Finite Group Databasebeta

← #368517

Subgroups — #368517

Order 512.

1724 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1869C211yesmaximal, characteristic
2C2×#1869C211yesmaximal, characteristic
3C2×#1752C211yesmaximal, characteristic, complemented
4C22×#336C211yesmaximal, characteristic, complemented
5C4×#386C211yesmaximal, characteristic
6#11298C212yesmaximal, complemented
7#11147C212yesmaximal, complemented
8#10922C212yesmaximal, complemented
9#10798C212yesmaximal, complemented
10#10714C211yesmaximal, characteristic, complemented
11#10712C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×#97C2212yes—
22C2×#386C411yescharacteristic
23C2×#386C411yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25C2×C4×C22⋊C4C2212yes—
26#1869—24no—
27#1869—24no—
28#1752—24no—
29C2×#336—24no—
30C2×#336C2211yescharacteristic
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1437C412yes—
41#1436C412yes—
42C22⋊#95C411yescharacteristic
43C22⋊#95C411yescharacteristic
44C4×#95C2212yes—
45C23×D4C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4C2×C411yescharacteristic
48C24×C4C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C23×C8—22no—
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—24no—
54C23×C8—24no—
55C23×C8—22no—
56C23×C8—22no—
57C23×C8—42no—
58C23×C8—42no—
59C23×C8D414yes—
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C23×C8C2312yes—
63C23×C8C2311yescharacteristic
64C23×C8C2312yes—
65C23×C8C2311yescharacteristic
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C22⋊C4C2×C411yescharacteristic
74C22×C22⋊C4C2×C411yescharacteristic
75C22×C22⋊C4C2×C411yescharacteristic
76C22×C22⋊C4C2×C411yescharacteristic
77C22×C22⋊C4C2×C412yes—
78C22×C22⋊C4C2×C412yes—
79C22×C42—22no—
80C22×C42—22no—
81C22×C42—42no—
82C22×C42—42no—
83C22×C42C2×C412yes—
84C22×C42C2312yes—
85C22×C42C2312yes—
86C22×C42C2311yescharacteristic
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8—24no—
97C2×C22⋊C8—42no—
98C2×C22⋊C8—42no—
99C2×C22⋊C8D414yes—
100C2×#97—24no—
101C2×#97—24no—
102C2×#97—42no—
103#386—24no—
104#386—24no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—42no—
108C4×C22⋊C4—42no—
109C2×#95C2×C412yes—
110C2×#95C2×C412yes—
111C2×#95C2×C412yes—
112C2×#95C2×C412yes—
113C2×#95C2×C412yes—
114C2×#95C2×C412yes—
115C2×#95C2×C412yes—
116#336—24no—
117#336—22no—
118#336—22no—
119#336—24no—
120#336—24no—
121#336—24no—
122#336—24no—
123#336—42no—
124#336—42no—
125#336—42no—
126C25C22×C411yescharacteristic
127C25C22×C411yescharacteristic
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4M1614yes—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—24no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4M1614yes—
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C412yes—
161C23×C4C22×C412yes—
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C412yes—
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C412yes—
166C23×C4C22×C411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4#4112yes—
170C23×C4#4112yes—
171C23×C4#4112yes—
172C23×C4#4112yes—
173C23×C4C2411yescharacteristic
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—28no—
178C22×C8—28no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—21no—
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×C8—22no—
194C22×C8—22no—
195C22×C8—28no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—44no—
207C22×C8—48no—
208C22×C8—48no—
209C22×C8—48no—
210C22×C8—48no—
211C22×C8—44no—
212C22×C8—44no—
213C22×C8C2×D412yes—
214C22×C8C2×D414yes—
215C22×C8C2×D412yes—
216C22×C8#4114yes—
217C22×C8#4114yes—
218C4×D4—44no—
219C4×D4—44no—
220C4×D4—44no—
221C4×D4—44no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C42—24no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42—44no—
265C2×C42—44no—
266C2×C42—48no—
267C22⋊C8—28no—
268C22⋊C8—48no—
269C22⋊C8—44no—
270C22⋊C8—44no—
271C22⋊C8—44no—
272#97—48no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—42no—
278#95—42no—
279#95—42no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24—22no—
288C24—22no—
289C24—21no—
290C24—22no—
291C24—21no—
292C24—21no—
293C24—22no—
294C24#11712yes—
295C24C4×D412yes—
296C24C4×D412yes—
297C24C2×M1612yes—
298C24C2×M1612yes—
299C24C2×M1612yes—
300C24#13112yes—
301C24#13112yes—
302C24C23×C411yescharacteristic
303C2×D4—28no—
304C2×D4—28no—
