Finite Group Databasebeta

← #365731

Subgroups — #365731

Order 512.

1682 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#124C211yesmaximal, characteristic
2C2×#1959C211yesmaximal, characteristic
3C2×#1959C211yesmaximal, characteristic
4C2×#1836C211yesmaximal, characteristic, complemented
5#14712C211yesmaximal, characteristic, complemented
6#11067C212yesmaximal, complemented
7#11066C212yesmaximal
8#10843C211yesmaximal, characteristic, complemented
9#10789C212yesmaximal, complemented
10#10788C212yesmaximal
11#10693C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×#97C2211yescharacteristic
20C22×#97C2211yescharacteristic
21C4×#124—24no—
22C4×#124—24no—
23C2×#391C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#389—21no—
26C2×#389—21no—
27C2×#389C2212yes—
28C2×#388—21no—
29C2×#388—21no—
30C2×#384C2211yescharacteristic
31C2×#384C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×C4×C22⋊C4—21no—
34C2×C4×C22⋊C4—21no—
35C2×C43C2211yescharacteristic
36#1959—24no—
37#1959C418yescomplemented
38#1836—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45C4⋊#95—21no—
46C4⋊#95—21no—
47C4⋊#95C2212yes—
48C23×Q8C2311yescharacteristic
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×#124—22no—
54C2×#124—22no—
55C2×#124—22no—
56C2×#124C2×C414yes—
57C2×#124D414yes—
58C2×#124Q814yes—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—24no—
62C2×C4×Q8—42no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—24no—
66C2×C4×D4—42no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C42—21no—
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—42no—
93C2×#103—42no—
94C2×#103—42no—
95C2×#103C2×C418yescomplemented
96C2×#102—24no—
97C2×#102—24no—
98C2×#102—42no—
99C2×#102—42no—
100C2×#102—42no—
101C2×#102C2×C418yes—
102C2×#97—22no—
103C2×#97—22no—
104C2×#97—22no—
105C2×#97—22no—
106C2×#97—22no—
107C2×#97C2×C414yescomplemented
108#391—22no—
109#391—22no—
110#390—22no—
111#390—22no—
112#389—42no—
113#389—42no—
114#389—44no—
115#389—44no—
116#388—28no—
117#388—28no—
118#384—22no—
119#384—22no—
120#384—22no—
121#384C2×C414yescomplemented
122#382—22no—
123#382D414yes—
124C4×C22⋊C4—28no—
125C4×C22⋊C4—28no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C43—22no—
134C43—22no—
135#340—24no—
136#340—24no—
137#340—42no—
138#340—42no—
139#336—24no—
140#336—24no—
141C25—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8C22×C412yes—
150C22×Q8C2×D412yes—
151C22×Q8C2×Q812yes—
152C22×D4—21no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4C22×C412yes—
160C22×D4C2×D412yes—
161C22×D4C2×Q812yes—
162C23×C4—21no—
163C23×C4—24no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2411yescharacteristic
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8C22×C414yes—
194C22×C8#4114yes—
195#124—28no—
196#124—28no—
197#124—28no—
198#124C4⋊C4116yescomplemented
199C4×Q8—48no—
200C4×Q8—48no—
201C4×D4—48no—
202C4×D4—48no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
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—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4C22×C414yes—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×D412yes—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—41no—
273C2×C42—41no—
274C2×C42—41no—
275C2×C42—41no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42C22×C412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2×Q812yes—
282C2×C42#4114yes—
283C2×C42#4112yes—
284C2×C42#4112yes—
285#103—48no—
286#103—48no—
287#102—48no—
288#102—48no—
289#97—28no—
290#97—44no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—44no—
296#95—44no—
297#95—44no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—41no—
302C24—41no—
303C24—41no—
304C24—41no—
305C24—41no—
306C24—41no—
307C24—41no—
308C24—41no—
309C24—41no—
310C24—41no—
311C24—41no—
312C24—41no—
313C24C22×D411yescharacteristic
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—24no—
318C2×Q8—24no—
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—24no—
322C2×Q8—24no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8C2×C4⋊C414yes—
330C2×D4—22no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4C2×C4⋊C414yes—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—48no—
383C22×C4—48no—
384C22×C4—48no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—84no—
415C22×C4—84no—
416C22×C4—84no—
417C22×C4—84no—
418C22×C4C4×D412yes—
419C22×C4C4×D412yes—
420C22×C4#12012yes—
421C22×C4#12012yes—
422C22×C4C22⋊Q812yes—
423C22×C4C22⋊Q812yes—
424C22×C4#12412yes—
