Finite Group Databasebeta

← #385939

Subgroups — #385939

Order 512.

1998 conjugacy classes of subgroups.

The subgroup lattice has 773 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2C2×#1954C212yesmaximal, complemented
3C2×#1916C212yesmaximal, complemented
4#11042C212yesmaximal, complemented
5#10964C212yesmaximal, complemented
6#10964C212yesmaximal, complemented
7#10921C212yesmaximal, complemented
8#10712C211yesmaximal, characteristic, complemented
9#10712C211yesmaximal, characteristic, complemented
10C22×#121C411yescharacteristic
11C22×C4×D4C2212yes—
12C22×C4×D4C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C22×C4⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2212yes—
21C22×C22⋊C8C2211yescharacteristic
22C22×C4×C8C2212yes—
23C4×#121—28no—
24C2×#390C411yescharacteristic
25C2×#389C411yescharacteristic
26C2×#388C412yes—
27C2×#386C412yes—
28C2×#380C411yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2212yes—
31#1954—28no—
32#1916—28no—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C23×D4C2×C412yes—
41C23×D4C2×C411yescharacteristic
42C24×C4C2×C411yescharacteristic
43C24×C4C2×C411yescharacteristic
44C24×C4C2312yes—
45C24×C4C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×#121—24no—
51C2×#121—24no—
52C2×#121C818yescomplemented
53C2×C4×D4—24no—
54C2×C4×D4—24no—
55C2×C4×D4—24no—
56C2×C4×D4—24no—
57C2×C4×D4—24no—
58C2×C4×D4—42no—
59C2×C4×D4—42no—
60C2×C4×D4—41no—
61C2×C4×D4—41no—
62C2×C4×D4—42no—
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C412yes—
68C22×C22⋊C4C2×C412yes—
69C22×C22⋊C4C2×C412yes—
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C412yes—
72C22×C42—24no—
73C22×C42—41no—
74C22×C42—41no—
75C22×C42—42no—
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81C2×C4⋊C8—24no—
82C2×C4⋊C8—24no—
83C2×C4⋊C8—42no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—41no—
92C2×C22⋊C8—41no—
93C2×C22⋊C8—42no—
94C2×C22⋊C8—42no—
95C2×C22⋊C8—42no—
96C2×C22⋊C8—42no—
97C2×C22⋊C8—42no—
98C2×C22⋊C8—42no—
99C2×C22⋊C8—41no—
100C2×C22⋊C8—41no—
101C2×C22⋊C8—42no—
102C2×C22⋊C8—42no—
103C2×C22⋊C8—42no—
104C2×C22⋊C8D414yescomplemented
105C2×C4×C8—24no—
106C2×C4×C8—24no—
107C2×C4×C8—42no—
108#390—24no—
109#389—24no—
110#388—28no—
111#386—28no—
112#380—24no—
113C4×C4⋊C4D418yes—
114C4×C22⋊C4—24no—
115C4×C22⋊C4—24no—
116C2×#95C2×C412yes—
117C2×#95C2×C412yes—
118C2×#95C2×C412yes—
119#336—24no—
120#336—24no—
121#336—24no—
122#336—24no—
123#336—41no—
124#336—42no—
125#336—41no—
126C25C22×C412yes—
127C25C22×C411yescharacteristic
128C22×D4—24no—
129C22×D4—24no—
130C22×D4—24no—
131C22×D4—22no—
132C22×D4—24no—
133C22×D4—22no—
134C22×D4—41no—
135C22×D4—41no—
136C22×D4—42no—
137C22×D4—41no—
138C22×D4—41no—
139C22×D4—42no—
140C22×D4C2×C818yes—
141C22×D4C2×C814yes—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—41no—
170C23×C4—41no—
171C23×C4—41no—
172C23×C4—41no—
173C23×C4—41no—
174C23×C4—41no—
175C23×C4—41no—
176C23×C4—41no—
177C23×C4C2×C814yes—
178C23×C4C22×C412yes—
179C23×C4C22×C412yes—
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C412yes—
185C23×C4C22×C411yescharacteristic
186C23×C4C2×D412yes—
187C23×C4#4112yes—
188C23×C4C2411yescharacteristic
