Finite Group Databasebeta

← #6567477

Subgroups — #6567477

Order 512.

1993 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3400C212yesmaximal, complemented
2#36595C211yesmaximal, characteristic
3#34337C211yesmaximal, characteristic, complemented
4#29801C212yesmaximal, complemented
5#29796C211yesmaximal, characteristic
6#28447C212yesmaximal, complemented
7#28441C211yesmaximal, characteristic
8#28161C211yesmaximal, characteristic, complemented
9#28157C212yesmaximal, complemented
10#25377C214yesmaximal, complemented
11#25369C212yesmaximal, complemented
12#24739C214yesmaximal, complemented
13#24733C212yesmaximal, complemented
14#23684C212yesmaximal, complemented
15#23679C211yesmaximal, characteristic, complemented
16#22736C211yesmaximal, characteristic, complemented
17#22546C212yesmaximal, complemented
18C2×#559C2212yes—
19C2×#557C2211yescharacteristic
20#3400—216no—
21C2×#497C2214yes—
22C2×#492C2212yes—
23C2×#464C2214yes—
24C2×#464C2214yes—
25C2×#464C2214yes—
26C2×#462C2212yes—
27C2×#460C2212yes—
28C2×#460C2214yes—
29C2×#461C2214yes—
30C2×#446C2212yes—
31C2×#440C2214yes—
32C2×C4×Q16C2212yes—
33#2867C2211yescharacteristic
34#2858C2212yes—
35#2548—21no—
36#2548—21no—
37#2548C2212yes—
38#2544C2214yes—
39#2445—22no—
40#2445—22no—
41#2370—21no—
42#2370—22no—
43#2366—21no—
44#2366—21no—
45#2365C2211yescharacteristic
46#2365C2212yes—
47Q8×C4⋊C4—21no—
48Q8×C4⋊C4C2211yescharacteristic
49Q8×C4⋊C4C2211yescharacteristic
50Q8×C4⋊C4C2212yes—
51#2330—21no—
52#2330—22no—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2212yes—
56C4×#127C2212yes—
57#2295C2211yescharacteristic
58#2295C2212yes—
59#2276—21no—
60C42×Q8C2211yescharacteristic
61#2083C2211yescharacteristic
62#2072C2212yes—
63#2072C2211yescharacteristic
64#2068C2212yes—
65#2067C2214yes—
66#2056C2212yes—
67#2055C2214yes—
68#2043C2211yescharacteristic
69#2042C2214yes—
70#2041C2212yes—
71#2041C2212yes—
72#2016C2212yes—
73#2016C2211yescharacteristic
74#2015C2214yes—
75#2014C2212yes—
76#2014C2212yes—
77#2014C2212yes—
78#2014C2212yes—
79#1983C2214yes—
80#1982C2212yes—
81#1959C2214yes—
82#1957C2212yes—
83#1940C2214yes—
84#1939C2212yes—
85#1880C2214yes—
86#1880C2214yes—
87#1878C2214yes—
88#1831C2211yescharacteristic
89#1831C2212yes—
90#1803C2212yes—
91#1802C2214yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94#1762C2214yes—
95C4×#103C2212yes—
96C22×Q16C2312yes—
97C22×Q16C2314yes—
98C22×SD16C2314yes—
99C22×SD16C2314yes—
100C22×SD16C2314yes—
101#559—28no—
102#559—28no—
103#557—24no—
104#557D418yescomplemented
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×#127—21no—
109C2×#127—21no—
110C2×#127—21no—
111C2×#127—21no—
112C2×#127C2312yes—
113C2×#127C2312yes—
114C2×#127C2312yes—
115C2×#125—21no—
116C2×#125—22no—
117C2×#124—22no—
118C2×#124C2314yes—
119C2×#124C2314yes—
120C2×#123—24no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8D411yescharacteristic
124C2×C4×Q8D411yescharacteristic
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2311yescharacteristic
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2312yes—
129C2×C4×Q8C2312yes—
130C2×C4×Q8C2312yes—
131C2×C4×Q8C2311yescharacteristic
132C2×C4×D4—22no—
133C2×C4×D4—22no—
134C2×C4×D4—22no—
135C2×C4×D4C2314yes—
136#497—216no—
137#492—28no—
138#464—216no—
139#464—48no—
140#464—48no—
141#462—28no—
142#461—216no—
143#460—28no—
144#460—48no—
145#446—28no—
146#440—216no—
147C4×Q16—28no—
148C2×#107C2312yes—
149C2×C4⋊C8C2314yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2314yes—
152C2×C4⋊C8C2312yes—
153C2×#103C2314yes—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#103C2312yes—
157C2×#103C2312yes—
158C2×#103C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#97C2314yes—
