Finite Group Databasebeta

← #369382

Subgroups — #369382

Order 512.

1814 conjugacy classes of subgroups.

The subgroup lattice has 743 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#489C211yesmaximal, characteristic
2C2×C4×#124C211yesmaximal, characteristic
3C2×#1993C211yesmaximal, characteristic, complemented
4C2×#1959C211yesmaximal, characteristic
5#14714C211yesmaximal, characteristic, complemented
6#11190C212yesmaximal, complemented
7#11189C212yesmaximal
8#10843C211yesmaximal, characteristic, complemented
9#10789C212yesmaximal, complemented
10#10788C212yesmaximal
11#10693C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C2×#489—24no—
17C2×#489—24no—
18C2×#489C418yescomplemented
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#102C2212yes—
22C22×#102C2212yes—
23C22×#97C2211yescharacteristic
24C22×#97C2211yescharacteristic
25C4×#124—24no—
26C4×#124—24no—
27C2×#391—21no—
28C2×#391—21no—
29C2×#391C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#388—21no—
32C2×#388—21no—
33C2×#384C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×C4×C22⋊C4—21no—
36C2×C4×C22⋊C4—21no—
37C2×C43C2211yescharacteristic
38#1993—24no—
39#1959—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46C4⋊#95—21no—
47C4⋊#95—21no—
48C4⋊#95C2212yes—
49C4⋊#95C2212yes—
50C23×Q8—21no—
51C23×Q8C2311yescharacteristic
52C23×D4C2311yescharacteristic
53C24×C4—21no—
54C23×C8C2312yes—
55C23×C8C2312yes—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8C2×C414yescomplemented
62C2×#124—22no—
63C2×#124—22no—
64C2×#124—22no—
65C2×#124C2×C414yes—
66C2×#124D414yes—
67C2×#124D414yes—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—24no—
71C2×C4×Q8—42no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—24no—
75C2×C4×D4—42no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4—21no—
89C22×C42—21no—
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42D411yescharacteristic
93C22×C42D411yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97#489—232no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—42no—
103C2×#103C2×C418yescomplemented
104C2×#102—24no—
105C2×#102—24no—
106C2×#102—24no—
107C2×#102—24no—
108C2×#102—42no—
109C2×#102C2×C418yes—
110C2×#97—22no—
111C2×#97—22no—
112C2×#97—22no—
113C2×#97—22no—
114C2×#97—22no—
115C2×#97C2×C414yescomplemented
116#391—22no—
117#391—22no—
118#391—44no—
119#391—44no—
120#390—22no—
121#390—22no—
122#388—28no—
123#388—28no—
124#384—22no—
125#384—22no—
126#382—22no—
127#382D414yes—
128C4×C22⋊C4—28no—
129C4×C22⋊C4—28no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138C43—22no—
139C43—22no—
140#340—24no—
141#340—24no—
142#340—42no—
143#340—42no—
144#336—24no—
145#336—24no—
146C25—21no—
147C22×Q8—24no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—42no—
156C22×Q8—42no—
157C22×Q8—42no—
158C22×Q8—42no—
159C22×Q8—42no—
160C22×Q8—42no—
161C22×Q8C22×C412yes—
162C22×Q8C2×D412yes—
163C22×Q8C2×D412yes—
164C22×D4—21no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4C22×C412yes—
172C22×D4C2×D412yes—
173C22×D4C2×D412yes—
174C23×C4—21no—
175C23×C4—24no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2411yescharacteristic
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8C22×C414yes—
206C22×C8#4114yes—
207C4⋊Q8—28no—
208C4⋊Q8—28no—
209C4⋊Q8—44no—
210C4⋊Q8—44no—
211#124—28no—
212#124—28no—
213#124—28no—
214#124C22⋊C4116yes—
215C4×Q8—48no—
216C4×Q8—48no—
217C4×D4—48no—
218C4×D4—48no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4C22×C414yes—
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4C2×D412yes—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—41no—
283C2×C42—41no—
284C2×C42—41no—
285C2×C42—41no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42C22×C412yes—
289C2×C42C2×D412yes—
290C2×C42C2×D412yes—
291C2×C42C2×D412yes—
292C2×C42#4114yes—
293C2×C42#4112yes—
294C2×C42#4112yes—
295#103—216no—
296#103—48no—
297#102—216no—
298#102—48no—
299#97—28no—
300#97—44no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—41no—
313C24—41no—
314C24—41no—
315C24—41no—
316C24—41no—
317C24—41no—
318C24—41no—
319C24—41no—
320C24—41no—
321C24—41no—
322C24—41no—
323C24—41no—
324C24C22×D411yescharacteristic
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—24no—
329C2×Q8—24no—
330C2×Q8—24no—
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—48no—
335C2×Q8—48no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—48no—
343C2×Q8—84no—
344C2×Q8—84no—
345C2×Q8—84no—
346C2×Q8—84no—
347C2×Q8C2×C22⋊C414yes—
348C2×D4—22no—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4C2×C22⋊C414yes—
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—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4—84no—
435C22×C4—84no—
436C22×C4C4×D412yes—
437C22×C4C4×D412yes—
438C22×C4#12012yes—
439C22×C4#12012yes—
440C22×C4#12112yes—
441C22×C4#12112yes—
442C22×C4#12412yes—
443C22×C4#12412yes—
