Finite Group Databasebeta

← #6970994

Subgroups — #6970994

Order 512.

1852 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36289C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic, complemented
3#32775C212yesmaximal, complemented
4#31947C214yesmaximal, complemented
5#31012C214yesmaximal, complemented
6#30203C211yesmaximal, characteristic, complemented
7#29950C212yesmaximal, complemented
8#29668C212yesmaximal, complemented
9#29148C211yesmaximal, characteristic, complemented
10#28386C211yesmaximal, characteristic, complemented
11#24814C212yesmaximal, complemented
12#24733C212yesmaximal, complemented
13#24240C212yesmaximal
14#24122C212yesmaximal, complemented
15#23933C212yesmaximal, complemented
16#23797C211yesmaximal, characteristic, complemented
17#22012C211yesmaximal, characteristic, complemented
18C2×#554C2211yescharacteristic
19C2×D42C2211yescharacteristic
20C2×#533C2212yescomplemented
21C2×#477C2212yescomplemented
22C2×#464C2214yescomplemented
23C2×#460C2212yes—
24C2×#450C2214yes—
25C2×C4×Q16C2212yescomplemented
26C2×C4×SD16C2212yes—
27#2868C2211yescharacteristic, complemented
28#2858C2211yescharacteristic
29#2857C2212yescomplemented
30#2850C2212yes—
31#2563C2211yescharacteristic
32#2552—21no—
33#2548C2211yescharacteristic
34#2542—21no—
35#2538C2212yes—
36#2504C2212yes—
37#2490—21no—
38#2490—21no—
39#2472—22no—
40#2472—22no—
41#2441—22no—
42#2435C2212yes—
43#2429—22no—
44#2423C2214yescomplemented
45#2365—21no—
46#2354—21no—
47Q8×C4⋊C4C2212yescomplemented
48D4×C22⋊C4—21no—
49D4×C22⋊C4—21no—
50C4×C4⋊Q8C2211yescharacteristic, complemented
51C4×#124C2211yescharacteristic
52C2×#390C2212yes—
53C42×D4C2211yescharacteristic
54#2097C2212yescomplemented
55#2096C2211yescharacteristic, complemented
56#2093C2211yescharacteristic
57#2093C2212yescomplemented
58#2085C2214yescomplemented
59#2071C2214yescomplemented
60#2067C2214yes—
61#2067C2214yescomplemented
62#2042C2214yescomplemented
63#2036C2214yescomplemented
64#2034C2214yes—
65#2024C2212yescomplemented
66#2023C2212yes—
67#2023C2212yescomplemented
68#2023C2212yescomplemented
69#2016C2211yescharacteristic, complemented
70#2015C2212yes—
71#2013C2211yescharacteristic
72#2013C2212yescomplemented
73#2001C2214yescomplemented
74#2000C2214yes—
75#1990C2212yescomplemented
76#1973C2214yescomplemented
77#1959C2212yes—
78#1957C2212yes—
79#1954C2212yes—
80#1939C2212yescomplemented
81#1924C2214yescomplemented
82#1880C2214yes—
83#1880C2214yescomplemented
84#1872C2214yescomplemented
85#1852C2214yescomplemented
86#1836C2212yescomplemented
87#1817C2212yescomplemented
88#1805C2214yescomplemented
89#1803C2212yescomplemented
90#1802C2212yes—
91#1797C2212yescomplemented
92#1786C2214yes—
93#1767C2211yescharacteristic, complemented
94#1744C2211yescharacteristic, complemented
95C23×D4—21no—
96C23×D4C2312yes—
97C22×Q16C2312yes—
98C22×SD16C2312yes—
99C22×SD16C2314yes—
100#554—44no—
101D42—44no—
102#533—48no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×#127C2311yescharacteristic
109C2×#125—21no—
110C2×#125—22no—
111C2×#124—21no—
112C2×#124—21no—
113C2×#124C2311yescharacteristic
114C2×#124C2312yes—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—22no—
118C2×C22⋊Q8C2314yes—
119C2×C22⋊Q8C2312yes—
120C2×#121—21no—
121C2×#121C2312yes—
122C2×#120—22no—
123C2×C4×Q8—22no—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2311yescharacteristic
126C2×C4×Q8C2314yes—
127C2×C4×D4—21no—
128C2×C4×D4D411yescharacteristic
129C2×C4×D4D411yescharacteristic
130C2×C4×D4C2311yescharacteristic
131C2×C4×D4C2311yescharacteristic
132C2×#117—22no—
133C22×C4⋊C4C2312yes—
134C22×C22⋊C4—22no—
135C22×C22⋊C4—22no—
136C22×C42C2312yes—
137#477—44no—
138#464—48no—
139#460—44no—
140#450—48no—
141C4×Q16—44no—
142C4×SD16—44no—
143C2×#107C2314yes—
144C2×#107C2312yes—
145C2×#106C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#102C2314yes—
154C2×#102C2312yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×C22⋊C8C2314yes—
159C2×C22⋊C8C2314yes—
160C2×#97C2312yes—
161C2×C4×C8C2312yes—
162#399—24no—
163#398—24no—
164#397—22no—
