Finite Group Databasebeta

← #878101

Subgroups — #878101

Order 512.

2678 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2804C212yesmaximal
2#16987C212yesmaximal, complemented
3#16823C212yesmaximal, complemented
4#15691C214yesmaximal, complemented
5#15568C214yesmaximal, complemented
6#15146C212yesmaximal, complemented
7#15125C212yesmaximal, complemented
8#15125C212yesmaximal, complemented
9#14697C212yesmaximal, complemented
10#14513C212yesmaximal
11#14130C212yesmaximal, complemented
12#14053C212yesmaximal, complemented
13#13906C211yesmaximal, characteristic
14#13872C211yesmaximal, characteristic
15C4×#386C211yesmaximal, characteristic
16C22×C4⋊Q8C2212yes—
17C22×C4×D4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2804—216no—
21C2×#400C2214yes—
22C2×#393C2214yes—
23C2×#393C2214yes—
24C2×#391C2214yes—
25C2×#390C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#386C2212yes—
29C2×#386C2211yescharacteristic
30C2×#386C2211yescharacteristic
31C2×#385C2214yes—
32C2×#385C2214yes—
33C2×#384C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2211yescharacteristic
37C2×#380C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2212yes—
41#1446C2214yes—
42#1446C2214yes—
43#1446C2214yes—
44#1444C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1441C2214yes—
51#1441C2212yes—
52#1441C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2212yes—
56#1439C2212yes—
57#1439C2212yes—
58#1437C2212yes—
59#1436C2212yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2214yes—
72C22⋊#95C2211yescharacteristic
73C22⋊#95C2214yes—
74C22⋊#95C2211yescharacteristic
75C4×#95C2212yes—
76C23×Q8C2314yes—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C2×C4⋊Q8—28no—
84C2×C4⋊Q8—44no—
85C2×C4⋊Q8—44no—
86C2×C4×D4—42no—
87C2×C4×D4—42no—
88C2×C4×D4Q818yescomplemented
89C22×C4⋊C4—44no—
90C22×C4⋊C4—42no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C42—44no—
113C22×C42—42no—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2312yes—
119#400—216no—
120#393—216no—
121#393—216no—
122#391—216no—
123#390—28no—
124#389—44no—
125#389—44no—
126#386—28no—
127#386D418yes—
128#386Q818yescomplemented
129#385—216no—
130#385—216no—
131#384—42no—
132#380—28no—
133#380—28no—
134#380—44no—
135#380—42no—
136C4×C4⋊C4—42no—
137C4×C22⋊C4—42no—
138C4×C22⋊C4—44no—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C25C2411yescharacteristic
166C25C2411yescharacteristic
167C22×Q8—28no—
168C22×Q8—28no—
169C22×Q8—44no—
170C22×Q8—44no—
171C22×Q8—44no—
172C22×Q8—44no—
173C22×Q8—44no—
174C22×D4—41no—
175C22×D4—41no—
176C22×D4—41no—
177C22×D4—41no—
178C22×D4C2×D414yes—
179C22×D4C2×Q814yes—
180C22×D4C2×Q814yes—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—41no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—41no—
194C23×C4—41no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—41no—
198C23×C4—44no—
199C23×C4—42no—
200C23×C4—44no—
201C23×C4—42no—
202C23×C4C2×D414yes—
203C23×C4C2×Q814yes—
204C23×C4C2×Q814yes—
205C23×C4C2×Q814yes—
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C4⋊Q8—432no—
220C4×D4—416no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—416no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4C2×Q814yes—
252C2×C4⋊C4#4118yes—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—41no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—41no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4C2×D418yes—
278C2×C22⋊C4C2×D414yes—
279C2×C22⋊C4C2×Q814yes—
280C2×C22⋊C4C2×Q814yes—
281C2×C22⋊C4C2×Q814yes—
282C2×C22⋊C4C2×Q818yes—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—416no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42C2×Q814yes—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—42no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—42no—
315#95—42no—
316#95—42no—
317#95—42no—
318#95—42no—
319#95—44no—
320#95—42no—
321#95—44no—
322#95—44no—
323#95C2×D418yes—
324#95C2×Q818yes—
325C24—22no—
326C24—22no—
327C24—21no—
328C24—21no—
329C24—22no—
330C24—22no—
331C24—21no—
332C24—21no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—21no—
337C24—22no—
338C24—21no—
339C24—21no—
340C24—21no—
341C24C22×D412yes—
342C24C22×D412yes—
343C24C22×Q812yes—
344C24C22×Q812yes—
345C24C22×Q812yes—
346C24C22×Q812yes—
347C24C2511yescharacteristic
