Finite Group Databasebeta

← #6606253

Subgroups — #6606253

Order 512.

2165 conjugacy classes of subgroups.

The subgroup lattice has 741 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3329C212yesmaximal
2D4×C4⋊Q8C211yesmaximal, characteristic, complemented
3C4×#555C211yesmaximal, characteristic
4#33162C211yesmaximal, characteristic, complemented
5#33114C212yesmaximal, complemented
6#28748C211yesmaximal, characteristic, complemented
7#28284C211yesmaximal, characteristic, complemented
8#25278C212yesmaximal, complemented
9#25099C214yesmaximal, complemented
10#24759C212yesmaximal, complemented
11#24630C212yesmaximal, complemented
12#24498C214yesmaximal, complemented
13#24209C212yesmaximal, complemented
14#22650C211yesmaximal, characteristic, complemented
15#22430C212yesmaximal, complemented
16C4×#461C211yesmaximal, characteristic, complemented
17#21259C212yesmaximal, complemented
18C2×#555C2211yescharacteristic
19C2×D4×Q8C2212yes—
20C22×C4⋊Q8C2212yes—
21#3329—216no—
22C2×#487C2212yes—
23C2×#485C2214yes—
24C2×#477C2212yes—
25C2×#468C2214yes—
26C2×#461C2212yes—
27C2×#461C2211yescharacteristic
28C2×#461C2212yes—
29C2×#461C2212yes—
30C2×#448C2214yes—
31C2×C4×SD16C2212yes—
32C2×#436C2212yes—
33C2×#423C2212yes—
34C2×#419C2214yes—
35C2×#419C2214yes—
36#2868C2211yescharacteristic
37#2857C2212yes—
38#2551—21no—
39#2551—21no—
40#2546C2212yes—
41#2440—22no—
42#2440—22no—
43#2434C2214yes—
44#2372—22no—
45#2369C2211yescharacteristic
46#2363—21no—
47#2363C2212yes—
48#2354C2211yescharacteristic
49#2354C2211yescharacteristic
50D4×C4⋊C4—21no—
51D4×C4⋊C4—21no—
52#2346—21no—
53#2346C2212yes—
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×#124—22no—
57C4×#121—21no—
58C4×#121C2212yes—
59C42×D4C2211yescharacteristic
60C42×D4C2211yescharacteristic
61C4×#117—21no—
62C4×#117C2212yes—
63#2068C2214yes—
64#2068C2214yes—
65#2055C2214yes—
66#2043C2214yes—
67#2043C2214yes—
68#2013C2214yes—
69#1981C2211yescharacteristic
70#1981C2214yes—
71#1981C2214yes—
72#1981C2212yes—
73#1981C2212yes—
74#1981C2212yes—
75#1981C2212yes—
76#1981C2212yes—
77#1981C2212yes—
78#1956C2212yes—
79#1948C2211yescharacteristic
80#1947C2212yes—
81#1939C2212yes—
82#1928C2214yes—
83#1927C2214yes—
84#1927C2212yes—
85#1894C2214yes—
86#1877C2212yes—
87#1873C2214yes—
88#1851C2214yes—
89#1842C2211yescharacteristic
90#1803C2212yes—
91#1802C2211yescharacteristic
92#1797C2212yes—
93#1787C2214yes—
94C4×C4⋊C8C2211yescharacteristic
95C4×#103C2212yes—
96C23×Q8C2314yes—
97C22×SD16C2312yes—
98C22×SD16C2314yes—
99C22×M16C2314yes—
100#555—24no—
101#555D418yes—
102D4×Q8—28no—
103D4×Q8—28no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—28no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8—22no—
108C2×C4⋊Q8—44no—
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2314yes—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2312yes—
114C2×C4⋊Q8C2311yescharacteristic
115C2×#124—22no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8C2314yes—
118C2×#121—21no—
119C2×#121C2312yes—
120C2×C4×Q8—21no—
121C2×C4×Q8C2312yes—
122C2×C4×D4—22no—
123C2×C4×D4—22no—
124C2×C4×D4D411yescharacteristic
125C2×C4×D4D411yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2311yescharacteristic
130C2×#117—21no—
131C2×#117—21no—
132C2×#117C2312yes—
133C2×#117C2312yes—
