Finite Group Databasebeta

← #415795

Subgroups — #415795

Order 512.

2322 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#474C211yesmaximal, characteristic, complemented
2C2×#2547C211yesmaximal, characteristic, complemented
3C2×#2537C211yesmaximal, characteristic
4C2×#1990C211yesmaximal, characteristic, complemented
5C2×#1954C211yesmaximal, characteristic, complemented
6C2×C4×#102C211yesmaximal, characteristic, complemented
7#11096C214yesmaximal, complemented
8#10969C214yesmaximal, complemented
9#10849C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×#124—21no—
12C22×#121—21no—
13C22×#121C2212yes—
14C22×C4×D4C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C2×#474—24no—
17C2×#474—24no—
18C2×#474—24no—
19C22×#107C2212yes—
20C22×C4⋊C8C2211yescharacteristic
21C22×#102C2212yes—
22C22×#102C2212yes—
23C22×C22⋊C8C2214yes—
24C22×C4×C8C2211yescharacteristic
25#2547—24no—
26#2547—24no—
27#2537—24no—
28#2537—24no—
29C2×#397—22no—
30C2×#394—22no—
31C2×#391—21no—
32C2×#384C2211yescharacteristic
33C2×#380—21no—
34C2×#380C2212yes—
35C2×#380C2214yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37#1990—24no—
38#1954—24no—
39C4×#102—24no—
40C2×#340C2214yes—
41C2×#340C2212yes—
42C2×#340C2214yes—
43C23×Q8—21no—
44C23×D4—22no—
45C23×D4D411yescharacteristic
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C24×C4C2312yes—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8D414yescomplemented
57C2×#124—28no—
58C2×#124—44no—
59C2×#124—44no—
60C2×#121—28no—
61C2×#121—24no—
62C2×#121—24no—
63C2×#121—28no—
64C2×#121—44no—
65C2×#121—44no—
66C2×#121—44no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4D414yes—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2312yes—
87C22×C22⋊C4—22no—
88C22×C22⋊C4—24no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—22no—
91C22×C22⋊C4D412yes—
92C22×C42—21no—
93C22×C42C2311yescharacteristic
94#474—232no—
95C2×#107—24no—
96C2×#107—24no—
97C2×#107—24no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8D414yescomplemented
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—24no—
104C2×#102—24no—
105C2×#102—24no—
106C2×#102—24no—
107C2×C22⋊C8—28no—
108C2×C22⋊C8—28no—
109C2×C22⋊C8—44no—
110C2×C4×C8—22no—
111C2×C4×C8—22no—
112C2×C4×C8—22no—
113#397—216no—
114#394—216no—
115#391—28no—
116#384—22no—
117#384D414yes—
118#380—28no—
119#380—24no—
120#380—24no—
121#380—28no—
122#380—28no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4D414yescomplemented
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—24no—
128#340—28no—
129#340—24no—
130#340—44no—
131C25—21no—
132C25C2×D412yes—
133C22×Q8—24no—
134C22×Q8—42no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×D4—28no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—48no—
156C22×D4—44no—
157C22×D4—44no—
158C22×D4—84no—
159C22×D4D814yes—
160C22×D4C2×D412yes—
161C22×D4#4114yes—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4C2×D412yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2411yescharacteristic
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—24no—
193C22×C8—22no—
194C22×C8—24no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8C2×D414yes—
198C22×C8#4114yes—
199C4⋊Q8—28no—
200C4⋊Q8—28no—
201C4⋊Q8—44no—
202C4⋊Q8—44no—
203#124—432no—
204#121—416no—
205#121—48no—
206#121—432no—
207#121—48no—
208C4×D4—28no—
209C4×D4—28no—
210C4×D4—44no—
211C4×D4—44no—
212C4×D4—44no—
213C4×D4—44no—
214C4×D4—44no—
215C4×D4—44no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—82no—
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2×D414yes—
247C2×C4⋊C4C2×D414yes—
248C2×C4⋊C4C2×D414yes—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—216no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—48no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—48no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C42—24no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42#4112yes—
272#107—216no—
273C4⋊C8—28no—
274#102—216no—
275#102—216no—
276C22⋊C8—416no—
277C4×C8—44no—
278#95—28no—
279#95—216no—
280#95—24no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—24no—
286C24—24no—
287C24—24no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—24no—
292C24—24no—
293C24—41no—
294C24—41no—
295C24—42no—
296C24—41no—
297C24—41no—
298C24—41no—
299C24—41no—
300C24—42no—
301C24—42no—
302C24C2×D814yes—
303C24C22×D411yescharacteristic
304C2×Q8—48no—
305C2×Q8—48no—
306C2×Q8—48no—
307C2×Q8—84no—
308C2×Q8—84no—
309C2×Q8—84no—
310C2×Q8—84no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—28no—
322C2×D4—28no—
323C2×D4—28no—
324C2×D4—416no—
325C2×D4—416no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—88no—
338C2×D4—816no—
339C2×D4—816no—
340C2×D4#12414yescomplemented
