Finite Group Databasebeta

← #414116

Subgroups — #414116

Order 512.

1290 conjugacy classes of subgroups.

The subgroup lattice has 758 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2066C211yesmaximal, characteristic, complemented
2C2×#2065C211yesmaximal, characteristic
3C2×#1958C211yesmaximal, characteristic
4C2×#1934C211yesmaximal, characteristic
5C2×#1933C211yesmaximal, characteristic, complemented
6C2×C8×C4⋊C4C211yesmaximal, characteristic, complemented
7#15234C211yesmaximal, characteristic, complemented
8#15217C211yesmaximal, characteristic
9#11372C211yesmaximal, characteristic, complemented
10#11122C211yesmaximal, characteristic, complemented
11#11121C211yesmaximal, characteristic
12#11009C211yesmaximal, characteristic
13#10926C211yesmaximal, characteristic, complemented
14#10925C211yesmaximal, characteristic
15#10861C211yesmaximal, characteristic, complemented
16C22×#124C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C22×#106C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×#103C2211yescharacteristic
21C22×#103C2211yescharacteristic
22C22×#103C2211yescharacteristic
23C22×#103C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29C22×C4×C8C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31C2×#391C2211yescharacteristic
32C2×#391C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#390C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#382C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39#2066—24no—
40#2065—24no—
41#1958—24no—
42#1934—24no—
43#1933—24no—
44C8×C4⋊C4—24no—
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#340C2211yescharacteristic
48C2×#340C2211yescharacteristic
49C2×#340C2211yescharacteristic
50C2×#340C2211yescharacteristic
51C2×#336C2211yescharacteristic
52C2×#336C2211yescharacteristic
53#1442—22no—
54#1441—22no—
55#1440—22no—
56#1439—22no—
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2211yescharacteristic
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2211yescharacteristic
61C23×Q8C2311yescharacteristic
62C23×D4C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C2×#124—22no—
68C2×#124—22no—
69C2×#124—22no—
70C2×#124—22no—
71C2×#124—41no—
72C2×#124—41no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4D412yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C22⋊C4—22no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—21no—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×#106—22no—
90C2×#106—22no—
91C2×#106—22no—
92C2×#106—22no—
93C2×#106—41no—
94C2×#106—41no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8D414yes—
98C2×#103—22no—
99C2×#103—22no—
100C2×#103—22no—
101C2×#103—22no—
102C2×#103—22no—
103C2×#103—22no—
104C2×#103—22no—
105C2×#103—22no—
106C2×#103—22no—
107C2×#103—22no—
108C2×#103—22no—
109C2×#103—41no—
110C2×#102—22no—
111C2×#102—22no—
112C2×#102—22no—
113C2×#102—22no—
114C2×#102—22no—
115C2×#102—22no—
116C2×#102—22no—
117C2×#102—22no—
118C2×#102—22no—
119C2×#102—22no—
120C2×#102—22no—
121C2×#102—41no—
122C2×C4×C8—22no—
123C2×C4×C8—22no—
124C2×C4×C8—22no—
125C2×C4×C8—41no—
126C2×C4×C8—41no—
127C2×C4×C8—41no—
128C2×C4×C8—41no—
129C2×C4×C8D414yes—
130C2×C4×C8D414yes—
131#391—22no—
132#391—22no—
133#391—22no—
134#391—41no—
135#390—22no—
136#390—22no—
137#390—22no—
138#390—41no—
139#389—22no—
140#389—22no—
141#389—22no—
142#389—22no—
143#382—22no—
144#382—22no—
145C4×C4⋊C4—22no—
146C4×C4⋊C4—22no—
147C2×#95—22no—
148C2×#95—21no—
149C2×#95—22no—
150C2×#95—21no—
151C2×#95—21no—
152C2×#95—22no—
153C2×#95—21no—
154C2×#95—22no—
155C2×#95—22no—
156C2×#95—22no—
157C2×#95D412yes—
158#340—22no—
159#340—22no—
160#340—22no—
161#340—22no—
162#340—22no—
163#340—22no—
164#336—22no—
