Finite Group Databasebeta

← #374376

Subgroups — #374376

Order 512.

1944 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#464C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#1959C211yesmaximal, characteristic
4C2×#1880C212yesmaximal
5C2×#1878C212yesmaximal, complemented
6C2×#1763C211yesmaximal, characteristic, complemented
7#14153C211yesmaximal, characteristic
8#11067C212yesmaximal, complemented
9#11066C212yesmaximal
10#10849C211yesmaximal, characteristic, complemented
11#10835C211yesmaximal, characteristic, complemented
12C23×Q16C2211yescharacteristic
13C23×SD16C2211yescharacteristic
14C22×#124C2211yescharacteristic
15C22×C4×Q8C2212yes—
16C22×C4×Q8C2211yescharacteristic
17C2×#464—24no—
18C2×#464—24no—
19C2×#464C418yescomplemented
20C22×C4⋊C8C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2211yescharacteristic
23C22×#103C2212yes—
24C22×#102C2212yes—
25C22×#102C2211yescharacteristic
26C22×#97C2211yescharacteristic
27Q8×C4⋊C4—24no—
28Q8×C4⋊C4—24no—
29C2×#391C2212yes—
30C2×#390C2212yes—
31C2×#389—22no—
32C2×#389C2212yes—
33C2×#384—21no—
34C2×#384C2211yescharacteristic
35C2×C4×C4⋊C4—21no—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38#1959—24no—
39#1880—28no—
40#1878—28no—
41#1763—24no—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#336C2212yes—
46C2×#336C2212yes—
47#1439—21no—
48#1439—21no—
49#1437—22no—
50C23×Q8C2311yescharacteristic
51C23×Q8C2311yescharacteristic
52C23×D4C2311yescharacteristic
53C22×Q16—24no—
54C22×Q16—22no—
55C22×Q16—22no—
56C22×Q16—24no—
57C22×Q16C2×C414yes—
58C22×SD16—24no—
59C22×SD16—22no—
60C22×SD16—22no—
61C22×SD16—24no—
62C22×SD16C2×C414yes—
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2312yes—
66C2×#124—22no—
67C2×#124—22no—
68C2×#124—22no—
69C2×#124—22no—
70C2×#124—22no—
71C2×#124C2×C414yes—
72C2×C4×Q8—24no—
73C2×C4×Q8—24no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×C4×Q8—24no—
78C2×C4×Q8—24no—
79C2×C4×Q8—42no—
80C2×C4×Q8—42no—
81C2×C4×Q8C2×C414yes—
82C2×C4×Q8D414yes—
83C2×C4×Q8D414yes—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—22no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4D411yescharacteristic
90C22×C4⋊C4D411yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C22⋊C4—21no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4—22no—
97C22×C22⋊C4—22no—
98C22×C42—21no—
99C22×C42—22no—
100C22×C42—21no—
101C22×C42C2312yes—
102C22×C42C2311yescharacteristic
103#464—232no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8C2×C414yes—
107C2×#103—24no—
108C2×#103—24no—
109C2×#103—22no—
110C2×#103—22no—
111C2×#103—24no—
112C2×#103—24no—
113C2×#103—42no—
114C2×#103C2×C414yes—
115C2×#103D418yes—
116C2×#102—22no—
117C2×#102—22no—
118C2×#102—24no—
119C2×#102—24no—
120C2×#102—42no—
121C2×#102C2×C414yes—
122C2×#97—22no—
123C2×#97—22no—
124C2×#97—22no—
125#391—24no—
126#391—42no—
127#390—24no—
128#390—42no—
129#389—216no—
130#389—24no—
131#389—24no—
132#384—28no—
133#384—22no—
134#384D414yes—
135C4×C4⋊C4—28no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—22no—
139C4×C4⋊C4—22no—
140C2×#95—22no—
141C2×#95—22no—
142C2×#95—22no—
143C2×#95—21no—
144C2×#95D412yes—
145#340—24no—
146#340—24no—
147#340—24no—
148#336—24no—
149#336—24no—
150C25—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—22no—
161C22×Q8—21no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8C22×C412yes—
