Finite Group Databasebeta

← #6435878

Subgroups — #6435878

Order 512.

1735 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic, complemented
2#35264C211yesmaximal, characteristic
3#30216C212yesmaximal, complemented
4#30212C212yesmaximal, complemented
5#30211C214yesmaximal, complemented
6#30209C214yesmaximal, complemented
7#30207C212yesmaximal, complemented
8#30203C212yesmaximal, complemented
9#23630C211yesmaximal, characteristic, complemented
10#23628C211yesmaximal, characteristic, complemented
11#23627C211yesmaximal, characteristic, complemented
12C2×#2071C211yesmaximal, characteristic, complemented
13#23458C212yesmaximal, complemented
14#23456C212yesmaximal, complemented
15#21326C212yesmaximal, complemented
16#21206C212yesmaximal
17#20368C211yesmaximal, characteristic, complemented
18C2×#544C2211yescharacteristic
19C2×#544C2212yes—
20C2×#535C2211yescharacteristic
21C2×#417C2212yes—
22C2×#418C2212yes—
23C2×#416C2212yes—
24C22×#102C2212yes—
25C2×#405C2212yes—
26#2868C2212yescomplemented
27#2861C2212yes—
28#2859C2212yescomplemented
29#2858C2212yes—
30#2660C2214yescomplemented
31#2641C2212yes—
32#2640C2212yes—
33#2552—21no—
34#2552—21no—
35#2551—21no—
36#2551—21no—
37#2491—21no—
38#2491—21no—
39#2490—21no—
40#2490—21no—
41#2472—22no—
42#2472—22no—
43#2440—22no—
44#2440—22no—
45#2317C2212yes—
46#2311C2211yescharacteristic
47#2309C2211yescharacteristic
48#2297C2211yescharacteristic
49C2×#390C2211yescharacteristic
50#2285C2211yescharacteristic
51C2×#384C2211yescharacteristic
52C4×#117C2211yescharacteristic
53#2074C2212yescomplemented
54#2073C2214yes—
55#2073C2212yescomplemented
56#2072C2212yescomplemented
57#2072C2211yescharacteristic, complemented
58#2072C2212yescomplemented
59#2072C2211yescharacteristic, complemented
60#2071C2214yes—
61#2071C2212yes—
62#2071C2212yes—
63#2071C2212yes—
64#2071C2212yes—
65#2071C2212yescomplemented
66#2071C2211yescharacteristic, complemented
67#2071C2212yescomplemented
68#2071C2211yescharacteristic, complemented
69#2026C2214yescomplemented
70#2025C2214yes—
71#2025C2214yes—
72#2025C2214yescomplemented
73#2024C2214yescomplemented
74#2023C2214yes—
75#2023C2214yes—
76#2023C2214yescomplemented
77#1993C2214yescomplemented
78#1992C2214yescomplemented
79#1991C2214yescomplemented
80#1990C2214yescomplemented
81#1961C2214yes—
82#1959C2214yes—
83#1958C2214yes—
84#1957C2214yes—
85#1746C2214yescomplemented
86#1745C2212yescomplemented
87#1744C2212yescomplemented
88#1736C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×D4C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#544—28no—
100#544—44no—
101#544—44no—
102#535—28no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2314yes—
109C2×#127C2312yes—
110C2×#125—21no—
111C2×#125—22no—
112C2×#125C2312yes—
113C2×#125C2314yes—
114C2×#124—22no—
115C2×#124—22no—
116C2×#124C2312yes—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8—22no—
119C2×#121—21no—
120C2×#121—21no—
121C2×#120—22no—
122C2×#117C2311yescharacteristic
123C2×#117C2312yes—
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C42C2311yescharacteristic
133#418—44no—
134#417—44no—
135#416—44no—
136C2×#103C2314yes—
137C2×#103C2312yes—
138C2×#103C2312yes—
139C2×#102—44no—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147#405—28no—
148C2×#97C2314yes—
149C2×C4×C8C2314yes—
150#398—24no—
151#398—24no—
152#397—22no—
153#397—22no—
154#395—24no—
155#395—24no—
156#394—22no—
157#394—22no—
158#394—22no—
159#394—22no—
160#393—22no—
161#393—22no—
162#391C2312yes—
163#391C2311yescharacteristic
164#391C2311yescharacteristic
