Finite Group Databasebeta

← #616163

Subgroups — #616163

Order 512.

3966 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2856C211yesmaximal, characteristic
2C2×#2420C212yesmaximal, complemented
3C2×C4×#127C211yesmaximal, characteristic, complemented
4#15224C212yesmaximal, complemented
5#15219C212yesmaximal, complemented
6#15138C214yesmaximal, complemented
7#15126C214yesmaximal, complemented
8#14713C211yesmaximal, characteristic, complemented
9#14712C211yesmaximal, characteristic, complemented
10#14687C211yesmaximal, characteristic, complemented
11#14686C211yesmaximal, characteristic, complemented
12#13979C212yesmaximal, complemented
13#13970C214yesmaximal, complemented
14#13916C214yesmaximal, complemented
15C42×C22⋊C4C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#121C2214yes—
18C22×#121C2212yes—
19C22×C4×D4C2214yes—
20C22×C4×D4C2212yes—
21C23×C42C2211yescharacteristic
22#2856—28no—
23#2420—216no—
24C2×#397C2214yes—
25C2×#394C2214yes—
26C4×#127—28no—
27C2×#390C2214yes—
28C2×#390C2212yes—
29C2×#390C2214yes—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#390C2214yes—
34C2×#390C2214yes—
35C2×#389C2212yes—
36C2×#389C2214yes—
37C2×#386C2214yes—
38C2×#384C2211yescharacteristic
39C2×#384C2211yescharacteristic
40C2×#384C2211yescharacteristic
41C2×#382C2212yes—
42C2×#382C2212yes—
43C2×#381C2214yes—
44C2×#380C2212yes—
45C2×#380C2214yes—
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×C43C2211yescharacteristic
51C2×C43C2211yescharacteristic
52#1440C2218yes—
53#1440C2218yes—
54#1440C2218yes—
55#1439C2218yes—
56#1438C2218yes—
57C4⋊#95C2214yes—
58C4⋊#95C2214yes—
59C4⋊#95C2214yes—
60C4⋊#95C2214yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C22⋊#95C2218yes—
64C4×#95C2212yes—
65C4×#95C2212yes—
66C4×#95C2211yescharacteristic
67C4×#95C2211yescharacteristic
68C23×D4C2312yes—
69C23×D4C2314yes—
70C23×D4C2314yes—
71C24×C4C2311yescharacteristic
72C24×C4C2312yes—
73C24×C4C2314yes—
74C2×#127—24no—
75C2×#127—24no—
76C2×#127D418yes—
77C2×#121—216no—
78C2×#121—28no—
79C2×#121—28no—
80C2×#121—216no—
81C2×#121—44no—
82C2×#121—48no—
83C2×C4×D4—216no—
84C2×C4×D4—28no—
85C2×C4×D4—28no—
86C2×C4×D4—216no—
87C2×C4×D4—48no—
88C2×C4×D4—44no—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2318yes—
100C22×C22⋊C4C2318yes—
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2318yes—
107C22×C42—24no—
108C22×C42—24no—
109C22×C42—24no—
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#397—216no—
122#394—216no—
123#390—216no—
124#390—28no—
125#390—216no—
126#390—28no—
127#390—28no—
128#390—28no—
129#390—48no—
130#390—48no—
131#389—28no—
132#389—48no—
133#386—216no—
134#384—24no—
135#384—24no—
136#384—24no—
137#382—28no—
138#382—44no—
139#381—216no—
140#380—28no—
141#380—48no—
142C4×C22⋊C4—28no—
143C4×C22⋊C4—28no—
144C4×C22⋊C4—24no—
145C4×C22⋊C4D418yescomplemented
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2318yes—
156C2×#95C2318yes—
157C2×#95C2318yes—
158C2×#95C2311yescharacteristic
159C2×#95C2311yescharacteristic
160C43—24no—
161C43—24no—
162C25C2414yes—
163C25C2411yescharacteristic
164C22×D4—28no—
165C22×D4—28no—
166C22×D4—216no—
167C22×D4—24no—
168C22×D4—28no—
169C22×D4—28no—
170C22×D4—28no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—28no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—28no—
177C22×D4—28no—
178C22×D4—24no—
179C22×D4—44no—
180C22×D4—48no—
181C22×D4C2×D418yes—
