Finite Group Databasebeta

← #6508971

Subgroups — #6508971

Order 512.

1714 conjugacy classes of subgroups.

The subgroup lattice has 688 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#27603C211yesmaximal, characteristic, complemented
6#27601C211yesmaximal, characteristic, complemented
7#27601C211yesmaximal, characteristic, complemented
8#27600C211yesmaximal, characteristic, complemented
9#24940C214yesmaximal, complemented
10#24938C214yesmaximal, complemented
11#24936C214yesmaximal, complemented
12#24934C214yesmaximal, complemented
13#21941C212yesmaximal, complemented
14#21738C211yesmaximal, characteristic
15#20699C212yesmaximal, complemented
16C22×C4×D4C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×C22⋊C8C2212yes—
19#2485—22no—
20#2485—22no—
21#2485C2214yes—
22#2485C2214yes—
23C2×#397—21no—
24C2×#394—21no—
25#2353—22no—
26#2353—22no—
27#2352C2212yes—
28#2352C2212yes—
29#2352C2212yes—
30#2352C2212yes—
31#2313—21no—
32#2312—21no—
33#2308—21no—
34C2×#389C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#388—21no—
37C2×#387—21no—
38#2283C2212yes—
39#2280C2212yes—
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45#2273—21no—
46#2101C2214yescomplemented
47#2101C2214yescomplemented
48#2085C2214yescomplemented
49#2085C2214yescomplemented
50#2077C2214yescomplemented
51#2077C2214yescomplemented
52#2076C2214yescomplemented
53#2075C2214yescomplemented
54#1921C2214yescomplemented
55#1920C2214yescomplemented
56#1919C2214yescomplemented
57#1918C2214yescomplemented
58#1869C2214yes—
59#1869C2214yes—
60#1854C2214yescomplemented
61#1854C2214yescomplemented
62#1853C2212yes—
63#1853C2212yes—
64#1853C2212yescomplemented
65#1853C2212yescomplemented
66#1852C2212yes—
67#1852C2212yes—
68#1852C2212yescomplemented
69#1852C2212yescomplemented
70#1825C2214yescomplemented
71#1825C2214yescomplemented
72#1824C2212yes—
73#1824C2212yes—
74#1824C2212yescomplemented
75#1824C2212yescomplemented
76#1823C2212yes—
77#1823C2212yes—
78#1823C2212yescomplemented
79#1823C2212yescomplemented
80#1817C2212yes—
81#1808C2214yescomplemented
82#1807C2214yescomplemented
83#1779C2211yescharacteristic
84#1779C2211yescharacteristic
85#1735C2214yescomplemented
86#1735C2214yescomplemented
87C23×D4D411yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C23×C8C2314yes—
91C2×C4×D4—28no—
92C2×C4×D4D412yes—
93C2×C4×D4D412yes—
94C2×C4×D4C2312yes—
95C2×C4×D4C2312yes—
96C2×#117—22no—
97C2×#117—28no—
98C2×#117C2314yes—
99C2×#117C2312yes—
100C2×#117C2312yes—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4D411yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4D411yescharacteristic
119C22×C22⋊C4D411yescharacteristic
120C22×C42—21no—
121C22×C42C2311yescharacteristic
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#106C2314yes—
125C2×#106C2314yes—
126C2×C22⋊C8—28no—
127C2×C22⋊C8C2312yes—
128C2×C22⋊C8C2312yes—
129C2×C22⋊C8C2314yes—
130C2×C22⋊C8C2312yes—
131C2×C22⋊C8C2312yes—
132C2×C22⋊C8C2312yes—
133C2×C22⋊C8C2312yes—
134#397—24no—
135#397—24no—
136#394—24no—
137#394—24no—
138#389—22no—
139#389—22no—
140#389C2314yes—
141#389C2314yes—
142#388—22no—
143#388—22no—
144#388—22no—
145#388—22no—
146#388—24no—
147#388—24no—
148#387—22no—
149#387—22no—
150#387—22no—
151#387—22no—
152#387—24no—
153#387—24no—
154#386—22no—
155#386—22no—
156#386—22no—
157#386—22no—
158#383—22no—
159#382C2312yes—
160#382C2312yes—
161#381—22no—
162#381—22no—
163#380—21no—
164#380—21no—
165#380—21no—
166#380—21no—
167#380—22no—
168#380—21no—
169#380—21no—
170#380—21no—
171#380—21no—
172#380C2312yes—
173#380C2314yes—
174#380C2312yes—
175#380C2312yes—
176#380C2312yes—
177#380C2312yes—
178#380C2312yes—
179#380C2312yes—
180#380C2312yes—
181#380C2312yes—
182#380C2312yes—
183#379—22no—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2312yes—
186C4×C22⋊C4—22no—
187C4×C22⋊C4—22no—
188C4×C22⋊C4—22no—
189C4×C22⋊C4—22no—
190C4×C22⋊C4—22no—
191C4×C22⋊C4—22no—
192C4×C22⋊C4—22no—
193#376—22no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#336C2314yes—
214#336C2314yes—
215C25C2×D411yescharacteristic
216C25C2×D411yescharacteristic
217C22×D4—24no—
218C22×D4—24no—
219C22×D4—42no—
220C22×D4C2×D412yes—
221C22×D4C2×D412yes—
222C23×C4—24no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D412yes—
242C23×C4#4118yes—
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2412yes—
247C22×C8—28no—
248C22×C8—44no—
249C22×C8—44no—
250C22×C8C2414yes—
251C22×C8C2414yes—
252C4×D4—28no—
253C4×D4—28no—
254C4×D4—24no—
255C4×D4—24no—
256C4×D4—24no—
257C4×D4—24no—
258#117—28no—
259#117—28no—
260#117—48no—
261#117—44no—
262#117—44no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—82no—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—82no—
315C2×C22⋊C4—82no—
