Finite Group Databasebeta

← #366973

Subgroups — #366973

Order 512.

1922 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#460C211yesmaximal, characteristic
2C2×#1968C212yesmaximal, complemented
3C2×#1957C211yesmaximal, characteristic
4C2×#1948C211yesmaximal, characteristic, complemented
5C2×C8×C4⋊C4C211yesmaximal, characteristic, complemented
6C2×C4×#103C211yesmaximal, characteristic, complemented
7#14154C211yesmaximal, characteristic
8#13926C211yesmaximal, characteristic, complemented
9#11121C212yesmaximal, complemented
10#11023C212yesmaximal, complemented
11#10925C212yesmaximal, complemented
12C23×SD16C2212yes—
13C22×#127C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C2×#460—24no—
16C2×#460—24no—
17C2×#460C418yescomplemented
18C22×#106C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×#103C2211yescharacteristic
21C22×#103C2211yescharacteristic
22C22×#102C2212yes—
23C22×#102C2212yes—
24C22×#102C2212yes—
25C22×C4×C8C2212yes—
26C22×C4×C8C2211yescharacteristic
27C2×#391C2212yes—
28C2×#390—22no—
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#384C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1968—28no—
35#1957—24no—
36#1948—24no—
37C8×C4⋊C4—24no—
38C4×#103—24no—
39C2×#340C2212yes—
40C2×#340C2211yescharacteristic
41C2×#340C2211yescharacteristic
42C2×#336C2212yes—
43#1441—22no—
44#1438—22no—
45C4⋊#95—22no—
46C4⋊#95C2212yes—
47C23×Q8C2311yescharacteristic
48C23×D4—21no—
49C23×D4—21no—
50C23×D4C2312yes—
51C22×SD16—24no—
52C22×SD16—24no—
53C22×SD16—24no—
54C22×SD16—24no—
55C22×SD16—42no—
56C22×SD16—42no—
57C22×SD16C2×C418yes—
58C23×C8C2311yescharacteristic
59C23×C8C2312yes—
60C23×C8C2311yescharacteristic
61C2×#127—22no—
62C2×#127—22no—
63C2×#127—22no—
64C2×#127—22no—
65C2×#127—22no—
66C2×#127C2×C414yes—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×Q8C2×C414yes—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4D412yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—24no—
82C22×C22⋊C4—24no—
83C22×C42—21no—
84C22×C42—21no—
85C22×C42C2312yes—
86C22×C42C2311yescharacteristic
87#460—232no—
88C2×#106—24no—
89C2×#106—24no—
90C2×#106—42no—
91C2×C4⋊C8C2×C414yes—
92C2×C4⋊C8D414yes—
93C2×C4⋊C8D414yes—
94C2×#103—22no—
95C2×#103—22no—
96C2×#103—22no—
97C2×#103—22no—
98C2×#103—22no—
99C2×#103D414yescomplemented
100C2×#102—24no—
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—24no—
104C2×#102—24no—
105C2×#102—24no—
106C2×#102—42no—
107C2×#102—42no—
108C2×#102C2×C418yes—
109C2×C4×C8—22no—
110C2×C4×C8—22no—
111C2×C4×C8—24no—
112C2×C4×C8—24no—
113C2×C4×C8—42no—
114C2×C4×C8D414yes—
115#391—24no—
116#391—42no—
117#390—24no—
118#390—24no—
119#390—42no—
120#390—42no—
121#390—48no—
122#384—22no—
123#384—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C2×#95—24no—
129C2×#95—21no—
130C2×#95—24no—
131C2×#95D412yes—
132#340—24no—
133#340—22no—
134#340—22no—
135#336—24no—
136C25—22no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8C22×C412yes—
149C22×Q8C2×D412yes—
150C22×Q8#4112yes—
151C22×D4—24no—
152C22×D4—24no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—42no—
167C22×D4—42no—
168C22×D4—42no—
169C22×D4—42no—
170C22×D4—44no—
171C22×D4—44no—
172C22×D4—82no—
173C22×D4—82no—
174C22×D4C22×C414yes—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—22no—
180C23×C4C2×D414yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2411yescharacteristic
