Finite Group Databasebeta

← #366986

Subgroups — #366986

Order 512.

1866 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#461C211yesmaximal, characteristic
2C2×#1981C211yesmaximal, characteristic, complemented
3C2×#1968C212yesmaximal
4C2×#1948C211yesmaximal, characteristic, complemented
5C2×#1802C211yesmaximal, characteristic
6C2×C4×C4⋊C8C211yesmaximal, characteristic, complemented
7#14714C211yesmaximal, characteristic, complemented
8C4×#390C211yesmaximal, characteristic
9#11122C212yesmaximal, complemented
10#11023C212yesmaximal
11#10926C212yesmaximal, complemented
12C23×SD16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×#461—24no—
16C2×#461—24no—
17C2×#461C418yescomplemented
18C22×#106C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×#102C2212yes—
25C22×C4×C8C2212yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4—22no—
33C2×C43C2211yescharacteristic
34#1981—24no—
35#1968—28no—
36#1948—24no—
37#1802—24no—
38C4×C4⋊C8—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42#1437—22no—
43C4⋊#95—22no—
44C4⋊#95—22no—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2211yescharacteristic
47C23×Q8C2312yes—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C22×SD16—24no—
51C22×SD16—24no—
52C22×SD16—24no—
53C22×SD16—24no—
54C22×SD16—42no—
55C22×SD16—42no—
56C22×SD16C2×C418yes—
57C23×C8C2312yes—
58C23×C8C2312yes—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8C2×C414yes—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4C2×C414yes—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C22⋊C4—21no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C42—21no—
83C22×C42D412yes—
84C22×C42D412yes—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#461—232no—
89C2×#106—24no—
90C2×#106—24no—
91C2×#106—42no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8C2×C414yes—
95C2×C4⋊C8D414yes—
96C2×C4⋊C8D414yes—
97C2×C4⋊C8D414yes—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—42no—
105C2×#103—42no—
106C2×#103C2×C418yes—
107C2×#102—24no—
108C2×#102—24no—
109C2×#102—42no—
110C2×C4×C8—24no—
111C2×C4×C8—24no—
112C2×C4×C8—42no—
113#391—24no—
114#391—24no—
115#391—42no—
116#391—42no—
117#390—22no—
118#390—22no—
119#390—22no—
120#390D414yes—
121#384—22no—
122#384—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C22⋊C4—48no—
126C2×#95—24no—
127C2×#95—24no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—21no—
131C43—22no—
132C43—22no—
133#340—24no—
134#340—24no—
135#336—24no—
136C25—21no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8C22×C414yes—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4C22×C412yes—
163C22×D4C2×D412yes—
164C22×D4#4112yes—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—24no—
170C23×C4—22no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4C2×D412yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D412yes—
178C23×C4C2×D411yescharacteristic
179C23×C4C2411yescharacteristic
180C2×SD16—216no—
181C2×SD16—216no—
182C2×SD16—48no—
183C2×SD16—48no—
184C2×SD16—48no—
185C2×SD16—48no—
186C2×SD16—48no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8C22×C414yes—
198C22×C8C2×D414yes—
199C22×C8C2×D414yes—
200C22×C8C2×D414yes—
201C4⋊Q8—28no—
202C4⋊Q8—28no—
203C4⋊Q8—44no—
204C4⋊Q8—44no—
205C4×D4—28no—
206C4×D4—28no—
207C4×D4—44no—
208C4×D4—44no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4C22×C412yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4#4112yes—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—44no—
269C2×C42SD1618yescomplemented
270C2×C42C22×C412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42#4112yes—
276C2×C42#4112yes—
277C2×C42#4112yes—
278#106—48no—
279C4⋊C8—28no—
280C4⋊C8C22⋊C4116yescomplemented
281#103—216no—
282#103—48no—
283#103—48no—
284#102—48no—
285C4×C8—48no—
286#95—48no—
287#95—48no—
288#95—42no—
289#95—44no—
290#95—44no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—41no—
295C24—41no—
296C24—41no—
297C24—41no—
298C24—42no—
299C24—42no—
300C24—42no—
301C24—42no—
302C24C22×D411yescharacteristic
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4C4×D414yes—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—48no—
394C22×C4—416no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—88no—
401C22×C4—84no—
402C22×C4—84no—
403C22×C4C4×D412yes—
404C22×C4C4×D412yes—
405C22×C4#12112yes—
406C22×C4#12114yes—
407C22×C4#12112yes—
408C22×C4#12712yes—
409C22×C4#12712yes—
410C22×C4C2×SD1614yes—
411C22×C4C2×SD1614yes—
412C22×C4C2×SD1614yes—
413C22×C4#13514yes—
414C22×C4#13714yes—
