Finite Group Databasebeta

← #874897

Subgroups — #874897

Order 512.

3418 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2734C211yesmaximal, characteristic, complemented
2C2×#2434C212yesmaximal
3C22×#391C211yesmaximal, characteristic
4#17078C212yesmaximal, complemented
5#16629C212yesmaximal, complemented
6#15852C214yesmaximal, complemented
7#15318C214yesmaximal, complemented
8#14821C212yesmaximal
9#14758C214yesmaximal
10#14697C212yesmaximal
11#14467C211yesmaximal, characteristic, complemented
12#14450C212yesmaximal, complemented
13#14047C214yesmaximal
14C24×Q8C2211yescharacteristic
15C22×C4⋊Q8C2212yes—
16C22×C4⋊Q8C2212yes—
17C22×C22⋊Q8C2212yes—
18C23×C4⋊C4C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2734—28no—
21#2434—216no—
22C2×#398C2212yes—
23C2×#398C2212yes—
24C2×#395C2214yes—
25C2×#395C2214yes—
26C2×#395C2214yes—
27C2×#391—28no—
28C2×#391C2211yescharacteristic
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2211yescharacteristic
32C2×#391C2214yes—
33C2×#391C2211yescharacteristic
34C2×#391C2211yescharacteristic
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#387C2214yes—
38C2×#385C2214yes—
39C2×#385C2214yes—
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×C4×C22⋊C4C2212yes—
44C22×#95C2214yes—
45#1446C2214yes—
46#1445C2214yes—
47#1444C2214yes—
48#1444C2214yes—
49#1444C2214yes—
50#1443C2214yes—
51#1442C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2212yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2212yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66#1434C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C22⋊#95C2212yes—
70C23×Q8—24no—
71C23×Q8—24no—
72C23×Q8—42no—
73C23×Q8—42no—
74C23×Q8—42no—
75C23×Q8—42no—
76C23×Q8D418yes—
77C23×Q8C2312yes—
78C23×Q8C2312yes—
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2312yes—
84C24×C4C2312yes—
85C2×C4⋊Q8—28no—
86C2×C4⋊Q8—28no—
87C2×C4⋊Q8—28no—
88C2×C4⋊Q8—28no—
89C2×C4⋊Q8—44no—
90C2×C4⋊Q8—44no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—44no—
93C2×C22⋊Q8—44no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—42no—
96C22×C4⋊C4—42no—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C42—24no—
115C22×C42—42no—
116C22×C42—42no—
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122#398—28no—
123#398—44no—
124#395—216no—
125#395—216no—
126#395—216no—
127#391—28no—
128#391—28no—
129#391—24no—
130#391—24no—
131#391—24no—
132#391—216no—
133#391D418yes—
134#389—28no—
135#389—44no—
136#387—216no—
137#385—216no—
138#385—216no—
139#380—44no—
140#380—44no—
141#380Q8116yescomplemented
142C4×C22⋊C4—28no—
143C2×#95—216no—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C25C2411yescharacteristic
168C22×Q8—232no—
169C22×Q8—24no—
170C22×Q8—22no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—22no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—416no—
179C22×Q8—41no—
180C22×Q8—41no—
181C22×Q8—42no—
182C22×Q8—41no—
183C22×Q8—41no—
184C22×Q8—41no—
185C22×Q8—41no—
186C22×Q8—42no—
187C22×Q8—41no—
188C22×Q8—41no—
189C22×Q8—42no—
190C22×Q8—42no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×Q8—416no—
194C22×Q8—416no—
195C22×Q8—416no—
196C22×Q8—416no—
197C22×Q8—416no—
198C22×Q8C2×D418yes—
199C22×Q8C2×D414yes—
200C22×Q8C2×D414yes—
201C22×Q8C2×D414yes—
202C22×Q8#4114yes—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—28no—
219C23×C4—24no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—44no—
226C23×C4C2×D414yes—
227C23×C4C2×D418yes—
228C23×C4C2×Q818yes—
229C23×C4C2414yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2414yes—
235C23×C4C2414yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
241C4⋊Q8—432no—
242C4⋊Q8—432no—
243C22⋊Q8—432no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4—41no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—416no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4C2×Q818yescomplemented
288C2×C4⋊C4C2×Q818yescomplemented
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4C2×Q8116yes—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—42no—
303C2×C42—42no—
304C2×C42—41no—
305C2×C42—41no—
306C2×C42—42no—
307C2×C42—41no—
308C2×C42—41no—
309C2×C42—416no—
310C2×C42C2×D414yes—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—24no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—42no—
331#95—44no—
332#95C2×D4116yes—