165#397—22no—
166#396—24no—
167#394—22no—
168#394—22no—
169#394—22no—
170#393—22no—
171#391—22no—
172#391C2312yes—
173#391C2311yescharacteristic
174#390—22no—
175#390—21no—
176#390—21no—
177#390—21no—
178#390—21no—
179#390—22no—
180#390—22no—
181#390C2312yes—
182#390C2312yes—
183#390C2312yes—
184#390C2312yes—
185#390C2311yescharacteristic
186#389—22no—
187#389—21no—
188#389—21no—
189#389—22no—
190#389—24no—
191#389C2314yes—
192#388—24no—
193#388—22no—
194#388—22no—
195#388—21no—
196#388—21no—
197#387—24no—
198#386—22no—
199#386—22no—
200#386—22no—
201#386—22no—
202#384—22no—
203#384—21no—
204#384—21no—
205#384C2311yescharacteristic
206#384C2312yes—
207#384C2312yes—
208#382C2312yes—
209#382C2311yescharacteristic
210#380—22no—
211#380—21no—
212#380—21no—
213#380C2314yes—
214#379—22no—
215#379—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C22⋊C4—21no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4C2312yes—
227C43C2311yescharacteristic
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#336C2311yescharacteristic
233#336C2312yes—
234#336C2312yes—
235#336C2311yescharacteristic
236#336C2314yes—
237C25—22no—
238C22×Q8—22no—
239C22×Q8—21no—
240C22×Q8C2411yescharacteristic
241C22×Q8C2412yes—
242C22×D4—24no—
243C22×D4—22no—
244C22×D4—21no—
245C22×D4—22no—
246C22×D4—21no—
247C22×D4—44no—
248C22×D4—42no—
249C22×D4—44no—
250C22×D4—44no—
251C22×D4—44no—
252C22×D4—44no—
253C22×D4C2×D411yescharacteristic
254C22×D4C2412yes—
255C22×D4C2411yescharacteristic
256C23×C4—22no—
257C23×C4—21no—
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2412yes—
261C2×Q16—24no—
262C2×Q16—42no—
263C2×Q16—42no—
264C2×SD16—24no—
265C2×SD16—24no—
266C2×SD16—28no—
267C2×SD16—42no—
268C2×SD16—44no—
269C2×SD16—44no—
270C22×C8C2412yes—
271C22×C8C2414yes—
272C22×C8C2412yes—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—44no—
276C4⋊Q8—44no—
277C4⋊Q8—82no—
278#127—24no—
279#125—48no—
280#125—44no—
281#124—22no—
282#124—41no—
283#124—44no—
284#124—82no—
285#124—82no—
286C22⋊Q8—28no—
287C22⋊Q8—48no—
288C22⋊Q8—44no—
289C22⋊Q8—44no—
290C22⋊Q8—48no—
291C22⋊Q8—84no—
292#121—44no—
293#121—44no—
294#120—48no—
295C4×Q8—28no—
296C4×Q8—24no—
297C4×Q8—24no—
298C4×Q8—24no—
299C4×Q8—44no—
300C4×Q8—48no—
301C4×D4—24no—
302C4×D4—24no—
303C4×D4—24no—
304C4×D4—42no—
305C4×D4—44no—
306#117—48no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2412yes—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—24no—
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—84no—
353C2×C22⋊C4—84no—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C22⋊C4C2×D412yes—
357C2×C42—24no—
358C2×C42—21no—
359C2×C42—22no—
360C2×C42—44no—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D412yes—
366C2×C42C2×D412yes—
367C2×C42C2412yes—
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371#107—42no—
372#107—44no—
373#106—42no—
374C4⋊C8—24no—
375C4⋊C8—44no—
376#103—42no—
377#103—42no—
378#103—42no—
379#103—44no—
380#103—44no—
381#102—42no—
382#102—44no—
383#102—44no—
384#102—44no—
385#102—44no—
386C22⋊C8—44no—
387C22⋊C8—44no—
388#97—42no—
389C4×C8—24no—
390#95—21no—
391#95—21no—
392#95—21no—
393#95—21no—
394#95—24no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—24no—
400#95C2×D412yes—
401#95C2×D412yes—
402#95C2×D412yes—
403C24—21no—
404C24—22no—
405C24—44no—
406C24—44no—
407C24—44no—
408C24—44no—
409C24—82no—
410C24—82no—
411C24—82no—
412C24C22×D412yes—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8—21no—
422C2×Q8—44no—
423C2×Q8—44no—
424C2×Q8—42no—
425C2×Q8—44no—
426C2×Q8—81no—
427C2×Q8—81no—
428C2×Q8H(5,𝔽2)14yes—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—22no—
436C2×D4—21no—
437C2×D4—22no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—44no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—82no—
448C2×D4—81no—
449C2×D4—81no—
450C2×D4—88no—
451C2×D4—88no—
452C2×D4—816no—
453C2×D4#13714yes—
454C2×D42−1+412yes—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C2511yescharacteristic
488Q16—48no—
489SD16—48no—
490SD16—88no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—24no—