165C2×C4×C8C2312yes—
166#394—24no—
167#394—24no—
168#394—24no—
169#391C2312yes—
170#391C2311yescharacteristic
171#391C2312yes—
172#391C2312yes—
173#391C2311yescharacteristic
174#390—22no—
175#390—22no—
176#390—22no—
177#390—22no—
178#390—22no—
179#390—22no—
180#390—22no—
181#390—22no—
182#390C2314yes—
183#389—21no—
184#389—22no—
185#389—21no—
186#389—22no—
187#389—22no—
188#389—21no—
189#389—21no—
190#389—22no—
191#389—21no—
192#389—21no—
193#389—22no—
194#389—22no—
195#389—22no—
196#389C2312yes—
197#389C2311yescharacteristic
198#389C2311yescharacteristic
199#387—22no—
200#387—21no—
201#387—22no—
202#387—21no—
203#387—22no—
204#387—22no—
205#387—22no—
206#386—24no—
207#386—24no—
208#384—21no—
209#384—22no—
210#384C2311yescharacteristic
211#384C2312yes—
212#384C2311yescharacteristic
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4D411yescharacteristic
226C4×C4⋊C4D412yes—
227C4×C4⋊C4D411yescharacteristic
228C4×C4⋊C4D412yes—
229C4×C4⋊C4C2311yescharacteristic
230C4×C4⋊C4C2312yes—
231C4×C4⋊C4C2311yescharacteristic
232C4×C4⋊C4C2312yes—
233#376—22no—
234#376—21no—
235C43C2312yes—
236C43C2311yescharacteristic
237#340C2312yes—
238#340C2312yes—
239#340C2314yes—
240#336C2311yescharacteristic
241#336C2312yes—
242#336C2311yescharacteristic
243#336C2312yes—
244#336C2314yes—
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2412yes—
247C22×Q8C2412yes—
248C22×Q8C2411yescharacteristic
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4C2414yes—
256C23×C4—24no—
257C2×Q16—24no—
258C2×Q16—28no—
259C2×Q16—24no—
260C2×Q16—42no—
261C2×Q16—44no—
262C2×Q16—44no—
263C2×SD16—28no—
264C2×SD16—28no—
265C2×SD16—28no—
266C2×SD16—28no—
267C2×SD16—44no—
268C2×SD16—44no—
269C2×SD16—44no—
270C2×SD16—44no—
271C2×SD16—44no—
272C22×C8C2412yes—
273C22×C8C2414yes—
274C22×C8C2412yes—
275C4⋊Q8—24no—
276C4⋊Q8—22no—
277C4⋊Q8—42no—
278C4⋊Q8C2×D418yes—
279C4⋊Q8C2×D414yes—
280#127—24no—
281#127—24no—
282#127—44no—
283#127—44no—
284#127—44no—
285#127—44no—
286#127—82no—
287#127—82no—
288#125—28no—
289#125—216no—
290#124—28no—
291#124—28no—
292#124—44no—
293#124—44no—
294#124—48no—
295#123—416no—
296C4×Q8—28no—
297C4×Q8—24no—
298C4×Q8—24no—
299C4×Q8—22no—
300C4×Q8—24no—
301C4×Q8—24no—
302C4×Q8—42no—
303C4×Q8—44no—
304C4×Q8—44no—
305C4×Q8—44no—
306C4×Q8—44no—
307C4×Q8C2×D414yes—
308C4×Q8C2×D418yes—
309C4×Q8C2×D414yes—
310C4×Q8#4114yes—
311C4×D4—28no—
312C4×D4—28no—
313C4×D4—48no—
314C4×D4—84no—
315C4×D4—84no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2411yescharacteristic
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C42—22no—
356C2×C42—21no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—21no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42C2×D412yes—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D412yes—
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D412yes—
368C2×C42C2×D412yes—
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2412yes—
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2412yes—
375C2×C42C2411yescharacteristic
376#107—24no—
377C4⋊C8—24no—
378C4⋊C8—28no—
379C4⋊C8—24no—
380C4⋊C8—28no—
381#103—24no—
382#103—28no—
383#103—24no—
384#103—24no—
385#103—42no—
386#103—44no—
387#102—28no—
388#102—28no—
389#102—44no—
390#102—44no—
391#102—44no—
392#97—28no—
393C4×C8—24no—
394#95—21no—
395#95—22no—
396#95—21no—
397#95—22no—
398#95—21no—
399#95—22no—
400#95—21no—
401#95—21no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—21no—
406#95—22no—
407#95—22no—
408#95—21no—
409#95C2×D412yes—
410#95C2×D412yes—
411#95C2×D412yes—
412C24—24no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—24no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—24no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8#12114yes—
