Finite Group Databasebeta

← #366813

Subgroups — #366813

Order 512.

2088 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2501C211yesmaximal, characteristic
2C2×#1893C211yesmaximal, characteristic
3C2×#1874C211yesmaximal, characteristic
4C2×#1850C211yesmaximal, characteristic, complemented
5C2×#1823C211yesmaximal, characteristic, complemented
6C22×#340C211yesmaximal, characteristic, complemented
7C22⋊C4×C4⋊C4C211yesmaximal, characteristic, complemented
8#11357C212yesmaximal, complemented
9#11076C212yesmaximal, complemented
10#10931C212yesmaximal, complemented
11#10849C212yesmaximal, complemented
12C24×C8C2211yescharacteristic
13C22×#123—21no—
14C22×#123—21no—
15C22×#121C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C22×C4⋊C8C2212yes—
19C22×#102C2211yescharacteristic
20C22×#102C2212yes—
21C22×#102C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23#2501—24no—
24#2501C418yescomplemented
25C2×#399—22no—
26C2×#390C2212yes—
27C2×#384C2212yes—
28C2×#380C2211yescharacteristic
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C22⋊C4—21no—
31C2×C4×C22⋊C4C2211yescharacteristic
32#1893—24no—
33#1874—24no—
34#1850—24no—
35#1823—24no—
36C2×#340—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2211yescharacteristic
43C2×#340C2212yes—
44C2×#336C2212yes—
45#1436—22no—
46C4⋊#95C2212yes—
47C22⋊#95—21no—
48C23×D4—21no—
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C23×C8—22no—
54C23×C8—24no—
55C23×C8—22no—
56C23×C8—22no—
57C23×C8—42no—
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C23×C8C2312yes—
61C2×#123—28no—
62C2×#123—28no—
63C2×#123—44no—
64C2×#123—44no—
65C2×#123—44no—
66C2×#123—44no—
67C2×#121—22no—
68C2×#121—22no—
69C2×#121—22no—
70C2×#121—22no—
71C2×#121—22no—
72C2×#121C2×C414yes—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—24no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—42no—
85C22×C4⋊C4C2×C414yes—
86C22×C4⋊C4D414yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C22⋊C4—21no—
93C22×C22⋊C4—21no—
94C22×C22⋊C4—22no—
95C22×C22⋊C4—22no—
96C22×C22⋊C4D411yescharacteristic
97C22×C22⋊C4D411yescharacteristic
98C22×C42—22no—
99C22×C42C2312yes—
100C2×C4⋊C8—24no—
101C2×C4⋊C8—24no—
102C2×C4⋊C8—42no—
103C2×#102—24no—
104C2×#102—22no—
105C2×#102—22no—
106C2×#102—24no—
107C2×#102—24no—
108C2×#102—42no—
109C2×#102—42no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113#399—216no—
114#390—24no—
115#390C2×C418yescomplemented
116#384—24no—
117#384C2×C418yescomplemented
118#380—22no—
119#380D414yes—
120C4×C4⋊C4—24no—
121C4×C4⋊C4—42no—
122C4×C22⋊C4—44no—
123C4×C22⋊C4C2×C414yes—
124C4×C22⋊C4D414yes—
125C2×#95—21no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95D412yes—
130#340—24no—
131#340—24no—
132#340—22no—
133#340—22no—
134#340—24no—
135#340—24no—
136#340—24no—
137#336—24no—
138C25—21no—
139C25C2×D411yescharacteristic
140C22×D4—24no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4—42no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4—42no—
156C22×D4C22×C412yes—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—21no—
167C23×C4—24no—
168C23×C4—44no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—82no—
178C23×C4C22×C412yes—
179C23×C4C2×D412yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D412yes—
182C23×C4C2×D412yes—
183C23×C4C2×D412yes—
184C23×C4C2×D412yes—
185C23×C4#4112yes—
186C23×C4#4112yes—
187C23×C4C2411yescharacteristic
188C22×C8—28no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—48no—
206C22×C8—48no—
207C22×C8—44no—
208C22×C8—44no—
209C22×C8#4114yes—
210#123—432no—
211#123—432no—
212#121—28no—
213#121—28no—
214#121—44no—
215#121—44no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—48no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4C22×C414yes—
255C2×C4⋊C4C22×C412yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4#4114yes—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4C22⋊C414yes—
275C2×C22⋊C4C22⋊C414yes—
276C2×C22⋊C4D814yes—
277C2×C22⋊C4SD1614yes—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—44no—
283C2×C42—44no—
284C2×C42—82no—
285C2×C42C22×C414yes—
286C2×C42C2×D414yes—
287C4⋊C8—48no—
288#102—48no—
289#102—44no—
290#102—44no—
291C22⋊C8—44no—
292#95—24no—
293#95—44no—
294#95—44no—
295#95—82no—
296#95C22⋊C418yes—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—41no—
303C24—41no—
304C24—42no—
305C24—42no—
306C24—41no—
307C24—41no—
308C24—41no—
309C24—41no—
310C24—42no—
311C24—42no—
312C24C2×C22⋊C412yes—
313C24#12112yes—
314C24#12112yes—
315C24C2×D812yes—
316C24C2×SD1612yes—
317C24AGL(1,ℤ/8)12yes—
318C24AGL(1,ℤ/8)12yes—
319C24C22×D411yescharacteristic
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—48no—
325C2×D4—48no—
326C2×D4—48no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—42no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—84no—
342C2×D4—84no—
343C2×D4—84no—
