Finite Group Databasebeta

← #365762

Subgroups — #365762

Order 512.

1716 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2352C211yesmaximal, characteristic
2C2×#1955C211yesmaximal, characteristic
3C2×#1947C211yesmaximal, characteristic, complemented
4C2×#1836C211yesmaximal, characteristic, complemented
5#13906C211yesmaximal, characteristic
6#11096C212yesmaximal, complemented
7#11096C212yesmaximal, complemented
8#10850C211yesmaximal, characteristic, complemented
9#10731C212yesmaximal, complemented
10#10730C212yesmaximal, complemented
11#10703C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×#97C2211yescharacteristic
20#2352—24no—
21#2352—24no—
22C2×#390—21no—
23C2×#388—21no—
24C2×#387—21no—
25C2×#386—21no—
26C2×#384C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4—21no—
35#1955C418yescomplemented
36#1947—24no—
37#1836—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45#1439—21no—
46#1439—21no—
47C4⋊#95—21no—
48C23×D4D411yescharacteristic
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C24×C4C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4C2×C414yes—
59C2×C4×D4Q814yes—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—24no—
63C2×#117—42no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—42no—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4D411yescharacteristic
84C22×C22⋊C4D411yescharacteristic
85C22×C42—21no—
86C22×C42—21no—
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89C2×#107—24no—
90C2×#107—42no—
91C2×#107—42no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8C2×C414yescomplemented
95C2×C22⋊C8—42no—
96C2×C22⋊C8—42no—
97C2×C22⋊C8—42no—
98C2×C22⋊C8—42no—
99C2×C22⋊C8C2×C418yes—
100C2×C22⋊C8C2×C418yescomplemented
101C2×#97—22no—
102C2×#97—22no—
103C2×#97C2×C414yescomplemented
104#390—44no—
105#388—28no—
106#387—28no—
107#386—28no—
108#384—22no—
109#384—22no—
110#382—22no—
111#382D414yes—
112#380—22no—
113#380—22no—
114#380—22no—
115#380—22no—
116#380—24no—
117#380—24no—
118#380—42no—
119#380—42no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4D414yes—
124C4×C22⋊C4—28no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—21no—
133#340—24no—
134#340—24no—
135#340—24no—
136#340—24no—
137#340—24no—
138#340—42no—
139#340—42no—
140C25C2×D411yescharacteristic
141C25C2×D411yescharacteristic
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—41no—
147C22×D4—41no—
148C22×D4C4⋊C414yes—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—21no—
170C23×C4—24no—
171C23×C4—22no—
172C23×C4—44no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—82no—
176C23×C4C22×C412yes—
177C23×C4C22×C412yes—
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×Q812yes—
183C23×C4C2×Q812yes—
184C23×C4C2411yescharacteristic
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—22no—
196C22×C8—22no—
197C22×C8C22×C414yes—
198C22×C8C22×C414yes—
199C4×D4—28no—
200C4×D4—28no—
201C4×D4—28no—
202C4×D4—28no—
203#117—48no—
204#117—44no—
205#117—44no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—48no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—82no—
240C2×C4⋊C4C4⋊C414yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4#4114yes—
246C2×C4⋊C4#4114yes—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—82no—
262C2×C22⋊C4—82no—
263C2×C22⋊C4—82no—
264C2×C22⋊C4C4⋊C414yes—
265C2×C22⋊C4C4⋊C414yes—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—44no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—82no—
273C2×C42C22×C412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×Q812yes—
277C2×C42#4112yes—
278C2×C42#4112yes—
279#107—48no—
280C4⋊C8—28no—
281C22⋊C8—48no—
282C22⋊C8—48no—
283#97—28no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—44no—
290#95—44no—
291#95—42no—
292C24—21no—
293C24—21no—
294C24—22no—
295C24—22no—
296C24—21no—
297C24—21no—
298C24—22no—
299C24—22no—
300C24—21no—
301C24—21no—
302C24—22no—
303C24—22no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—22no—
309C24—22no—
310C24C2×C4⋊C412yes—
311C24C2×C4⋊C412yes—
312C24AGL(1,ℤ/8)12yes—
313C24#13712yes—
314C24C22×D411yescharacteristic
315C2×D4—28no—