348C2×Q8—416no—
349C2×Q8—416no—
350C2×Q8—88no—
351C2×Q8—88no—
352C2×Q8—88no—
353C2×Q8—88no—
354C2×Q8—88no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4C4⋊Q8116yescomplemented
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—216no—
411C22×C4—216no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—48no—
434C22×C4—44no—
435C22×C4—48no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—48no—
441C22×C4—44no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—48no—
450C22×C4—88no—
451C22×C4—84no—
452C22×C4—84no—
453C22×C4—88no—
454C22×C4C22×D414yes—
455C22×C4C22×D414yes—
456C22×C4C22×D412yes—
457C22×C4C22×Q814yes—
458C22×C4C22×Q814yes—
459C22×C4C22×Q812yes—
460C22×C4C22×Q812yes—
461C22×C4C22×Q812yes—
462C22×C4C22×Q812yes—
463C22×C4C22×Q812yes—
464C22×C4C22×Q812yes—
465C22×C4C2×#4118yes—
466C22×C4C2×#4114yes—
467C22×C4H(5,𝔽2)12yes—
468C22×C4H(5,𝔽2)14yes—
469C22×C4H(5,𝔽2)12yes—
470C4⋊C4—216no—
471C4⋊C4—44no—
472C4⋊C4—416no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—84no—
478C4⋊C4—88no—
479C4⋊C4—88no—
480C4⋊C4—84no—
481C22⋊C4—48no—
482C22⋊C4—44no—
483C22⋊C4—48no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—416no—
488C22⋊C4—416no—
489C22⋊C4—88no—
490C22⋊C4—84no—
491C42—28no—
492C42—48no—
493C42—48no—
494C42—48no—
495C42—84no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—22no—
508C23—24no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—22no—
515C23—24no—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—24no—
520C23—42no—
521C23—42no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—42no—
533C23—42no—
534C23—41no—
535C23—42no—
536C23—41no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23C2×C4⋊Q814yes—
541C23C2×C4⋊Q814yes—
542C23C23×D411yescharacteristic
543C23C23×Q811yescharacteristic
544C23C23×Q811yescharacteristic
545C23C22×#4112yes—
546C23C2×H(5,𝔽2)11yescharacteristic
547C23C2×H(5,𝔽2)12yes—
548C23C2×H(5,𝔽2)11yescharacteristic
549C23#58511yescharacteristic
550C23#58511yescharacteristic
551C23#58512yes—
552C23#58512yes—
553Q8—832no—
554D4—416no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—416no—
583C2×C4—416no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—416no—
614C2×C4—416no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
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—44no—
636C2×C4—816no—
637C2×C4C2×C4⋊Q814yes—
638C2×C4#53818yes—
639C2×C4#54314yes—
640C2×C4#55418yes—
641C2×C4#55914yes—
642C2×C4H(3,𝔽4)14yes—
643C22—28no—
644C22—28no—
645C22—24no—
646C22—24no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—44no—
656C22—44no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22C22×C4⋊Q811yescharacteristic
666C22C2×#53812yes—
667C22C2×#54311yescharacteristic
668C22#345811yescharacteristic
669C22D4×#4112yes—
670C22C2×#55412yes—
671C22Q8×#4112yes—
672C22C2×#55911yescharacteristic
673C22#349612yes—
674C22#349611yescharacteristic
675C22#349611yescharacteristic
676C22#350112yes—
677C22#350111yescharacteristic
678C22#350111yescharacteristic
679C22#350412yes—
680C22#352112yes—
681C22#352412yes—
682C22C2×H(3,𝔽4)11yescharacteristic
683C22#354011yescharacteristic
684C22#354011yescharacteristic
685C22#354012yes—
686C22#354012yes—
687C22#355112yes—
688C4—24no—
689C4—42no—
690C4—42no—
691C4—42no—
692C4—44no—
693C4—44no—
694C4—48no—
695C4—44no—
696C4—44no—
697C4—84no—
698C4—84no—
699C4—84no—
700C4#280418yescomplemented
701C2—44no—
702C2—44no—
703C2#3500211yescharacteristic
704C2#3626112yes—
705C2#3742611yescharacteristic
706C2#3810612yes—
707C2#3940812yes—
708C2#3966612yes—
709C2#4547911yescharacteristic
710C2#4549411yescharacteristic
711C2C2×#280411yescharacteristic
712C2#5962811yescharacteristic
713C2#5962911yescharacteristic
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. #16987C2semidirect
3. #16823C2semidirect
4. #15691C2semidirect
5. #15568C2semidirect
6. #15146C2semidirect
7. #15125C2semidirect
8. #15125C2semidirect
9. #14697C2semidirect
11. #14130C2semidirect
12. #14053C2semidirect
88. C2×C4×D4Q8semidirect
128. #386Q8semidirect
361. C2×D4C4⋊Q8semidirect
700. C4#2804semidirect

Maximal subgroups

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

subgroupindex
C2×#28042
#169872
#168232
#156912
#155682
#151462
#151252
#146972
#145132
#141302
#140532
#139062
#138722
C4×#3862