333#95C2×Q8116yescomplemented
334C24—21no—
335C24—21no—
336C24—21no—
337C24—21no—
338C24—22no—
339C24—22no—
340C24C22×D412yes—
341C24C22×Q814yes—
342C24H(5,𝔽2)12yes—
343C24H(5,𝔽2)12yes—
344C242−1+414yes—
345C24C2511yescharacteristic
346C2×Q8—216no—
347C2×Q8—28no—
348C2×Q8—216no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8—48no—
352C2×Q8—44no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×Q8—48no—
362C2×Q8—4128no—
363C2×Q8—44no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×Q8—48no—
368C2×Q8—48no—
369C2×Q8—48no—
370C2×Q8—84no—
371C2×Q8—84no—
372C2×Q8—84no—
373C2×Q8—84no—
374C2×Q8#121116yes—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—216no—
399C22×C4—216no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—216no—
429C22×C4—216no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
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—24no—
448C22×C4—24no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—416no—
461C22×C4—48no—
462C22×C4—84no—
463C22×C4—88no—
464C22×C4C22×D414yes—
465C22×C4C22×D414yes—
466C22×C4C22×D412yes—
467C22×C4C22×D414yes—
468C22×C4C22×D414yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D414yes—
472C22×C4C22×Q814yes—
473C22×C4C22×Q818yes—
474C22×C4C22×Q818yes—
475C22×C4C22×Q814yes—
476C22×C4C2×#4114yes—
477C22×C4C2×#4112yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C42−1+414yes—
481C4⋊C4—216no—
482C4⋊C4—44no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—44no—
486C4⋊C4—48no—
487C4⋊C4—416no—
488C4⋊C4—416no—
489C4⋊C4—416no—
490C4⋊C4—416no—
491C4⋊C4—416no—
492C4⋊C4—84no—
493C4⋊C4—84no—
494C4⋊C4—84no—
495C4⋊C4—84no—
496C22⋊C4—232no—
497C22⋊C4—416no—
498C42—44no—
499C42—48no—
500C42—44no—
501C42—48no—
502C42—48no—
503C23—22no—
504C23—22no—
505C23—24no—
506C23—24no—
507C23—22no—
508C23—22no—
509C23—24no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23#53414yes—
522C23#54418yes—
523C23D4×Q818yes—
524C23Q8214yes—
525C23#56314yes—
526C23C23×D411yescharacteristic
527C23C23×D411yescharacteristic
528C23C23×Q812yes—
529C23C22×#4111yescharacteristic
530C23C2×H(5,𝔽2)11yescharacteristic
531C23C2×H(5,𝔽2)11yescharacteristic
532C23C2×H(5,𝔽2)11yescharacteristic
533C23C2×H(5,𝔽2)11yescharacteristic
534C23C2×2−1+412yes—
535C23C2×2−1+412yes—
536C23#58512yes—
537Q8—416no—
538Q8—432no—
539Q8—416no—
540Q8—816no—
541C2×C4—232no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—432no—
581C2×C4—432no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—816no—
621C2×C4C2×#12118yes—
622C2×C4C2×#12114yes—
623C2×C4#53414yes—
624C2×C4D4×Q818yes—
625C22—24no—
626C22—24no—
627C22—28no—
628C22—28no—
629C22—24no—
630C22—24no—
631C22—24no—
632C22—24no—
633C22—28no—
634C22—28no—
635C22#273418yescomplemented
636C22C22×#12111yescharacteristic
637C22C2×#53411yescharacteristic
638C22C2×#53411yescharacteristic
639C22C2×#54412yes—
640C22C2×D4×Q812yes—
641C22C2×D4×Q812yes—
642C22C2×Q8211yescharacteristic
643C22Q8×#4112yes—
644C22#348411yescharacteristic
645C22#348411yescharacteristic
646C22#348912yes—
647C22#349011yescharacteristic
648C22#349011yescharacteristic
649C22#349712yes—
650C22#350112yes—
651C22#350312yes—
652C22#352312yes—
653C22C2×#56311yescharacteristic
654C22#353512yes—
655C22#353812yes—
656C22#354411yescharacteristic
657C22#354511yescharacteristic
658C22#356112yes—
659C4—28no—
660C4—24no—
661C4—24no—
662C4—28no—
663C4—44no—
664C4—44no—
665C4—44no—
666C4—48no—
667C4—48no—
668C4—82no—
669C4—84no—
670C4—84no—
671C2—28no—
672C2#3498111yescharacteristic
673C2Q8×#12112yes—
674C2#3719011yescharacteristic
675C2#3768512yes—
676C2#3844612yes—
677C2#3961512yes—
678C2#4613611yescharacteristic
679C2#4620511yescharacteristic
680C2C2×#273411yescharacteristic
681C2#5915111yescharacteristic
682C2#5992311yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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×#2734C2semidirect
4. #17078C2semidirect
5. #16629C2semidirect
6. #15852C2semidirect
7. #15318C2semidirect
11. #14467C2semidirect
12. #14450C2semidirect
141. #380Q8semidirect
287. C2×C4⋊C4C2×Q8semidirect
288. C2×C4⋊C4C2×Q8semidirect
333. #95C2×Q8semidirect
635. C22#2734semidirect

Maximal subgroups

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

subgroupindex
C2×#27342
C2×#24342
C22×#3912
#170782
#166292
#158522
#153182
#148212
#147582
#146972
#144672
#144502
#140472