Finite Group Databasebeta

← #381862

Subgroups — #381862

Order 512.

1848 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1999C211yesmaximal, characteristic
2C2×#1970C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊C8C211yesmaximal, characteristic, complemented
4#14698C212yesmaximal, complemented
5#11256C211yesmaximal, characteristic, complemented
6#11189C212yesmaximal, complemented
7#11096C211yesmaximal, characteristic, complemented
8#11022C211yesmaximal, characteristic, complemented
9#10969C212yesmaximal, complemented
10#10925C212yesmaximal, complemented
11#10820C211yesmaximal, characteristic, complemented
12C23×D8C2211yescharacteristic
13C22×#121C2212yes—
14C23×C42C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23C2×#390C2212yes—
24C2×#390C2212yes—
25C2×#386—22no—
26C2×#386—22no—
27C2×#380C2212yes—
28#1999C418yescomplemented
29#1970—24no—
30C4×C22⋊C8—24no—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1437—22no—
39#1437—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C23×D4—22no—
43C23×D4C2312yes—
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4C2311yescharacteristic
47C22×D8—22no—
48C22×D8—22no—
49C22×D8—42no—
50C22×D8—42no—
51C22×D8C2×C414yescomplemented
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×#121—24no—
57C2×#121—42no—
58C2×#121—42no—
59C2×#121—42no—
60C2×#121—42no—
61C2×#121C2×C418yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C42—22no—
72C22×C42—22no—
73C22×C42—42no—
74C22×C42—42no—
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×#107—22no—
80C2×#107—42no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—42no—
84C2×#102—42no—
85C2×#102—42no—
86C2×#102—42no—
87C2×#102—42no—
88C2×#102—42no—
89C2×#102C2×C418yescomplemented
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8C2×C414yes—
96C2×C4×C8—22no—
97C2×C4×C8—42no—
98C2×C4×C8—42no—
99C2×C4×C8D414yescomplemented
100#390—24no—
101#390—24no—
102#390—42no—
103#390—42no—
104#386—48no—
105#386—48no—
106#380—42no—
107#380—42no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113#340—22no—
114#340—42no—
115#340—42no—
116#340—42no—
117#340C2×C414yescomplemented
118#336—22no—
119#336—22no—
120C25—22no—
121C25C2×D411yescharacteristic
122C22×D4—28no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—44no—
137C22×D4—44no—
138C22×D4—44no—
139C22×D4—44no—
140C22×D4—44no—
141C22×D4—44no—
142C22×D4C22×C414yes—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—24no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4D814yescomplemented
164C23×C4C22×C412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4#4112yes—
171C23×C4C2411yescharacteristic
172C2×D8—28no—
173C2×D8—48no—
174C2×D8—44no—
175C2×D8—44no—
176C2×D8—48no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8C22×C412yes—
192C22×C8C22×C412yes—
193C22×C8C2×D412yes—
194C22×C8C2×D412yes—
195C22×C8#4112yes—
196C22×C8#4112yes—
197#121—48no—
198#121—48no—
199#121—48no—
200#121—48no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
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⋊C4C22×C414yes—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—44no—
246C2×C42—44no—
247C2×C42—48no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42C2×D412yes—
251C2×C42#4112yes—
252#107—44no—
253#102—48no—
254#102—48no—
255#102—48no—
256C22⋊C8—28no—
257C22⋊C8—28no—
258C4×C8—44no—
259C4×C8—44no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265C24—21no—
266C24—21no—
267C24—22no—
268C24—22no—
269C24—21no—
270C24—21no—
271C24—22no—
272C24—22no—
273C24—24no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24—42no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24—42no—
288C24C4×D412yes—
289C24C2×D812yes—
290C24#13512yes—
291C24C22×D411yescharacteristic
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—416no—
299C2×D4—416no—
300C2×D4—416no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
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—88no—
324C2×D4—88no—
325C2×D4—88no—
326C2×D4—88no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—28no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—28no—
354C22×C4—24no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—48no—
361C22×C4—44no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—44no—
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—42no—
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×C4C4×D412yes—
394C22×C4#12112yes—
395C22×C4#12112yes—