166C22×Q8C22×C412yes—
167C22×Q8C2×D412yes—
168C22×Q8C2×D412yes—
169C22×Q8C2×D414yes—
170C22×Q8#4114yes—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4—22no—
175C22×D4—21no—
176C22×D4—22no—
177C22×D4—21no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4C22×C412yes—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—22no—
184C23×C4—21no—
185C23×C4—22no—
186C23×C4C2×D412yes—
187C23×C4C2×D412yes—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2411yescharacteristic
191C2×Q16—28no—
192C2×Q16—216no—
193C2×Q16—48no—
194C2×Q16—44no—
195C2×Q16—44no—
196C2×SD16—28no—
197C2×SD16—216no—
198C2×SD16—48no—
199C2×SD16—44no—
200C2×SD16—44no—
201C22×C8—22no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—22no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8C22×C412yes—
216C22×C8C22×C412yes—
217C22×C8C2×D414yes—
218#124—28no—
219#124—28no—
220#124—44no—
221#124—44no—
222C4×Q8—28no—
223C4×Q8—28no—
224C4×Q8—28no—
225C4×Q8—48no—
226C4×Q8—48no—
227C4×Q8—48no—
228C4×Q8—48no—
229C4×Q8C22⋊C4116yescomplemented
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—81no—
258C2×C4⋊C4—81no—
259C2×C4⋊C4—82no—
260C2×C4⋊C4—82no—
261C2×C4⋊C4C22×C412yes—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4C2×D414yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D412yes—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—82no—
277C2×C22⋊C4—82no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—28no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42—42no—
291C2×C42—81no—
292C2×C42—81no—
293C2×C42C22×C412yes—
294C2×C42C2×D412yes—
295C2×C42C2×D412yes—
296C2×C42C2×D412yes—
297C2×C42#4112yes—
298C2×C42#4112yes—
299C4⋊C8—28no—
300#103—216no—
301#103—28no—
302#103—48no—
303#102—28no—
304#102—48no—
305#97—44no—
306#95—28no—
307#95—24no—
308#95—44no—
309#95—44no—
310#95—42no—
311C24—22no—
312C24—41no—
313C24—41no—
314C24—42no—
315C24—42no—
316C24—42no—
317C24—42no—
318C24—41no—
319C24—41no—
320C24—41no—
321C24—41no—
322C24C22×D411yescharacteristic
323C2×Q8—22no—
324C2×Q8—24no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—24no—
330C2×Q8—24no—
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—24no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—24no—
339C2×Q8—24no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—42no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8C2×C22⋊C414yes—
355C2×Q8C4×D418yes—
356C2×Q8#12118yes—
357C2×Q8#12414yes—
358C2×Q8#12714yes—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—42no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—42no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—42no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4C4×D412yes—
428C22×C4C4×D412yes—
429C22×C4#12012yes—
430C22×C4#12012yes—
431C22×C4#12112yes—
432C22×C4#12112yes—
433C22×C4#13714yes—
434C22×C4C23×C411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C2×#4112yes—
440C22×C4C2×#4111yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4H(5,𝔽2)12yes—
443C22×C42−1+412yes—
444Q16—416no—
445SD16—416no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C4⋊C4—28no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—216no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—416no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—84no—
480C4⋊C4—88no—
