Finite Group Databasebeta

← #616203

Subgroups — #616203

Order 512.

3326 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2485C212yesmaximal
2C2×#2285C211yesmaximal, characteristic, complemented
3C2×C42×D4C211yesmaximal, characteristic
4#15234C212yesmaximal, complemented
5#15233C212yesmaximal, complemented
6#15132C214yesmaximal, complemented
7#15129C214yesmaximal
8#14712C211yesmaximal, characteristic, complemented
9#14712C211yesmaximal, characteristic, complemented
10#14686C211yesmaximal, characteristic, complemented
11#14686C211yesmaximal, characteristic, complemented
12#14686C211yesmaximal, characteristic, complemented
13#14042C214yesmaximal, complemented
14#13994C214yesmaximal, complemented
15#13939C212yesmaximal, complemented
16C22×C4×D4C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×#117C2212yes—
19C23×C42C2211yescharacteristic
20C23×C42C2211yescharacteristic
21#2485—216no—
22C2×#397C2214yes—
23C2×#394C2214yes—
24C2×#389C2214yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2214yes—
28C2×#388C2214yes—
29C2×#387C2214yes—
30#2285—28no—
31C2×#384C2211yescharacteristic
32C2×#384C2211yescharacteristic
33C2×#384C2211yescharacteristic
34C2×#384C2211yescharacteristic
35C2×#382C2212yes—
36C2×#380C2214yes—
37C2×#380C2214yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2214yes—
42C2×#380C2214yes—
43C42×D4—28no—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2214yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51C2×C43C2211yescharacteristic
52#1443C2218yes—
53#1442C2218yes—
54#1441C2218yes—
55#1440C2218yes—
56#1440C2218yes—
57C4⋊#95C2214yes—
58C4⋊#95C2214yes—
59C4⋊#95C2214yes—
60C4⋊#95C2214yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67#1434C2218yes—
68C23×D4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C24×C4C2312yes—
71C24×C4C2312yes—
72C24×C4C2311yescharacteristic
73C2×C4×D4—28no—
74C2×C4×D4—24no—
75C2×C4×D4—28no—
76C2×C4×D4—24no—
77C2×C4×D4—28no—
78C2×C4×D4D418yes—
79C2×#117—28no—
80C2×#117—28no—
81C2×#117—28no—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2311yescharacteristic
99C22×C22⋊C4C2311yescharacteristic
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C42—24no—
103C22×C42—24no—
104C22×C42—24no—
105C22×C42—24no—
106C22×C42—24no—
107C22×C42D418yes—
108C22×C42C2312yes—
109C22×C42C2311yescharacteristic
110C22×C42C2314yes—
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118#397—216no—
119#394—216no—
120#389—28no—
121#389—28no—
122#389—216no—
123#389—48no—
124#388—216no—
125#387—216no—
126#384—24no—
127#384—24no—
128#384—24no—
129#384—24no—
130#382—28no—
131#380—28no—
132#380—28no—
133#380—28no—
134#380—216no—
135#380—216no—
136#380—48no—
137#380—48no—
138C4×C4⋊C4—28no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4—216no—
141C4×C22⋊C4—28no—
142C4×C22⋊C4—24no—
143C4×C22⋊C4—28no—
144C4×C22⋊C4—24no—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2318yes—
152C2×#95C2318yes—
153C2×#95C2318yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2318yes—
157C2×#95C2318yes—
158C2×#95C2318yes—
159C43—24no—
160C25C2411yescharacteristic
161C25C2411yescharacteristic
162C22×D4—22no—
163C22×D4—24no—
164C22×D4—22no—
165C22×D4—24no—
166C22×D4C2×D414yes—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4C2×D418yes—
190C23×C4C2×D414yes—
191C23×C4C2×D414yes—
192C23×C4C2×D414yes—
193C23×C4#4118yes—
194C23×C4#4114yes—
195C23×C4C2414yes—
196C23×C4C2414yes—
197C23×C4C2414yes—
198C23×C4C2414yes—
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C4×D4—264no—
209C4×D4—232no—
210#117—432no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—216no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—48no—
244C2×C4⋊C4C2×D414yes—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—216no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—48no—
263C2×C22⋊C4—48no—
264C2×C22⋊C4C2×D414yes—
265C2×C22⋊C4C2×D414yes—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—232no—
275C2×C42—28no—
276C2×C42—28no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42—24no—
281C2×C42—28no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—416no—
291C2×C42C2×D414yes—
292C2×C42C2×D418yes—
293C2×C42C2×D414yes—
294C2×C42C2×D414yes—
295C2×C42#4118yes—
296C2×C42#4114yes—
297C2×C42#4114yes—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—216no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—216no—
308#95—216no—
309#95—48no—
310#95—48no—
311#95—48no—
312C24—21no—
313C24—21no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—21no—
318C24—21no—
319C24—21no—
320C24—21no—
321C24—22no—
322C24—22no—