165#390—21no—
166#390—21no—
167#390—22no—
168#390—22no—
169#390—21no—
170#390—21no—
171#390C2312yes—
172#390C2312yes—
173#390C2312yes—
174#390C2312yes—
175#390C2312yes—
176#390C2311yescharacteristic
177#390C2311yescharacteristic
178#389—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#386—21no—
189#386—21no—
190#386—22no—
191#385—24no—
192#384—22no—
193#384—22no—
194#384—21no—
195#384—21no—
196#384C2314yes—
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2311yescharacteristic
200#384C2312yes—
201#382C2314yes—
202#382C2312yes—
203#380—21no—
204#380—21no—
205#380—21no—
206#380—21no—
207#379—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C22⋊C4—21no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4D412yes—
218#376C2312yes—
219C43C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2312yes—
233#336C2312yes—
234#336C2312yes—
235#336C2314yes—
236#336C2314yes—
237C25—21no—
238C2×#41—24no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—24no—
242C22×Q8—22no—
243C22×Q8—24no—
244C22×Q8C2411yescharacteristic
245C22×D4—21no—
246C22×D4—21no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—24no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—24no—
253C22×D4C2412yes—
254C22×D4C2411yescharacteristic
255C23×C4—21no—
256C23×C4—22no—
257C23×C4—21no—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2411yescharacteristic
261C2×M16—24no—
262C2×M16—24no—
263C2×M16—24no—
264C22×C8—24no—
265C22×C8—24no—
266C22×C8—24no—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C22×C8C2414yes—
270C4⋊Q8—28no—
271C4⋊Q8—44no—
272C4⋊Q8—44no—
273C4⋊Q8—48no—
274C4⋊Q8—44no—
275C4⋊Q8—84no—
276C4⋊Q8—84no—
277#127—28no—
278#125—28no—
279#125—44no—
280#125—44no—
281#125—48no—
282#125—44no—
283#124—28no—
284#124—84no—
285#124—84no—
286C22⋊Q8—416no—
287C22⋊Q8—48no—
288#121—44no—
289#121—44no—
290#120—48no—
291#117—22no—
292#117—22no—
293#117—24no—
294#117—24no—
295#117—24no—
296#117#4118yes—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—82no—
319C2×C4⋊C4—82no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
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⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—82no—
349C2×C22⋊C4—82no—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C42—24no—
353C2×C42—21no—
354C2×C42—22no—
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364C2×C42C2412yes—
365#103—42no—
366#103—42no—
367#103—44no—
368#102—44no—
369#102—44no—
370#102—42no—
371#102—42no—
372#102—44no—
373#102—44no—
374#102—44no—
375#97—28no—
376C4×C8—28no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—24no—
386#95—24no—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24—22no—
392C24—21no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24—81no—
397C24—81no—
398C24—81no—
399C24—81no—
400C24C22×D411yescharacteristic
401#41—48no—
402#41—48no—
403C2×Q8—22no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—48no—
408C2×Q8—44no—
409C2×Q8—82no—
410C2×Q8—82no—
411C2×Q8—84no—
412C2×Q8—84no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—48no—
430C2×D4—48no—
431C2×D4—82no—
432C2×D4—82no—
433C2×D4—82no—
434C2×D4—82no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—81no—
454C22×C4—81no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C2×#4114yes—
467C22×C4H(5,𝔽2)12yes—
468C22×C4H(5,𝔽2)12yes—
469C22×C42−1+414yes—
470C22×C4C2511yescharacteristic
471M16—48no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—48no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—84no—
510C4⋊C4—84no—
511C4⋊C4#12414yes—
512C4⋊C4#12414yes—
513C22⋊C4—24no—
