Finite Group Databasebeta

← #119995

Subgroups — #119995

Order 512.

1270 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1797C211yesmaximal, characteristic
2C2×#1436C211yesmaximal, characteristic
3#10712C211yesmaximal, characteristic
4C2×#1280C212yesmaximal
5C2×#1280C212yesmaximal
6C23×C4⋊C4C411yescharacteristic
7C23×C42C2211yescharacteristic
8C22×C4⋊C8—21no—
9C22×C22⋊C8C2212yes—
10C22×C22⋊C8C2212yes—
11C22×#95C411yescharacteristic
12C22×#95C411yescharacteristic
13C22×#95C411yescharacteristic
14C22×#95C2211yescharacteristic
15C22×#95C2211yescharacteristic
16#1797—24no—
17C2×#336—21no—
18C2×#336—21no—
19C2×#336—21no—
20#1436—21no—
21#1436—21no—
22#1436—21no—
23#1436—21no—
24#1280—24no—
25#1280—24no—
26C24×C4C2×C411yescharacteristic
27C24×C4C2×C411yescharacteristic
28C24×C4C2×C411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4C2×C411yescharacteristic
31C24×C4C2×C411yescharacteristic
32C24×C4C2311yescharacteristic
33C23×C8—22no—
34C23×C8—22no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—42no—
39C22×C4⋊C4—42no—
40C22×C4⋊C4C814yescomplemented
41C22×C42—22no—
42C22×C42—22no—
43C22×C42—42no—
44C22×C42—42no—
45C22×C42D411yescharacteristic
46C22×C42D411yescharacteristic
47C22×C42D411yescharacteristic
48C22×C42D411yescharacteristic
49C2×C4⋊C8—24no—
50C2×C4⋊C8—42no—
51C2×C4⋊C8—42no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—22no—
54C2×C22⋊C8—22no—
55C2×C22⋊C8—22no—
56C2×C22⋊C8—22no—
57C2×C22⋊C8—22no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8—22no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8D414yes—
63C2×C22⋊C8Q814yes—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95—22no—
67C2×#95—22no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—22no—
77C2×#95—21no—
78C2×#95—21no—
79C2×#95—22no—
80C2×#95—42no—
81C2×#95—42no—
82C2×#95D412yes—
83C2×#95Q812yes—
84#336—24no—
85#336—42no—
86#336—42no—
87C25C22×C411yescharacteristic
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—22no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—22no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—41no—
115C23×C4—41no—
116C23×C4—41no—
117C23×C4—41no—
118C23×C4—41no—
119C23×C4—41no—
120C23×C4—41no—
121C23×C4—41no—
122C23×C4—41no—
123C23×C4—41no—
124C23×C4—41no—
125C23×C4—41no—
126C23×C4—41no—
127C23×C4—41no—
128C23×C4—41no—
129C23×C4—41no—
130C23×C4—41no—
131C23×C4—41no—
132C23×C4—41no—
133C23×C4—41no—
134C23×C4—41no—
135C23×C4—41no—
136C23×C4—41no—
137C23×C4—41no—
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C4⋊C412yes—
143C23×C4C4⋊C412yes—
144C23×C4C2×C812yes—
145C23×C4C2×C812yes—
146C23×C4C2×C812yes—
147C23×C4M1612yes—
148C23×C4M1612yes—
149C23×C4M1612yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×Q812yes—
155C23×C4C2×Q811yescharacteristic
156C23×C4#4111yescharacteristic
157C23×C4#4111yescharacteristic
158C23×C4#4112yes—
159C23×C4#4112yes—
160C22×C8—24no—
161C22×C8—24no—
162C22×C8—42no—
163C22×C8—42no—
164C22×C8—42no—
165C22×C8—42no—
166C22×C8—42no—
167C22×C8—42no—
168C22×C8—42no—
169C22×C8—42no—
170C22×C8—42no—
171C22×C8—42no—
172C22×C8—42no—
173C22×C8—42no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—48no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—48no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—21no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—21no—
192C2×C42—48no—
193C2×C42—48no—
194C2×C42—41no—
195C2×C42—41no—
196C2×C42—41no—
197C2×C42—41no—
198C4⋊C8—48no—
199C22⋊C8—24no—
200C22⋊C8—24no—
201C22⋊C8—24no—
202C22⋊C8—24no—
203C22⋊C8—42no—
204C22⋊C8—42no—
205C22⋊C8—42no—
206C22⋊C8—42no—
207#95—24no—
208#95—24no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—42no—
214#95—42no—
215#95—42no—
216#95—42no—
217#95—41no—
218#95—42no—
219#95—41no—
220C24—21no—
221C24—21no—
222C24—21no—
223C24—21no—
224C24—21no—
225C24—21no—
226C24—21no—
227C24—21no—
228C24C2×C22⋊C411yescharacteristic
229C24C2×C4⋊C411yescharacteristic
230C24#11711yescharacteristic
231C24C4×D412yes—
232C24C4×Q812yes—
233C24C22×C811yescharacteristic
234C24C2×M1611yescharacteristic
235C24C2×M1611yescharacteristic
236C24C2×M1611yescharacteristic
237C24#13112yes—
238C24#13112yes—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—82no—
367C22×C4—82no—
368C22×C4—82no—
369C22×C4—82no—
370C22×C4—82no—
371C22×C4—82no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4—82no—