184C2×SD16—216no—
185C2×SD16—216no—
186C2×SD16—48no—
187C2×SD16—48no—
188C2×SD16—48no—
189C2×SD16—48no—
190C2×SD16—48no—
191C22×C8—22no—
192C22×C8—21no—
193C22×C8—22no—
194C22×C8—21no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8C22×C414yes—
206C22×C8C2×D412yes—
207C22×C8C2×D414yes—
208C22×C8C2×D414yes—
209C22×C8C2×D412yes—
210C22×C8C2×D412yes—
211C22×C8#4112yes—
212#127—28no—
213#127—28no—
214#127—44no—
215#127—44no—
216C4×Q8—28no—
217C4×Q8—28no—
218C4×Q8—44no—
219C4×Q8—44no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—82no—
250C2×C4⋊C4SD1618yescomplemented
251C2×C4⋊C4C22×C412yes—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4#4112yes—
254C2×C22⋊C4—48no—
255C2×C22⋊C4—48no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4—48no—
258C2×C22⋊C4—84no—
259C2×C22⋊C4—84no—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—44no—
271C2×C42C22×C412yes—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42#4112yes—
276C2×C42#4112yes—
277#106—48no—
278C4⋊C8C22⋊C4116yescomplemented
279#103—28no—
280#103—28no—
281#102—216no—
282#102—48no—
283#102—48no—
284C4×C8—28no—
285C4×C8—48no—
286#95—24no—
287#95—48no—
288#95—42no—
289#95—84no—
290C24—24no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24—42no—
295C24—42no—
296C24—42no—
297C24—44no—
298C24—44no—
299C24—44no—
300C24—44no—
301C24C22×D411yescharacteristic
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8C4×D414yes—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—416no—
361C2×D4—88no—
362C2×D4—84no—
363C2×D4—84no—
364C2×D4—88no—
365C2×D4—88no—
366C2×D4—84no—
367C2×D4—84no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
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—44no—
402C22×C4—44no—
403C22×C4C4×D412yes—
404C22×C4C4×D412yes—
405C22×C4#12112yes—
406C22×C4#12112yes—
407C22×C4#12712yes—
408C22×C4#12712yes—
409C22×C4C2×SD1618yes—
410C22×C4C2×SD1614yes—
411C22×C4#13514yes—
412C22×C4C23×C411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4H(5,𝔽2)11yescharacteristic
423C22×C4H(5,𝔽2)11yescharacteristic
424C22×C4H(5,𝔽2)11yescharacteristic
425C22×C42−1+411yescharacteristic
426SD16—432no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8C2×C22⋊C418yes—
447C2×C8#12114yes—
448C4⋊C4—216no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—48no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—84no—
467C4⋊C4—84no—
468C22⋊C4—816no—
469C22⋊C4—816no—
470C42—42no—
471C42—42no—
472C42—42no—
473C42—42no—
474C42—48no—
475C42—84no—
476C42C2×C22⋊C414yes—
477C42C4×D414yes—
478C42#12114yes—
479C42#12414yes—
480C23—44no—
481C23—44no—
482C23—48no—
483C23—48no—
484C23—44no—
485C23—84no—
486C23—84no—
487C23—84no—
488C23—84no—
489C23—82no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23—82no—
494C23—82no—
495C23—84no—
496C23—84no—
497C23—84no—
498C23—84no—
499C23—84no—
500C23—82no—
501C23—82no—
502C23—84no—
503C23C2×C4×D411yescharacteristic
504C23C2×#12111yescharacteristic
505C23C2×#12711yescharacteristic
506C23D4211yescharacteristic
507C23#54611yescharacteristic
508C23#54811yescharacteristic
509C23#55011yescharacteristic
510C23C22×SD1612yes—
511C23C2×#13512yes—
512C23#57612yes—
513C23#57712yes—
514Q8—24no—
515Q8—24no—
516Q8—42no—
517Q8—42no—
518Q8—42no—
519Q8—42no—
520Q8—42no—
521Q8—42no—
522D4—44no—
523D4—44no—
524D4—44no—
525D4—44no—
526D4—816no—
527D4—82no—