344C2×D4—84no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—48no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—84no—
408C22×C4—88no—
409C22×C4—88no—
410C22×C4—84no—
411C22×C4—84no—
412C22×C4—84no—
413C22×C4—84no—
414C22×C4C2×C22⋊C414yes—
415C22×C4C2×C22⋊C412yes—
416C22×C4C2×C22⋊C414yes—
417C22×C4C4×D414yes—
418C22×C4C4×D414yes—
419C22×C4#12112yes—
420C22×C4#12112yes—
421C22×C4#12414yes—
422C22×C4#12714yes—
423C22×C4C2×D814yes—
424C22×C4C2×SD1614yes—
425C22×C4C23×C411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4112yes—
432C22×C4C2×#4112yes—
433C22×C4H(5,𝔽2)12yes—
434C22×C42−1+412yes—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—416no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—48no—
464C4⋊C4—416no—
465C4⋊C4—42no—
466C4⋊C4—41no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—41no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—48no—
473C4⋊C4—416no—
474C4⋊C4#12314yescomplemented
475C22⋊C4—28no—
476C22⋊C4—416no—
477C22⋊C4—48no—
478C22⋊C4—48no—
479C22⋊C4—416no—
480C22⋊C4#102116yescomplemented
481C42—24no—
482C42—24no—
483C42—24no—
484C42—24no—
485C42—88no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—24no—
496C23—22no—
497C23—24no—
498C23—22no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—44no—
503C23—42no—
504C23—42no—
505C23—44no—
506C23—42no—
507C23—82no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—81no—
512C23—81no—
513C23—82no—
514C23—81no—
515C23—81no—
516C23—81no—
517C23—81no—
518C23—82no—
519C23—81no—
520C23—82no—
521C23—81no—
522C23—82no—
523C23—82no—
524C23—82no—
525C23#39014yes—
526C23C2×#10214yes—
527C23#41814yes—
528C23#45914yes—
529C23#45914yes—
530C23#46014yes—
531C23#46014yes—
532C23C22×C22⋊C411yescharacteristic
533C23C2×#12111yescharacteristic
534C23C2×#12111yescharacteristic
535C23#54612yes—
536C23#54712yes—
537C23C22×D811yescharacteristic
538C23C22×SD1611yescharacteristic
539C23C2×AGL(1,ℤ/8)11yescharacteristic
540C23C2×AGL(1,ℤ/8)11yescharacteristic
541C23#57712yes—
542C23#57812yes—
543D4—42no—
544D4—44no—
545D4—42no—
546D4—816no—
547D4—82no—
548D4—81no—
549D4—81no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—816no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—816no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4#39014yes—
607C2×C4C2×#10218yes—
608C2×C4C2×C4×D411yescharacteristic
609C2×C4C2×C4×D411yescharacteristic
610C2×C4C4×#4112yes—
611C2×C4#52012yes—
612C2×C4C2×#12311yescharacteristic
613C2×C4#52511yescharacteristic
614C2×C4C2×#12411yescharacteristic
615C2×C4#52912yes—
616C2×C4C2×#12711yescharacteristic
617C2×C4#53411yescharacteristic
618C2×C4#53611yescharacteristic
619C2×C4#53912yes—
620C2×C4#54612yes—
621C2×C4#54612yes—
622C2×C4#54712yes—
623C2×C4#54712yes—
624C2×C4#55112yes—
625C2×C4#55212yes—
626C8—44no—
627C8—81no—
628C8—81no—
629C22—24no—
630C22—24no—
631C22—24no—
632C22—28no—
633C22—28no—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—28no—
638C22—28no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—84no—
643C22—84no—
644C22—161no—
645C22—161no—
646C22—162no—
647C22—162no—
648C22—161no—
649C22—161no—
650C22#198218yes—
651C22C2×#39011yescharacteristic
652C22#233912yes—
653C22#241211yescharacteristic
654C22#241311yescharacteristic
655C22#242512yes—
656C22C22×#10211yescharacteristic
657C22C2×#41811yescharacteristic
658C22#289812yes—
659C22C2×#45911yescharacteristic
660C22C2×#45911yescharacteristic
661C22C2×#46011yescharacteristic
662C22C2×#46011yescharacteristic
663C22#304211yescharacteristic
664C22#304411yescharacteristic
665C22#304612yes—
666C22#329412yes—
667C22#329812yes—
668C22#330612yes—
669C22#330812yes—
670C22#333412yes—
671C22#333512yes—
672C22#334312yes—
673C22#334512yes—
674C4—24no—
675C4—28no—
676C4—44no—
677C4—44no—
678C4—84no—
679C4—82no—
680C4—84no—
681C4—82no—
682C4C4×#12311yescharacteristic
683C4#232211yescharacteristic
684C4#235912yes—
685C4#248812yes—
686C4#250111yescharacteristic
687C4#253911yescharacteristic
688C2—28no—
689C2—162no—
690C2#1399611yescharacteristic
691C2C2×#198211yescharacteristic
692C2#2126211yescharacteristic
693C2#2248512yes—
694C2#2249512yes—
695C2#2452611yescharacteristic
696C2#2452811yescharacteristic
697C2#2457611yescharacteristic
698C2#2457811yescharacteristic
699C2#2520012yes—
700C2#2520412yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1850C2semidirect
5. C2×#1823C2semidirect
6. C22×#340C2semidirect
7. C22⋊C4×C4⋊C4C2semidirect
8. #11357C2semidirect
9. #11076C2semidirect
10. #10931C2semidirect
11. #10849C2semidirect
24. #2501C4semidirect
115. #390C2×C4semidirect
117. #384C2×C4semidirect
474. C4⋊C4#123semidirect
480. C22⋊C4#102semidirect

Maximal subgroups

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

subgroupindex
C2×#25012
C2×#18932
C2×#18742
C2×#18502
C2×#18232
C22×#3402
C22⋊C4×C4⋊C42
#113572
#110762
#109312
#108492