Finite Group Databasebeta

← #385833

Subgroups — #385833

Order 512.

2138 conjugacy classes of subgroups.

The subgroup lattice has 778 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2C2×#1869C212yesmaximal, complemented
3C2×#1869C212yesmaximal, complemented
4#11346C212yesmaximal, complemented
5#11147C212yesmaximal, complemented
6#10964C212yesmaximal, complemented
7#10714C212yesmaximal, complemented
8#10712C211yesmaximal, characteristic, complemented
9#10712C211yesmaximal, characteristic, complemented
10C24×C8C2212yes—
11C22×#121C411yescharacteristic
12C22×C4×D4C2212yes—
13C22×C4×D4C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C4×#121—28no—
22C2×#390C411yescharacteristic
23C2×#389C411yescharacteristic
24C2×#388C412yes—
25C2×#386C412yes—
26C2×#380C411yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2212yes—
29#1869—28no—
30#1869—28no—
31C2×#336C2212yes—
32C2×#336C2212yes—
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×C8—24no—
47C23×C8—24no—
48C23×C8—24no—
49C23×C8—42no—
50C23×C8—42no—
51C23×C8—42no—
52C23×C8—42no—
53C23×C8C2312yes—
54C23×C8C2312yes—
55C23×C8C2312yes—
56C23×C8C2312yes—
57C2×#121—24no—
58C2×#121—24no—
59C2×#121C818yescomplemented
60C2×C4×D4—24no—
61C2×C4×D4—24no—
62C2×C4×D4—24no—
63C2×C4×D4—24no—
64C2×C4×D4—41no—
65C2×C4×D4—42no—
66C2×C4×D4—41no—
67C2×C4×D4—42no—
68C2×C4×D4—42no—
69C2×C4×D4D418yes—
70C22×C4⋊C4C2×C412yes—
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C22⋊C4C2×C412yes—
75C22×C22⋊C4C2×C412yes—
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2×C412yes—
78C22×C22⋊C4C2×C412yes—
79C22×C42—24no—
80C22×C42—41no—
81C22×C42—41no—
82C22×C42—42no—
83C22×C42C2×C411yescharacteristic
84C22×C42C2×C411yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2312yes—
87C22×C42C2311yescharacteristic
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8—41no—
97C2×C22⋊C8—41no—
98C2×C22⋊C8—42no—
99C2×C22⋊C8—42no—
100C2×C22⋊C8—42no—
101C2×C22⋊C8—42no—
102C2×C22⋊C8—42no—
103C2×C22⋊C8—41no—
104C2×C22⋊C8—41no—
105C2×C22⋊C8D414yescomplemented
106#390—24no—
107#389—24no—
108#388—28no—
109#386—28no—
110#380—24no—
111C4×C4⋊C4—24no—
112C4×C22⋊C4—24no—
113C4×C22⋊C4—24no—
114C2×#95C2×C412yes—
115C2×#95C2×C412yes—
116C2×#95C2×C412yes—
117#336—24no—
118#336—24no—
119#336—24no—
120#336—24no—
121#336—42no—
122#336—42no—
123#336—41no—
124#336—42no—
125#336—41no—
126C25C22×C412yes—
127C25C22×C411yescharacteristic
128C22×D4—24no—
129C22×D4—24no—
130C22×D4—24no—
131C22×D4—24no—
132C22×D4—41no—
133C22×D4—41no—
134C22×D4—42no—
135C22×D4—41no—
136C22×D4—41no—
137C22×D4—42no—
138C22×D4C22⋊C414yes—
139C22×D4C2×C814yes—
140C22×D4C2×C818yescomplemented
141C22×D4M1614yes—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—41no—
168C23×C4—41no—
169C23×C4—41no—
170C23×C4—41no—
171C23×C4—41no—
172C23×C4—41no—
173C23×C4—41no—
174C23×C4—41no—
175C23×C4C2×C814yescomplemented
176C23×C4C22×C412yes—
177C23×C4C22×C412yes—
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C412yes—
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C2×D414yes—
185C23×C4C2×D414yes—
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×C8—48no—
216C22×C8—48no—
217C22×C8—48no—
218C22×C8—48no—
219C22×C8—48no—
220C22×C8—48no—
221C22×C8—48no—
222C22×C8C2×D414yes—
223C22×C8#4114yes—
224#121—232no—
225C4×D4—48no—
226C4×D4—48no—
227C4×D4—48no—
228C4×D4—48no—
229C4×D4—48no—
230C4×D4—48no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4C22⋊C414yes—