316C2×C22⋊C4C2×D412yes—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—44no—
327C2×C42C2×D412yes—
328C2×C42C2412yes—
329#107—44no—
330#107—44no—
331#106—44no—
332#106—44no—
333C22⋊C8—24no—
334C22⋊C8—24no—
335C22⋊C8—28no—
336C22⋊C8—42no—
337C22⋊C8—42no—
338C22⋊C8—42no—
339C22⋊C8—42no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—24no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—24no—
350#95—24no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—41no—
364C24—41no—
365C24—41no—
366C24—41no—
367C24—41no—
368C24—41no—
369C24—41no—
370C24—41no—
371C24#12314yes—
372C24#13514yes—
373C24C22×D412yes—
374C24C22×D411yescharacteristic
375C24C22×D411yescharacteristic
376C24C22×D411yescharacteristic
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—416no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
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—416no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—88no—
439C22×C4—81no—
440C22×C4—81no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—81no—
446C22×C4—81no—
447C22×C4#12314yes—
448C22×C4#12318yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C2×#4114yes—
456C22×C4C2×#4114yes—
457C22×C4C2×#4114yes—
458C22×C4C2511yescharacteristic
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—88no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—48no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—81no—
482C4⋊C4—81no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4—84no—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—44no—
491C22⋊C4—48no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C22⋊C4—84no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C42—24no—
505C42—22no—
506C42—22no—
507C42—48no—
508C42—82no—
509C42—82no—
510C42—82no—
511C42—82no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—22no—
517C23—21no—
518C23—21no—
519C23—22no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—22no—
525C23—22no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—22no—
531C23—21no—
532C23—21no—
533C23—22no—
534C23—28no—
535C23—28no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—82no—
541C23—82no—
542C23—82no—
543C23—82no—
544C23—82no—
545C23—82no—
546C23—82no—
547C23—82no—
548C23C2×#12314yes—
549C23C2×#12312yes—
550C23C2×#13512yes—
551C23C2×#13512yes—
552C23C2×#13512yes—
553C23#57512yes—
554C23C23×D411yescharacteristic
555D4—48no—
556D4—48no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—48no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—162no—
614C2×C4—162no—
615C2×C4—162no—
616C2×C4—162no—
617C2×C4C2×#12312yes—
618C2×C4C2×#12314yes—
619C2×C4C22×#4112yes—
620C2×C4#58512yes—
621C2×C4#58512yes—
622C8—44no—
623C8—44no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—24no—
635C22—22no—
636C22—22no—
637C22—24no—
638C22—24no—
639C22—42no—
640C22—42no—
641C22—41no—
642C22—42no—
643C22—41no—
644C22—41no—
645C22—42no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—88no—
652C22—88no—
653C22#309114yes—
654C22C22×#12311yescharacteristic
655C22#348511yescharacteristic
656C22#348511yescharacteristic
657C22C22×#13511yescharacteristic
658C22C2×#57511yescharacteristic
659C22#358511yescharacteristic
660C22#358611yescharacteristic
661C4—22no—
662C4—24no—
663C4—42no—
664C4—42no—
665C4—41no—
666C4—41no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4#349114yes—
676C2—24no—
677C2—42no—
678C2—42no—
679C2—44no—
680C2#3544811yescharacteristic
681C2C2×#309111yescharacteristic
682C2#6052011yescharacteristic
683C2#6169611yescharacteristic
684C2#6169911yescharacteristic
685C2#6171011yescharacteristic
686C2#6171211yescharacteristic
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. #27603C2semidirect
6. #27601C2semidirect
7. #27601C2semidirect
8. #27600C2semidirect
9. #24940C2semidirect
10. #24938C2semidirect
11. #24936C2semidirect
12. #24934C2semidirect
13. #21941C2semidirect
15. #20699C2semidirect
46. #2101C22semidirect
47. #2101C22semidirect
48. #2085C22semidirect
49. #2085C22semidirect
50. #2077C22semidirect
51. #2077C22semidirect
52. #2076C22semidirect
53. #2075C22semidirect
54. #1921C22semidirect
55. #1920C22semidirect
56. #1919C22semidirect
57. #1918C22semidirect
60. #1854C22semidirect
61. #1854C22semidirect
64. #1853C22semidirect
65. #1853C22semidirect
68. #1852C22semidirect
69. #1852C22semidirect
70. #1825C22semidirect
71. #1825C22semidirect
74. #1824C22semidirect
75. #1824C22semidirect
78. #1823C22semidirect
79. #1823C22semidirect
81. #1808C22semidirect
82. #1807C22semidirect
85. #1735C22semidirect
86. #1735C22semidirect

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
#276032
#276012
#276002
#249402
#249382
#249362
#249342
#219412
#217382
#206992