Finite Group Databasebeta

← #406299

Subgroups — #406299

Order 512.

1896 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic
2C2×#1893C211yesmaximal, characteristic
3C2×#1893C211yesmaximal, characteristic
4C2×#1807C211yesmaximal, characteristic, complemented
5C22×#336C211yesmaximal, characteristic, complemented
6#14685C211yesmaximal, characteristic, complemented
7#11293C212yesmaximal, complemented
8#11220C212yesmaximal, complemented
9#11189C212yesmaximal, complemented
10#11121C212yesmaximal, complemented
11#10721C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#123—21no—
15C22×#121C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C22×#106C2212yes—
18C22×#102C2211yescharacteristic
19C22×#102C2212yes—
20C22×#102C2211yescharacteristic
21#2854—24no—
22#2854—24no—
23C2×#390C2212yes—
24C2×#389C2212yes—
25C2×#388—22no—
26C2×#386—21no—
27C2×#381—21no—
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×#380C2211yescharacteristic
31#1893—24no—
32#1893—24no—
33#1807—24no—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#336—24no—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45#1434—22no—
46C22⋊#95—21no—
47C23×D4—21no—
48C23×D4C2311yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4D411yescharacteristic
51C24×C4C2311yescharacteristic
52C23×C8—22no—
53C23×C8—22no—
54C23×C8—22no—
55C23×C8—24no—
56C23×C8—24no—
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—42no—
60C23×C8—42no—
61C23×C8D414yes—
62C23×C8C2311yescharacteristic
63C23×C8C2312yes—
64C23×C8C2311yescharacteristic
65C2×#123—28no—
66C2×#123—44no—
67C2×#123—44no—
68C2×#121—22no—
69C2×#121—22no—
70C2×#121—22no—
71C2×#121—22no—
72C2×#121—22no—
73C2×#121D414yes—
74C22×C4⋊C4—22no—
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⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C42—22no—
87C22×C42—22no—
88C22×C42—22no—
89C22×C42—22no—
90C22×C42—22no—
91C22×C42—22no—
92C22×C42D412yes—
93C22×C42C2312yes—
94C2×#106—24no—
95C2×#106—24no—
96C2×#106—42no—
97C2×#102—24no—
98C2×#102—22no—
99C2×#102—22no—
100C2×#102—24no—
101C2×#102—24no—
102C2×#102—42no—
103C2×#102—42no—
104#390—24no—
105#390—24no—
106#389—24no—
107#389—24no—
108#388—216no—
109#386—28no—
110#381—44no—
111#380—22no—
112#380—22no—
113#380—22no—
114#380—22no—
115#380—22no—
116#380D414yes—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125#340—22no—
126#340—22no—
127#340—24no—
128#340—24no—
129#340—24no—
130#340—42no—
131#336—24no—
132#336—24no—
133C25—21no—
134C25C2×D411yescharacteristic
135C22×D4—24no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4C2×D412yes—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—24no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—42no—
172C23×C4SD1614yes—
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C23×C4C2×D412yes—
179C23×C4#4112yes—
180C23×C4#4112yes—
181C23×C4#4112yes—
182C23×C4C2411yescharacteristic
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—28no—
187C22×C8—28no—
188C22×C8—22no—
189C22×C8—21no—
190C22×C8—22no—
191C22×C8—21no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—28no—
199C22×C8—44no—
200C22×C8—48no—
201C22×C8—48no—
202C22×C8—48no—
203C22×C8—48no—
204C22×C8—44no—
205C22×C8—44no—
206C22×C8C2×D412yes—
207C22×C8C2×D412yes—
208C22×C8#4114yes—
209C22×C8#4114yes—
210#123—432no—
211#121—28no—
212#121—28no—
213#121—44no—
214#121—44no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D414yes—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—44no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42—44no—
275#106—48no—
276#102—48no—
277#102—44no—
278#102—44no—
279#95—28no—
280#95—24no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—42no—
285#95—42no—
286#95—44no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—41no—
293C24—41no—
294C24—42no—
295C24—42no—
296C24—41no—
297C24—42no—
298C24—41no—
299C24—41no—
300C24—41no—
301C24—42no—
302C24#12012yes—
303C24#12112yes—
304C24#12312yes—
305C24C2×SD1612yes—
306C24#13512yes—
307C24AGL(1,ℤ/8)12yes—
308C24AGL(1,ℤ/8)12yes—
309C24C22×D411yescharacteristic
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—48no—
315C2×D4—48no—
316C2×D4—48no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—42no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—84no—
332C2×D4—84no—
333C2×D4—84no—
