Finite Group Databasebeta

← #397165

Subgroups — #397165

Order 512.

2140 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#466C211yesmaximal, characteristic
2C2×#2054C211yesmaximal, characteristic, complemented
3C2×#1874C212yesmaximal, complemented
4C2×#1850C211yesmaximal, characteristic, complemented
5C22×#340C211yesmaximal, characteristic, complemented
6#14137C212yesmaximal, complemented
7#11256C211yesmaximal, characteristic, complemented
8#11232C211yesmaximal, characteristic, complemented
9#11189C212yesmaximal, complemented
10#11076C212yesmaximal, complemented
11#11022C211yesmaximal, characteristic, complemented
12C23×D8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#121C2212yes—
15C23×C4⋊C4C2212yes—
16C2×#466—24no—
17C2×#466—28no—
18C22×#107C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#102C2212yes—
22C22×#102C2212yes—
23C22×#102C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25C2×#390C2212yes—
26C2×#390C2212yes—
27C2×#386—22no—
28C2×#386—22no—
29C2×C4×C22⋊C4C2211yescharacteristic
30#2054—24no—
31#1874—28no—
32#1850—24no—
33C2×#340—24no—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41#1439—22no—
42#1439—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C23×D4—22no—
46C23×D4C2312yes—
47C24×C4—22no—
48C24×C4C2311yescharacteristic
49C22×D8—24no—
50C22×D8—22no—
51C22×D8—22no—
52C22×D8—22no—
53C22×D8—42no—
54C23×C8—24no—
55C23×C8—22no—
56C23×C8—22no—
57C23×C8—42no—
58C23×C8D414yes—
59C23×C8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×#121—24no—
64C2×#121—24no—
65C2×#121—24no—
66C2×#121—24no—
67C2×#121—24no—
68C2×#121D418yes—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—24no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—42no—
75C22×C4⋊C4—42no—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4D411yescharacteristic
85C22×C22⋊C4D411yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#466—232no—
89C2×#107—22no—
90C2×#107—24no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—42no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—42no—
104C2×#102—42no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—24no—
107#390—24no—
108#390—24no—
109#390—42no—
110#390—42no—
111#386—48no—
112#386—48no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95D411yescharacteristic
120C2×#95D411yescharacteristic
121#340—22no—
122#340—24no—
123#340—24no—
124#340—24no—
125#340D414yescomplemented
126#336—22no—
127#336—22no—
128C25—22no—
129C25C2×D411yescharacteristic
130C22×D4—28no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—44no—
145C22×D4—44no—
146C22×D4—44no—
147C22×D4—44no—
148C22×D4—44no—
149C22×D4—44no—
150C22×D4C2×D414yes—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—82no—
166C23×C4—82no—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D414yes—
172C23×C4C2×D412yes—
173C23×C4C2×D411yescharacteristic
174C23×C4#4112yes—
175C23×C4#4114yes—
176C23×C4C2411yescharacteristic
177C2×D8—28no—
178C2×D8—48no—
179C2×D8—48no—
180C2×D8—44no—
181C2×D8—44no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—28no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8—21no—
188C22×C8—22no—
189C22×C8—28no—
190C22×C8—21no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—22no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—22no—
199C22×C8—21no—
200C22×C8—44no—
201C22×C8—48no—
202C22×C8—48no—
203C22×C8C2×D412yes—
204C22×C8C2×D412yes—
205C22×C8C2×D412yes—
206C22×C8#4112yes—
207#121—216no—
208#121—216no—
209#121—48no—
210#121—48no—
211C2×C4⋊C4—216no—
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—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—48no—
228C2×C4⋊C4—48no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—48no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4C2×D414yes—
234C2×C4⋊C4C2×D414yes—
235C2×C4⋊C4#4114yes—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—82no—
252C2×C22⋊C4—82no—
253C2×C22⋊C4—82no—
254C2×C22⋊C4—82no—
255C2×C22⋊C4D814yescomplemented
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—22no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—22no—
265C2×C42—21no—
266#107—28no—
267C4⋊C8—44no—
268C4⋊C8—44no—
269#102—216no—
270#102—48no—
271#102—48no—
272C22⋊C8—44no—
273#95—22no—
274#95—22no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279C24—21no—
280C24—22no—
281C24—21no—
282C24—21no—
283C24—22no—
284C24—21no—
285C24—24no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24—42no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24—42no—
300C24#12114yes—
301C24#12112yes—
302C24C2×D812yes—
303C24#13512yes—
304C24AGL(1,ℤ/8)14yes—
305C24C22×D411yescharacteristic
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—24no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—416no—
313C2×D4—416no—
314C2×D4—416no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—88no—