514C22⋊C4—42no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—42no—
520C22⋊C4—44no—
521C22⋊C4—42no—
522C22⋊C4—82no—
523C22⋊C4—81no—
524C22⋊C4—82no—
525C22⋊C4—81no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4#12418yes—
533C42—22no—
534C42—22no—
535C42—24no—
536C42—24no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—42no—
542C42—42no—
543C42—42no—
544C42—44no—
545C42C2×#4118yes—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—41no—
551C23—42no—
552C23—41no—
553C23—42no—
554C23—42no—
555C23—41no—
556C23—41no—
557C23—42no—
558C23—41no—
559C23—42no—
560C23—41no—
561C23—42no—
562C23—41no—
563C23—42no—
564C23—84no—
565C23—84no—
566C23—84no—
567C23—162no—
568C23—162no—
569C23—162no—
570C23—162no—
571C23C2×#12412yes—
572C23#53512yes—
573C23#53612yes—
574C23C23×D411yescharacteristic
575Q8—42no—
576Q8—42no—
577Q8—164no—
578Q8—168no—
579D4—44no—
580D4—44no—
581D4—42no—
582D4—42no—
583D4—84no—
584D4—84no—
585D4—164no—
586D4—164no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—84no—
631C2×C4—164no—
632C2×C4—162no—
633C2×C4C2×#12414yes—
634C2×C4C2×#12414yes—
635C2×C4C2×#12412yes—
636C2×C4#53512yes—
637C2×C4#53612yes—
638C2×C4#54218yes—
639C2×C4#54218yes—
640C2×C4#57612yes—
641C2×C4#57712yes—
642C2×C4C22×#4112yes—
643C2×C4C2×H(5,𝔽2)11yescharacteristic
644C2×C4C2×H(5,𝔽2)11yescharacteristic
645C2×C4C2×2−1+412yes—
646C8—42no—
647C8—42no—
648C8—44no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—81no—
656C22—82no—
657C22—81no—
658C22—82no—
659C22—81no—
660C22—82no—
661C22—81no—
662C22—82no—
663C22—81no—
664C22—82no—
665C22—81no—
666C22—82no—
667C22—81no—
668C22—82no—
669C22—168no—
670C22#264014yes—
671C22C22×#12411yescharacteristic
672C22C2×#53511yescharacteristic
673C22C2×#53611yescharacteristic
674C22C2×#57611yescharacteristic
675C22C2×#57711yescharacteristic
676C22#358511yescharacteristic
677C22#358611yescharacteristic
678C4—22no—
679C4—22no—
680C4—24no—
681C4—24no—
682C4—42no—
683C4—42no—
684C4—41no—
685C4—41no—
686C4—81no—
687C4—84no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—81no—
692C4—81no—
693C4#264014yes—
694C4C2×#54212yes—
695C4C2×#54212yes—
696C2—24no—
697C2—161no—
698C2—162no—
699C2C2×#264011yescharacteristic
700C2#6062211yescharacteristic
701C2#6062511yescharacteristic
702C2#6083211yescharacteristic
703C2#6083611yescharacteristic
704C2#6087111yescharacteristic
705C2#6087311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35275C2semidirect
3. #30216C2semidirect
4. #30212C2semidirect
5. #30211C2semidirect
6. #30209C2semidirect
7. #30207C2semidirect
8. #30203C2semidirect
9. #23630C2semidirect
10. #23628C2semidirect
11. #23627C2semidirect
12. C2×#2071C2semidirect
13. #23458C2semidirect
14. #23456C2semidirect
15. #21326C2semidirect
17. #20368C2semidirect
26. #2868C22semidirect
28. #2859C22semidirect
30. #2660C22semidirect
53. #2074C22semidirect
55. #2073C22semidirect
56. #2072C22semidirect
57. #2072C22semidirect
58. #2072C22semidirect
59. #2072C22semidirect
65. #2071C22semidirect
66. #2071C22semidirect
67. #2071C22semidirect
68. #2071C22semidirect
69. #2026C22semidirect
72. #2025C22semidirect
73. #2024C22semidirect
76. #2023C22semidirect
77. #1993C22semidirect
78. #1992C22semidirect
79. #1991C22semidirect
80. #1990C22semidirect
85. #1746C22semidirect
86. #1745C22semidirect
87. #1744C22semidirect

Maximal subgroups

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

subgroupindex
#352752
#352642
#302162
#302122
#302112
#302092
#302072
#302032
#236302
#236282
#236272
C2×#20712
#234582
#234562
#213262
#212062
#203682