Finite Group Databasebeta

← #6508994

Subgroups — #6508994

Order 512.

1692 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2485C211yesmaximal, characteristic
2#34185C211yesmaximal, characteristic
3#27617C212yesmaximal, complemented
4#27616C212yesmaximal, complemented
5#27604C212yesmaximal, complemented
6#27602C212yesmaximal, complemented
7#24963C214yesmaximal, complemented
8#24961C214yesmaximal, complemented
9#24959C214yesmaximal, complemented
10#24957C214yesmaximal, complemented
11#21942C211yesmaximal, characteristic, complemented
12#21942C211yesmaximal, characteristic, complemented
13#21742C211yesmaximal, characteristic
14#20702C211yesmaximal, characteristic, complemented
15#20702C211yesmaximal, characteristic, complemented
16C22×C4×D4C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C2×#407C2211yescharacteristic
19C2×#407C2211yescharacteristic
20#2485—22no—
21#2485—22no—
22#2485C2214yes—
23#2485C2214yes—
24C2×#397—21no—
25C2×#394—21no—
26#2353—22no—
27#2353—22no—
28#2352C2212yes—
29#2352C2212yes—
30#2352C2212yes—
31#2352C2212yes—
32#2313—21no—
33#2312—21no—
34#2308—21no—
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#388—21no—
38C2×#387—21no—
39#2283C2212yes—
40#2280C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46#2273—21no—
47#2101C2214yescomplemented
48#2101C2214yescomplemented
49#2085C2214yescomplemented
50#2085C2214yescomplemented
51#2077C2214yescomplemented
52#2077C2214yescomplemented
53#2076C2214yescomplemented
54#2075C2214yescomplemented
55#1921C2214yescomplemented
56#1920C2214yescomplemented
57#1919C2214yescomplemented
58#1918C2214yescomplemented
59#1870C2214yes—
60#1870C2214yes—
61#1854C2214yescomplemented
62#1854C2214yescomplemented
63#1853C2212yes—
64#1853C2212yes—
65#1853C2212yescomplemented
66#1853C2212yescomplemented
67#1852C2212yes—
68#1852C2212yes—
69#1852C2212yescomplemented
70#1852C2212yescomplemented
71#1825C2214yescomplemented
72#1825C2214yescomplemented
73#1825C2214yescomplemented
74#1825C2214yescomplemented
75#1824C2212yes—
76#1824C2212yes—
77#1823C2212yes—
78#1823C2212yes—
79#1817C2212yes—
80#1809C2214yescomplemented
81#1809C2214yescomplemented
82#1779C2212yes—
83#1736C2212yescomplemented
84#1736C2212yescomplemented
85#1736C2212yescomplemented
86#1736C2212yescomplemented
87C23×D4D411yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C22×M16C2312yes—
91C22×M16C2312yes—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4D412yes—
95C2×C4×D4D412yes—
96C2×C4×D4C2312yes—
97C2×C4×D4C2312yes—
98C2×#117—21no—
99C2×#117—21no—
100C2×#117—24no—
101C2×#117—24no—
102C2×#117C2312yes—
103C2×#117C2312yes—
104C2×#117C2312yes—
105C2×#117C2312yes—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4D411yescharacteristic
124C22×C22⋊C4D411yescharacteristic
125C22×C42—21no—
126C22×C42C2311yescharacteristic
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#106C2314yes—
130C2×#106C2314yes—
131#407—24no—
132#407—24no—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2314yes—
138C2×C22⋊C8C2314yes—
139#397—24no—
140#397—24no—
141#394—24no—
142#394—24no—
143#389—22no—
144#389—22no—
145#389C2314yes—
146#389C2314yes—
147#388—22no—
148#388—22no—
149#388—22no—
150#388—22no—
151#388—24no—
152#388—24no—
153#387—22no—
154#387—22no—
155#387—22no—
156#387—22no—
157#387—24no—
158#387—24no—
159#386—22no—
160#386—22no—
161#386—22no—
162#386—22no—
163#383—22no—
164#382C2312yes—
165#382C2312yes—
166#381—22no—
167#381—22no—
168#380—21no—
169#380—21no—
170#380—21no—
171#380—21no—
172#380—21no—
173#380—21no—
174#380—22no—
175#380—22no—
176#380C2312yes—
177#380C2312yes—
178#380C2312yes—
179#380C2312yes—
180#380C2312yes—
181#380C2314yes—
182#380C2314yes—
183#380C2312yes—
184#380C2312yes—
185#380C2312yes—
186#379—22no—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C4×C22⋊C4—22no—
190C4×C22⋊C4—22no—
191C4×C22⋊C4—22no—
192C4×C22⋊C4—22no—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—22no—
195C4×C22⋊C4—22no—
196#376—22no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#336C2314yes—
217#336C2314yes—
218C25C2×D411yescharacteristic
219C25C2×D411yescharacteristic
220C22×D4—24no—
221C22×D4—42no—
222C22×D4—42no—
223C22×D4C2×D412yes—
224C22×D4C2×D412yes—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4—42no—
238C23×C4—42no—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D412yes—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4#4114yes—
247C23×C4#4114yes—
248C23×C4C2412yes—
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C2×M16—24no—
253C2×M16—24no—
254C2×M16—42no—
