Finite Group Databasebeta

← #6606242

Subgroups — #6606242

Order 512.

2195 conjugacy classes of subgroups.

The subgroup lattice has 745 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3331C212yesmaximal, complemented
2#36127C211yesmaximal, characteristic
3C4×#554C211yesmaximal, characteristic, complemented
4#33163C211yesmaximal, characteristic, complemented
5#33108C212yesmaximal, complemented
6#28748C211yesmaximal, characteristic, complemented
7#28280C211yesmaximal, characteristic, complemented
8#25273C212yesmaximal, complemented
9#25098C214yesmaximal, complemented
10#24760C212yesmaximal, complemented
11#24629C212yesmaximal, complemented
12#24491C214yesmaximal, complemented
13#24213C212yesmaximal, complemented
14#22650C211yesmaximal, characteristic, complemented
15#22431C212yesmaximal, complemented
16C4×#460C211yesmaximal, characteristic, complemented
17#21258C212yesmaximal, complemented
18C2×#554C2211yescharacteristic
19C2×#547C2212yes—
20C2×#532C2212yes—
21#3331—216no—
22C2×#487C2212yes—
23C2×#483C2214yes—
24C2×#476C2212yes—
25C2×#470C2214yes—
26C2×#462C2212yes—
27C2×#462C2212yes—
28C2×#462C2212yes—
29C2×#460C2211yescharacteristic
30C2×#452C2214yes—
31C2×C4×SD16C2212yes—
32C2×#436C2212yes—
33C2×#423C2212yes—
34C2×#417C2214yes—
35C2×#417C2214yes—
36#2868C2211yescharacteristic
37#2857C2212yes—
38#2547C2212yes—
39#2537—21no—
40#2537—21no—
41#2432C2214yes—
42#2422—22no—
43#2422—22no—
44#2370—21no—
45#2369C2211yescharacteristic
46#2368—22no—
47#2360C2212yes—
48#2355C2211yescharacteristic
49#2349—21no—
50D4×C4⋊C4—21no—
51D4×C4⋊C4—21no—
52D4×C4⋊C4C2211yescharacteristic
53#2345C2212yes—
54C4×C4⋊Q8C2211yescharacteristic
55C4×#127C2211yescharacteristic
56C4×#125—21no—
57C4×C22⋊Q8—22no—
58C4×C22⋊Q8C2212yes—
59C42×Q8C2211yescharacteristic
60C42×D4—21no—
61C42×D4C2211yescharacteristic
62C2×C4×C4⋊C4C2212yes—
63#2066C2214yes—
64#2065C2214yes—
65#2056C2214yes—
66#2043C2214yes—
67#2042C2214yes—
68#2015C2214yes—
69#1983C2214yes—
70#1983C2214yes—
71#1982C2211yescharacteristic
72#1982C2212yes—
73#1981C2212yes—
74#1981C2212yes—
75#1981C2212yes—
76#1981C2212yes—
77#1981C2212yes—
78#1956C2212yes—
79#1948C2211yescharacteristic
80#1948C2212yes—
81#1939C2212yes—
82#1927C2214yes—
83#1927C2214yes—
84#1926C2212yes—
85#1894C2214yes—
86#1880C2212yes—
87#1872C2214yes—
88#1851C2214yes—
89#1842C2211yescharacteristic
90#1803C2211yescharacteristic
91#1802C2212yes—
92#1797C2212yes—
93#1788C2214yes—
94C4×C4⋊C8C2211yescharacteristic
95C4×#102C2212yes—
96C22×#41C2314yes—
97C22×Q16C2314yes—
98C22×SD16C2312yes—
99C22×M16C2314yes—
100#554—24no—
101#554D418yescomplemented
102#547—28no—
103#547—28no—
104#532—28no—
105#532—44no—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×#127C2311yescharacteristic
111C2×#125—21no—
112C2×#124C2314yes—
113C2×#123—24no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8C2314yes—
116C2×C22⋊Q8C2312yes—
117C2×#121—22no—
118C2×#121—22no—
119C2×#121—22no—
120C2×C4×Q8—21no—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4—22no—
129C2×C4×D4—22no—
130C2×C4×D4D411yescharacteristic
131C2×C4×D4D411yescharacteristic
132C2×C4×D4C2312yes—
133C2×C4×D4C2311yescharacteristic
134C22×C4⋊C4—22no—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C42—22no—
138C22×C42C2312yes—
139#487—28no—
140#483—216no—
141#476—28no—
142#470—216no—