134C22×C4⋊C4—22no—
135C22×C4⋊C4—22no—
136C22×C42—21no—
137C22×C42C2312yes—
138#487—28no—
139#485—216no—
140#477—28no—
141#468—216no—
142#461—24no—
143#461—28no—
144#461—44no—
145#461—44no—
146#448—216no—
147C4×SD16—28no—
148#436—28no—
149C2×#107C2314yes—
150C2×#106C2312yes—
151#423—44no—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2311yescharacteristic
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2311yescharacteristic
156C2×C4⋊C8C2314yes—
157C2×C4⋊C8C2312yes—
158#419—216no—
159#419—48no—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2312yes—
165C2×#103C2312yes—
166C2×#103C2314yes—
167C2×#103C2312yes—
168C2×#103C2314yes—
169C2×#103C2314yes—
170C2×#103C2314yes—
171C2×#103C2314yes—
172C2×#102C2312yes—
173C2×C22⋊C8C2314yes—
174C2×C4×C8C2312yes—
175#398—24no—
176#395—24no—
177#395—24no—
178#391—21no—
179#391—22no—
180#391C2312yes—
181#391C2314yes—
182#390—21no—
183#390—22no—
184#390C2312yes—
185#389—21no—
186#389—22no—
187#389—22no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—22no—
192#389—22no—
193#389C2312yes—
194#389C2312yes—
195#388—22no—
196#388—22no—
197#388—22no—
198#388—22no—
199#388—22no—
200#388—22no—
201#386—22no—
202#386—22no—
203#385—24no—
204#385—24no—
205#384—21no—
206#384—21no—
207#384—21no—
208#384—21no—
209#384C2311yescharacteristic
210#384C2311yescharacteristic
211#382—22no—
212#380—22no—
213#380—22no—
214#380—21no—
215#380C2312yes—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4D411yescharacteristic
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2311yescharacteristic
229C4×C4⋊C4C2311yescharacteristic
230C4×C22⋊C4—22no—
231C4×C22⋊C4—22no—
232C4×C22⋊C4—21no—
233C4×C22⋊C4—21no—
234C4×C22⋊C4—22no—
235C4×C22⋊C4—21no—
236C4×C22⋊C4—22no—
237C4×C22⋊C4D412yes—
238C4×C22⋊C4D412yes—
239C4×C22⋊C4C2312yes—
240#376—22no—
241#376—22no—
242C43—22no—
243C43C2311yescharacteristic
244#340C2314yes—
245#340C2312yes—
246#340C2312yes—
247#336C2314yes—
248#336C2312yes—
249C22×Q8—21no—
250C22×Q8—24no—
251C22×Q8—28no—
252C22×Q8—28no—
253C22×Q8—28no—
254C22×Q8—44no—
255C22×Q8—44no—
256C22×Q8—44no—
257C22×Q8C2414yes—
258C22×Q8C2412yes—
259C22×D4—22no—
260C22×D4C2×D411yescharacteristic
261C22×D4C2411yescharacteristic
262C23×C4—21no—
263C23×C4—24no—
264C23×C4—21no—
265C23×C4—21no—
266C23×C4C2×D412yes—
267C23×C4C2×D412yes—
268C23×C4C2412yes—
269C2×SD16—24no—
270C2×SD16—24no—
271C2×SD16—28no—
272C2×SD16—42no—
273C2×SD16—44no—
274C2×SD16—44no—
275C2×M16—28no—
276C2×M16—44no—
277C2×M16—44no—
278C22×C8C2412yes—
279C22×C8C2414yes—
280C22×C8C2412yes—
281C4⋊Q8—28no—
282C4⋊Q8—24no—
283C4⋊Q8—22no—
284C4⋊Q8—24no—
285C4⋊Q8—24no—
286C4⋊Q8—22no—
287C4⋊Q8—24no—
288C4⋊Q8—48no—
289C4⋊Q8—48no—
290C4⋊Q8—44no—
291C4⋊Q8—42no—
292C4⋊Q8—42no—
293C4⋊Q8—48no—
294C4⋊Q8C2×D414yes—
295C4⋊Q8#4114yes—
296#124—216no—
297C22⋊Q8—28no—
298C22⋊Q8—28no—
299C22⋊Q8—416no—
300#121—28no—
301#121—24no—
302#121C2×D418yes—
303C4×Q8—24no—
304C4×Q8—24no—
305C4×Q8—82no—
306C4×D4—24no—
307C4×D4—22no—
308C4×D4—24no—
309C4×D4—22no—
310C4×D4—22no—
311C4×D4—22no—
312C4×D4—41no—
313C4×D4—48no—
314C4×D4—84no—
315C4×D4C2×D414yes—
316C4×D4C2×D414yes—
317C4×D4#4114yes—
318#117—24no—
319#117—28no—
320#117—24no—
321#117—42no—
322#117—44no—
323#117C2×D418yes—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—48no—
344C2×C4⋊C4—48no—
