Finite Group Databasebeta

← #892328

Subgroups — #892328

Order 512.

3282 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2733C212yesmaximal, complemented
2#17080C211yesmaximal, characteristic, complemented
3#17068C211yesmaximal, characteristic, complemented
4#16551C212yesmaximal, complemented
5#15763C214yesmaximal, complemented
6#15709C214yesmaximal, complemented
7#14762C214yesmaximal, complemented
8#14722C212yesmaximal, complemented
9#14708C212yesmaximal, complemented
10#14589C212yesmaximal, complemented
11#14227C212yesmaximal, complemented
12#14098C214yesmaximal, complemented
13C4×#390C211yesmaximal, characteristic, complemented
14C22×#124C2214yes—
15C22×#124C2212yes—
16C22×#121C2212yes—
17C22×C4×D4C2211yescharacteristic
18#2733—216no—
19C2×#394C2214yes—
20C2×#394C2212yes—
21C2×#394C2212yes—
22C2×#394C2214yes—
23C2×#392C2214yes—
24C2×#392C2214yes—
25C2×#392C2212yes—
26C2×#392C2214yes—
27C2×#392C2214yes—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#390C2212yes—
31C2×#390C2211yescharacteristic
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#390C2211yescharacteristic
35C2×#388C2214yes—
36C2×#388C2214yes—
37C2×#386C2214yes—
38C2×#382C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2214yes—
41C2×C43C2211yescharacteristic
42#1446C2214yes—
43#1444C2214yes—
44#1442C2212yes—
45#1441C2212yes—
46#1441C2212yes—
47#1440C2214yes—
48#1439C2214yes—
49#1439C2212yes—
50#1439C2214yes—
51#1438C2214yes—
52#1438C2214yes—
53#1438C2212yes—
54#1438C2214yes—
55#1437C2214yes—
56#1437C2214yes—
57#1437C2214yes—
58#1437C2214yes—
59#1437C2214yes—
60#1437C2214yes—
61#1437C2214yes—
62#1437C2214yes—
63C4⋊#95C2211yescharacteristic
64C4⋊#95C2211yescharacteristic
65#1434C2214yes—
66C22⋊#95C2212yes—
67C4×#95C2212yes—
68C23×Q8C2312yes—
69C23×D4C2312yes—
70C23×D4C2314yes—
71C23×D4C2311yescharacteristic
72C23×D4C2312yes—
73C23×D4C2314yes—
74C23×D4C2312yes—
75C24×C4C2312yes—
76C2×#124—28no—
77C2×#124—216no—
78C2×#124—216no—
79C2×#124—44no—
80C2×#124—44no—
81C2×#124—48no—
82C2×#121—48no—
83C2×#121—44no—
84C2×C4×D4—24no—
85C2×C4×D4—42no—
86C2×C4×D4—42no—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C42C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2314yes—
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116#394—28no—
117#394—28no—
118#394—216no—
119#394—48no—
120#392—216no—
121#392—216no—
122#392—48no—
123#392—44no—
124#392—48no—
125#391—28no—
126#391—44no—
127#390—24no—
128#390—28no—
129#390—44no—
130#390—44no—
131#390D418yescomplemented
132#388—216no—
133#388—216no—
134#386—48no—
135#382—44no—
136C4×C4⋊C4—42no—
137C4×C22⋊C4—48no—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C43—42no—
155C25C2414yes—
156C25C2412yes—
157C22×Q8—24no—
158C22×Q8—24no—
159C22×Q8—42no—
160C22×Q8—42no—
161C22×Q8—42no—
162C22×Q8—42no—
163C22×Q8#4118yes—
164C22×D4—28no—
165C22×D4—28no—
166C22×D4—24no—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—24no—
170C22×D4—28no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—41no—
175C22×D4—41no—
176C22×D4—41no—
177C22×D4—41no—
178C22×D4—44no—
179C22×D4—42no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4—44no—
183C22×D4—42no—
184C22×D4—42no—
185C22×D4—44no—
186C22×D4—42no—
187C22×D4—42no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—42no—
191C22×D4—42no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4—42no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4C2×D414yes—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—28no—
205C23×C4—42no—
206C23×C4—44no—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218#124—432no—
219#124—464no—
220#121—432no—
221C4×D4—416no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4C2×D414yes—
239C2×C4⋊C4#4114yes—
240C2×C4⋊C4#4114yes—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4C2×D4116yes—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—28no—
293C2×C42—28no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—44no—
299C2×C42—41no—
300C2×C42—42no—
301C2×C42—41no—
302C2×C42—41no—
303C2×C42—41no—
304C2×C42—44no—
305C2×C42C2×D414yes—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—42no—
315#95—44no—
316#95—42no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—42no—
322C24—22no—
323C24—24no—
324C24—22no—
325C24—24no—
326C24—24no—
327C24—22no—
328C24—22no—
329C24—24no—
330C24—22no—
331C24—22no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24C22×D414yes—
348C24C2511yescharacteristic
349C2×Q8—216no—
