
50#2285C2212yescomplemented
51C2×#384C2211yescharacteristic
52C2×#384C2212yescomplemented
53#2101C2214yescomplemented
54#2101C2214yescomplemented
55#2100C2214yescomplemented
56#2099C2214yescomplemented
57#2094C2214yes—
58#2093C2214yes—
59#2092C2214yes—
60#2092C2214yes—
61#2085C2212yescomplemented
62#2085C2212yescomplemented
63#2084C2211yescharacteristic, complemented
64#2084C2211yescharacteristic, complemented
65#2084C2212yescomplemented
66#2084C2212yescomplemented
67#2083C2211yescharacteristic, complemented
68#2083C2211yescharacteristic, complemented
69#2083C2212yescomplemented
70#2083C2212yescomplemented
71#2073C2214yes—
72#2071C2212yes—
73#2071C2212yes—
74#2071C2214yes—
75#2071C2212yes—
76#2071C2212yes—
77#1875C2214yes—
78#1875C2214yes—
79#1874C2214yes—
80#1874C2214yes—
81#1854C2214yescomplemented
82#1854C2214yescomplemented
83#1853C2214yescomplemented
84#1852C2214yescomplemented
85#1818C2214yescomplemented
86#1817C2212yescomplemented
87#1817C2212yescomplemented
88#1736C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×D4C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#535—28no—
100C2×#127—21no—
101C2×#124—21no—
102C2×#123—22no—
103C2×#123—22no—
104C2×#121C2312yes—
105C2×#121C2312yes—
106C2×#120—22no—
107#520—44no—
108#520—44no—
109C4×#41—28no—
110C2×C4×Q8—22no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D411yescharacteristic
113C2×C4×D4—24no—
114C2×C4×D4—22no—
115C2×C4×D4D412yes—
116C2×C4×D4D412yes—
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C2×#117C2312yes—
122C2×#117C2314yes—
123C2×#117C2311yescharacteristic
124C2×#117C2312yes—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2314yes—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136#428—44no—
137#427—44no—
138C2×#107C2314yes—
139C2×#107C2312yes—
140C2×#107C2312yes—
141C2×#106C2314yes—
142C2×#106C2312yes—
143C2×#106C2312yes—
144#418—44no—
145C2×#102—44no—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150#408—28no—
151C2×C22⋊C8C2314yes—
152C2×C22⋊C8C2314yes—
153#399—22no—
154#399—22no—
155#397—22no—
156#397—22no—
157#394—22no—
158#394—22no—
159#394—22no—
160#394—22no—
161#391—21no—
162#390—21no—
163#390—21no—
164#390—21no—
165#390—21no—
166#390—21no—
167#390C2312yes—
168#390C2312yes—
169#390C2312yes—
170#390C2312yes—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389—24no—
176#389—24no—
177#389C2312yes—
178#389C2312yes—
179#389C2311yescharacteristic
180#389C2311yescharacteristic
181#389C2311yescharacteristic
182#389C2311yescharacteristic
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#386—22no—
188#386—22no—
189#386—22no—
190#385—21no—
191#385—21no—
192#385—24no—
193#385—24no—
194#385—24no—
195#384—21no—
196#384—21no—
197#384—22no—
198#384—22no—
199#384C2314yes—
200#384C2312yes—
201#384C2312yes—
202#384C2311yescharacteristic
203#384C2311yescharacteristic
204#382C2314yes—
205#382C2312yes—
206#380—21no—
207#380—21no—
208#380—22no—
209#380—22no—
210#380—22no—
211#380C2312yes—
212#380C2312yes—
213#379—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—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—
237C25—21no—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—24no—
241C2×#41#4118yes—
242C22×Q8C2×D411yescharacteristic
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—21no—
246C22×D4—24no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—24no—
250C22×D4C2×D412yes—
251C22×D4C2×D411yescharacteristic
252C22×D4C2412yes—
253C23×C4—21no—
254C23×C4—24no—
255C23×C4—22no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2412yes—
261C23×C4C2411yescharacteristic
262C2×M16—24no—
263C2×M16—24no—
264C2×M16—24no—
265C22×C8—24no—
266C22×C8—24no—
267C22×C8—24no—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2414yes—
271#127—44no—
272#124—44no—
273#123—84no—
274#123—84no—
275#121—28no—
276#121—28no—
277#120—48no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—48no—
281C4×D4—28no—
282C4×D4—28no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—416no—
286C4×D4—48no—
287#117—28no—
288#117—24no—
289#117—24no—
290#117—24no—
291#117—24no—
292#117—22no—
293#117—22no—
294#117—44no—
295#117—44no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—82no—
321C2×C4⋊C4—84no—
322C2×C4⋊C4—82no—
323C2×C4⋊C4—84no—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2414yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—82no—
353C2×C22⋊C4—82no—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C22⋊C4C2×D412yes—
357C2×C42—28no—
358C2×C42—24no—
359C2×C42—24no—
360C2×C42—22no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D412yes—
366C2×C42C2×D412yes—
367C2×C42C2412yes—
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370#107—42no—
