Finite Group Databasebeta

← #443464

Subgroups — #443464

Order 512.

1772 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#500C211yesmaximal, characteristic, complemented
2C2×#2491C211yesmaximal, characteristic
3C2×#2033C212yesmaximal
4C2×#2023C211yesmaximal, characteristic, complemented
5#15125C211yesmaximal, characteristic, complemented
6#11384C211yesmaximal, characteristic, complemented
7#11123C212yesmaximal, complemented
8#11096C212yesmaximal, complemented
9#10925C211yesmaximal, characteristic, complemented
10#10921C211yesmaximal, characteristic, complemented
11#10907C212yesmaximal, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×#124—21no—
15C22×#121C2211yescharacteristic
16C2×#500—28no—
17C2×#500—24no—
18C22×#107C2212yes—
19C22×#107C2212yes—
20C22×#102C2212yes—
21C22×#102C2212yes—
22C22×C22⋊C8C2212yes—
23C22×C4×C8C2211yescharacteristic
24#2491—24no—
25#2491—24no—
26C2×#394—21no—
27C2×#393—21no—
28C2×#393—21no—
29C2×#390C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#388—21no—
32C2×#380C2211yescharacteristic
33C2×#380C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2211yescharacteristic
36#2033—28no—
37#2023—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2211yescharacteristic
44#1446—21no—
45#1442—21no—
46#1439—21no—
47C4⋊#95C2212yes—
48C4⋊#95C2211yescharacteristic
49C23×Q8—21no—
50C23×Q8—21no—
51C23×D4—21no—
52C23×D4C2311yescharacteristic
53C24×C4C2311yescharacteristic
54C23×C8C2312yes—
55C23×C8C2312yes—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—24no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—42no—
62C2×C4⋊Q8—42no—
63C2×C4⋊Q8D414yes—
64C2×#124—28no—
65C2×#124—44no—
66C2×#124—44no—
67C2×#121—22no—
68C2×#121—22no—
69C2×#121—24no—
70C2×#121—42no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4D411yescharacteristic
86C22×C22⋊C4D411yescharacteristic
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#500—232no—
91C2×#107—24no—
92C2×#107—24no—
93C2×#107—24no—
94C2×#107—24no—
95C2×#107—24no—
96C2×#107D418yes—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—24no—
102C2×#102—42no—
103C2×C22⋊C8—24no—
104C2×C22⋊C8—24no—
105C2×C22⋊C8—24no—
106C2×C4×C8—24no—
107C2×C4×C8—22no—
108#394—28no—
109#393—28no—
110#393—44no—
111#390—22no—
112#390—22no—
113#390—22no—
114#390D414yes—
115#388—28no—
116#380—22no—
117#380—22no—
118#380—24no—
119#380—42no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—22no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95D411yescharacteristic
132C2×#95D411yescharacteristic
133#340—24no—
134#340—24no—
135#340—24no—
136#340—42no—
137#336—24no—
138#336—22no—
139C25—21no—
140C25C2×D411yescharacteristic
141C22×Q8—24no—
142C22×Q8—44no—
143C22×Q8—44no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×Q8—42no—
147C22×Q8—42no—
148C22×Q8—42no—
149C22×Q8—82no—
150C22×Q8—82no—
151C22×D4—24no—
152C22×D4—22no—
153C22×D4—21no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—44no—
160C22×D4—42no—
161C22×D4—42no—
162C22×D4—82no—
163C22×D4C2×D412yes—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—22no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2411yescharacteristic
183C22×C8—22no—
184C22×C8—22no—
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×C8C2×D414yes—
195C22×C8#4114yes—
196C22×C8#4114yes—
197C4⋊Q8—28no—
198C4⋊Q8—28no—
199C4⋊Q8—44no—
200C4⋊Q8—48no—
201C4⋊Q8—48no—
202C4⋊Q8—44no—
203#124—432no—
204#121—48no—
205#121—44no—
206#121—44no—
207C2×C4⋊C4—21no—
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—24no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
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—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4C2×D414yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D414yes—
234C2×C4⋊C4C2×D412yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—82no—
258C2×C22⋊C4—82no—
259C2×C22⋊C4D814yes—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—24no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271#107—216no—
272#107—216no—
273#102—48no—
274#102—48no—
275C22⋊C8—48no—
276C4×C8—28no—
277#95—24no—
278#95—24no—
279#95—22no—
280#95—22no—
281#95—44no—
282#95—44no—
283#95—42no—
284#95—42no—
285#95—82no—
286C24—21no—
287C24—21no—
288C24—21no—
289C24—22no—
290C24—22no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—41no—
298C24—41no—
299C24—42no—
300C24—42no—
301C24—41no—
302C24—41no—
303C24—41no—
304C24—41no—
305C24—42no—
306C24—42no—
307C24C2×D812yes—
308C24C22×D411yescharacteristic