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
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×C8—22no—
206C22×C8—22no—
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×C8C2×D414yes—
216C22×C8#4114yes—
217#121—232no—
218C4×D4—48no—
219C4×D4—48no—
220C4×D4—48no—
221C4×D4—48no—
222C4×D4—48no—
223C4×D4—48no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4C22⋊C414yes—
233C2×C4⋊C4C22⋊C414yes—
234C2×C4⋊C4C2×C814yes—
235C2×C4⋊C4M1614yes—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4C2×C818yes—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—22no—
264C2×C42—48no—
265C2×C42—48no—
266C2×C42—41no—
267C2×C42—41no—
268C2×C42—41no—
269C2×C42—41no—
270C2×C42C2×D414yes—
271C2×C42C2×D414yes—
272C2×C42C2×D414yes—
273C4⋊C8—48no—
274C22⋊C8—44no—
275C22⋊C8—48no—
276C22⋊C8—48no—
277C22⋊C8—48no—
278C22⋊C8—48no—
279C22⋊C8—44no—
280C22⋊C8—48no—
281C4×C8—48no—
282#95—24no—
283#95—24no—
284#95C22⋊C418yes—
285C24—21no—
286C24—21no—
287C24—21no—
288C24—21no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24C4×D412yes—
312C24C22×C814yes—
313C24C22×C812yes—
314C24#13112yes—
315C24C23×C411yescharacteristic
316C2×D4—216no—
317C2×D4—216no—
318C2×D4—28no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—48no—
322C2×D4—48no—
323C2×D4—48no—
324C2×D4—48no—
325C2×D4—48no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—44no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—28no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
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×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4C2×C22⋊C414yes—
434C22×C4C2×C22⋊C412yes—
435C22×C4C2×C22⋊C414yes—
436C22×C4C2×C22⋊C412yes—
437C22×C4C4×D412yes—
438C22×C4C22×C812yes—
439C22×C4C22×C814yes—
440C22×C4C22×C812yes—
441C22×C4C2×M1614yes—
442C22×C4C2×M1612yes—
443C22×C4#13112yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C2×#4111yescharacteristic
447C22×C4C2×#4112yes—
448C22×C4H(5,𝔽2)11yescharacteristic
449C22×C4H(5,𝔽2)11yescharacteristic
450C22×C4H(5,𝔽2)11yescharacteristic
451C22×C4H(5,𝔽2)12yes—
452C22×C4H(5,𝔽2)12yes—
453C22×C42−1+411yescharacteristic
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—42no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8—42no—
479C2×C8—42no—
480C2×C8—42no—
481C2×C8—42no—
482C4⋊C4—44no—
483C4⋊C4—48no—
484C4⋊C4—44no—
485C4⋊C4C22⋊C8116yescomplemented
486C22⋊C4—216no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C42—24no—
492C42—24no—
493C42—24no—
494C42—24no—
495C42—24no—
496C42—24no—
497C42—44no—
498C42—44no—
499C42—42no—
500C42—42no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23C8×D414yes—
571C23C22×C22⋊C411yescharacteristic
572C23C2×C4×D411yescharacteristic
573C23C4×#4111yescharacteristic
574C23#51811yescharacteristic
575C23#51811yescharacteristic
576C23#51811yescharacteristic
577C23#52011yescharacteristic
578C23C23×C811yescharacteristic
579C23C22×M1611yescharacteristic
580C23C2×#13111yescharacteristic
581C23C2×#13111yescharacteristic
582C23#56811yescharacteristic
583C23#56811yescharacteristic
584C23#56811yescharacteristic
585C23#56811yescharacteristic
586D4—432no—
587D4—416no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—416no—