481C4⋊C4C2×C22⋊C414yes—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—88no—
485C22⋊C4—88no—
486C42—216no—
487C42—416no—
488C42—42no—
489C42—42no—
490C42—42no—
491C42—42no—
492C42—84no—
493C42C2×C22⋊C414yes—
494C42C4×D414yes—
495C42#12114yes—
496C42#12314yes—
497C23—44no—
498C23—44no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—82no—
503C23—82no—
504C23—82no—
505C23—82no—
506C23—82no—
507C23—81no—
508C23—81no—
509C23—81no—
510C23—81no—
511C23—82no—
512C23—81no—
513C23—82no—
514C23—81no—
515C23—82no—
516C23—82no—
517C23—82no—
518C23—81no—
519C23—81no—
520C23C2×C4×D411yescharacteristic
521C23C2×#12011yescharacteristic
522C23C2×#12111yescharacteristic
523C23D4212yes—
524C23#54812yes—
525C23C2×#13712yes—
526C23#57512yes—
527C23#57611yescharacteristic
528C23#57711yescharacteristic
529C23#57711yescharacteristic
530C23#57811yescharacteristic
531Q8—24no—
532Q8—28no—
533Q8—28no—
534Q8—24no—
535Q8—24no—
536Q8—42no—
537Q8—44no—
538Q8—42no—
539Q8—82no—
540Q8—81no—
541Q8—81no—
542Q8#39018yes—
543D4—42no—
544D4—44no—
545D4—42no—
546D4—82no—
547D4—81no—
548D4—81no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4#38614yes—
605C2×C4#38614yes—
606C2×C4C22×C22⋊C411yescharacteristic
607C2×C4C2×C4×D411yescharacteristic
608C2×C4C2×C4×D412yes—
609C2×C4#52012yes—
610C2×C4C2×#12111yescharacteristic
611C2×C4C2×#12112yes—
612C2×C4C2×#12311yescharacteristic
613C2×C4C2×#12411yescharacteristic
614C2×C4C2×#12711yescharacteristic
615C2×C4#53511yescharacteristic
616C2×C4#53611yescharacteristic
617C2×C4#53712yes—
618C2×C4D4212yes—
619C2×C4#54712yes—
620C2×C4#54712yes—
621C2×C4#54812yes—
622C2×C4#55312yes—
623C2×C4#55312yes—
624C8—44no—
625C8—42no—
626C8—42no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—84no—
631C22—84no—
632C22—161no—
633C22—161no—
634C22—162no—
635C22—162no—
636C22—161no—
637C22—161no—
638C22C2×#38611yescharacteristic
639C22#236112yes—
640C22#241612yes—
641C22#243311yescharacteristic
642C22#243611yescharacteristic
643C22C4×#13712yes—
644C22#295111yescharacteristic
645C22#295211yescharacteristic
646C22#300411yescharacteristic
647C22#300511yescharacteristic
648C22#301211yescharacteristic
649C22#301311yescharacteristic
650C22#303712yes—
651C22#304511yescharacteristic
652C22#304611yescharacteristic
653C22#326612yes—
654C22#326812yes—
655C22#327012yes—
656C22#327112yes—
657C22#336912yes—
658C22#337112yes—
659C22#337512yes—
660C22#337712yes—
661C4—28no—
662C4—24no—
663C4—24no—
664C4—44no—
665C4—44no—
666C4—82no—
667C4—84no—
668C4—82no—
669C4—84no—
670C4C2×#39011yescharacteristic
671C4#231111yescharacteristic
672C4#236112yes—
673C4#241612yes—
674C4#250111yescharacteristic
675C4#250411yescharacteristic
676C2—162no—
677C2#1413711yescharacteristic
678C2#2092611yescharacteristic
679C2#2093711yescharacteristic
680C2#2300912yes—
681C2#2301312yes—
682C2#2473912yes—
683C2#2474112yes—
684C2#2576511yescharacteristic
685C2#2576611yescharacteristic
686C2#2594111yescharacteristic
687C2#2594211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×Q8×C4⋊C4C2semidirect
5. C2×#1878C2semidirect
6. C2×#1763C2semidirect
8. #11067C2semidirect
10. #10849C2semidirect
11. #10835C2semidirect
19. C2×#464C4semidirect
229. C4×Q8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4642
C2×Q8×C4⋊C42
C2×#19592
C2×#18802
C2×#18782
C2×#17632
#141532
#110672
#110662
#108492
#108352