528D4—82no—
529D4—816no—
530D4—82no—
531D4—82no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
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—82no—
557C2×C4—82no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—82no—
565C2×C4—84no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—84no—
569C2×C4—82no—
570C2×C4#39014yes—
571C2×C4#39014yes—
572C2×C4C4×SD1618yes—
573C2×C4#46518yes—
574C2×C4#49214yes—
575C2×C4#49514yes—
576C2×C4C22×C22⋊C411yescharacteristic
577C2×C4C2×C4×D411yescharacteristic
578C2×C4C2×C4×D411yescharacteristic
579C2×C4C4×#4111yescharacteristic
580C2×C4#51811yescharacteristic
581C2×C4#52011yescharacteristic
582C2×C4C2×#12111yescharacteristic
583C2×C4C2×#12111yescharacteristic
584C2×C4#52511yescharacteristic
585C2×C4C2×#12411yescharacteristic
586C2×C4#52911yescharacteristic
587C2×C4#53211yescharacteristic
588C2×C4#53511yescharacteristic
589C2×C4#53711yescharacteristic
590C2×C4#54011yescharacteristic
591C2×C4#54211yescharacteristic
592C2×C4D4211yescharacteristic
593C2×C4D4211yescharacteristic
594C2×C4#54611yescharacteristic
595C2×C4#54611yescharacteristic
596C2×C4#54711yescharacteristic
597C2×C4#54811yescharacteristic
598C2×C4#54811yescharacteristic
599C2×C4#55011yescharacteristic
600C2×C4#55011yescharacteristic
601C2×C4#55311yescharacteristic
602C2×C4#55311yescharacteristic
603C2×C4#55511yescharacteristic
604C8—28no—
605C8—24no—
606C8—42no—
607C22—88no—
608C22—84no—
609C22—88no—
610C22—84no—
611C22—84no—
612C22—164no—
613C22—162no—
614C22—162no—
615C22—164no—
616C22—162no—
617C22—162no—
618C22C2×#39011yescharacteristic
619C22#236111yescharacteristic
620C22#236311yescharacteristic
621C22#242611yescharacteristic
622C22#242811yescharacteristic
623C22#254011yescharacteristic
624C22#254111yescharacteristic
625C22C2×C4×SD1612yes—
626C22#294612yes—
627C22C2×#46512yes—
628C22#306212yes—
629C22C2×#49211yescharacteristic
630C22C2×#49511yescharacteristic
631C22#315411yescharacteristic
632C22#315511yescharacteristic
633C22D4×SD1612yes—
634C22#328712yes—
635C22#329512yes—
636C22#330412yes—
637C22#336212yes—
638C22#336512yes—
639C22#339212yes—
640C22#339612yes—
641C4—24no—
642C4—48no—
643C4—44no—
644C4—82no—
645C4—82no—
646C4—82no—
647C4—162no—
648C4#196818yescomplemented
649C4C4×#12111yescharacteristic
650C4#232411yescharacteristic
651C4D4×C22⋊C411yescharacteristic
652C4#234111yescharacteristic
653C4#241411yescharacteristic
654C4#241611yescharacteristic
655C4#249011yescharacteristic
656C4#249111yescharacteristic
657C2—164no—
658C2#1399811yescharacteristic
659C2C2×#196811yescharacteristic
660C2#2123511yescharacteristic
661C2#2299712yes—
662C2#2309512yes—
663C2#2529512yes—
664C2#2538712yes—
665C2#2840211yescharacteristic
666C2#2840411yescharacteristic
667C2#2846111yescharacteristic
668C2#2846311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1968C2semidirect
4. C2×#1948C2semidirect
5. C2×C8×C4⋊C4C2semidirect
6. C2×C4×#103C2semidirect
8. #13926C2semidirect
9. #11121C2semidirect
10. #11023C2semidirect
11. #10925C2semidirect
17. C2×#460C4semidirect
99. C2×#103D4semidirect
250. C2×C4⋊C4SD16semidirect
278. C4⋊C8C22⋊C4semidirect
648. C4#1968semidirect

Maximal subgroups

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

subgroupindex
C22×#4602
C2×#19682
C2×#19572
C2×#19482
C2×C8×C4⋊C42
C2×C4×#1032
#141542
#139262
#111212
#110232
#109252