182C22×D4C2×D4116yes—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—28no—
196C23×C4—28no—
197C23×C4—216no—
198C23×C4—44no—
199C23×C4—48no—
200C23×C4C2×D414yes—
201C23×C4#4118yes—
202C23×C4C2414yes—
203C23×C4C2414yes—
204C23×C4C2414yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215#127—232no—
216#121—432no—
217#121—464no—
218C4×D4—432no—
219C4×D4—464no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—48no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—216no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—216no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—216no—
252C2×C22⋊C4—216no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—216no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—48no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—48no—
261C2×C22⋊C4—48no—
262C2×C22⋊C4—48no—
263C2×C22⋊C4—48no—
264C2×C22⋊C4—48no—
265C2×C22⋊C4—48no—
266C2×C22⋊C4C2×D414yes—
267C2×C22⋊C4C2×D414yes—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—24no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—416no—
295C2×C42C2×D414yes—
296C2×C42C2×D414yes—
297C2×C42C2×D414yes—
298C2×C42#4118yes—
299C2×C42#4114yes—
300C2×C42#4114yes—
301C2×C42#4114yes—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—48no—
312#95—48no—
313#95—48no—
314#95C2×D414yes—
315#95C2×D414yes—
316C24—21no—
317C24—22no—
318C24—21no—
319C24—21no—
320C24—21no—
321C24—22no—
322C24—21no—
323C24—22no—
324C24—21no—
325C24—24no—
326C24—24no—
327C24—28no—
328C24—28no—
329C24—24no—
330C24—24no—
331C24—28no—
332C24—28no—
333C24—24no—
334C24—24no—
335C24C22×D418yes—
336C24C22×D412yes—
337C24C2×#4114yes—
338C24C2511yescharacteristic
339C2×D4—232no—
340C2×D4—216no—
341C2×D4—216no—
342C2×D4—232no—
343C2×D4—232no—
344C2×D4—216no—
345C2×D4—416no—
346C2×D4—432no—
347C2×D4—416no—
348C2×D4—416no—
349C2×D4—48no—
350C2×D4—416no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—416no—
354C2×D4—48no—
355C2×D4—416no—
356C2×D4—88no—
357C2×D4—816no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—216no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—216no—
401C22×C4—28no—
402C22×C4—216no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—416no—
419C22×C4—432no—
420C22×C4—48no—
421C22×C4—44no—
422C22×C4—48no—
423C22×C4—44no—
424C22×C4—48no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—48no—
428C22×C4—416no—
429C22×C4—88no—
430C22×C4—816no—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D414yes—
438C22×C4C22×D412yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4114yes—
442C22×C4C2×#4114yes—
443C22×C4C2×#4114yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4114yes—
447C22×C4C2×#4114yes—
448C22×C4C2×#4114yes—
449C22×C4C2×#4112yes—
450C22×C4H(5,𝔽2)14yes—
451C22×C4H(5,𝔽2)14yes—
452C22×C4H(5,𝔽2)12yes—
453C22×C42−1+412yes—
454C4⋊C4—48no—
455C4⋊C4—416no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—416no—
459C4⋊C4—416no—
460C4⋊C4—88no—
461C22⋊C4—216no—
462C22⋊C4—28no—
463C22⋊C4—28no—
464C22⋊C4—416no—
465C22⋊C4—432no—
466C22⋊C4—416no—
467C22⋊C4—432no—
468C22⋊C4—88no—
469C22⋊C4—88no—
470C22⋊C4—816no—
471C22⋊C4—88no—
472C42—28no—
473C42—216no—
474C42—28no—
475C42—28no—
476C42—48no—
477C42—44no—
478C42—48no—
479C42—44no—
480C42—48no—
481C42—48no—
482C42#123116yes—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—22no—
488C23—24no—
489C23—22no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—24no—