415C22×C4C23×C411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4H(5,𝔽2)11yescharacteristic
426C22×C4H(5,𝔽2)11yescharacteristic
427C22×C4H(5,𝔽2)11yescharacteristic
428C22×C42−1+411yescharacteristic
429SD16—432no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8C2×C22⋊C418yes—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—84no—
463C4⋊C4—84no—
464C22⋊C4—416no—
465C22⋊C4—48no—
466C22⋊C4—416no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—216no—
472C42—48no—
473C42—41no—
474C42—41no—
475C42—41no—
476C42—41no—
477C42—84no—
478C42C2×C22⋊C414yes—
479C42C4×D414yes—
480C42#12114yes—
481C42#12414yes—
482C23—24no—
483C23—42no—
484C23—44no—
485C23—44no—
486C23—42no—
487C23—42no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—81no—
496C23—81no—
497C23—81no—
498C23—81no—
499C23—82no—
500C23—82no—
501C23—82no—
502C23—82no—
503C23—82no—
504C23—82no—
505C23C2×C4×D411yescharacteristic
506C23C2×#12111yescharacteristic
507C23C2×#12111yescharacteristic
508C23#52511yescharacteristic
509C23C2×#12711yescharacteristic
510C23#53511yescharacteristic
511C23#53711yescharacteristic
512C23C22×SD1612yes—
513C23C2×#13512yes—
514C23C2×#13712yes—
515C23#57512yes—
516Q8—44no—
517Q8—44no—
518Q8—44no—
519Q8—44no—
520Q8—82no—
521Q8—82no—
522Q8—82no—
523Q8—82no—
524D4—24no—
525D4—24no—
526D4—42no—
527D4—42no—
528D4—42no—
529D4—42no—
530D4—42no—
531D4—42no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—816no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4—82no—
577C2×C4#39014yes—
578C2×C4#39014yes—
579C2×C4C4×SD1618yes—
580C2×C4#46118yescomplemented
581C2×C4#46518yes—
582C2×C4#49214yes—
583C2×C4#49514yes—
584C2×C4C22×C22⋊C411yescharacteristic
585C2×C4C2×C4×D411yescharacteristic
586C2×C4C2×C4×D411yescharacteristic
587C2×C4C4×#4111yescharacteristic
588C2×C4#51811yescharacteristic
589C2×C4#52011yescharacteristic
590C2×C4C2×#12111yescharacteristic
591C2×C4C2×#12411yescharacteristic
592C2×C4#52911yescharacteristic
593C2×C4#53211yescharacteristic
594C2×C4#54011yescharacteristic
595C2×C4#54211yescharacteristic
596C2×C4D4211yescharacteristic
597C2×C4D4211yescharacteristic
598C2×C4D4211yescharacteristic
599C2×C4#54611yescharacteristic
600C2×C4#54611yescharacteristic
601C2×C4#54611yescharacteristic
602C2×C4#54711yescharacteristic
603C2×C4#54811yescharacteristic
604C2×C4#54811yescharacteristic
605C2×C4#54811yescharacteristic
606C2×C4#55011yescharacteristic
607C2×C4#55011yescharacteristic
608C2×C4#55011yescharacteristic
609C2×C4#55311yescharacteristic
610C2×C4#55311yescharacteristic
611C2×C4#55511yescharacteristic
612C8—28no—
613C8—44no—
614C22—44no—
615C22—48no—
616C22—82no—
617C22—82no—
618C22—84no—
619C22—84no—
620C22—84no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22C2×#39011yescharacteristic
626C22C4×#12111yescharacteristic
627C22#232411yescharacteristic
628C22#241411yescharacteristic
629C22#241611yescharacteristic
630C22#254011yescharacteristic
631C22#254111yescharacteristic
632C22C2×C4×SD1612yes—
633C22C4×#13712yes—
634C22C2×#46112yes—
635C22#303512yes—
636C22C2×#46512yes—
637C22#305612yes—
638C22#310112yes—
639C22#310712yes—
640C22C2×#49211yescharacteristic
641C22C2×#49511yescharacteristic
642C22#315211yescharacteristic
643C22#315311yescharacteristic
644C22#320612yes—
645C22#321212yes—
646C22#324612yes—
647C22#324812yes—
648C4—28no—
649C4—24no—
650C4—44no—
651C4—82no—
652C4—84no—
653C4—82no—
654C4—82no—
655C4—162no—
656C4#196818yescomplemented
657C4D4×C22⋊C411yescharacteristic
658C4#234111yescharacteristic
659C4#236111yescharacteristic
660C4#236311yescharacteristic
661C4#242611yescharacteristic
662C4#242811yescharacteristic
663C4#249011yescharacteristic
664C4#249111yescharacteristic
665C2—84no—
666C2#1399811yescharacteristic
667C2C2×#196811yescharacteristic
668C2#2123711yescharacteristic
669C2C4×#46112yes—
670C2#2214612yes—
671C2#2465112yes—
672C2#2475112yes—
673C2#2837011yescharacteristic
674C2#2837211yescharacteristic
675C2#2843811yescharacteristic
676C2#2843911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1981C2semidirect
4. C2×#1948C2semidirect
6. C2×C4×C4⋊C8C2semidirect
7. #14714C2semidirect
9. #11122C2semidirect
11. #10926C2semidirect
17. C2×#461C4semidirect
269. C2×C42SD16semidirect
280. C4⋊C8C22⋊C4semidirect
580. C2×C4#461semidirect
656. C4#1968semidirect

Maximal subgroups

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

subgroupindex
C22×#4612
C2×#19812
C2×#19682
C2×#19482
C2×#18022
C2×C4×C4⋊C82
#147142
C4×#3902
#111222
#110232
#109262