378C22×C4C22⋊C812yes—
379C22×C4C22⋊C812yes—
380C22×C4#9912yes—
381C22×C4#9912yes—
382C22×C4#10012yes—
383C22×C4#10112yes—
384C22×C4C4⋊C814yescomplemented
385C22×C4#12011yescharacteristic
386C22×C4#12111yescharacteristic
387C22×C4#12111yescharacteristic
388C22×C4#12112yes—
389C22×C4#12112yes—
390C22×C4C22⋊Q811yescharacteristic
391C22×C4C22⋊Q812yes—
392C22×C4C22⋊Q811yescharacteristic
393C22×C4C22⋊Q812yes—
394C22×C4C22⋊Q812yes—
395C22×C4#12311yescharacteristic
396C22×C4#12312yes—
397C22×C4#12411yescharacteristic
398C22×C4#12512yes—
399C22×C4#12512yes—
400C22×C4#12612yes—
401C22×C4C4⋊Q811yescharacteristic
402C22×C4C4⋊Q812yes—
403C2×C8—44no—
404C2×C8—44no—
405C2×C8—44no—
406C2×C8—44no—
407C2×C8—44no—
408C2×C8—44no—
409C2×C8—82no—
410C2×C8—82no—
411C2×C8—82no—
412C2×C8—82no—
413C2×C8—82no—
414C2×C8—82no—
415C2×C8—82no—
416C2×C8—82no—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C42—24no—
420C42—24no—
421C42—44no—
422C42—44no—
423C42—42no—
424C42—42no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23#38011yescharacteristic
466C23#38611yescharacteristic
467C23#38712yes—
468C23#38912yes—
469C23#39111yescharacteristic
470C23C2×C22⋊C811yescharacteristic
471C23#40711yescharacteristic
472C23#40811yescharacteristic
473C23#40811yescharacteristic
474C23C2×#9911yescharacteristic
475C23C2×#9911yescharacteristic
476C23#41011yescharacteristic
477C23#41011yescharacteristic
478C23C2×#10011yescharacteristic
479C23C2×#10111yescharacteristic
480C23#41311yescharacteristic
481C23#41311yescharacteristic
482C23C2×C4⋊C811yescharacteristic
483C23#42311yescharacteristic
484C23#42412yes—
485C23#43111yescharacteristic
486C23#43211yescharacteristic
487C23#43312yes—
488C23C8×D412yes—
489C23#43512yes—
490C23#43512yes—
491C23#43612yes—
492C23C8×Q812yes—
493C23#44612yes—
494C23#44612yes—
495C23#44612yes—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—48no—
541C2×C4—48no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—48no—
546C2×C4—48no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4#32314yes—
594C2×C4#39411yescharacteristic
595C2×C4#39511yescharacteristic
596C2×C4#39612yes—
597C2×C4#39811yescharacteristic
598C2×C4#39812yes—
599C2×C4#40011yescharacteristic
600C8—84no—
601C8—84no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22#144111yescharacteristic
635C22C2×#32311yescharacteristic
636C22#145811yescharacteristic
637C22#145911yescharacteristic
638C22#146111yescharacteristic
639C22#146311yescharacteristic
640C22#146411yescharacteristic
641C22#146511yescharacteristic
642C22#146611yescharacteristic
643C22#146811yescharacteristic
644C22#146811yescharacteristic
645C22#146911yescharacteristic
646C22#147011yescharacteristic
647C22#147111yescharacteristic
648C22#147211yescharacteristic
649C22#147311yescharacteristic
650C22#147311yescharacteristic
651C22#179711yescharacteristic
652C22#179811yescharacteristic
653C22#179911yescharacteristic
654C22#179911yescharacteristic
655C22#180011yescharacteristic
656C22#180111yescharacteristic
657C22#186911yescharacteristic
658C22#187011yescharacteristic
659C22#189811yescharacteristic
660C22#189911yescharacteristic
661C22#190011yescharacteristic
662C22#190111yescharacteristic
663C22#191612yes—
664C22#191712yes—
665C22#193912yes—
666C22#194012yes—
667C22#198411yescharacteristic
668C22#198511yescharacteristic
669C22#199611yescharacteristic
670C22#199611yescharacteristic
671C22#199711yescharacteristic
672C22#199811yescharacteristic
673C4—24no—
674C4—21no—
675C4—22no—
676C4—21no—
677C4—21no—
678C4—22no—
679C4—21no—
680C4—44no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—82no—
685C4—82no—
686C4—82no—
687C2—22no—
688C2—22no—
689C2—22no—
690C2—22no—
691C2#855711yescharacteristic
692C2#856011yescharacteristic
693C2#856111yescharacteristic
694C2#856211yescharacteristic
695C2#907911yescharacteristic
696C2#908211yescharacteristic
697C2#908411yescharacteristic
698C2#908511yescharacteristic
699C2#987811yescharacteristic
700C2#988011yescharacteristic
701C2#988111yescharacteristic
702C2#988311yescharacteristic
703C2#1120211yescharacteristic
704C2#1120311yescharacteristic
705C2#1120411yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC24C22×C8
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
40. C22×C4⋊C4C8semidirect
384. C22×C4C4⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×#17972
C2×#14362
#107122
C2×#12802