Finite Group Databasebeta

← #405893

Subgroups — #405893

Order 512.

1744 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1893C211yesmaximal, characteristic
2C2×#1874C211yesmaximal, characteristic
3C2×#1850C211yesmaximal, characteristic, complemented
4C22×#340C211yesmaximal, characteristic, complemented
5#14137C211yesmaximal, characteristic
6#13907C211yesmaximal, characteristic, complemented
7#11316C212yesmaximal, complemented
8#11228C212yesmaximal, complemented
9#11076C212yesmaximal, complemented
10#11066C212yesmaximal, complemented
11#10730C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C2×#390C2212yes—
22C2×#389C2212yes—
23C2×#386—21no—
24C2×#386—21no—
25C2×#380—21no—
26C2×#380C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28#1893—24no—
29#1874—24no—
30#1850—24no—
31C2×#340—24no—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#336C2212yes—
40#1441—22no—
41#1439—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C22⋊#95—21no—
45C23×D4—21no—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C24×C4—21no—
49C24×C4C2311yescharacteristic
50C23×C8—22no—
51C23×C8—22no—
52C23×C8—42no—
53C23×C8—42no—
54C23×C8D414yes—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C23×C8C2312yes—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121—22no—
61C2×#121—22no—
62C2×#121—22no—
63C2×#121Q814yes—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—42no—
73C22×C4⋊C4—42no—
74C22×C4⋊C4D414yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4D411yescharacteristic
85C22×C22⋊C4D411yescharacteristic
86C22×C42C2312yes—
87C2×C4⋊C8—24no—
88C2×C4⋊C8—42no—
89C2×C4⋊C8—42no—
90C2×#102—24no—
91C2×#102—22no—
92C2×#102—22no—
93C2×#102—24no—
94C2×#102—42no—
95C2×#102—42no—
96C2×#102—42no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100#390—24no—
101#390—42no—
102#389—24no—
103#389—42no—
104#386—44no—
105#386—44no—
106#380—22no—
107#380—22no—
108#380—44no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95D412yes—
118#340—24no—
119#340—22no—
120#340—22no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—42no—
125#336—24no—
126C25—21no—
127C25C2×D411yescharacteristic
128C22×D4—24no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—42no—
139C22×D4—42no—
140C22×D4—42no—
141C22×D4—42no—
142C22×D4—42no—
143C22×D4—42no—
144C22×D4C2×Q812yes—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—24no—
154C23×C4—22no—
155C23×C4—44no—
156C23×C4—44no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—82no—
163C23×C4—82no—
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×Q812yes—
170C23×C4#4112yes—
171C23×C4#4112yes—
172C23×C4#4112yes—
173C23×C4C2411yescharacteristic
174C22×C8—28no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—48no—
192C22×C8—44no—
193C22×C8—44no—
194C22×C8—48no—
195C22×C8C2×D414yes—
196#121—28no—
197#121—28no—
198#121—44no—
199#121—44no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—48no—
226C2×C4⋊C4—48no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—82no—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×Q812yes—
237C2×C4⋊C4#4114yes—
238C2×C4⋊C4#4114yes—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—82no—
255C2×C22⋊C4—82no—
256C2×C22⋊C4SD1614yes—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C4⋊C8—48no—
264#102—44no—
265#102—48no—
266#102—44no—
267C22⋊C8—44no—
268#95—24no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—42no—
273#95—44no—
274#95—82no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—41no—
281C24—41no—
282C24—42no—
283C24—42no—
284C24—41no—
285C24—41no—
286C24—41no—
287C24—41no—
288C24—42no—
289C24—42no—
290C24#12112yes—
291C24#12112yes—
292C24C22⋊Q812yes—
293C24C2×SD1612yes—
294C24#13512yes—
295C24AGL(1,ℤ/8)12yes—
296C24AGL(1,ℤ/8)12yes—
297C24C22×D411yescharacteristic
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—48no—
303C2×D4—48no—
304C2×D4—48no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—84no—
320C2×D4—84no—
321C2×D4—84no—
322C2×D4—84no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—48no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—48no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—48no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—84no—
387C22×C4—88no—
388C22×C4—88no—
389C22×C4—88no—
390C22×C4—88no—