165#336—22no—
166C25—21no—
167C22×Q8—21no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×Q8—21no—
171C22×Q8—21no—
172C22×Q8—21no—
173C22×Q8—21no—
174C22×Q8—21no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8—21no—
179C22×Q8—21no—
180C22×Q8#4112yes—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—21no—
188C22×D4—21no—
189C22×D4—21no—
190C22×D4—21no—
191C22×D4—21no—
192C22×D4—21no—
193C22×D4—21no—
194C22×D4#4112yes—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4C2×D412yes—
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D412yes—
205C23×C4C2×D411yescharacteristic
206C23×C4C2411yescharacteristic
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8C2×D412yes—
228C22×C8C2×D412yes—
229C22×C8C2×D412yes—
230C22×C8C2×D412yes—
231C22×C8C2×D412yes—
232C22×C8C2×D412yes—
233C22×C8#4112yes—
234C22×C8#4112yes—
235#124—28no—
236#124—44no—
237#124—44no—
238#124—44no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—82no—
280C2×C4⋊C4#4112yes—
281C2×C4⋊C4#4112yes—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—82no—
291C2×C22⋊C4—82no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42#4112yes—
310#106—44no—
311#106—44no—
312C4⋊C8—28no—
313#103—28no—
314#103—44no—
315#103—44no—
316#103—44no—
317#102—28no—
318#102—44no—
319#102—44no—
320#102—44no—
321C4×C8—28no—
322C4×C8—44no—
323C4×C8—44no—
324#95—24no—
325#95—44no—
326#95—44no—
327#95—42no—
328#95—42no—
329#95—42no—
330#95—42no—
331#95—44no—
332#95—82no—
333#95—82no—
334#95—82no—
335C24—22no—
336C24—41no—
337C24—41no—
338C24—41no—
339C24—41no—
340C24—41no—
341C24—41no—
342C24—42no—
343C24—42no—
344C24—42no—
345C24—42no—
346C24C22×D411yescharacteristic
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×Q8—41no—
357C2×Q8—41no—
358C2×Q8—41no—
359C2×Q8—41no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8—41no—
363C2×Q8—41no—
364C2×Q8—41no—
365C2×Q8—41no—
366C2×Q8—41no—
367C2×Q8—41no—
368C2×Q8—41no—
369C2×Q8—41no—
370C2×Q8—41no—
371C2×Q8—41no—
372C2×Q8—41no—
373C2×Q8—41no—
374C2×Q8—41no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—41no—
385C2×D4—41no—
386C2×D4—41no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4#12114yes—
460C22×C4#12712yes—
461C22×C4#12712yes—
462C22×C4#13514yes—
463C22×C4#13514yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C2×#4111yescharacteristic
468C22×C4C2×#4111yescharacteristic
469C22×C4C2×#4111yescharacteristic
470C22×C4C2×#4111yescharacteristic
471C22×C4C2×#4111yescharacteristic
472C22×C4C2×#4111yescharacteristic
473C22×C4C2×#4111yescharacteristic
474C22×C4H(5,𝔽2)11yescharacteristic
475C22×C4H(5,𝔽2)11yescharacteristic
476C22×C42−1+411yescharacteristic
477C22×C42−1+411yescharacteristic
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—41no—
497C2×C8—41no—
498C2×C8—41no—
499C2×C8—41no—
500C2×C8—41no—
501C2×C8—41no—
502C2×C8—41no—
503C2×C8—41no—
504C2×C8#12114yes—
505C2×C8#12114yes—
506C4⋊C4—216no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—41no—
521C4⋊C4—41no—
522C4⋊C4—41no—
523C4⋊C4—41no—
524C4⋊C4—41no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4—84no—
532C4⋊C4—84no—
533C22⋊C4—416no—
534C22⋊C4—88no—
535C22⋊C4—84no—
536C22⋊C4—84no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—41no—
546C42—41no—
547C42—41no—
548C42—41no—
549C23—44no—
550C23—42no—
551C23—44no—
552C23—42no—
553C23—42no—
554C23—82no—
555C23—82no—
556C23—82no—
557C23—82no—
558C23—81no—
559C23—81no—
560C23—81no—
561C23—81no—
562C23—81no—
563C23—82no—
564C23—82no—
565C23—82no—
566C23—82no—
567C23—82no—
568C23—81no—
569C23—81no—
570C23—82no—
571C23—81no—
572C23C2×#12111yescharacteristic
573C23C2×#12711yescharacteristic
574C23#53211yescharacteristic
575C23#54611yescharacteristic