444C22×C4AGL(1,ℤ/8)12yes—
445C22×C4AGL(1,ℤ/8)12yes—
446C22×C4#13712yes—
447C22×C4#13712yes—
448C22×C4C23×C411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C2×#4111yescharacteristic
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4111yescharacteristic
456C22×C4H(5,𝔽2)12yes—
457C22×C42−1+412yes—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—42no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—48no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—48no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—84no—
487C4⋊C4—84no—
488C22⋊C4—216no—
489C22⋊C4—216no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C42—24no—
494C42—24no—
495C42—44no—
496C42—44no—
497C42—44no—
498C42—44no—
499C42—48no—
500C42—42no—
501C42—42no—
502C42C2×C22⋊C414yes—
503C42C4×D414yes—
504C42#12114yes—
505C42#12314yes—
506C23—24no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—81no—
526C23—81no—
527C23—81no—
528C23—81no—
529C23—81no—
530C23—81no—
531C23—81no—
532C23—81no—
533C23—81no—
534C23—81no—
535C23—81no—
536C23—81no—
537C23—81no—
538C23—81no—
539C23—81no—
540C23—81no—
541C23C2×C4×D411yescharacteristic
542C23C2×#12011yescharacteristic
543C23C2×#12111yescharacteristic
544C23C2×#12411yescharacteristic
545C23#53211yescharacteristic
546C23#53511yescharacteristic
547C23#53611yescharacteristic
548C23C2×AGL(1,ℤ/8)11yescharacteristic
549C23C2×AGL(1,ℤ/8)11yescharacteristic
550C23C2×#13711yescharacteristic
551C23C2×#13711yescharacteristic
552C23#57511yescharacteristic
553C23#57511yescharacteristic
554C23#57511yescharacteristic
555C23#57511yescharacteristic
556Q8—28no—
557Q8—44no—
558Q8—44no—
559Q8—44no—
560Q8—816no—
561D4—28no—
562D4—44no—
563D4—44no—
564D4—44no—
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—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—816no—
621C2×C4—81no—
622C2×C4—81no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4—81no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4—82no—
637C2×C4—82no—
638C2×C4—82no—
639C2×C4—82no—
640C2×C4#38614yes—
641C2×C4#38614yes—
642C2×C4#48914yescomplemented
643C2×C4C22×C22⋊C411yescharacteristic
644C2×C4C2×C4×D411yescharacteristic
645C2×C4C4×#4112yes—
646C2×C4#52012yes—
647C2×C4C2×#12111yescharacteristic
648C2×C4C2×#12311yescharacteristic
649C2×C4#52912yes—
650C2×C4#54212yes—
651C2×C4#54612yes—
652C2×C4#54612yes—
653C2×C4#54612yes—
654C2×C4#54712yes—
655C2×C4#54712yes—
656C2×C4#54812yes—
657C2×C4#55612yes—
658C2×C4#55912yes—
659C8—44no—
660C8—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22C2×#38611yescharacteristic
677C22C4×#12411yescharacteristic
678C22#232611yescharacteristic
679C22#243211yescharacteristic
680C22#243511yescharacteristic
681C22#254411yescharacteristic
682C22#254711yescharacteristic
683C22C4×AGL(1,ℤ/8)11yescharacteristic
684C22C4×AGL(1,ℤ/8)11yescharacteristic
685C22C4×#13711yescharacteristic
686C22C4×#13711yescharacteristic
687C22#300611yescharacteristic
688C22#300611yescharacteristic
689C22#300711yescharacteristic
690C22#300711yescharacteristic
691C22#303811yescharacteristic
692C22#303911yescharacteristic
693C22#304011yescharacteristic
694C22#304011yescharacteristic
695C22C2×#48911yescharacteristic
696C22#313511yescharacteristic
697C22#314011yescharacteristic
698C22#314111yescharacteristic
699C22#317111yescharacteristic
700C22#317211yescharacteristic
701C22#318011yescharacteristic
702C22#318111yescharacteristic
703C22#321311yescharacteristic
704C22#321411yescharacteristic
705C22#321511yescharacteristic
706C22#321611yescharacteristic
707C22#323811yescharacteristic
708C22#323911yescharacteristic
709C22#324011yescharacteristic
710C22#324011yescharacteristic
711C4—24no—
712C4—24no—
713C4—24no—
714C4—42no—
715C4—42no—
716C4—42no—
717C4—44no—
718C4—82no—
719C4—82no—
720C4—84no—
721C4—162no—
722C4#233912yes—
723C4#236712yes—
724C4#242712yes—
725C4#244712yes—
726C2—84no—
727C2#1402711yescharacteristic
728C2#2092711yescharacteristic
729C2#2092811yescharacteristic
730C2C4×#48911yescharacteristic
731C2#2165511yescharacteristic
732C2#2222311yescharacteristic
733C2#2222411yescharacteristic
734C2#2567811yescharacteristic
735C2#2567911yescharacteristic
736C2#2586311yescharacteristic
737C2#2586411yescharacteristic
738C2#2862011yescharacteristic
739C2#2862111yescharacteristic
740C2#2872611yescharacteristic
741C2#2872711yescharacteristic
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
3. C2×#1993C2semidirect
5. #14714C2semidirect
6. #11190C2semidirect
8. #10843C2semidirect
9. #10789C2semidirect
11. #10693C2semidirect
18. C2×#489C4semidirect
61. C2×C4⋊Q8C2×C4semidirect
103. C2×#103C2×C4semidirect
115. C2×#97C2×C4semidirect
642. C2×C4#489semidirect

Maximal subgroups

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

subgroupindex
C22×#4892
C2×C4×#1242
C2×#19932
C2×#19592
#147142
#111902
#111892
#108432
#107892
#107882
#106932