Finite Group Databasebeta

← #6493629

Subgroups — #6493629

Order 512.

1922 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2#34012C211yesmaximal, characteristic
3#33114C214yesmaximal, complemented
4#29923C212yesmaximal, complemented
5#29922C212yesmaximal, complemented
6#27980C212yesmaximal, complemented
7#27916C212yesmaximal, complemented
8#27896C214yesmaximal, complemented
9#27586C212yesmaximal, complemented
10#27583C212yesmaximal, complemented
11#24010C214yesmaximal
12#21987C212yesmaximal, complemented
13#21811C212yesmaximal, complemented
14#20715C211yesmaximal, characteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×C22⋊Q8C2212yes—
17C23×C42C2211yescharacteristic
18C2×#485C2218yes—
19C2×#432C2212yes—
20#2857—21no—
21#2857—21no—
22#2857—21no—
23#2857—21no—
24#2857C2212yes—
25#2857C2212yes—
26#2857C2212yes—
27#2857C2212yes—
28C2×#391C2212yes—
29C2×#389—21no—
30C2×#389—21no—
31#2285C2211yescharacteristic
32#2285C2211yescharacteristic
33#2285C2211yescharacteristic
34#2285C2211yescharacteristic
35#2285C2211yescharacteristic
36#2285C2211yescharacteristic
37C2×#384C2211yescharacteristic
38#2283—21no—
39#2283—21no—
40#2283—21no—
41#2283—21no—
42#2283C2212yes—
43#2283C2212yes—
44#2280—21no—
45#2280—21no—
46#2280C2212yes—
47C2×#380—21no—
48C2×#380—21no—
49C4×#117—21no—
50C4×#117—21no—
51C4×#117C2212yes—
52#2096C2214yescomplemented
53#2095C2214yescomplemented
54#2072C2214yescomplemented
55#2072C2214yescomplemented
56#2036C2218yes—
57#2035C2218yes—
58#1981C2214yescomplemented
59#1981C2214yescomplemented
60#1960C2214yescomplemented
61#1960C2214yescomplemented
62#1948C2214yescomplemented
63#1947C2214yescomplemented
64#1928C2218yescomplemented
65#1927C2218yescomplemented
66#1873C2218yes—
67#1853C2214yes—
68#1852C2214yes—
69#1847C2214yescomplemented
70#1836C2214yescomplemented
71#1817C2212yes—
72#1797C2212yes—
73#1797C2212yes—
74#1787C2214yes—
75#1787C2214yes—
76#1779C2212yes—
77#1749C2212yes—
78#1749C2212yes—
79C23×Q8C2312yes—
80C24×C4D411yescharacteristic
81C24×C4D411yescharacteristic
82C24×C4C2311yescharacteristic
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—28no—
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2312yes—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—48no—
91C2×C22⋊Q8C2314yes—
92C2×C22⋊Q8C2314yes—
93C2×#117—21no—
94C2×#117—21no—
95C2×#117—21no—
96C2×#117—21no—
97C2×#117C2312yes—
98C2×#117C2312yes—
99C22×C4⋊C4—22no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4C2312yes—
105C22×C22⋊C4—22no—
106C22×C42—28no—
107C22×C42D411yescharacteristic
108C22×C42D411yescharacteristic
109C22×C42D411yescharacteristic
110C22×C42D411yescharacteristic
111C22×C42D411yescharacteristic
112C22×C42D411yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119#485—416no—
120#432—28no—
121C2×#107C2318yes—
122C2×#106C2318yes—
123C2×C4⋊C8C2314yes—
124C2×#103C2318yes—
125C2×#103C2318yes—
126C2×#103C2318yes—
127C2×#103C2318yes—
128C2×C22⋊C8C2314yes—
129C2×C22⋊C8C2314yes—
130C2×C22⋊C8C2314yes—
131C2×#97C2314yes—
132#391—22no—
133#391—22no—
134#391C2314yes—
135#391C2314yes—
136#389—22no—
137#389—22no—
138#389—22no—
139#389—22no—
140#389—22no—
141#389—22no—
142#389—22no—
143#389—22no—
144#384—21no—
145#384—21no—
146#384C2311yescharacteristic
147#384C2311yescharacteristic
148#384C2312yes—
149#384C2312yes—
150#384C2311yescharacteristic
151#384C2311yescharacteristic
152#383—22no—
153#383—22no—
154#382—21no—
155#382—21no—
156#382—21no—
157#382—21no—
158#382C2312yes—
159#382C2312yes—
160#381—22no—
161#381—22no—
162#381—22no—
163#380—22no—
164#380—22no—
165#380—22no—
166#380—22no—
167#380—22no—
168#380—22no—
169#380—22no—
170#380—22no—
171#380—21no—
172#380—21no—
173#380—21no—
174#380—21no—
175#380—21no—
176#380—21no—
177#380—21no—
178#380—21no—
179#380C2312yes—
180#380C2312yes—
181#380C2312yes—
182#380C2312yes—
183#379—22no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200C4×C22⋊C4—22no—
201C4×C22⋊C4—22no—
202#376—22no—
203#376—22no—
204#376—22no—
205C2×#95—22no—
206C2×#95—22no—
207C43—22no—
208#340C2318yes—
209#340C2318yes—
210#336C2314yes—
211#336C2314yes—
212C25C2×D411yescharacteristic
213C22×Q8—22no—
214C22×Q8—28no—
215C22×Q8—44no—
216C22×Q8—44no—
217C22×Q8C2414yes—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C22×C8C2418yes—