323C24—21no—
324C24—21no—
325C24—22no—
326C24—21no—
327C24—21no—
328C24—22no—
329C24C22×D412yes—
330C24C22×D412yes—
331C24C22×D412yes—
332C24C2×#4112yes—
333C24C2×#4114yes—
334C24C2511yescharacteristic
335C2×D4—216no—
336C2×D4—216no—
337C2×D4—28no—
338C2×D4—28no—
339C2×D4—28no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—28no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—28no—
382C22×C4—216no—
383C22×C4—216no—
384C22×C4—216no—
385C22×C4—216no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—216no—
389C22×C4—216no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—48no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4#121116yes—
423C22×C4C22×D412yes—
424C22×C4C22×D414yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D414yes—
432C22×C4C2×#4114yes—
433C22×C4C2×#4114yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4114yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4114yes—
442C22×C4C2×#4114yes—
443C22×C4H(5,𝔽2)14yes—
444C22×C42−1+412yes—
445C22×C42−1+412yes—
446C22×C42−1+414yes—
447C4⋊C4—216no—
448C4⋊C4—28no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—416no—
452C4⋊C4—416no—
453C4⋊C4—416no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—416no—
457C4⋊C4—416no—
458C4⋊C4—88no—
459C22⋊C4—216no—
460C22⋊C4—216no—
461C22⋊C4—28no—
462C22⋊C4—28no—
463C22⋊C4—416no—
464C22⋊C4—416no—
465C22⋊C4—88no—
466C22⋊C4—88no—
467C42—28no—
468C42—216no—
469C42—28no—
470C42—216no—
471C42—416no—
472C42—48no—
473C42—44no—
474C42—44no—
475C42—48no—
476C42#123116yes—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—24no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—24no—
486C23—22no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23—41no—
508C23—41no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23C2×#12114yes—
522C23#53214yes—
523C23C23×D411yescharacteristic
524C23C23×D411yescharacteristic
525C23C22×#4111yescharacteristic
526C23C22×#4111yescharacteristic
527C23C22×#4111yescharacteristic
528C23C22×#4112yes—
529C23C22×#4112yes—
530C23C2×H(5,𝔽2)12yes—
531C23C2×2−1+411yescharacteristic
532C23C2×2−1+411yescharacteristic
533C23C2×2−1+412yes—
534D4—232no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—232no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—432no—
580C2×C4—432no—
581C2×C4—416no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—48no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—48no—
594C2×C4—48no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—48no—
600C2×C4—48no—
601C2×C4—48no—
602C2×C4C2×#12114yes—
603C2×C4C2×#12114yes—
604C2×C4C2×#12318yes—
605C2×C4C2×#12314yes—
606C2×C4#52518yes—
607C2×C4#53214yes—
608C2×C4#53614yes—
609C2×C4#53718yes—
610C2×C4#54714yes—
611C2×C4#54714yes—
612C2×C4#55218yes—
613C2×C4#55214yes—
614C2×C4#55214yes—
615C2×C4#55318yes—
616C2×C4#55518yes—
617C2×C4#55618yes—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—42no—
635C22—42no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22C22×#12111yescharacteristic
641C22C22×#12311yescharacteristic
642C22C2×#52512yes—
643C22C2×#53211yescharacteristic
644C22C2×#53211yescharacteristic
645C22#344412yes—
646C22C2×#53611yescharacteristic
647C22C2×#53712yes—
648C22C2×#54711yescharacteristic
649C22C2×#54711yescharacteristic
650C22C2×#55212yes—
651C22C2×#55211yescharacteristic
652C22C2×#55211yescharacteristic
653C22C2×#55312yes—
654C22C2×#55512yes—
655C22C2×#55612yes—
656C22#349811yescharacteristic
657C22#349911yescharacteristic
658C22#350912yes—
659C22#350912yes—
660C22#351112yes—
661C22#351112yes—
662C22#351112yes—
663C4—24no—
664C4—24no—
665C4—42no—
666C4—42no—
667C4—44no—
668C4—44no—
669C4—44no—
670C4—84no—
671C4—84no—
672C4—84no—
673C4—84no—
674C4#2485116yescomplemented
675C4#248618yes—
676C2—44no—
677C2—44no—
678C2C2×#248611yescharacteristic
679C2C2×#248512yes—
680C2#3527311yescharacteristic
681C2#3529712yes—
682C2#3618111yescharacteristic
683C2#3618211yescharacteristic
684C2#3618711yescharacteristic
685C2#3618811yescharacteristic
686C2#3620912yes—
687C2#3621012yes—
688C2#3662411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2285C2semidirect
4. #15234C2semidirect
5. #15233C2semidirect
6. #15132C2semidirect
8. #14712C2semidirect
9. #14712C2semidirect
10. #14686C2semidirect
11. #14686C2semidirect
12. #14686C2semidirect
13. #14042C2semidirect
14. #13994C2semidirect
15. #13939C2semidirect
674. C4#2485semidirect

Maximal subgroups

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

subgroupindex
C2×#24852
C2×#22852
C2×C42×D42
#152342
#152332
#151322
#151292
#147122
#146862
#140422
#139942
#139392