242C2×C4⋊C4C2×C814yescomplemented
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4C22⋊C418yes—
257C2×C22⋊C4C2×C818yescomplemented
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—22no—
273C2×C42—48no—
274C2×C42—48no—
275C2×C42—41no—
276C2×C42—41no—
277C2×C42—41no—
278C2×C42—41no—
279C2×C42C2×D414yes—
280C22⋊C8—44no—
281C22⋊C8—48no—
282C22⋊C8—48no—
283C22⋊C8—48no—
284C22⋊C8—48no—
285C22⋊C8—44no—
286#95—24no—
287#95—24no—
288#95—24no—
289C24—21no—
290C24—22no—
291C24—22no—
292C24—21no—
293C24—21no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—22no—
301C24—21no—
302C24—22no—
303C24—21no—
304C24—22no—
305C24—22no—
306C24—21no—
307C24—21no—
308C24—21no—
309C24—21no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24C2×C22⋊C414yes—
314C24C4×D412yes—
315C24C22×C814yes—
316C24C22×C812yes—
317C24C2×M1614yes—
318C24#13112yes—
319C24C23×C411yescharacteristic
320C2×D4—216no—
321C2×D4—216no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—48no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—48no—
330C2×D4—48no—
331C2×D4—48no—
332C2×D4—48no—
333C2×D4C22⋊C8116yescomplemented
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—28no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
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×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4C2×C22⋊C412yes—
440C22×C4C2×C22⋊C414yes—
441C22×C4C2×C22⋊C412yes—
442C22×C4C4×D412yes—
443C22×C4C22×C812yes—
444C22×C4C22×C814yes—
445C22×C4C22×C812yes—
446C22×C4C2×M1612yes—
447C22×C4#13112yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4C2×#4112yes—
452C22×C4H(5,𝔽2)11yescharacteristic
453C22×C4H(5,𝔽2)11yescharacteristic
454C22×C4H(5,𝔽2)12yes—
455C22×C4H(5,𝔽2)11yescharacteristic
456C22×C4H(5,𝔽2)12yes—
457C22×C42−1+411yescharacteristic
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—432no—
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—
482C2×C8—42no—
483C2×C8—42no—
484C2×C8—42no—
485C2×C8—42no—
486C2×C8—42no—
487C4⋊C4—28no—
488C4⋊C4—44no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C22⋊C4—216no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C42—24no—
497C42—24no—
498C42—44no—
499C42—44no—
500C42—42no—
501C42—42no—
502C42—42no—
503C42—42no—
504C42—42no—
505C42—42no—
506C23—24no—
507C23—22no—
508C23—22no—
509C23—24no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—22no—
516C23—24no—
517C23—24no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—22no—
525C23—24no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—22no—
535C23—24no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—24no—
540C23—24no—
541C23—22no—
542C23—41no—
543C23—42no—
544C23—42no—
545C23—41no—
546C23—41no—
547C23—42no—
548C23—41no—
549C23—41no—
550C23—42no—
551C23—41no—
552C23—42no—
553C23—42no—
554C23—41no—
555C23—42no—
556C23—42no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—41no—
565C23—42no—
566C23—41no—
567C23—42no—
568C23—41no—
569C23—42no—
570C23—41no—
571C23—42no—
572C23—41no—
573C23—42no—
574C23C2×C22⋊C818yes—
575C23C8×D414yes—
576C23C22×C22⋊C411yescharacteristic
577C23C2×C4×D411yescharacteristic
578C23C4×#4111yescharacteristic
579C23#51811yescharacteristic
580C23#51811yescharacteristic
581C23#51811yescharacteristic
582C23#52011yescharacteristic
583C23C23×C811yescharacteristic
584C23C22×M1611yescharacteristic
585C23C2×#13111yescharacteristic