350C2×Q8—48no—
351C2×Q8—48no—
352C2×Q8—84no—
353C2×Q8—84no—
354C2×Q8—84no—
355C2×Q8—84no—
356C2×D4—28no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—416no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—416no—
368C2×D4—48no—
369C2×D4—416no—
370C2×D4—48no—
371C2×D4—84no—
372C2×D4—84no—
373C2×D4—84no—
374C2×D4—84no—
375C2×D4—88no—
376C2×D4—84no—
377C2×D4—88no—
378C2×D4—88no—
379C2×D4—88no—
380C2×D4—88no—
381C2×D4—88no—
382C2×D4—88no—
383C2×D4—88no—
384C2×D4—88no—
385C2×D4—88no—
386C2×D4—84no—
387C2×D4—84no—
388C2×D4—84no—
389C2×D4—88no—
390C2×D4—88no—
391C2×D4—88no—
392C2×D4—88no—
393C2×D4—88no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—48no—
448C22×C4—416no—
449C22×C4—48no—
450C22×C4—88no—
451C22×C4—84no—
452C22×C4C22×D414yes—
453C22×C4C22×D412yes—
454C22×C4C22×D414yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4114yes—
457C22×C4C2×#4118yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4114yes—
461C22×C4H(5,𝔽2)14yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C4H(5,𝔽2)14yes—
464C22×C4H(5,𝔽2)14yes—
465C22×C4H(5,𝔽2)12yes—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—48no—
470C4⋊C4—84no—
471C4⋊C4—84no—
472C4⋊C4—84no—
473C22⋊C4—232no—
474C22⋊C4—416no—
475C22⋊C4—48no—
476C22⋊C4—416no—
477C22⋊C4—416no—
478C22⋊C4—416no—
479C22⋊C4—416no—
480C22⋊C4—416no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C22⋊C4—88no—
484C22⋊C4—88no—
485C22⋊C4—88no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C42—28no—
489C42—216no—
490C42—44no—
491C42—48no—
492C42—416no—
493C42—44no—
494C42—84no—
495C23—24no—
496C23—28no—
497C23—24no—
498C23—24no—
499C23—28no—
500C23—42no—
501C23—44no—
502C23—42no—
503C23—42no—
504C23—44no—
505C23—44no—
506C23—42no—
507C23—44no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—42no—
513C23—44no—
514C23—42no—
515C23—42no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23C23×D411yescharacteristic
554C23C22×#4111yescharacteristic
555C23C22×#4111yescharacteristic
556C23C22×#4112yes—
557C23C2×H(5,𝔽2)11yescharacteristic
558C23C2×H(5,𝔽2)12yes—
559C23C2×H(5,𝔽2)11yescharacteristic
560C23C2×H(5,𝔽2)12yes—
561C23C2×H(5,𝔽2)12yes—
562C23#58511yescharacteristic
563C23#58511yescharacteristic
564Q8—816no—
565D4—416no—
566D4—816no—
567D4—832no—
568D4—816no—
569D4—816no—
570D4—832no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—28no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—24no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—816no—
636C2×C4#53214yes—
637C2×C4#54618yes—
638C2×C4H(3,𝔽4)14yes—
639C22—44no—
640C22—48no—
641C22—44no—
642C22—44no—
643C22—48no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—84no—
649C22—84no—
650C22—84no—
651C22—84no—
652C22—84no—
653C22—84no—
654C22—84no—
655C22—84no—
656C22—84no—
657C22—84no—
658C22—84no—
659C22—84no—
660C22—84no—
661C22—84no—
662C22—84no—
663C22—84no—
664C22—84no—
665C22C2×#53211yescharacteristic
666C22#345911yescharacteristic
667C22#345912yes—
668C22#346011yescharacteristic
669C22C2×#54612yes—
670C22#348311yescharacteristic
671C22#348312yes—
672C22#348312yes—
673C22#349011yescharacteristic
674C22#349012yes—
675C22#349311yescharacteristic
676C22#349611yescharacteristic
677C22#349611yescharacteristic
678C22#349812yes—
679C22#351712yes—
680C22#351912yes—
681C22#351912yes—
682C22C2×H(3,𝔽4)11yescharacteristic
683C22#353712yes—
684C22#354311yescharacteristic
685C22#354911yescharacteristic
686C22#355812yes—
687C22#355812yes—
688C4—24no—
689C4—24no—
690C4—28no—
691C4—44no—
692C4—44no—
693C4—44no—
694C4—42no—
695C4—44no—
696C4—48no—
697C4—84no—
698C4—84no—
699C2—84no—
700C2—164no—
701C2#3500311yescharacteristic
702C2#3641212yes—
703C2#3724911yescharacteristic
704C2#3882212yes—
705C2#3883112yes—
706C2#3941212yes—
707C2#4599611yescharacteristic
708C2#4608911yescharacteristic
709C2#5230711yescharacteristic
710C2#5941911yescharacteristic
711C2#6004811yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2733C2semidirect
2. #17080C2semidirect
3. #17068C2semidirect
4. #16551C2semidirect
5. #15763C2semidirect
6. #15709C2semidirect
7. #14762C2semidirect
8. #14722C2semidirect
9. #14708C2semidirect
10. #14589C2semidirect
11. #14227C2semidirect
12. #14098C2semidirect
13. C4×#390C2semidirect
131. #390D4semidirect

Maximal subgroups

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

subgroupindex
C2×#27332
#170802
#170682
#165512
#157632
#157092
#147622
#147222
#147082
#145892
#142272
#140982
C4×#3902