396C22×C4C2×D812yes—
397C22×C4C2×D812yes—
398C22×C4#13512yes—
399C22×C4#13512yes—
400C22×C4AGL(1,ℤ/8)14yes—
401C22×C4C23×C411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4H(5,𝔽2)12yes—
408C22×C4H(5,𝔽2)12yes—
409C22×C4H(5,𝔽2)12yes—
410C22×C4H(5,𝔽2)12yes—
411D8—416no—
412C2×C8—24no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—24no—
416C2×C8—22no—
417C2×C8—24no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—24no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8C4×D414yes—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C22⋊C4—416no—
445C22⋊C4—88no—
446C22⋊C4—88no—
447C22⋊C4—88no—
448C22⋊C4—88no—
449C22⋊C4—88no—
450C42—22no—
451C42—22no—
452C42—24no—
453C42—44no—
454C42—44no—
455C42—41no—
456C42—41no—
457C42—42no—
458C23—24no—
459C23—22no—
460C23—22no—
461C23—24no—
462C23—24no—
463C23—22no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—41no—
471C23—41no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—41no—
479C23—41no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23—82no—
494C23—82no—
495C23—82no—
496C23—82no—
497C23—82no—
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—
515C23C4×D814yes—
516C23C2×C4×D411yescharacteristic
517C23C2×#12111yescharacteristic
518C23#53211yescharacteristic
519C23#53412yes—
520C23#53512yes—
521C23C22×D811yescharacteristic
522C23C2×#13511yescharacteristic
523C23C2×#13511yescharacteristic
524C23C2×#13511yescharacteristic
525C23C2×AGL(1,ℤ/8)12yes—
526C23#57512yes—
527D4—44no—
528D4—44no—
529D4—44no—
530D4—44no—
531D4—832no—
532D4—82no—
533D4—82no—
534D4—82no—
535D4—82no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—416no—
551C2×C4—416no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4#46614yes—
589C2×C4#46914yes—
590C2×C4C2×C4×D411yescharacteristic
591C2×C4C4×#4111yescharacteristic
592C2×C4#51812yes—
593C2×C4#51812yes—
594C2×C4#52511yescharacteristic
595C2×C4#52911yescharacteristic
596C2×C4#53411yescharacteristic
597C2×C4#53511yescharacteristic
598C2×C4#53511yescharacteristic
599C2×C4#53511yescharacteristic
600C2×C4#53812yes—
601C2×C4#53812yes—
602C2×C4#53812yes—
603C2×C4#53812yes—
604C2×C4#53812yes—
605C2×C4#54012yes—
606C2×C4#54012yes—
607C2×C4#54012yes—
608C8—24no—
609C8—42no—
610C8—42no—
611C8—42no—
612C22—24no—
613C22—28no—
614C22—42no—
615C22—44no—
616C22—42no—
617C22—42no—
618C22—44no—
619C22—42no—
620C22—44no—
621C22—42no—
622C22—42no—
623C22—84no—
624C22—84no—
625C22—84no—
626C22—84no—
627C22—84no—
628C22—84no—
629C22—84no—
630C22—162no—
631C22—162no—
632C22—162no—
633C22—162no—
634C22—162no—
635C22—162no—
636C22—162no—
637C22—162no—
638C22C4×#12111yescharacteristic
639C22#232112yes—
640C22#261711yescharacteristic
641C22#261911yescharacteristic
642C22#266112yes—
643C22C2×C4×D811yescharacteristic
644C22C4×#13511yescharacteristic
645C22C4×AGL(1,ℤ/8)12yes—
646C22C2×#46611yescharacteristic
647C22#305011yescharacteristic
648C22C2×#46911yescharacteristic
649C22#305411yescharacteristic
650C22#316811yescharacteristic
651C22#316911yescharacteristic
652C22#317112yes—
653C22#318612yes—
654C22#318812yes—
655C22#318812yes—
656C22#319112yes—
657C22#320512yes—
658C22#320712yes—
659C22#321112yes—
660C22#321212yes—
661C4—24no—
662C4—24no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—84no—
667C4—162no—
668C4—162no—
669C4#232111yescharacteristic
670C4#232811yescharacteristic
671C4#233312yes—
672C4#263111yescharacteristic
673C4#263511yescharacteristic
674C4#264812yes—
675C2—44no—
676C2—164no—
677C2#1431111yescharacteristic
678C2C4×#46611yescharacteristic
679C2C4×#46911yescharacteristic
680C2#2204012yes—
681C2#2204512yes—
682C2#2973011yescharacteristic
683C2#2973211yescharacteristic
684C2#2980611yescharacteristic
685C2#2980911yescharacteristic
686C2#3031612yes—
687C2#3032012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#1999C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1970C2semidirect
3. C2×C4×C22⋊C8C2semidirect
4. #14698C2semidirect
5. #11256C2semidirect
6. #11189C2semidirect
7. #11096C2semidirect
8. #11022C2semidirect
9. #10969C2semidirect
10. #10925C2semidirect
11. #10820C2semidirect
28. #1999C4semidirect
51. C22×D8C2×C4semidirect
89. C2×#102C2×C4semidirect
99. C2×C4×C8D4semidirect
117. #340C2×C4semidirect
163. C23×C4D8semidirect

Maximal subgroups

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

subgroupindex
C2×#19992
C2×#19702
C2×C4×C22⋊C82
#146982
#112562
#111892
#110962
#110222
#109692
#109252
#108202