316C2×D4—28no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—44no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—48no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—48no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—42no—
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—84no—
409C22×C4—88no—
410C22×C4—88no—
411C22×C4C2×C4⋊C412yes—
412C22×C4C2×C4⋊C412yes—
413C22×C4C2×C4⋊C412yes—
414C22×C4C2×C4⋊C412yes—
415C22×C4#12012yes—
416C22×C4#12012yes—
417C22×C4#12312yes—
418C22×C4#12312yes—
419C22×C4C23×C411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×Q811yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4H(5,𝔽2)12yes—
428C22×C4H(5,𝔽2)12yes—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—28no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—48no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—84no—
465C4⋊C4—84no—
466C4⋊C4#12414yes—
467C22⋊C4—28no—
468C22⋊C4—28no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—84no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C42—48no—
475C42—84no—
476C42—84no—
477C42C4×D414yes—
478C42C4×D414yes—
479C42C22⋊Q814yes—
480C42C22⋊Q814yes—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—22no—
485C23—24no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—22no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—42no—
499C23—42no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23#42814yes—
527C23C22×C4⋊C411yescharacteristic
528C23C2×#12011yescharacteristic
529C23C2×#12311yescharacteristic
530C23#54612yes—
531C23#54612yes—
532C23C2×AGL(1,ℤ/8)11yescharacteristic
533C23C2×#13711yescharacteristic
534C23#57511yescharacteristic
535C23#57511yescharacteristic
536C23#57612yes—
537C23#57812yes—
538D4—416no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—816no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4#38514yes—
607C2×C4#38514yes—
608C2×C4C2×C4×D411yescharacteristic
609C2×C4C2×C4×D411yescharacteristic
610C2×C4C4×#4112yes—
611C2×C4#51812yes—
612C2×C4C2×C22⋊Q811yescharacteristic
613C2×C4C2×C22⋊Q811yescharacteristic
614C2×C4C2×#12411yescharacteristic
615C2×C4#52912yes—
616C2×C4#53211yescharacteristic
617C2×C4#53511yescharacteristic
618C2×C4#53511yescharacteristic
619C2×C4#54012yes—
620C2×C4#54612yes—
621C2×C4#55012yes—
622C2×C4#55312yes—
623C2×C4#55412yes—
624C2×C4#55412yes—
625C2×C4#55512yes—
626C8—44no—
627C8—44no—
628C22—24no—
629C22—28no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—44no—
634C22—44no—
635C22—42no—
636C22—44no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—44no—
643C22—44no—
644C22—81no—
645C22—81no—
646C22—82no—
647C22—82no—
648C22—81no—
649C22—81no—
650C22C2×#38511yescharacteristic
651C22#234912yes—
652C22#242012yes—
653C22#243111yescharacteristic
654C22#243511yescharacteristic
655C22C2×#42811yescharacteristic
656C22#291111yescharacteristic
657C22#291312yes—
658C22#300611yescharacteristic
659C22#300711yescharacteristic
660C22#300811yescharacteristic
661C22#301111yescharacteristic
662C22#309411yescharacteristic
663C22#309511yescharacteristic
664C22#309812yes—
665C22#330812yes—
666C22#330912yes—
667C22#331212yes—
668C22#331312yes—
669C22#331412yes—
670C22#331512yes—
671C22#331812yes—
672C22#331912yes—
673C4—24no—
674C4—24no—
675C4—44no—
676C4—42no—
677C4—42no—
678C4—82no—
679C4—82no—
680C4—84no—
681C4—82no—
682C4—162no—
683C4C4×#12411yescharacteristic
684C4#232811yescharacteristic
685C4#236312yes—
686C4#245412yes—
687C4#256311yescharacteristic
688C4#256611yescharacteristic
689C2—44no—
690C2—82no—
691C2#1397911yescharacteristic
692C2#2084611yescharacteristic
693C2#2084711yescharacteristic
694C2#2270912yes—
695C2#2271212yes—
696C2#2498112yes—
697C2#2498412yes—
698C2#2558111yescharacteristic
699C2#2558411yescharacteristic
700C2#2589611yescharacteristic
701C2#2589711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1947C2semidirect
4. C2×#1836C2semidirect
6. #11096C2semidirect
7. #11096C2semidirect
8. #10850C2semidirect
9. #10731C2semidirect
10. #10730C2semidirect
11. #10703C2semidirect
35. #1955C4semidirect
94. C2×C4⋊C8C2×C4semidirect
100. C2×C22⋊C8C2×C4semidirect
103. C2×#97C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23522
C2×#19552
C2×#19472
C2×#18362
#139062
#110962
#108502
#107312
#107302
#107032