586C23C2×#13111yescharacteristic
587C23#56811yescharacteristic
588C23#56811yescharacteristic
589C23#56811yescharacteristic
590C23#56811yescharacteristic
591D4—416no—
592D4—432no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—416no—
618C2×C4—416no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4C2×C22⋊C814yes—
663C2×C4#53411yescharacteristic
664C2×C4#53412yes—
665C2×C4#53412yes—
666C2×C4#53711yescharacteristic
667C2×C4#53812yes—
668C2×C4#53811yescharacteristic
669C2×C4#53911yescharacteristic
670C2×C4#54012yes—
671C2×C4D4212yes—
672C2×C4#54612yes—
673C2×C4#54612yes—
674C2×C4#54612yes—
675C2×C4#54712yes—
676C2×C4#55312yes—
677C2×C4#56012yes—
678C2×C4#56212yes—
679C8—44no—
680C8—44no—
681C8—82no—
682C8—82no—
683C22—24no—
684C22—24no—
685C22—28no—
686C22—24no—
687C22—28no—
688C22—28no—
689C22—42no—
690C22—44no—
691C22—44no—
692C22—42no—
693C22—42no—
694C22—44no—
695C22—42no—
696C22—44no—
697C22—44no—
698C22—42no—
699C22—44no—
700C22—44no—
701C22—42no—
702C22—42no—
703C22—44no—
704C22—44no—
705C22—42no—
706C22—44no—
707C22—42no—
708C22—44no—
709C22—42no—
710C22—42no—
711C22—42no—
712C22—44no—
713C22#230811yescharacteristic
714C22#232511yescharacteristic
715C22#233311yescharacteristic
716C22D4×C22⋊C411yescharacteristic
717C22#233911yescharacteristic
718C22#236411yescharacteristic
719C22#237711yescharacteristic
720C22C22×C22⋊C811yescharacteristic
721C22#287911yescharacteristic
722C22#288011yescharacteristic
723C22#288011yescharacteristic
724C22C2×C8×D411yescharacteristic
725C22#293211yescharacteristic
726C22D4×M1611yescharacteristic
727C22#293511yescharacteristic
728C22C8×#4111yescharacteristic
729C22#296711yescharacteristic
730C22#296811yescharacteristic
731C22#297011yescharacteristic
732C22#297211yescharacteristic
733C22#297211yescharacteristic
734C22#297211yescharacteristic
735C22#297311yescharacteristic
736C22#297311yescharacteristic
737C22#297411yescharacteristic
738C22#297611yescharacteristic
739C22#297611yescharacteristic
740C22#297711yescharacteristic
741C22#297811yescharacteristic
742C22#298011yescharacteristic
743C22#298111yescharacteristic
744C22#298811yescharacteristic
745C22#299011yescharacteristic
746C22#299211yescharacteristic
747C22#299511yescharacteristic
748C4—24no—
749C4—44no—
750C4—42no—
751C4—44no—
752C4—42no—
753C4—42no—
754C4—42no—
755C4—82no—
756C4#266812yes—
757C4#267112yes—
758C4#268812yes—
759C4#270812yes—
760C2—44no—
761C2—48no—
762C2#1441711yescharacteristic
763C2#2173811yescharacteristic
764C2#2174411yescharacteristic
765C2#2216811yescharacteristic
766C2#2217011yescharacteristic
767C2#2232911yescharacteristic
768C2#2233911yescharacteristic
769C2D4×C22⋊C811yescharacteristic
770C2#2240311yescharacteristic
771C2#2246911yescharacteristic
772C2#2247511yescharacteristic
773C2#2311611yescharacteristic
774C2#2311911yescharacteristic
775C2#2332611yescharacteristic
776C2#2333311yescharacteristic
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×#1869C2semidirect
3. C2×#1869C2semidirect
4. #11346C2semidirect
5. #11147C2semidirect
6. #10964C2semidirect
7. #10714C2semidirect
8. #10712C2semidirect
9. #10712C2semidirect
59. C2×#121C8semidirect
105. C2×C22⋊C8D4semidirect
140. C22×D4C2×C8semidirect
175. C23×C4C2×C8semidirect
242. C2×C4⋊C4C2×C8semidirect
257. C2×C22⋊C4C2×C8semidirect
333. C2×D4C22⋊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×#18692
#113462
#111472
#109642
#107142
#107122