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2411yescharacteristic
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2411yescharacteristic
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4C2×D412yes—
365C2×C22⋊C4C2×D412yes—
366C2×C22⋊C4C2×D412yes—
367C2×C42—22no—
368C2×C42—24no—
369C2×C42—24no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—21no—
375C2×C42—21no—
376C2×C42—21no—
377C2×C42—22no—
378C2×C42—44no—
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D412yes—
382C2×C42C2×D412yes—
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2412yes—
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389#107—28no—
390#106—24no—
391C4⋊C8—24no—
392C4⋊C8—22no—
393C4⋊C8—22no—
394C4⋊C8—24no—
395C4⋊C8—44no—
396C4⋊C8—42no—
397#103—24no—
398#103—24no—
399#103—28no—
400#103—28no—
401#103—28no—
402#103—28no—
403#103—24no—
404#103—28no—
405#103—44no—
406#103—44no—
407#103—44no—
408#103—44no—
409#102—24no—
410C22⋊C8—28no—
411C4×C8—24no—
412#95—21no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—21no—
420#95—21no—
421#95—22no—
422#95—22no—
423#95—24no—
424#95—24no—
425#95C2×D412yes—
426#95C2×D412yes—
427#95C2×D412yes—
428C24—21no—
429C24C22×D412yes—
430C2×Q8—24no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—24no—
434C2×Q8—22no—
435C2×Q8—24no—
436C2×Q8—24no—
437C2×Q8—22no—
438C2×Q8—24no—
439C2×Q8—24no—
440C2×Q8—22no—
441C2×Q8—24no—
442C2×Q8—48no—
443C2×Q8—416no—
444C2×Q8—416no—
445C2×Q8—48no—
446C2×Q8—81no—
447C2×Q8—81no—
448C2×Q8—84no—
449C2×Q8—88no—
450C2×Q8—88no—
451C2×Q8C2×#4114yes—
452C2×D4—28no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—82no—
461C2×D4—82no—
462C2×D4#12114yes—
463C2×D4C22×D412yes—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—24no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—48no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—48no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—81no—
493C22×C4—84no—
494C22×C4C22×D412yes—
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D412yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D414yes—
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D412yes—
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C2×#4114yes—
505C22×C4C2511yescharacteristic
506SD16—48no—
507SD16—88no—
508M16—88no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—24no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—24no—
515C2×C8—24no—
516C2×C8—22no—
517C2×C8—22no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—21no—
530C4⋊C4—22no—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C4⋊C4—42no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—48no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—82no—
543C4⋊C4—81no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4#12114yes—
548C4⋊C4#13714yes—
549C4⋊C4C22×D412yes—
550C4⋊C4C22×D414yes—
551C4⋊C4H(5,𝔽2)12yes—
552C22⋊C4—28no—
553C22⋊C4—28no—
554C22⋊C4—24no—
555C22⋊C4—24no—
556C22⋊C4—24no—
557C22⋊C4—42no—
558C22⋊C4—44no—
559C22⋊C4—48no—
560C22⋊C4—42no—
561C22⋊C4—44no—
562C22⋊C4—82no—
563C22⋊C4—82no—
564C22⋊C4#12118yes—
565C42—28no—
566C42—24no—
567C42—24no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—21no—
573C42—21no—
574C42—41no—
575C42—44no—
576C42—44no—
577C42—42no—
578C42—42no—
579C42—42no—
580C42—44no—
581C42—44no—
582C42—82no—