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—84no—
396C22×C4—84no—
397C22×C4—84no—
398C22×C4—84no—
399C22×C4#12012yes—
400C22×C4#12012yes—
401C22×C4#12012yes—
402C22×C4#12012yes—
403C22×C4#12112yes—
404C22×C4#12112yes—
405C22×C4C2×D812yes—
406C22×C4#13512yes—
407C22×C4AGL(1,ℤ/8)12yes—
408C22×C4AGL(1,ℤ/8)12yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4112yes—
416C22×C4C2×#4112yes—
417C22×C4H(5,𝔽2)12yes—
418C22×C42−1+412yes—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—28no—
423C2×C8—24no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—44no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—28no—
438C4⋊C4—41no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—41no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4—48no—
448C4⋊C4—48no—
449C4⋊C4—84no—
450C22⋊C4—416no—
451C22⋊C4—416no—
452C22⋊C4—432no—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C42—48no—
456C42#12014yes—
457C42#12114yes—
458C42#12114yes—
459C42#12714yes—
460C23—28no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—28no—
465C23—24no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—42no—
472C23—42no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—82no—
497C23—82no—
498C23—82no—
499C23—82no—
500C23—81no—
501C23—81no—
502C23—82no—
503C23—81no—
504C23—81no—
505C23—82no—
506C23C2×#12011yescharacteristic
507C23C2×#12011yescharacteristic
508C23C2×#12111yescharacteristic
509C23#54612yes—
510C23#54712yes—
511C23C22×D811yescharacteristic
512C23C2×#13511yescharacteristic
513C23C2×AGL(1,ℤ/8)11yescharacteristic
514C23C2×AGL(1,ℤ/8)11yescharacteristic
515C23#57712yes—
516C23#57812yes—
517Q8—816no—
518D4—24no—
519D4—24no—
520D4—42no—
521D4—44no—
522D4—42no—
523D4—416no—
524D4—44no—
525D4—832no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—81no—
551C2×C4—82no—
552C2×C4—81no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—84no—
556C2×C4—81no—
557C2×C4—82no—
558C2×C4—81no—
559C2×C4—82no—
560C2×C4—84no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—84no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—816no—
567C2×C4—84no—
568C2×C4#39214yes—
569C2×C4#39214yes—
570C2×C4#44714yes—
571C2×C4#44714yes—
572C2×C4#44914yes—
573C2×C4#44914yes—
574C2×C4#46614yes—
575C2×C4#46914yes—
576C2×C4C2×#12011yescharacteristic
577C2×C4C2×#12111yescharacteristic
578C2×C4C2×#12111yescharacteristic
579C2×C4#52512yes—
580C2×C4C2×#12411yescharacteristic
581C2×C4#52912yes—
582C2×C4C2×#12711yescharacteristic
583C2×C4#53211yescharacteristic
584C2×C4#53511yescharacteristic
585C2×C4#53611yescharacteristic
586C2×C4#53712yes—
587C2×C4#54212yes—
588C2×C4#54612yes—
589C2×C4#54612yes—
590C2×C4#54812yes—
591C2×C4#54812yes—
592C2×C4#54812yes—
593C2×C4#55012yes—
594C8—44no—
595C8—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—48no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—48no—
604C22—84no—
605C22—84no—
606C22—82no—
607C22—84no—
608C22—84no—
609C22—84no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—84no—
618C22C2×#39211yescharacteristic
619C22#240712yes—
620C22#242712yes—
621C22#243211yescharacteristic
622C22#243511yescharacteristic
623C22C2×#44711yescharacteristic
624C22C2×#44711yescharacteristic
625C22C2×#44911yescharacteristic
626C22C2×#44911yescharacteristic
627C22#300911yescharacteristic
628C22#301111yescharacteristic
629C22#301312yes—
630C22C2×#46611yescharacteristic
631C22C2×#46911yescharacteristic
632C22#306312yes—
633C22#329412yes—
634C22#330212yes—
635C22#332412yes—
636C22#333012yes—
637C22#333412yes—
638C22#334012yes—
639C22#335312yes—
640C22#335412yes—
641C4—24no—
642C4—84no—
643C4—84no—
644C4—82no—
645C4—84no—
646C4—82no—
647C4#199918yes—
648C4#242612yes—
649C4#243111yescharacteristic
650C4#243611yescharacteristic
651C4#254012yes—
652C4#262011yescharacteristic
653C4#285011yescharacteristic
654C2—84no—
655C2—84no—
656C2#1521511yescharacteristic
657C2C2×#199911yescharacteristic
658C2#2339411yescharacteristic
659C2#2411612yes—
660C2#2411912yes—
661C2#2533812yes—
662C2#2534712yes—
663C2#2570011yescharacteristic
664C2#2570311yescharacteristic
665C2#2588211yescharacteristic
666C2#2588511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC2×#2537C2
LCSG ▷ C23×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
1. C22×#474C2semidirect
2. C2×#2547C2semidirect
4. C2×#1990C2semidirect
5. C2×#1954C2semidirect
6. C2×C4×#102C2semidirect
7. #11096C2semidirect
8. #10969C2semidirect
9. #10849C2semidirect
56. C2×C4⋊Q8D4semidirect
100. C2×C4⋊C8D4semidirect
124. C4×C4⋊C4D4semidirect
340. C2×D4#124semidirect

Maximal subgroups

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

subgroupindex
C22×#4742
C2×#25472
C2×#25372
C2×#19902
C2×#19542
C2×C4×#1022
#110962
#109692
#108492