371#107—42no—
372#107—44no—
373#106—42no—
374#106—42no—
375#106—44no—
376#102—44no—
377#102—44no—
378#102—44no—
379#102—44no—
380C22⋊C8—28no—
381C22⋊C8—28no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394C24—22no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—81no—
399C24—81no—
400C24—81no—
401C24—81no—
402C24C22×D412yes—
403C24C22×D411yescharacteristic
404#41—216no—
405#41—216no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8#12314yes—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—24no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—48no—
423C2×D4—48no—
424C2×D4—82no—
425C2×D4—82no—
426C2×D4—82no—
427C2×D4—82no—
428C2×D4#12318yes—
429C2×D4#12314yes—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—42no—
450C22×C4—41no—
451C22×C4—42no—
452C22×C4—41no—
453C22×C4—48no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—81no—
457C22×C4—81no—
458C22×C4—82no—
459C22×C4—84no—
460C22×C4—82no—
461C22×C4—84no—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C2×#4118yes—
471C22×C4C2×#4114yes—
472C22×C4H(5,𝔽2)12yes—
473C22×C4H(5,𝔽2)12yes—
474C22×C42−1+414yes—
475C22×C4C2511yescharacteristic
476M16—48no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—48no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—24no—
504C4⋊C4—42no—
505C4⋊C4—48no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—81no—
516C4⋊C4—81no—
517C4⋊C4—84no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—84no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—48no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C22⋊C4—82no—
539C22⋊C4—82no—
540C22⋊C4—82no—
541C22⋊C4—82no—
542C42—24no—
543C42—24no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—24no—
548C42—22no—
549C42—48no—
550C42—42no—
551C42—42no—
552C42—44no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—84no—
571C23—84no—
572C23—84no—
573C23—162no—
574C23—162no—
575C23—162no—
576C23—162no—
577C23C2×#12314yes—
578C23C2×#12312yes—
579C23#53512yes—
580C23#53612yes—
581C23C23×D411yescharacteristic
582Q8—28no—
583D4—216no—
584D4—28no—
585D4—44no—
586D4—44no—
587D4—84no—
588D4—164no—
589D4—164no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—48no—
614C2×C4—41no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—164no—
639C2×C4—162no—
640C2×C4—168no—
641C2×C4C2×#12314yes—
642C2×C4C2×#12312yes—
643C2×C4#53512yes—
644C2×C4#53612yes—
645C2×C4#53918yes—
646C2×C4#53918yes—
647C2×C4#57612yes—
648C2×C4#57712yes—
649C2×C4C22×#4112yes—
650C2×C4C2×H(5,𝔽2)11yescharacteristic
651C2×C4C2×H(5,𝔽2)11yescharacteristic
652C2×C4C2×2−1+412yes—
653C8—42no—
654C8—42no—
655C8—44no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—24no—
663C22—24no—
664C22—24no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—168no—
677C22#262314yes—
678C22C22×#12311yescharacteristic
679C22C2×#53511yescharacteristic
680C22C2×#53611yescharacteristic
681C22C2×#57611yescharacteristic
682C22C2×#57711yescharacteristic
683C22#358511yescharacteristic
684C22#358611yescharacteristic
685C4—24no—
686C4—22no—
687C4—22no—
688C4—42no—
689C4—41no—
690C4—42no—
691C4—41no—
692C4—81no—
693C4—81no—
694C4—81no—
695C4—84no—
696C4—82no—
697C4—81no—
698C4—82no—
699C4—82no—
700C4#262314yes—
701C4C2×#53912yes—
702C4C2×#53912yes—
703C2—24no—
704C2—44no—
705C2—162no—
706C2C2×#262311yescharacteristic
707C2#6052111yescharacteristic
708C2#6052511yescharacteristic
709C2#6083211yescharacteristic
710C2#6083611yescharacteristic
711C2#6087711yescharacteristic
712C2#6088511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
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. #29917C2semidirect
4. #29916C2semidirect
5. #29910C2semidirect
6. #29910C2semidirect
7. #29907C2semidirect
8. #29906C2semidirect
9. #23749C2semidirect
10. #23735C2semidirect
11. #23658C2semidirect
12. #23656C2semidirect
13. #23628C2semidirect
14. C2×#2071C2semidirect
16. #20849C2semidirect
17. #20684C2semidirect
43. #2317C22semidirect
50. #2285C22semidirect
52. C2×#384C22semidirect
53. #2101C22semidirect
54. #2101C22semidirect
55. #2100C22semidirect
56. #2099C22semidirect
61. #2085C22semidirect
62. #2085C22semidirect
63. #2084C22semidirect
64. #2084C22semidirect
65. #2084C22semidirect
66. #2084C22semidirect
67. #2083C22semidirect
68. #2083C22semidirect
69. #2083C22semidirect
70. #2083C22semidirect
81. #1854C22semidirect
82. #1854C22semidirect
83. #1853C22semidirect
84. #1852C22semidirect
85. #1818C22semidirect
86. #1817C22semidirect
87. #1817C22semidirect

Maximal subgroups

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

subgroupindex
#352542
#341832
#299172
#299162
#299102
#299072
#299062
#237492
#237352
#236582
#236562
#236282
C2×#20712
#209142
#208492
#206842