334C2×D4—84no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—28no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4#12012yes—
407C22×C4#12114yes—
408C22×C4#12114yes—
409C22×C4#12112yes—
410C22×C4#12314yes—
411C22×C4#12312yes—
412C22×C4#12414yes—
413C22×C4C2×SD1614yes—
414C22×C4#13514yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4112yes—
422C22×C4C2×#4112yes—
423C22×C4H(5,𝔽2)12yes—
424C22×C42−1+412yes—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—24no—
432C2×C8—22no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—416no—
436C2×C8—416no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8#12314yescomplemented
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—416no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—42no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C22⋊C4—416no—
464C22⋊C4—416no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—84no—
468C22⋊C4—84no—
469C42—42no—
470C42—42no—
471C42—42no—
472C42—42no—
473C42—48no—
474C23—24no—
475C23—22no—
476C23—22no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—42no—
491C23—44no—
492C23—42no—
493C23—42no—
494C23—44no—
495C23—81no—
496C23—81no—
497C23—82no—
498C23—81no—
499C23—81no—
500C23—82no—
501C23—82no—
502C23—82no—
503C23—82no—
504C23—82no—
505C23—82no—
506C23—82no—
507C23—82no—
508C23—82no—
509C23—81no—
510C23—81no—
511C23—81no—
512C23—81no—
513C23#39414yes—
514C23#44914yes—
515C23#45014yes—
516C23#46514yes—
517C23#46914yes—
518C23#48114yes—
519C23#48214yes—
520C23C2×#12011yescharacteristic
521C23C2×#12111yescharacteristic
522C23C2×#12311yescharacteristic
523C23#53212yes—
524C23#53712yes—
525C23C22×SD1611yescharacteristic
526C23C2×#13511yescharacteristic
527C23C2×#13512yes—
528C23C2×AGL(1,ℤ/8)11yescharacteristic
529C23C2×AGL(1,ℤ/8)11yescharacteristic
530C23#57512yes—
531D4—44no—
532D4—42no—
533D4—42no—
534D4—816no—
535D4—81no—
536D4—81no—
537D4—82no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—416no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—416no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4#39414yes—
596C2×C4C2×#12111yescharacteristic
597C2×C4C2×#12111yescharacteristic
598C2×C4C2×#12311yescharacteristic
599C2×C4C2×#12311yescharacteristic
600C2×C4#52511yescharacteristic
601C2×C4C2×#12411yescharacteristic
602C2×C4#52912yes—
603C2×C4#53411yescharacteristic
604C2×C4#53611yescharacteristic
605C2×C4#53912yes—
606C2×C4#54612yes—
607C2×C4#54612yes—
608C2×C4#54712yes—
609C2×C4#54712yes—
610C2×C4#55112yes—
611C2×C4#55112yes—
612C2×C4#55212yes—
613C2×C4#55212yes—
614C8—24no—
615C8—42no—
616C8—44no—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—24no—
622C22—28no—
623C22—28no—
624C22—24no—
625C22—28no—
626C22—28no—
627C22—82no—
628C22—84no—
629C22—84no—
630C22—82no—
631C22—82no—
632C22—161no—
633C22—161no—
634C22—161no—
635C22—161no—
636C22—162no—
637C22—162no—
638C22#203418yes—
639C22C2×#39411yescharacteristic
640C22#243012yes—
641C22#249812yes—
642C22#261811yescharacteristic
643C22#285411yescharacteristic
644C22C2×#45011yescharacteristic
645C22C2×#44911yescharacteristic
646C22#300212yes—
647C22C2×#46511yescharacteristic
648C22#305011yescharacteristic
649C22C2×#46911yescharacteristic
650C22#305411yescharacteristic
651C22C2×#48211yescharacteristic
652C22C2×#48111yescharacteristic
653C22#309112yes—
654C22#316712yes—
655C22#316812yes—
656C22#318012yes—
657C22#318012yes—
658C22#324612yes—
659C22#324712yes—
660C22#324712yes—
661C22#325012yes—
662C4—24no—
663C4—44no—
664C4—44no—
665C4—84no—
666C4—82no—
667C4—161no—
668C4—161no—
669C4—162no—
670C4#241211yescharacteristic
671C4#241311yescharacteristic
672C4#241912yes—
673C4#248611yescharacteristic
674C4#248812yes—
675C4#253411yescharacteristic
676C2—28no—
677C2—162no—
678C2#1492711yescharacteristic
679C2C2×#203411yescharacteristic
680C2#2347911yescharacteristic
681C2#2549512yes—
682C2#2550012yes—
683C2#2785712yes—
684C2#2785912yes—
685C2#2977211yescharacteristic
686C2#2977411yescharacteristic
687C2#3312811yescharacteristic
688C2#3313511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1807C2semidirect
5. C22×#336C2semidirect
6. #14685C2semidirect
7. #11293C2semidirect
8. #11220C2semidirect
9. #11189C2semidirect
10. #11121C2semidirect
11. #10721C2semidirect
443. C2×C8#123semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
C2×#18932
C2×#18072
C22×#3362
#146852
#112932
#112202
#111892
#111212
#107212