338C2×D4—88no—
339C2×D4—88no—
340C2×D4—88no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—24no—
372C22×C4—21no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—416no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—88no—
402C22×C4—88no—
403C22×C4—88no—
404C22×C4—88no—
405C22×C4—88no—
406C22×C4—88no—
407C22×C4#12014yes—
408C22×C4#12112yes—
409C22×C4#12114yes—
410C22×C4#12318yes—
411C22×C4#12414yes—
412C22×C4C2×D812yes—
413C22×C4C2×D812yes—
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D412yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C4H(5,𝔽2)12yes—
422C22×C4H(5,𝔽2)11yescharacteristic
423C22×C42−1+411yescharacteristic
424D8—416no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—416no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8#12114yes—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—416no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C22⋊C4—28no—
460C22⋊C4—44no—
461C22⋊C4—416no—
462C22⋊C4—88no—
463C22⋊C4—88no—
464C22⋊C4—88no—
465C22⋊C4—88no—
466C42—41no—
467C42—41no—
468C42—41no—
469C42—41no—
470C42—41no—
471C42—41no—
472C42—42no—
473C23—22no—
474C23—22no—
475C23—24no—
476C23—22no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—82no—
499C23—82no—
500C23—82no—
501C23—82no—
502C23—82no—
503C23—82no—
504C23—82no—
505C23—82no—
506C23—82no—
507C23—82no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—82no—
512C23—82no—
513C23—82no—
514C23—82no—
515C23—82no—
516C23—82no—
517C23—82no—
518C23—82no—
519C23—82no—
520C23—82no—
521C23—82no—
522C23—82no—
523C23—82no—
524C23—82no—
525C23—82no—
526C23#39414yes—
527C23#45918yes—
528C23#46318yes—
529C23#46614yes—
530C23#46914yes—
531C23C2×#12111yescharacteristic
532C23C2×#12112yes—
533C23D4211yescharacteristic
534C23#54612yes—
535C23#54711yescharacteristic
536C23C22×D811yescharacteristic
537C23C2×#13511yescharacteristic
538C23C2×AGL(1,ℤ/8)12yes—
539C23#57611yescharacteristic
540C23#57712yes—
541C23#57811yescharacteristic
542D4—44no—
543D4—44no—
544D4—44no—
545D4—44no—
546D4—832no—
547D4—82no—
548D4—82no—
549D4—82no—
550D4—82no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—832no—
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#39414yes—
606C2×C4C2×#12011yescharacteristic
607C2×C4C2×#12111yescharacteristic
608C2×C4C2×#12312yes—
609C2×C4#52512yes—
610C2×C4C2×#12411yescharacteristic
611C2×C4#52912yes—
612C2×C4#53211yescharacteristic
613C2×C4#53411yescharacteristic
614C2×C4#53711yescharacteristic
615C2×C4#53812yes—
616C2×C4#53912yes—
617C2×C4D4212yes—
618C2×C4#54612yes—
619C2×C4#54612yes—
620C2×C4#54712yes—
621C2×C4#54811yescharacteristic
622C2×C4#55312yes—
623C2×C4#55311yescharacteristic
624C8—24no—
625C8—42no—
626C8—81no—
627C8—81no—
628C22—28no—
629C22—28no—
630C22—28no—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—28no—
637C22—24no—
638C22—84no—
639C22—84no—
640C22—84no—
641C22—84no—
642C22—84no—
643C22—84no—
644C22—84no—
645C22—162no—
646C22—162no—
647C22—162no—
648C22—162no—
649C22—162no—
650C22—162no—
651C22—162no—
652C22—162no—
653C22#205418yescomplemented
654C22C2×#39411yescharacteristic
655C22#241611yescharacteristic
656C22#242412yes—
657C22#242512yes—
658C22#254311yescharacteristic
659C22C2×#45912yes—
660C22C2×#46312yes—
661C22#304512yes—
662C22#304612yes—
663C22C2×#46611yescharacteristic
664C22C2×#46911yescharacteristic
665C22#306011yescharacteristic
666C22#306111yescharacteristic
667C22#326512yes—
668C22D4×D811yescharacteristic
669C22#328511yescharacteristic
670C22#329412yes—
671C22#330112yes—
672C22#330812yes—
673C22#331812yes—
674C22#333411yescharacteristic
675C22#333911yescharacteristic
676C22#334512yes—
677C4—24no—
678C4—42no—
679C4—42no—
680C4—81no—
681C4—81no—
682C4—84no—
683C4—84no—
684C4—84no—
685C4#241011yescharacteristic
686C4#243011yescharacteristic
687C4#244312yes—
688C4#244412yes—
689C4#263911yescharacteristic
690C4#285011yescharacteristic
691C2—28no—
692C2—164no—
693C2#1474711yescharacteristic
694C2C2×#205411yescharacteristic
695C2#2354311yescharacteristic
696C2#2478011yescharacteristic
697C2#2478211yescharacteristic
698C2#2507612yes—
699C2#2509012yes—
700C2#2520012yes—
701C2#2521412yes—
702C2#2855111yescharacteristic
703C2#2856111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#466C2
LCSG ▷ C23×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
2. C2×#2054C2semidirect
3. C2×#1874C2semidirect
4. C2×#1850C2semidirect
5. C22×#340C2semidirect
6. #14137C2semidirect
7. #11256C2semidirect
8. #11232C2semidirect
9. #11189C2semidirect
10. #11076C2semidirect
11. #11022C2semidirect
125. #340D4semidirect
255. C2×C22⋊C4D8semidirect
653. C22#2054semidirect

Maximal subgroups

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

subgroupindex
C22×#4662
C2×#20542
C2×#18742
C2×#18502
C22×#3402
#141372
#112562
#112322
#111892
#110762
#110222