495C23—28no—
496C23—28no—
497C23—216no—
498C23—216no—
499C23—28no—
500C23—41no—
501C23—42no—
502C23—41no—
503C23—42no—
504C23—41no—
505C23—41no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—48no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—48no—
515C23—48no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—48no—
520C23—48no—
521C23—44no—
522C23—44no—
523C23—48no—
524C23—48no—
525C23—48no—
526C23—48no—
527C23—48no—
528C23#53214yes—
529C23C23×D411yescharacteristic
530C23C23×D411yescharacteristic
531C23C22×#4111yescharacteristic
532C23C22×#4111yescharacteristic
533C23C22×#4112yes—
534C23C22×#4112yes—
535C23C22×#4111yescharacteristic
536C23C2×H(5,𝔽2)12yes—
537C23C2×H(5,𝔽2)12yes—
538C23C2×H(5,𝔽2)11yescharacteristic
539C23C2×2−1+411yescharacteristic
540D4—464no—
541D4—432no—
542D4—832no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—432no—
591C2×C4—416no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—48no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—48no—
607C2×C4—832no—
608C2×C4C2×#12318yes—
609C2×C4C2×#12314yes—
610C2×C4#52918yes—
611C2×C4#53214yes—
612C2×C4#53214yes—
613C2×C4#54018yes—
614C2×C4#54214yes—
615C2×C4D4214yes—
616C2×C4#54618yes—
617C2×C4#54618yes—
618C2×C4#54714yes—
619C2×C4#54714yes—
620C2×C4#55318yes—
621C2×C4#55314yes—
622C2×C4#55518yes—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—42no—
627C22—44no—
628C22—42no—
629C22—44no—
630C22—42no—
631C22—44no—
632C22—42no—
633C22—44no—
634C22—48no—
635C22—48no—
636C22—416no—
637C22—48no—
638C22—416no—
639C22—84no—
640C22—84no—
641C22—84no—
642C22—88no—
643C22—84no—
644C22—88no—
645C22C22×#12311yescharacteristic
646C22C2×#52912yes—
647C22C2×#53211yescharacteristic
648C22C2×#53211yescharacteristic
649C22C2×#53211yescharacteristic
650C22#344412yes—
651C22C2×#54012yes—
652C22C2×#54211yescharacteristic
653C22C2×D4211yescharacteristic
654C22C2×#54612yes—
655C22C2×#54612yes—
656C22C2×#54711yescharacteristic
657C22C2×#54711yescharacteristic
658C22C2×#55312yes—
659C22C2×#55311yescharacteristic
660C22C2×#55512yes—
661C22#349811yescharacteristic
662C22#349811yescharacteristic
663C22#350812yes—
664C22#350812yes—
665C22#350812yes—
666C22#351112yes—
667C22#351112yes—
668C4—24no—
669C4—42no—
670C4—44no—
671C4—42no—
672C4—42no—
673C4—44no—
674C4—42no—
675C4—44no—
676C4—84no—
677C4—84no—
678C4—84no—
679C4#2420116yescomplemented
680C4#242218yes—
681C2—44no—
682C2—88no—
683C2C2×#242211yescharacteristic
684C2C2×#242012yes—
685C2#3527811yescharacteristic
686C2#3528812yes—
687C2#3612111yescharacteristic
688C2#3612111yescharacteristic
689C2#3614912yes—
690C2#3618211yescharacteristic
691C2#3618711yescharacteristic
692C2#3621112yes—
693C2#3663111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2856C2
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×#2420C2semidirect
3. C2×C4×#127C2semidirect
4. #15224C2semidirect
5. #15219C2semidirect
6. #15138C2semidirect
7. #15126C2semidirect
8. #14713C2semidirect
9. #14712C2semidirect
10. #14687C2semidirect
11. #14686C2semidirect
12. #13979C2semidirect
13. #13970C2semidirect
14. #13916C2semidirect
15. C42×C22⋊C4C2semidirect
145. C4×C22⋊C4D4semidirect
679. C4#2420semidirect

Maximal subgroups

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

subgroupindex
C2×#28562
C2×#24202
C2×C4×#1272
#152242
#152192
#151382
#151262
#147132
#147122
#146872
#146862
#139792
#139702
#139162
C42×C22⋊C42