583C42#13714yes—
584C42C22×D412yes—
585C42C2×#4112yes—
586C42C2×#4112yes—
587C42H(5,𝔽2)12yes—
588C23—22no—
589C23—22no—
590C23—22no—
591C23—22no—
592C23—22no—
593C23—22no—
594C23—22no—
595C23—41no—
596C23—41no—
597C23—41no—
598C23—41no—
599C23—41no—
600C23—41no—
601C23C2×#12114yes—
602C23C23×D411yescharacteristic
603Q8—24no—
604Q8—24no—
605Q8—42no—
606Q8—42no—
607Q8—44no—
608Q8—44no—
609Q8—44no—
610Q8—44no—
611Q8—84no—
612Q8—816no—
613D4—22no—
614D4—22no—
615D4—28no—
616D4—41no—
617D4—41no—
618D4—88no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—22no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—22no—
634C2×C4—24no—
635C2×C4—22no—
636C2×C4—24no—
637C2×C4—21no—
638C2×C4—22no—
639C2×C4—21no—
640C2×C4—22no—
641C2×C4—21no—
642C2×C4—22no—
643C2×C4—48no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—48no—
653C2×C4—48no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—41no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—41no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—41no—
665C2×C4—42no—
666C2×C4—84no—
667C2×C4—816no—
668C2×C4—84no—
669C2×C4C2×#12114yes—
670C2×C4C2×#12112yes—
671C2×C4C2×#12112yes—
672C2×C4#54612yes—
673C2×C4#54612yes—
674C2×C4C2×#13712yes—
675C2×C4C2×#13714yes—
676C2×C4C2×#13712yes—
677C2×C4C2×#13712yes—
678C2×C4C2×#13712yes—
679C2×C4#57712yes—
680C2×C4C23×D411yescharacteristic
681C2×C4C22×#4111yescharacteristic
682C2×C4C22×#4111yescharacteristic
683C2×C4C2×H(5,𝔽2)11yescharacteristic
684C2×C4C2×H(5,𝔽2)11yescharacteristic
685C2×C4#58511yescharacteristic
686C8—42no—
687C8—42no—
688C8—44no—
689C22—24no—
690C22—24no—
691C22—24no—
692C22—42no—
693C22—42no—
694C22—42no—
695C22—42no—
696C22—42no—
697C22—42no—
698C22—42no—
699C22—81no—
700C22—81no—
701C22—81no—
702C22—81no—
703C22C22×#12111yescharacteristic
704C22C2×#54611yescharacteristic
705C22#348811yescharacteristic
706C22C22×#13711yescharacteristic
707C22C22×#13711yescharacteristic
708C22C2×#57711yescharacteristic
709C22#358611yescharacteristic
710C4—22no—
711C4—22no—
712C4—24no—
713C4—22no—
714C4—24no—
715C4—42no—
716C4—44no—
717C4—41no—
718C4—44no—
719C4—42no—
720C4—42no—
721C4—82no—
722C4—82no—
723C4—84no—
724C4—81no—
725C4#303914yes—
726C4#332914yescomplemented
727C4D4×#4111yescharacteristic
728C4#348111yescharacteristic
729C4#349811yescharacteristic
730C4#351411yescharacteristic
731C2—44no—
732C2—82no—
733C2#3625811yescharacteristic
734C2C2×#303911yescharacteristic
735C2#6042911yescharacteristic
736C2C2×#332911yescharacteristic
737C2#6115711yescharacteristic
738C2#6196011yescharacteristic
739C2#6200211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#555C22
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
2. D4×C4⋊Q8C2semidirect
4. #33162C2semidirect
5. #33114C2semidirect
6. #28748C2semidirect
7. #28284C2semidirect
8. #25278C2semidirect
9. #25099C2semidirect
10. #24759C2semidirect
11. #24630C2semidirect
12. #24498C2semidirect
13. #24209C2semidirect
14. #22650C2semidirect
15. #22430C2semidirect
16. C4×#461C2semidirect
17. #21259C2semidirect
726. C4#3329semidirect

Maximal subgroups

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

subgroupindex
C2×#33292
D4×C4⋊Q82
C4×#5552
#331622
#331142
#287482
#282842
#252782
#250992
#247592
#246302
#244982
#242092
#226502
#224302
C4×#4612
#212592