576C23#54611yescharacteristic
577C23#54811yescharacteristic
578C23#54811yescharacteristic
579C23C2×#13512yes—
580C23C2×#13512yes—
581C23#57712yes—
582C23#57712yes—
583Q8—42no—
584Q8—42no—
585Q8—42no—
586Q8—42no—
587Q8—81no—
588Q8—81no—
589Q8—81no—
590Q8—81no—
591D4—42no—
592D4—42no—
593D4—42no—
594D4—42no—
595D4—81no—
596D4—81no—
597D4—81no—
598D4—81no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—81no—
630C2×C4—81no—
631C2×C4—81no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4—81no—
637C2×C4—81no—
638C2×C4—81no—
639C2×C4—82no—
640C2×C4—82no—
641C2×C4—81no—
642C2×C4—81no—
643C2×C4—81no—
644C2×C4—81no—
645C2×C4—81no—
646C2×C4—81no—
647C2×C4—81no—
648C2×C4—81no—
649C2×C4—81no—
650C2×C4—81no—
651C2×C4—82no—
652C2×C4—82no—
653C2×C4—82no—
654C2×C4—82no—
655C2×C4—82no—
656C2×C4—81no—
657C2×C4—81no—
658C2×C4—82no—
659C2×C4#49514yes—
660C2×C4C2×#12111yescharacteristic
661C2×C4C2×#12111yescharacteristic
662C2×C4#52511yescharacteristic
663C2×C4#52511yescharacteristic
664C2×C4#52511yescharacteristic
665C2×C4#52911yescharacteristic
666C2×C4#52911yescharacteristic
667C2×C4#53211yescharacteristic
668C2×C4#53211yescharacteristic
669C2×C4#53211yescharacteristic
670C2×C4#53411yescharacteristic
671C2×C4#53611yescharacteristic
672C2×C4#53711yescharacteristic
673C2×C4#53711yescharacteristic
674C2×C4#53911yescharacteristic
675C2×C4#53911yescharacteristic
676C2×C4#54611yescharacteristic
677C2×C4#54611yescharacteristic
678C2×C4#54811yescharacteristic
679C2×C4#54811yescharacteristic
680C2×C4#55111yescharacteristic
681C2×C4#55111yescharacteristic
682C2×C4#55211yescharacteristic
683C2×C4#55211yescharacteristic
684C2×C4#55311yescharacteristic
685C2×C4#55311yescharacteristic
686C2×C4#55611yescharacteristic
687C2×C4#55611yescharacteristic
688C8—24no—
689C8—24no—
690C8—42no—
691C8—42no—
692C22—82no—
693C22—84no—
694C22—84no—
695C22—82no—
696C22—82no—
697C22—162no—
698C22—161no—
699C22—161no—
700C22—162no—
701C22—161no—
702C22—161no—
703C22#253411yescharacteristic
704C22#254011yescharacteristic
705C22#254011yescharacteristic
706C22#254511yescharacteristic
707C22#255811yescharacteristic
708C22#255811yescharacteristic
709C22#285611yescharacteristic
710C22#305012yes—
711C22#306012yes—
712C22C2×#49511yescharacteristic
713C22#314711yescharacteristic
714C22#315411yescharacteristic
715C22#315411yescharacteristic
716C22#316812yes—
717C22#317312yes—
718C22#329612yes—
719C22#330112yes—
720C22#330312yes—
721C22#330412yes—
722C22#336012yes—
723C22#336012yes—
724C22#336512yes—
725C22#336512yes—
726C4—24no—
727C4—42no—
728C4—44no—
729C4—44no—
730C4—82no—
731C4—82no—
732C4—82no—
733C4—161no—
734C4—161no—
735C4—161no—
736C4—161no—
737C4#239211yescharacteristic
738C4#253611yescharacteristic
739C4#254311yescharacteristic
740C4#256011yescharacteristic
741C4#256011yescharacteristic
742C4#260111yescharacteristic
743C4#260111yescharacteristic
744C4#261811yescharacteristic
745C2—162no—
746C2#1516311yescharacteristic
747C2#2809912yes—
748C2#2839811yescharacteristic
749C2#2840211yescharacteristic
750C2#2840611yescharacteristic
751C2#2840811yescharacteristic
752C2#2864212yes—
753C2#2901212yes—
754C2#2901412yes—
755C2#3317311yescharacteristic
756C2#3318111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×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
1. C2×#2066C2semidirect
5. C2×#1933C2semidirect
6. C2×C8×C4⋊C4C2semidirect
7. #15234C2semidirect
9. #11372C2semidirect
10. #11122C2semidirect
13. #10926C2semidirect
15. #10861C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20662
C2×#20652
C2×#19582
C2×#19342
C2×#19332
C2×C8×C4⋊C42
#152342
#152172
#113722
#111222
#111212
#110092
#109262
#109252
#108612