143#462—28no—
144#462—44no—
145#462—44no—
146#460—24no—
147#452—216no—
148C4×SD16—28no—
149#436—28no—
150C2×#106C2312yes—
151C2×#106C2314yes—
152#423—44no—
153C2×C4⋊C8C2311yescharacteristic
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2311yescharacteristic
157C2×C4⋊C8C2314yes—
158C2×C4⋊C8C2312yes—
159#417—216no—
160#417—48no—
161C2×#103C2312yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#102C2314yes—
170C2×#102C2312yes—
171C2×#102C2312yes—
172C2×#102C2312yes—
173C2×#102C2314yes—
174C2×C22⋊C8C2314yes—
175C2×C4×C8C2312yes—
176#397—24no—
177#394—24no—
178#394—24no—
179#391—21no—
180#391—22no—
181#391C2312yes—
182#391C2314yes—
183#390—21no—
184#390—21no—
185#390C2312yes—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—22no—
190#389—22no—
191#389—22no—
192#389C2312yes—
193#389C2312yes—
194#387—22no—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#385—22no—
200#385—22no—
201#385—22no—
202#385—24no—
203#385—24no—
204#384—21no—
205#384—21no—
206#384—21no—
207#384—21no—
208#384C2311yescharacteristic
209#384C2311yescharacteristic
210#382—21no—
211#380—22no—
212#380—22no—
213#380—22no—
214#380—22no—
215#380—22no—
216#380C2312yes—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4D411yescharacteristic
227C4×C4⋊C4D411yescharacteristic
228C4×C4⋊C4C2312yes—
229C4×C4⋊C4C2311yescharacteristic
230C4×C4⋊C4C2312yes—
231C4×C4⋊C4C2311yescharacteristic
232C4×C4⋊C4C2312yes—
233C4×C4⋊C4C2311yescharacteristic
234C4×C4⋊C4C2311yescharacteristic
235C4×C22⋊C4—22no—
236C4×C22⋊C4—22no—
237C4×C22⋊C4—22no—
238C4×C22⋊C4D412yes—
239C4×C22⋊C4D412yes—
240C4×C22⋊C4C2312yes—
241C2×#95—22no—
242C43—21no—
243C43C2311yescharacteristic
244#340C2314yes—
245#340C2312yes—
246#340C2312yes—
247#336C2314yes—
248#336C2312yes—
249C2×#41—28no—
250C2×#41—28no—
251C2×#41—28no—
252C2×#41—44no—
253C2×#41—44no—
254C2×#41—44no—
255C22×Q8—21no—
256C22×Q8C2414yes—
257C22×Q8C2411yescharacteristic
258C22×D4—24no—
259C22×D4—21no—
260C22×D4—21no—
261C22×D4C2×D411yescharacteristic
262C22×D4C2412yes—
263C23×C4—22no—
264C23×C4—22no—
265C23×C4—22no—
266C23×C4—24no—
267C23×C4C2×D412yes—
268C23×C4C2×D412yes—
269C23×C4C2412yes—
270C2×Q16—28no—
271C2×Q16—44no—
272C2×Q16—44no—
273C2×SD16—24no—
274C2×SD16—24no—
275C2×SD16—42no—
276C2×M16—28no—
277C2×M16—44no—
278C2×M16—44no—
279C22×C8C2412yes—
280C22×C8C2414yes—
281C22×C8C2412yes—
282C4⋊Q8—24no—
283C4⋊Q8—22no—
284C4⋊Q8—24no—
285C4⋊Q8—24no—
286C4⋊Q8—24no—
287C4⋊Q8—42no—
288C4⋊Q8—42no—
289C4⋊Q8C2×D414yes—
290#127—22no—
291#127#4114yes—
292#125—28no—
293#124—28no—
294#124—44no—
295#123—416no—
296C22⋊Q8—24no—
297C22⋊Q8—216no—
298C22⋊Q8—28no—
299C22⋊Q8—28no—
300C22⋊Q8C2×D418yes—
301#121—48no—
302#121—48no—
303#121—48no—
304C4×Q8—22no—
305C4×Q8—22no—
306C4×Q8—22no—
307C4×Q8—22no—
308C4×Q8—41no—
309C4×Q8—82no—
310C4×Q8C2×D414yes—
311C4×D4—28no—
312C4×D4—24no—
313C4×D4—24no—
314C4×D4—24no—
315C4×D4—42no—
316C4×D4—48no—
317C4×D4—48no—
318C4×D4—44no—
319C4×D4—44no—
320C4×D4—44no—
321C4×D4C2×D414yes—
322C4×D4#4114yes—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—42no—
346C2×C4⋊C4—48no—
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D418yes—
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2411yescharacteristic
354C2×C4⋊C4C2411yescharacteristic
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2411yescharacteristic
