Finite Group Databasebeta

← #363642

Subgroups — #363642

Order 512.

1378 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C42×C22⋊C4C211yesmaximal, characteristic
2#11298C212yesmaximal, complemented
3#11147C212yesmaximal
4#10921C212yesmaximal
5#10797C212yesmaximal, complemented
6#10795C211yesmaximal, characteristic, complemented
7#10795C211yesmaximal, characteristic, complemented
8#10795C211yesmaximal, characteristic, complemented
9#10712C211yesmaximal, characteristic
10#10712C211yesmaximal, characteristic
11C4×#336C211yesmaximal, characteristic, complemented
12C23×C42C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14C22×C22⋊C8C2212yes—
15C22×C4×C8C2212yes—
16C2×C4×C22⋊C4C411yescharacteristic
17C2×C4×C22⋊C4C411yescharacteristic
18C2×C4×C22⋊C4C411yescharacteristic
19C2×C4×C22⋊C4C411yescharacteristic
20C2×C4×C22⋊C4C2211yescharacteristic
21C2×C4×C22⋊C4C2211yescharacteristic
22C2×C43C2211yescharacteristic
23C2×C43C2211yescharacteristic
24C2×#336C2212yes—
25C2×#336C2212yes—
26C2×#336C2211yescharacteristic
27C2×#336C2211yescharacteristic
28C2×#336C2212yes—
29C2×#336C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C4×#95C411yescharacteristic
38C4×#95C411yescharacteristic
39C4×#95C411yescharacteristic
40C4×#95C411yescharacteristic
41C4×#95C2211yescharacteristic
42C4×#95C2211yescharacteristic
43C24×C4C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C22×C22⋊C4C2×C411yescharacteristic
51C22×C22⋊C4C2×C411yescharacteristic
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C42—22no—
55C22×C42—22no—
56C22×C42—42no—
57C22×C42—42no—
58C22×C42C2×C411yescharacteristic
59C22×C42C2×C411yescharacteristic
60C22×C42C2×C411yescharacteristic
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42C2×C411yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8—24no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—42no—
79C2×C22⋊C8—42no—
80C2×C4×C8—24no—
81C2×C4×C8—24no—
82C2×C4×C8—42no—
83C4×C22⋊C4—22no—
84C4×C22⋊C4—22no—
85C4×C22⋊C4—24no—
86C4×C22⋊C4—24no—
87C4×C22⋊C4—22no—
88C4×C22⋊C4—22no—
89C4×C22⋊C4—24no—
90C4×C22⋊C4C818yescomplemented
91C2×#95C2×C411yescharacteristic
92C2×#95C2×C411yescharacteristic
93C2×#95C2×C411yescharacteristic
94C2×#95C2×C411yescharacteristic
95C2×#95C2×C411yescharacteristic
96C2×#95C2×C411yescharacteristic
97C2×#95C2×C411yescharacteristic
98C2×#95C2×C411yescharacteristic
99C2×#95C2×C411yescharacteristic
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C43—42no—
104C43—42no—
105#336—24no—
106#336—24no—
107#336—24no—
108#336—24no—
109#336—22no—
110#336—22no—
111#336—22no—
112#336—22no—
113#336—42no—
114#336—42no—
115#336—42no—
116#336—42no—
117#336—42no—
118C25C22×C411yescharacteristic
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—41no—
135C23×C4—41no—
136C23×C4—41no—
137C23×C4—41no—
138C23×C4—41no—
139C23×C4—41no—
140C23×C4C2×C814yes—
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4#4112yes—
155C23×C4#4112yes—
156C23×C4C2411yescharacteristic
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8#4114yes—
184C22×C8#4114yes—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4C2×C814yes—
195C2×C22⋊C4C2×C814yes—
196C2×C22⋊C4M1614yes—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—48no—
229C2×C42—48no—
230C2×C42—41no—
231C2×C42—41no—
232C2×C42—41no—
233C2×C42—41no—
234C2×C42—41no—
235C2×C42—41no—
236C2×C42—41no—
237C2×C42—41no—
238C2×C42—41no—
239C2×C42—41no—
240C2×C42—41no—
241C2×C42—41no—
242C2×C42—41no—
243C2×C42—41no—
244C2×C42—41no—
245C2×C42—41no—
246C2×C42C2×C814yes—
247C2×C42C2×C814yes—
248C2×C42M1614yes—
249C2×C42#4112yes—
250C2×C42#4112yes—
251C2×C42#4112yes—
252C2×C42#4112yes—
253C22⋊C8—48no—
254C22⋊C8—48no—
255C4×C8—48no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95C2×C814yes—
266#95C2×C814yes—
267#95M1614yes—
268C24—21no—
269C24—21no—
270C24—21no—
271C24—21no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24#11712yes—
281C24C22×C812yes—
282C24C2×M1612yes—
283C24C23×C411yescharacteristic
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—28no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—28no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4#11712yes—
385C22×C4#11712yes—
386C22×C4#11712yes—
387C22×C4#11712yes—
388C22×C4#11712yes—
389C22×C4C22×C812yes—
390C22×C4C22×C812yes—
391C22×C4C22×C812yes—
392C22×C4C22×C812yes—
393C22×C4C22×C812yes—