616C2×C4—416no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—82no—
649C2×C4—82no—
650C2×C4—82no—
651C2×C4—82no—
652C2×C4—82no—
653C2×C4—82no—
654C2×C4—82no—
655C2×C4—82no—
656C2×C4C2×C22⋊C818yes—
657C2×C4C2×C22⋊C814yes—
658C2×C4#53411yescharacteristic
659C2×C4#53512yes—
660C2×C4#53512yes—
661C2×C4#53711yescharacteristic
662C2×C4#53812yes—
663C2×C4#53811yescharacteristic
664C2×C4#53911yescharacteristic
665C2×C4#54012yes—
666C2×C4D4212yes—
667C2×C4#54612yes—
668C2×C4#54612yes—
669C2×C4#54812yes—
670C2×C4#55012yes—
671C2×C4#55312yes—
672C2×C4#56012yes—
673C2×C4#56212yes—
674C8—44no—
675C8—44no—
676C8—82no—
677C8—82no—
678C22—24no—
679C22—24no—
680C22—24no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—42no—
693C22—44no—
694C22—44no—
695C22—44no—
696C22—44no—
697C22—44no—
698C22—44no—
699C22—44no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22#230911yescharacteristic
709C22#232511yescharacteristic
710C22#233311yescharacteristic
711C22D4×C22⋊C411yescharacteristic
712C22#234111yescharacteristic
713C22#236411yescharacteristic
714C22#237711yescharacteristic
715C22C22×C22⋊C811yescharacteristic
716C22#287911yescharacteristic
717C22#288011yescharacteristic
718C22#288011yescharacteristic
719C22C2×C8×D411yescharacteristic
720C22#293211yescharacteristic
721C22D4×M1611yescharacteristic
722C22#293511yescharacteristic
723C22C8×#4111yescharacteristic
724C22#296711yescharacteristic
725C22#296811yescharacteristic
726C22#297111yescharacteristic
727C22#297211yescharacteristic
728C22#297211yescharacteristic
729C22#297211yescharacteristic
730C22#297411yescharacteristic
731C22#297511yescharacteristic
732C22#297511yescharacteristic
733C22#297611yescharacteristic
734C22#297611yescharacteristic
735C22#297711yescharacteristic
736C22#297811yescharacteristic
737C22#298011yescharacteristic
738C22#298111yescharacteristic
739C22#298811yescharacteristic
740C22#299111yescharacteristic
741C22#299211yescharacteristic
742C22#299511yescharacteristic
743C4—24no—
744C4—28no—
745C4—42no—
746C4—42no—
747C4—42no—
748C4—42no—
749C4—44no—
750C4—82no—
751C4#267312yes—
752C4#267612yes—
753C4#271112yes—
754C4#271412yes—
755C2—44no—
756C2—84no—
757C2#1441811yescharacteristic
758C2#2178611yescharacteristic
759C2#2179211yescharacteristic
760C2#2216811yescharacteristic
761C2#2217811yescharacteristic
762C2#2232911yescharacteristic
763C2#2233111yescharacteristic
764C2D4×C22⋊C811yescharacteristic
765C2#2240311yescharacteristic
766C2#2253011yescharacteristic
767C2#2253311yescharacteristic
768C2#2311611yescharacteristic
769C2#2312111yescharacteristic
770C2#2332611yescharacteristic
771C2#2332711yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C23×C8
socleC24C23×C4
prime radicalC22×#121C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1954C2semidirect
3. C2×#1916C2semidirect
4. #11042C2semidirect
5. #10964C2semidirect
6. #10964C2semidirect
7. #10921C2semidirect
8. #10712C2semidirect
9. #10712C2semidirect
52. C2×#121C8semidirect
104. C2×C22⋊C8D4semidirect
485. C4⋊C4C22⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
C2×#19542
C2×#19162
#110422
#109642
#109212
#107122