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4C2×D412yes—
366C2×C22⋊C4C2×D412yes—
367C2×C22⋊C4C2×D412yes—
368C2×C42—21no—
369C2×C42—24no—
370C2×C42—24no—
371C2×C42—21no—
372C2×C42—22no—
373C2×C42—21no—
374C2×C42—21no—
375C2×C42—21no—
376C2×C42—21no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—84no—
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D412yes—
382C2×C42C2×D412yes—
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2411yescharacteristic
387C2×C42C2412yes—
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390#106—24no—
391#106—28no—
392C4⋊C8—24no—
393C4⋊C8—22no—
394C4⋊C8—22no—
395C4⋊C8—24no—
396C4⋊C8—44no—
397C4⋊C8—42no—
398#103—24no—
399#103—28no—
400#103—28no—
401#103—28no—
402#103—28no—
403#103—44no—
404#103—44no—
405#103—44no—
406#102—24no—
407#102—24no—
408#102—28no—
409#102—24no—
410#102—44no—
411C22⋊C8—28no—
412C4×C8—24no—
413#95—21no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—21no—
421#95—21no—
422#95—22no—
423#95—22no—
424#95—24no—
425#95—24no—
426#95C2×D412yes—
427#95C2×D412yes—
428#95C2×D412yes—
429C24—22no—
430C24C22×D412yes—
431#41—416no—
432#41—416no—
433#41—88no—
434#41—88no—
435C2×Q8—24no—
436C2×Q8—21no—
437C2×Q8—24no—
438C2×Q8—21no—
439C2×Q8—24no—
440C2×Q8—24no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—24no—
444C2×Q8—24no—
445C2×Q8—21no—
446C2×Q8—24no—
447C2×Q8—81no—
448C2×Q8—81no—
449C2×Q8C22×D412yes—
450C2×D4—24no—
451C2×D4—24no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—48no—
462C2×D4—48no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—84no—
466C2×D4#12114yes—
467C2×D4C2×#4114yes—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—28no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—24no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—44no—
488C22×C4—48no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—48no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—84no—
495C22×C4—82no—
496C22×C4—82no—
497C22×C4—82no—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D412yes—
503C22×C4C22×D414yes—
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C2×#4114yes—
509C22×C4C2511yescharacteristic
510Q16—88no—
511SD16—48no—
512M16—88no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—24no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—24no—
519C2×C8—24no—
520C2×C8—22no—
521C2×C8—22no—
522C4⋊C4—28no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—28no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—22no—
530C4⋊C4—21no—
531C4⋊C4—22no—
532C4⋊C4—21no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—21no—
536C4⋊C4—22no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—44no—
541C4⋊C4—48no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—82no—
549C4⋊C4—81no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4#12114yes—
554C4⋊C4C22×D412yes—
555C4⋊C4C22×D414yes—
556C4⋊C4H(5,𝔽2)12yes—
557C22⋊C4—28no—
558C22⋊C4—24no—
559C22⋊C4—42no—
560C22⋊C4—48no—
561C22⋊C4—48no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—48no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4#12118yes—
568C42—24no—
569C42—24no—
570C42—24no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—21no—
576C42—21no—
577C42—41no—
578C42—42no—
579C42—42no—
580C42—42no—
581C42—42no—
582C42—44no—
583C42—81no—
584C42—82no—
585C42—82no—
586C42#13714yes—