394C22×C4C22×C812yes—
395C22×C4C2×M1612yes—
396C22×C4C2×M1612yes—
397C22×C4C2×M1612yes—
398C22×C4C2×M1612yes—
399C22×C4C2×M1612yes—
400C22×C4C2×M1612yes—
401C22×C4C2×M1612yes—
402C22×C4C2×M1612yes—
403C22×C4#13112yes—
404C22×C4#13112yes—
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C22×C4C2×#4111yescharacteristic
413C22×C4H(5,𝔽2)12yes—
414C22×C4H(5,𝔽2)12yes—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
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—42no—
442C2×C8—42no—
443C22⋊C4—28no—
444C22⋊C4—28no—
445C22⋊C4—28no—
446C22⋊C4—28no—
447C42—24no—
448C42—24no—
449C42—24no—
450C42—24no—
451C42—44no—
452C42—44no—
453C42—44no—
454C42—44no—
455C42—44no—
456C42—44no—
457C42—44no—
458C42—44no—
459C42—44no—
460C42—44no—
461C42—42no—
462C42—42no—
463C42—42no—
464C42—42no—
465C42—42no—
466C42—42no—
467C42—42no—
468C42—42no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23#43114yes—
504C23C2×#11711yescharacteristic
505C23C2×#11711yescharacteristic
506C23C2×#11711yescharacteristic
507C23C4×#4111yescharacteristic
508C23C4×#4111yescharacteristic
509C23#51811yescharacteristic
510C23#51811yescharacteristic
511C23C23×C811yescharacteristic
512C23C22×M1611yescharacteristic
513C23C22×M1611yescharacteristic
514C23C22×M1611yescharacteristic
515C23C2×#13111yescharacteristic
516C23C2×#13111yescharacteristic
517C23#56811yescharacteristic
518C23#56811yescharacteristic
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—416no—
571C2×C4—416no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4#43114yes—
613C2×C4#43114yes—
614C2×C4#43214yes—
615C2×C4#43314yes—
616C2×C4#52911yescharacteristic
617C2×C4#52912yes—
618C2×C4#52911yescharacteristic
619C2×C4#52912yes—
620C2×C4#52911yescharacteristic
621C2×C4#52911yescharacteristic
622C2×C4#53812yes—
623C2×C4#53812yes—
624C2×C4#55112yes—
625C2×C4#55112yes—
626C2×C4#55112yes—
627C2×C4#55312yes—
628C2×C4#55312yes—
629C2×C4#55312yes—
630C2×C4#55312yes—
631C2×C4#55912yes—
632C8—44no—
633C8—44no—
634C8—82no—
635C8—82no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22#228311yescharacteristic
652C22#229811yescharacteristic
653C22#231211yescharacteristic
654C22#235311yescharacteristic
655C22#235311yescharacteristic
656C22#235311yescharacteristic
657C22#235711yescharacteristic
658C22C2×#43111yescharacteristic
659C22C2×#43111yescharacteristic
660C22C2×#43111yescharacteristic
661C22C2×#43211yescharacteristic
662C22C2×#43311yescharacteristic
663C22#292011yescharacteristic
664C22#292011yescharacteristic
665C22#292111yescharacteristic
666C22#292511yescharacteristic
667C22#292511yescharacteristic
668C22#292511yescharacteristic
669C22#292511yescharacteristic
670C22C8×#4111yescharacteristic
671C22C8×#4111yescharacteristic
672C22#296511yescharacteristic
673C22#296511yescharacteristic
674C22#297011yescharacteristic
675C22#297011yescharacteristic
676C22#297011yescharacteristic
677C22#297111yescharacteristic
678C22#297211yescharacteristic
679C22#297211yescharacteristic
680C22#297311yescharacteristic
681C22#297411yescharacteristic
682C22#297411yescharacteristic
683C22#297411yescharacteristic
684C22#297411yescharacteristic
685C22#297511yescharacteristic
686C4—24no—
687C4—24no—
688C4—24no—
689C4—24no—
690C4—42no—
691C4—42no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—42no—
696C4—42no—
697C4—42no—
698C4—42no—
699C4#257612yes—
700C4#257812yes—
701C4#259512yes—
702C4#259712yes—
703C2—44no—
704C2#1394711yescharacteristic
705C2#2077711yescharacteristic
706C2#2078611yescharacteristic
707C2#2143011yescharacteristic
708C2#2143511yescharacteristic
709C2#2190211yescharacteristic
710C2#2190411yescharacteristic
711C2#2278511yescharacteristic
712C2#2278511yescharacteristic
713C2#2278511yescharacteristic
714C2#2278611yescharacteristic
715C2#2278711yescharacteristic
716C2#2278811yescharacteristic
717C2#2283511yescharacteristic
718C2#2283911yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C23×C8
socleC24C23×C4
prime radicalC25C22×C4
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. #11298C2semidirect
5. #10797C2semidirect
6. #10795C2semidirect
7. #10795C2semidirect
8. #10795C2semidirect
11. C4×#336C2semidirect
90. C4×C22⋊C4C8semidirect

Maximal subgroups

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

subgroupindex
C42×C22⋊C42
#112982
#111472
#109212
#107972
#107952
#107122
C4×#3362