495C2×C8—22no—
496C2×C8—24no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8—22no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—24no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—84no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—42no—
530C22⋊C4—48no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—44no—
538C22⋊C4—42no—
539C22⋊C4—44no—
540C22⋊C4—82no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C22⋊C4—82no—
545C22⋊C4—82no—
546C22⋊C4—82no—
547C22⋊C4—82no—
548C22⋊C4—84no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—24no—
554C42—22no—
555C42—21no—
556C42—21no—
557C42—42no—
558C42—42no—
559C42—48no—
560C42—42no—
561C42—44no—
562C42H(5,𝔽2)14yes—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—88no—
584C23—88no—
585C23—88no—
586C23—88no—
587C23—164no—
588C23—164no—
589C23—164no—
590C23C2×#13714yes—
591C23C23×D411yescharacteristic
592Q8—24no—
593Q8—24no—
594Q8—42no—
595Q8—42no—
596Q8—42no—
597Q8—42no—
598Q8—88no—
599Q8—162no—
600D4—28no—
601D4—24no—
602D4—44no—
603D4—42no—
604D4—44no—
605D4—88no—
606D4—88no—
607D4—84no—
608D4—162no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—22no—
627C2×C4—21no—
628C2×C4—22no—
629C2×C4—21no—
630C2×C4—22no—
631C2×C4—48no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—44no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—41no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—84no—
655C2×C4—84no—
656C2×C4—164no—
657C2×C4#53512yes—
658C2×C4#53512yes—
659C2×C4C2×#13712yes—
660C2×C4#57512yes—
661C2×C4#57612yes—
662C2×C4#57712yes—
663C2×C4C2×H(5,𝔽2)12yes—
664C2×C4C2×H(5,𝔽2)12yes—
665C2×C4C2×2−1+411yescharacteristic
666C2×C4#58511yescharacteristic
667C8—42no—
668C8—44no—
669C8—42no—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—1616no—
692C22C2×#53511yescharacteristic
693C22#348411yescharacteristic
694C22#348911yescharacteristic
695C22C22×#13711yescharacteristic
696C22C2×#57511yescharacteristic
697C22C2×#57611yescharacteristic
698C22C2×#57711yescharacteristic
699C4—22no—
700C4—22no—
701C4—22no—
702C4—42no—
703C4—42no—
704C4—41no—
705C4—41no—
706C4—42no—
707C4—44no—
708C4—81no—
709C4—82no—
710C4—81no—
711C4—82no—
712C4—84no—
713C4—81no—
714C4#321514yes—
715C4#353612yes—
716C4#354212yes—
717C2—44no—
718C2—82no—
719C2—162no—
720C2#4247811yescharacteristic
721C2C2×#321511yescharacteristic
722C2#6082211yescharacteristic
723C2#6159311yescharacteristic
724C2#6160511yescharacteristic
725C2#6212711yescharacteristic
726C2#6214411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×D42C22
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. #32775C2semidirect
4. #31947C2semidirect
5. #31012C2semidirect
6. #30203C2semidirect
7. #29950C2semidirect
8. #29668C2semidirect
9. #29148C2semidirect
10. #28386C2semidirect
11. #24814C2semidirect
12. #24733C2semidirect
14. #24122C2semidirect
15. #23933C2semidirect
16. #23797C2semidirect
17. #22012C2semidirect
20. C2×#533C22semidirect
21. C2×#477C22semidirect
22. C2×#464C22semidirect
25. C2×C4×Q16C22semidirect
27. #2868C22semidirect
29. #2857C22semidirect
44. #2423C22semidirect
47. Q8×C4⋊C4C22semidirect
50. C4×C4⋊Q8C22semidirect
54. #2097C22semidirect
55. #2096C22semidirect
57. #2093C22semidirect
58. #2085C22semidirect
59. #2071C22semidirect
61. #2067C22semidirect
62. #2042C22semidirect
63. #2036C22semidirect
65. #2024C22semidirect
67. #2023C22semidirect
68. #2023C22semidirect
69. #2016C22semidirect
72. #2013C22semidirect
73. #2001C22semidirect
75. #1990C22semidirect
76. #1973C22semidirect
80. #1939C22semidirect
81. #1924C22semidirect
83. #1880C22semidirect
84. #1872C22semidirect
85. #1852C22semidirect
86. #1836C22semidirect
87. #1817C22semidirect
88. #1805C22semidirect
89. #1803C22semidirect
91. #1797C22semidirect
93. #1767C22semidirect
94. #1744C22semidirect

Maximal subgroups

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

subgroupindex
#362892
#361342
#327752
#319472
#310122
#302032
#299502
#296682
#291482
#283862
#248142
#247332
#242402
#241222
#239332
#237972
#220122