587C42C22×D412yes—
588C42C2×#4112yes—
589C42C2×#4112yes—
590C42H(5,𝔽2)12yes—
591C23—22no—
592C23—22no—
593C23—22no—
594C23—22no—
595C23—22no—
596C23—22no—
597C23—24no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23C2×#12114yes—
605C23C23×D411yescharacteristic
606Q8—22no—
607Q8—22no—
608Q8—41no—
609Q8—41no—
610Q8—44no—
611Q8—44no—
612Q8—44no—
613Q8—44no—
614Q8—84no—
615D4—24no—
616D4—24no—
617D4—28no—
618D4—48no—
619D4—48no—
620D4—42no—
621D4—42no—
622D4—816no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—22no—
627C2×C4—21no—
628C2×C4—22no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—24no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—21no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—21no—
645C2×C4—22no—
646C2×C4—21no—
647C2×C4—22no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—42no—
656C2×C4—48no—
657C2×C4—41no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—41no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—41no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—84no—
670C2×C4—816no—
671C2×C4—88no—
672C2×C4—88no—
673C2×C4—88no—
674C2×C4C2×#12112yes—
675C2×C4C2×#12114yes—
676C2×C4C2×#12112yes—
677C2×C4#54612yes—
678C2×C4#54612yes—
679C2×C4C2×#13712yes—
680C2×C4C2×#13712yes—
681C2×C4C2×#13712yes—
682C2×C4#57512yes—
683C2×C4#57712yes—
684C2×C4C23×D411yescharacteristic
685C2×C4C22×#4111yescharacteristic
686C2×C4C22×#4111yescharacteristic
687C2×C4C2×H(5,𝔽2)11yescharacteristic
688C2×C4C2×H(5,𝔽2)11yescharacteristic
689C2×C4#58511yescharacteristic
690C8—42no—
691C8—42no—
692C8—44no—
693C22—24no—
694C22—24no—
695C22—24no—
696C22—42no—
697C22—42no—
698C22—42no—
699C22—42no—
700C22—44no—
701C22—44no—
702C22—44no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22C22×#12111yescharacteristic
708C22C2×#54611yescharacteristic
709C22#348811yescharacteristic
710C22C22×#13711yescharacteristic
711C22C2×#57511yescharacteristic
712C22C2×#57711yescharacteristic
713C22#358611yescharacteristic
714C4—22no—
715C4—22no—
716C4—24no—
717C4—22no—
718C4—42no—
719C4—44no—
720C4—41no—
721C4—42no—
722C4—42no—
723C4—42no—
724C4—42no—
725C4—82no—
726C4—84no—
727C4—81no—
728C4—82no—
729C4#303714yes—
730C4#333114yescomplemented
731C4D4×#4111yescharacteristic
732C4#348111yescharacteristic
733C4#349811yescharacteristic
734C4#351411yescharacteristic
735C2—44no—
736C2—84no—
737C2#3625811yescharacteristic
738C2C2×#303711yescharacteristic
739C2#6042911yescharacteristic
740C2C2×#333111yescharacteristic
741C2#6115911yescharacteristic
742C2#6196611yescharacteristic
743C2#6199611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36127C2
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
1. C2×#3331C2semidirect
3. C4×#554C2semidirect
4. #33163C2semidirect
5. #33108C2semidirect
6. #28748C2semidirect
7. #28280C2semidirect
8. #25273C2semidirect
9. #25098C2semidirect
10. #24760C2semidirect
11. #24629C2semidirect
12. #24491C2semidirect
13. #24213C2semidirect
14. #22650C2semidirect
15. #22431C2semidirect
16. C4×#460C2semidirect
17. #21258C2semidirect
101. #554D4semidirect
730. C4#3331semidirect

Maximal subgroups

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

subgroupindex
C2×#33312
#361272
C4×#5542
#331632
#331082
#287482
#282802
#252732
#250982
#247602
#246292
#244912
#242132
#226502
#224312
C4×#4602
#212582