255C2×M16—42no—
256C2×M16—42no—
257C2×M16—42no—
258C22×C8C2414yes—
259C22×C8C2414yes—
260C4×D4—28no—
261C4×D4—28no—
262C4×D4—24no—
263C4×D4—24no—
264C4×D4—24no—
265C4×D4—24no—
266#117—24no—
267#117—24no—
268#117—24no—
269#117—24no—
270#117—44no—
271#117—44no—
272#117—44no—
273#117—44no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—82no—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—82no—
328C2×C22⋊C4—82no—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—44no—
342C2×C42C2×D412yes—
343C2×C42C2412yes—
344#107—44no—
345#107—44no—
346#106—44no—
347#106—44no—
348C22⋊C8—24no—
349C22⋊C8—24no—
350C22⋊C8—24no—
351C22⋊C8—24no—
352C22⋊C8—44no—
353C22⋊C8—44no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—41no—
378C24—41no—
379C24—41no—
380C24—41no—
381C24—41no—
382C24—41no—
383C24—41no—
384C24—41no—
385C24#12314yes—
386C24C22×D412yes—
387C24C22×D411yescharacteristic
388C24C22×D411yescharacteristic
389C24C22×D411yescharacteristic
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—416no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—84no—
452C22×C4—84no—
453C22×C4—81no—
454C22×C4—81no—
455C22×C4—81no—
456C22×C4—81no—
457C22×C4—81no—
458C22×C4—81no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4#12314yes—
462C22×C4#12314yes—
463C22×C4#12314yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4C2511yescharacteristic
477M16—84no—
478M16—84no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—48no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—81no—
503C4⋊C4—81no—
504C4⋊C4—81no—
505C4⋊C4—81no—
506C4⋊C4—84no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—48no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—82no—
519C22⋊C4—82no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C22⋊C4—84no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C42—24no—
526C42—22no—
527C42—22no—
528C42—44no—
529C42—44no—
530C42—82no—
531C42—82no—
532C42—82no—
533C42—82no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—28no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—82no—
564C23—82no—
565C23—82no—
566C23—82no—
567C23—82no—
568C23—82no—
569C23—82no—
570C23—82no—
571C23C2×#12312yes—
572C23C2×#12312yes—
573C23C2×#12312yes—
574C23#57512yes—
575C23#57512yes—
576C23C23×D411yescharacteristic
577D4—48no—
578D4—48no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—84no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4—84no—
637C2×C4—162no—
638C2×C4—162no—
639C2×C4—162no—
640C2×C4—162no—
641C2×C4C2×#12312yes—
642C2×C4C2×#12312yes—
643C2×C4C2×#12312yes—
644C2×C4C22×#4111yescharacteristic
645C2×C4C22×#4111yescharacteristic
646C2×C4#58512yes—
647C2×C4#58512yes—
648C8—44no—
649C8—44no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—88no—
678C22—88no—
679C22#309314yes—
680C22C22×#12311yescharacteristic
681C22#348511yescharacteristic
682C22#348511yescharacteristic
683C22C2×#57511yescharacteristic
684C22C2×#57511yescharacteristic
685C22#358511yescharacteristic
686C22#358611yescharacteristic
687C4—22no—
688C4—22no—
689C4—22no—
690C4—42no—
691C4—42no—
692C4—42no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4#349112yes—
702C4#349112yes—
703C2—24no—
704C2—42no—
705C2—42no—
706C2—82no—
707C2#3544811yescharacteristic
708C2C2×#309311yescharacteristic
709C2#6052011yescharacteristic
710C2#6171011yescharacteristic
711C2#6171211yescharacteristic
712C2#6171611yescharacteristic
713C2#6171811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27617C2semidirect
4. #27616C2semidirect
5. #27604C2semidirect
6. #27602C2semidirect
7. #24963C2semidirect
8. #24961C2semidirect
9. #24959C2semidirect
10. #24957C2semidirect
11. #21942C2semidirect
12. #21942C2semidirect
14. #20702C2semidirect
15. #20702C2semidirect
47. #2101C22semidirect
48. #2101C22semidirect
49. #2085C22semidirect
50. #2085C22semidirect
51. #2077C22semidirect
52. #2077C22semidirect
53. #2076C22semidirect
54. #2075C22semidirect
55. #1921C22semidirect
56. #1920C22semidirect
57. #1919C22semidirect
58. #1918C22semidirect
61. #1854C22semidirect
62. #1854C22semidirect
65. #1853C22semidirect
66. #1853C22semidirect
69. #1852C22semidirect
70. #1852C22semidirect
71. #1825C22semidirect
72. #1825C22semidirect
73. #1825C22semidirect
74. #1825C22semidirect
80. #1809C22semidirect
81. #1809C22semidirect
83. #1736C22semidirect
84. #1736C22semidirect
85. #1736C22semidirect
86. #1736C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24852
#341852
#276172
#276162
#276042
#276022
#249632
#249612
#249592
#249572
#219422
#217422
#207022