391C22×C4—84no—
392C22×C4—84no—
393C22×C4#12114yes—
394C22×C4#12112yes—
395C22×C4#12112yes—
396C22×C4C22⋊Q814yes—
397C22×C4C22⋊Q814yes—
398C22×C4C22⋊Q812yes—
399C22×C4#12614yes—
400C22×C4C2×SD1614yes—
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×Q811yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4112yes—
408C22×C4C2×#4112yes—
409C22×C4H(5,𝔽2)12yes—
410C22×C4H(5,𝔽2)12yes—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—416no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—41no—
426C2×C8—42no—
427C2×C8—41no—
428C2×C8—42no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—88no—
449C4⋊C4#12314yes—
450C22⋊C4—28no—
451C22⋊C4—44no—
452C22⋊C4—48no—
453C22⋊C4—88no—
454C22⋊C4—88no—
455C22⋊C4—84no—
456C42—42no—
457C42—42no—
458C42—42no—
459C42—42no—
460C23—24no—
461C23—22no—
462C23—22no—
463C23—24no—
464C23—24no—
465C23—22no—
466C23—24no—
467C23—24no—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—42no—
477C23—44no—
478C23—42no—
479C23—44no—
480C23—42no—
481C23—81no—
482C23—81no—
483C23—81no—
484C23—81no—
485C23—82no—
486C23—81no—
487C23—82no—
488C23—82no—
489C23—81no—
490C23—82no—
491C23—81no—
492C23—81no—
493C23—82no—
494C23—82no—
495C23—82no—
496C23—82no—
497C23—82no—
498C23—82no—
499C23#39614yes—
500C23#46014yes—
501C23#46314yes—
502C23#46514yes—
503C23#46914yes—
504C23#47614yes—
505C23#47814yes—
506C23C2×#12111yescharacteristic
507C23C2×#12111yescharacteristic
508C23C2×C22⋊Q811yescharacteristic
509C23#54612yes—
510C23#54612yes—
511C23C22×SD1611yescharacteristic
512C23C2×#13511yescharacteristic
513C23C2×AGL(1,ℤ/8)11yescharacteristic
514C23C2×AGL(1,ℤ/8)11yescharacteristic
515C23#57612yes—
516C23#57812yes—
517D4—44no—
518D4—42no—
519D4—42no—
520D4—816no—
521D4—81no—
522D4—81no—
523D4—82no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—816no—
563C2×C4—816no—
564C2×C4—81no—
565C2×C4—82no—
566C2×C4—81no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4#39614yes—
582C2×C4C2×#12111yescharacteristic
583C2×C4C2×C22⋊Q811yescharacteristic
584C2×C4C2×C22⋊Q811yescharacteristic
585C2×C4C2×#12311yescharacteristic
586C2×C4#52511yescharacteristic
587C2×C4C2×#12611yescharacteristic
588C2×C4#52912yes—
589C2×C4#53212yes—
590C2×C4#53412yes—
591C2×C4#53411yescharacteristic
592C2×C4#53511yescharacteristic
593C2×C4#53812yes—
594C2×C4#55112yes—
595C2×C4#55112yes—
596C2×C4#55312yes—
597C2×C4#55312yes—
598C2×C4#55412yes—
599C2×C4#55412yes—
600C8—44no—
601C8—81no—
602C8—81no—
603C22—28no—
604C22—28no—
605C22—24no—
606C22—24no—
607C22—28no—
608C22—28no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—84no—
614C22—84no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—162no—
619C22—162no—
620C22—161no—
621C22—161no—
622C22—161no—
623C22—161no—
624C22#206518yes—
625C22C2×#39611yescharacteristic
626C22#241111yescharacteristic
627C22#241211yescharacteristic
628C22#245012yes—
629C22#254312yes—
630C22C2×#46311yescharacteristic
631C22C2×#46011yescharacteristic
632C22#304111yescharacteristic
633C22#304411yescharacteristic
634C22C2×#46511yescharacteristic
635C22C2×#46911yescharacteristic
636C22#306112yes—
637C22C2×#47611yescharacteristic
638C22C2×#47811yescharacteristic
639C22#308312yes—
640C22#329512yes—
641C22#329712yes—
642C22#329812yes—
643C22#330012yes—
644C22#330812yes—
645C22#330812yes—
646C22#331812yes—
647C22#331812yes—
648C4—24no—
649C4—44no—
650C4—44no—
651C4—82no—
652C4—82no—
653C4—84no—
654C4—82no—
655C4—162no—
656C4#248711yescharacteristic
657C4#251012yes—
658C4#256412yes—
659C4#257911yescharacteristic
660C4#261711yescharacteristic
661C4#285411yescharacteristic
662C2—28no—
663C2—162no—
664C2#1492111yescharacteristic
665C2C2×#206511yescharacteristic
666C2#2361111yescharacteristic
667C2#2447811yescharacteristic
668C2#2448111yescharacteristic
669C2#2452811yescharacteristic
670C2#2453111yescharacteristic
671C2#2641012yes—
672C2#2642012yes—
673C2#2855512yes—
674C2#2856112yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1850C2semidirect
4. C22×#340C2semidirect
6. #13907C2semidirect
7. #11316C2semidirect
8. #11228C2semidirect
9. #11076C2semidirect
10. #11066C2semidirect
11. #10730C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18932
C2×#18742
C2×#18502
C22×#3402
#141372
#139072
#113162
#112282
#110762
#110662
#107302