305C2×D4—28no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—44no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—28no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—28no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
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—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—48no—
377C22×C4—44no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4#11712yes—
397C22×C4C4×D412yes—
398C22×C4C4×D412yes—
399C22×C4#12114yes—
400C22×C4#12314yes—
401C22×C4#12414yes—
402C22×C4#12614yes—
403C22×C4C2×M1612yes—
404C22×C4C2×M1614yes—
405C22×C4#13114yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4112yes—
412C22×C4C2×#4112yes—
413C22×C4C2×#4111yescharacteristic
414C22×C4H(5,𝔽2)12yes—
415C22×C4H(5,𝔽2)12yes—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—22no—
421C2×C8—24no—
422C2×C8—22no—
423C2×C8—24no—
424C2×C8—22no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—416no—
430C2×C8—416no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8#12314yes—
452C4⋊C4—44no—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C42—48no—
460C42—41no—
461C42—41no—
462C42—41no—
463C42—41no—
464C42—42no—
465C42—42no—
466C42—42no—
467C42—42no—
468C42—42no—
469C42—42no—
470C42—42no—
471C42—42no—
472C23—22no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—24no—
493C23—41no—
494C23—42no—
495C23—42no—
496C23—41no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—41no—
510C23—41no—
511C23#38814yes—
512C23#43214yes—
513C23#43314yes—
514C23#43514yes—
515C23#43514yes—
516C23#43514yes—
517C23#43614yes—
518C23C2×#11711yescharacteristic
519C23C2×C4×D411yescharacteristic
520C23C2×C4×D411yescharacteristic
521C23C4×#4112yes—
522C23#51812yes—
523C23C22×M1611yescharacteristic
524C23C22×M1611yescharacteristic
525C23C2×#13111yescharacteristic
526C23C2×#13111yescharacteristic
527C23C2×#13112yes—
528C23#56812yes—
529D4—416no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—416no—
550C2×C4—416no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—82no—
591C2×C4—81no—
592C2×C4—82no—
593C2×C4#38814yes—
594C2×C4C2×#12111yescharacteristic
595C2×C4C2×#12311yescharacteristic
596C2×C4C2×#12311yescharacteristic
597C2×C4#52511yescharacteristic
598C2×C4C2×#12411yescharacteristic
599C2×C4C2×#12611yescharacteristic
600C2×C4#52912yes—
601C2×C4#53411yescharacteristic
602C2×C4#53511yescharacteristic
603C2×C4#53812yes—
604C2×C4#54612yes—
605C2×C4#54612yes—
606C2×C4#55112yes—
607C2×C4#55112yes—
608C2×C4#55312yes—
609C2×C4#55312yes—
610C2×C4#55312yes—
611C2×C4#55312yes—
612C8—24no—
613C8—44no—
614C8—42no—
615C8—44no—
616C8—81no—
617C8—81no—
618C22—28no—
619C22—28no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—28no—
626C22—28no—
627C22—24no—
628C22—44no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—44no—
633C22—82no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—82no—
638C22—81no—
639C22#192418yes—
640C22C2×#38811yescharacteristic
641C22C4×#12311yescharacteristic
642C22#232111yescharacteristic
643C22#235312yes—
644C22#236412yes—
645C22C2×#43211yescharacteristic
646C22C2×#43311yescharacteristic
647C22#292512yes—
648C22C2×#43511yescharacteristic
649C22C2×#43511yescharacteristic
650C22C2×#43511yescharacteristic
651C22C2×#43611yescharacteristic
652C22#292911yescharacteristic
653C22#293011yescharacteristic
654C22#293512yes—
655C22#296512yes—
656C22#296612yes—
657C22#296812yes—
658C22#297012yes—
659C22#297312yes—
660C22#297412yes—
661C22#297612yes—
662C22#297712yes—
663C4—24no—
664C4—44no—
665C4—44no—
666C4—42no—
667C4—81no—
668C4—82no—
669C4—82no—
670C4—81no—
671C4#241111yescharacteristic
672C4#241211yescharacteristic
673C4#244412yes—
674C4#250411yescharacteristic
675C4#251012yes—
676C4#259111yescharacteristic
677C2—28no—
678C2—82no—
679C2#1402011yescharacteristic
680C2C2×#192411yescharacteristic
681C2#2105211yescharacteristic
682C2#2156111yescharacteristic
683C2#2156311yescharacteristic
684C2#2203011yescharacteristic
685C2#2203411yescharacteristic
686C2#2278712yes—
687C2#2279112yes—
688C2#2311712yes—
689C2#2312012yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1752C2semidirect
4. C22×#336C2semidirect
6. #11298C2semidirect
7. #11147C2semidirect
8. #10922C2semidirect
9. #10798C2semidirect
10. #10714C2semidirect
11. #10712C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18692
C2×#17522
C22×#3362
C4×#3862
#112982
#111472
#109222
#107982
#107142
#107122