428C2×Q8C22×D414yes—
429C2×Q8C22×D412yes—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—24no—
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—24no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—44no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—44no—
447C2×D4—44no—
448C2×D4—82no—
449C2×D4—82no—
450C2×D4—82no—
451C2×D4—82no—
452C2×D4—82no—
453C2×D4—82no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—48no—
463C22×C4—84no—
464C22×C4—84no—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C2511yescharacteristic
476Q16—48no—
477Q16—88no—
478SD16—416no—
479SD16—88no—
480SD16—88no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—22no—
489C2×C8H(5,𝔽2)14yes—
490C4⋊C4—28no—
491C4⋊C4—24no—
492C4⋊C4—28no—
493C4⋊C4—24no—
494C4⋊C4—28no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—82no—
515C4⋊C4—81no—
516C4⋊C4—82no—
517C4⋊C4—81no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4#12114yes—
522C4⋊C4#12118yes—
523C4⋊C4C22×D412yes—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—84no—
528C22⋊C4—84no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C42—28no—
532C42—24no—
533C42—24no—
534C42—22no—
535C42—24no—
536C42—22no—
537C42—24no—
538C42—22no—
539C42—22no—
540C42—42no—
541C42—42no—
542C42—41no—
543C42—44no—
544C42—44no—
545C42—42no—
546C42—82no—
547C42—82no—
548C42—82no—
549C42#13714yes—
550C42C22×D412yes—
551C42C22×D414yes—
552C42C2×#4114yes—
553C42C2×#4112yes—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23C23×D411yescharacteristic
562Q8—28no—
563Q8—28no—
564Q8—24no—
565Q8—44no—
566Q8—42no—
567Q8—42no—
568Q8—42no—
569Q8—42no—
570Q8—42no—
571D4—44no—
572D4—44no—
573D4—44no—
574D4—44no—
575D4—88no—
576D4—88no—
577D4—88no—
578D4—164no—
579D4—164no—
580D4—164no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—168no—
627C2×C4C2×#12114yes—
628C2×C4C2×#12114yes—
629C2×C4C2×#12112yes—
630C2×C4C2×#12112yes—
631C2×C4C2×#13712yes—
632C2×C4C2×#13712yes—
633C2×C4C2×#13712yes—
634C2×C4#57512yes—
635C2×C4#57712yes—
636C2×C4C23×D411yescharacteristic
637C2×C4C22×#4111yescharacteristic
638C2×C4C2×H(5,𝔽2)12yes—
639C2×C4#58512yes—
640C8—24no—
641C8—42no—
642C8—44no—
643C22—84no—
644C22—84no—
645C22—84no—
646C22—84no—
647C22—84no—
648C22—84no—
649C22—84no—
650C22C22×#12111yescharacteristic
651C22#348711yescharacteristic
652C22#349811yescharacteristic
653C22C22×#13711yescharacteristic
654C22C2×#57511yescharacteristic
655C22C2×#57711yescharacteristic
656C22#358511yescharacteristic
657C4—24no—
658C4—22no—
659C4—22no—
660C4—22no—
661C4—22no—
662C4—24no—
663C4—42no—
664C4—44no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—41no—
669C4—82no—
670C4—82no—
671C4—81no—
672C4—82no—
673C4—81no—
674C4#303714yes—
675C4#348812yes—
676C4#351712yes—
677C2—164no—
678C2#3602911yescharacteristic
679C2C2×#303711yescharacteristic
680C2#6043111yescharacteristic
681C2#6189211yescharacteristic
682C2#6190411yescharacteristic
683C2#6242211yescharacteristic
684C2#6245111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2867C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3400C2semidirect
3. #34337C2semidirect
4. #29801C2semidirect
6. #28447C2semidirect
8. #28161C2semidirect
9. #28157C2semidirect
10. #25377C2semidirect
11. #25369C2semidirect
12. #24739C2semidirect
13. #24733C2semidirect
14. #23684C2semidirect
15. #23679C2semidirect
16. #22736C2semidirect
17. #22546C2semidirect
104. #557D4semidirect

Maximal subgroups

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

subgroupindex
C2×#34002
#365952
#343372
#298012
#297962
#284472
#284412
#281612
#281572
#253772
#253692
#247392
#247332
#236842
#236792
#227362
#225462