425C22×C4#12412yes—
426C22×C4AGL(1,ℤ/8)12yes—
427C22×C4AGL(1,ℤ/8)12yes—
428C22×C4#13712yes—
429C22×C4#13712yes—
430C22×C4C23×C411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×Q811yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4111yescharacteristic
438C22×C4H(5,𝔽2)12yes—
439C22×C4H(5,𝔽2)12yes—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—84no—
470C4⋊C4—84no—
471C4⋊C4—88no—
472C22⋊C4—216no—
473C22⋊C4—216no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C42—24no—
478C42—24no—
479C42—44no—
480C42—44no—
481C42—44no—
482C42—44no—
483C42—48no—
484C42—42no—
485C42—42no—
486C42C2×C4⋊C414yes—
487C42C4×D414yes—
488C42C22⋊Q814yes—
489C42#12314yes—
490C23—24no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—81no—
510C23—81no—
511C23—81no—
512C23—81no—
513C23—81no—
514C23—81no—
515C23—81no—
516C23—81no—
517C23—81no—
518C23—81no—
519C23—81no—
520C23—81no—
521C23—81no—
522C23—81no—
523C23—81no—
524C23—81no—
525C23C2×C4×D411yescharacteristic
526C23C2×#12011yescharacteristic
527C23C2×C22⋊Q811yescharacteristic
528C23C2×#12411yescharacteristic
529C23#53211yescharacteristic
530C23#53511yescharacteristic
531C23#53511yescharacteristic
532C23C2×AGL(1,ℤ/8)11yescharacteristic
533C23C2×AGL(1,ℤ/8)11yescharacteristic
534C23C2×#13711yescharacteristic
535C23C2×#13711yescharacteristic
536C23#57511yescharacteristic
537C23#57511yescharacteristic
538C23#57511yescharacteristic
539C23#57511yescharacteristic
540Q8—28no—
541Q8—44no—
542Q8—44no—
543Q8—44no—
544D4—28no—
545D4—44no—
546D4—44no—
547D4—44no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—816no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—82no—
609C2×C4—81no—
610C2×C4—82no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4#38514yes—
624C2×C4#38514yes—
625C2×C4#48914yes—
626C2×C4C22×C4⋊C411yescharacteristic
627C2×C4C2×C4×D411yescharacteristic
628C2×C4C4×#4112yes—
629C2×C4#51812yes—
630C2×C4C2×C22⋊Q811yescharacteristic
631C2×C4C2×#12311yescharacteristic
632C2×C4#52912yes—
633C2×C4#54012yes—
634C2×C4#54612yes—
635C2×C4#54612yes—
636C2×C4#54612yes—
637C2×C4#55012yes—
638C2×C4#55312yes—
639C2×C4#55412yes—
640C2×C4#55412yes—
641C2×C4#55512yes—
642C8—44no—
643C8—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22C2×#38511yescharacteristic
660C22C4×#12411yescharacteristic
661C22#232811yescharacteristic
662C22#243111yescharacteristic
663C22#243511yescharacteristic
664C22#256311yescharacteristic
665C22#256611yescharacteristic
666C22C4×AGL(1,ℤ/8)11yescharacteristic
667C22C4×AGL(1,ℤ/8)11yescharacteristic
668C22C4×#13711yescharacteristic
669C22C4×#13711yescharacteristic
670C22#300611yescharacteristic
671C22#300611yescharacteristic
672C22#300711yescharacteristic
673C22#300711yescharacteristic
674C22#307911yescharacteristic
675C22#307911yescharacteristic
676C22#308011yescharacteristic
677C22#308011yescharacteristic
678C22C2×#48911yescharacteristic
679C22#313511yescharacteristic
680C22#314011yescharacteristic
681C22#314111yescharacteristic
682C22#317111yescharacteristic
683C22#317211yescharacteristic
684C22#318011yescharacteristic
685C22#318111yescharacteristic
686C22#321311yescharacteristic
687C22#321411yescharacteristic
688C22#321511yescharacteristic
689C22#321611yescharacteristic
690C22#321711yescharacteristic
691C22#321811yescharacteristic
692C22#321911yescharacteristic
693C22#321911yescharacteristic
694C4—24no—
695C4—24no—
696C4—24no—
697C4—42no—
698C4—42no—
699C4—42no—
700C4—44no—
701C4—84no—
702C4—82no—
703C4—82no—
704C4—162no—
705C4#234912yes—
706C4#236312yes—
707C4#242012yes—
708C4#245412yes—
709C2—84no—
710C2#1397911yescharacteristic
711C2#2087511yescharacteristic
712C2#2087511yescharacteristic
713C2C4×#48911yescharacteristic
714C2#2165511yescharacteristic
715C2#2227211yescharacteristic
716C2#2227311yescharacteristic
717C2#2560411yescharacteristic
718C2#2560511yescharacteristic
719C2#2586311yescharacteristic
720C2#2586411yescharacteristic
721C2#2914611yescharacteristic
722C2#2914811yescharacteristic
723C2#2919711yescharacteristic
724C2#2919811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1836C2semidirect
5. #14712C2semidirect
6. #11067C2semidirect
8. #10843C2semidirect
9. #10789C2semidirect
11. #10693C2semidirect
37. #1959C4semidirect
95. C2×#103C2×C4semidirect
107. C2×#97C2×C4semidirect
121. #384C2×C4semidirect
198. #124C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1242
C2×#19592
C2×#18362
#147122
#110672
#110662
#108432
#107892
#107882
#106932