248C4⋊Q8—28no—
249C4⋊Q8—28no—
250C4⋊Q8—44no—
251C4⋊Q8—44no—
252C22⋊Q8—216no—
253C22⋊Q8—48no—
254C22⋊Q8—48no—
255C22⋊Q8—48no—
256#117—28no—
257#117—44no—
258#117—44no—
259#117—44no—
260#117—44no—
261#117—44no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—48no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4C2414yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—48no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42#4118yes—
331C2×C42C2411yescharacteristic
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335#107—48no—
336#106—48no—
337C4⋊C8—28no—
338#103—48no—
339#103—48no—
340#103—48no—
341#103—48no—
342C22⋊C8—28no—
343C22⋊C8—28no—
344C22⋊C8—44no—
345#97—28no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—41no—
368C24—41no—
369C24C22×D411yescharacteristic
370C24C22×D411yescharacteristic
371C24C22×D411yescharacteristic
372C2×Q8—28no—
373C2×Q8—28no—
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×Q8—816no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
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—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
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—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—416no—
435C22×C4—416no—
436C22×C4—416no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4#12314yes—
444C22×C4#12314yes—
445C22×C4#13714yes—
446C22×C4#13714yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C2×#4114yes—
459C22×C4H(5,𝔽2)14yes—
460C22×C4C2511yescharacteristic
461C2×C8—28no—
462C2×C8—28no—
463C2×C8—28no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—28no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—48no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C22⋊C4—48no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—48no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—44no—
509C42—44no—
510C42—44no—
511C42—44no—
512C42—44no—
513C42—44no—
514C42#12314yes—
515C42#12314yes—
516C42C2×#4114yes—
517C42C2×#4114yes—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—28no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23C2×#12312yes—
542C23#53512yes—
543C23C2×#13712yes—
544C23C2×#13712yes—
545C23#57512yes—
546C23C23×D411yescharacteristic
547Q8—48no—
548Q8—48no—
549Q8—88no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—88no—
609C2×C4—88no—
610C2×C4C2×#12312yes—
611C2×C4C2×#12312yes—
612C2×C4C2×#12312yes—
613C2×C4C2×#12312yes—
614C2×C4C2×#12312yes—
615C2×C4#53512yes—
616C2×C4C2×#13712yes—
617C2×C4C2×#13712yes—
618C2×C4C2×#13712yes—
619C2×C4C2×#13712yes—
620C2×C4#57512yes—
621C2×C4C22×#4112yes—
622C2×C4C2×H(5,𝔽2)12yes—
623C2×C4#58512yes—
624C8—48no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—416no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22#309514yes—
647C22C22×#12311yescharacteristic
648C22C2×#53511yescharacteristic
649C22#344911yescharacteristic
650C22C22×#13711yescharacteristic
651C22C22×#13711yescharacteristic
652C22C2×#57511yescharacteristic
653C22C2×#57511yescharacteristic
654C4—22no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—22no—
659C4—22no—
660C4—22no—
661C4—22no—
662C4—41no—
663C4—41no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—84no—
668C4—82no—
669C4—84no—
670C4#309514yes—
671C4#349414yes—
672C2—24no—
673C2—42no—
674C2—42no—
675C2#3526211yescharacteristic
676C2C2×#309511yescharacteristic
677C2C2×#309511yescharacteristic
678C2#6079011yescharacteristic
679C2#6080011yescharacteristic
680C2#6095211yescharacteristic
681C2#6096311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33114C2semidirect
4. #29923C2semidirect
5. #29922C2semidirect
6. #27980C2semidirect
7. #27916C2semidirect
8. #27896C2semidirect
9. #27586C2semidirect
10. #27583C2semidirect
12. #21987C2semidirect
13. #21811C2semidirect
52. #2096C22semidirect
53. #2095C22semidirect
54. #2072C22semidirect
55. #2072C22semidirect
58. #1981C22semidirect
59. #1981C22semidirect
60. #1960C22semidirect
61. #1960C22semidirect
62. #1948C22semidirect
63. #1947C22semidirect
64. #1928C22semidirect
65. #1927C22semidirect
69. #1847C22semidirect
70. #1836C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
#340122
#331142
#299232
#299222
#279802
#279162
#278962
#275862
#275832
#240102
#219872
#218112
#207152