309C2×Q8—48no—
310C2×Q8—48no—
311C2×Q8—48no—
312C2×Q8—84no—
313C2×Q8—88no—
314C2×Q8—88no—
315C2×Q8—88no—
316C2×Q8—88no—
317C2×Q8—84no—
318C2×Q8—84no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—48no—
324C2×D4—48no—
325C2×D4—41no—
326C2×D4—42no—
327C2×D4—41no—
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—84no—
339C2×D4—88no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—41no—
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—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4#12012yes—
406C22×C4#12012yes—
407C22×C4C2×D812yes—
408C22×C4C2×D812yes—
409C22×C4#13512yes—
410C22×C4AGL(1,ℤ/8)12yes—
411C22×C4#13712yes—
412C22×C4C22×D412yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C2×#4112yes—
416C22×C4C2×#4112yes—
417C22×C4H(5,𝔽2)12yes—
418C22×C42−1+412yes—
419C22×C42−1+412yes—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8#12418yescomplemented
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4#12014yes—
452C22⋊C4—28no—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—44no—
458C22⋊C4—48no—
459C22⋊C4—84no—
460C22⋊C4—88no—
461C42—22no—
462C42—22no—
463C42—24no—
464C42—48no—
465C42—41no—
466C42—41no—
467C42—42no—
468C23—24no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—41no—
477C23—42no—
478C23—41no—
479C23—42no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—41no—
485C23—42no—
486C23—41no—
487C23—42no—
488C23—41no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—82no—
499C23—82no—
500C23—82no—
501C23—82no—
502C23—81no—
503C23—82no—
504C23—81no—
505C23—82no—
506C23—81no—
507C23—81no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—82no—
512C23—82no—
513C23—82no—
514C23C2×#12011yescharacteristic
515C23#53211yescharacteristic
516C23#53611yescharacteristic
517C23D4212yes—
518C23#54712yes—
519C23C22×D811yescharacteristic
520C23C2×#13511yescharacteristic
521C23C2×AGL(1,ℤ/8)11yescharacteristic
522C23C2×#13711yescharacteristic
523C23#57612yes—
524C23#57812yes—
525Q8—816no—
526Q8—816no—
527D4—44no—
528D4—44no—
529D4—816no—
530D4—82no—
531D4—82no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—41no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
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—82no—
589C2×C4—82no—
590C2×C4#44714yes—
591C2×C4#45214yes—
592C2×C4C2×#12011yescharacteristic
593C2×C4C2×#12412yes—
594C2×C4#53211yescharacteristic
595C2×C4#53411yescharacteristic
596C2×C4#53611yescharacteristic
597C2×C4#53612yes—
598C2×C4#54212yes—
599C2×C4#54212yes—
600C2×C4D4212yes—
601C2×C4#54712yes—
602C2×C4#54812yes—
603C2×C4#54812yes—
604C2×C4#54812yes—
605C2×C4#54812yes—
606C2×C4#55512yes—
607C2×C4#56312yes—
608C8—28no—
609C8—82no—
610C22—42no—
611C22—44no—
612C22—42no—
613C22—42no—
614C22—44no—
615C22—81no—
616C22—82no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—162no—
629C22—162no—
630C22—162no—
631C22—162no—
632C22#240311yescharacteristic
633C22#243211yescharacteristic
634C22#243211yescharacteristic
635C22#249112yes—
636C22#267412yes—
637C22C2×#44711yescharacteristic
638C22C2×#45211yescharacteristic
639C22#300811yescharacteristic
640C22#301111yescharacteristic
641C22#316911yescharacteristic
642C22#317512yes—
643C22#318211yescharacteristic
644C22#322611yescharacteristic
645C22#322811yescharacteristic
646C22#324312yes—
647C22#326412yes—
648C22#327012yes—
649C22D4×D812yes—
650C22#328512yes—
651C22#333412yes—
652C22#334112yes—
653C22#335312yes—
654C22#335712yes—
655C4—24no—
656C4—24no—
657C4—44no—
658C4—42no—
659C4—82no—
660C4—84no—
661C4—82no—
662C4—81no—
663C4—81no—
664C4—162no—
665C4#243612yes—
666C4#249112yes—
667C4#267412yes—
668C4#273112yes—
669C2—82no—
670C2—162no—
671C2#1576111yescharacteristic
672C2#2384811yescharacteristic
673C2#2385911yescharacteristic
674C2#2567311yescharacteristic
675C2#2568111yescharacteristic
676C2#2570311yescharacteristic
677C2#2570411yescharacteristic
678C2#2779112yes—
679C2#2780212yes—
680C2#3077312yes—
681C2#3077812yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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
1. C22×#500C2semidirect
4. C2×#2023C2semidirect
5. #15125C2semidirect
6. #11384C2semidirect
7. #11123C2semidirect
8. #11096C2semidirect
9. #10925C2semidirect
10. #10921C2semidirect
11. #10907C2semidirect
433. C2×C8#124semidirect

Maximal subgroups

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

subgroupindex
C22×#5002
C2×#24912
C2×#20332
C2×#20232
#151252
#113842
#111232
#110962
#109252
#109212
#109072