Finite Group Databasebeta

← #822345

Subgroups — #822345

Order 512.

3198 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2415C212yesmaximal
2C2×#2429C212yesmaximal, complemented
3C2×#2362C212yesmaximal, complemented
4#16514C212yesmaximal
5#15780C212yesmaximal
6#15778C212yesmaximal, complemented
7#15774C214yesmaximal, complemented
8#15182C212yesmaximal, complemented
9#15180C211yesmaximal, characteristic, complemented
10#15081C214yesmaximal, complemented
11#14151C212yesmaximal, complemented
12#13951C212yesmaximal, complemented
13#13935C212yesmaximal, complemented
14#13910C211yesmaximal, characteristic, complemented
15C4×#394C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×#117C2212yes—
22#2429—216no—
23#2415—216no—
24C2×#398C2214yes—
25C2×#397C2214yes—
26C2×#396C2214yes—
27C2×#394C2211yescharacteristic
28C2×#394C2214yes—
29#2362—216no—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#389C2214yes—
34C2×#389C2214yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#388C2214yes—
39C2×#388C2214yes—
40C2×#388C2214yes—
41C2×#388C2214yes—
42C2×#387C2214yes—
43C2×#384C2212yes—
44C2×#384C2211yescharacteristic
45C2×#380C2212yes—
46C2×C4×C4⋊C4C2214yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2212yes—
51#1444C2212yes—
52#1444C2212yes—
53#1442C2211yescharacteristic
54#1442C2212yes—
55#1442C2212yes—
56#1441C2211yescharacteristic
57#1441C2212yes—
58#1441C2214yes—
59#1441C2212yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1440C2212yes—
64#1439C2212yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1438C2211yescharacteristic
70C4⋊#95C2214yes—
71C4⋊#95C2214yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75#1434C2214yes—
76#1434C2214yes—
77C4×#95C2211yescharacteristic
78C23×Q8C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2312yes—
81C2×C4⋊Q8—28no—
82C2×C4⋊Q8—42no—
83C2×C22⋊Q8—28no—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8—28no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—44no—
88C2×C22⋊Q8D4116yes—
89C2×C4×Q8—28no—
90C2×C4×Q8—28no—
91C2×C4×Q8—28no—
92C2×C4×D4—28no—
93C2×C4×D4—42no—
94C2×#117—28no—
95C2×#117—28no—
96C2×#117—28no—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C42C2314yes—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2314yes—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129#398—216no—
130#397—216no—
131#396—216no—
132#394—24no—
133#394—216no—
134#391—28no—
135#391—28no—
136#391—28no—
137#389—28no—
138#389—216no—
139#389—28no—
140#389—216no—
141#389—28no—
142#388—216no—
143#388—216no—
144#388—216no—
145#388—216no—
146#387—216no—
147#384—28no—
148#384—42no—
149#380—28no—
150C4×C4⋊C4—216no—
151C4×C4⋊C4—42no—
152C4×C4⋊C4D418yescomplemented
153C4×C22⋊C4—28no—
154C4×C22⋊C4—28no—
155C2×#95C2311yescharacteristic
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2311yescharacteristic
177C2×#95C2311yescharacteristic
178C2×#95C2312yes—
179C25C2412yes—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8C2×D418yes—
187C22×D4—24no—
188C22×D4—22no—
189C22×D4—41no—
190C22×D4—41no—
191C22×D4—42no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4C2×D418yes—
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2414yes—
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C4⋊Q8—416no—
214C22⋊Q8—264no—
215C22⋊Q8—432no—
216C4×Q8—432no—
217C4×D4—416no—
218#117—432no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4C2×D4116yes—
259C2×C4⋊C4C2×D418yes—
260C2×C4⋊C4C2×D418yes—
261C2×C4⋊C4#4114yes—
262C2×C4⋊C4#4118yes—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4C2×D4116yes—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—28no—
295C2×C42—28no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—28no—
300C2×C42—28no—
301C2×C42—24no—
302C2×C42—28no—
303C2×C42—28no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42C2×D414yes—
308C2×C42C2×D418yes—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—22no—
324#95—24no—
325#95—24no—
326#95—41no—
327#95—44no—
328#95—42no—
329#95—44no—
330#95—41no—
331#95—41no—
332#95C2×D418yes—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24C22×D414yes—
348C24C2511yescharacteristic
349C2×Q8—216no—
350C2×Q8—216no—
351C2×Q8—216no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—48no—
356C2×D4—48no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—48no—
360C2×D4—44no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—216no—
404C22×C4—216no—
405C22×C4—216no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4C22×D412yes—
437C22×C4C22×D418yes—
438C22×C4C22×D414yes—
439C22×C4C22×D418yes—
440C22×C4C22×D418yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D414yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4118yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4114yes—
449C22×C4H(5,𝔽2)12yes—
450C22×C42−1+412yes—
451C22×C42−1+414yes—
452C22×C42−1+412yes—
453C22×C42−1+412yes—
454C4⋊C4—232no—
455C4⋊C4—44no—
456C4⋊C4—416no—
457C4⋊C4—416no—
458C4⋊C4—416no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—416no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—416no—
466C4⋊C4—48no—
467C4⋊C4—84no—
468C4⋊C4—84no—
469C4⋊C4#121132yes—
470C22⋊C4—216no—
471C22⋊C4—232no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—416no—
476C22⋊C4—416no—
477C22⋊C4—416no—
478C22⋊C4—48no—
479C42—28no—
480C42—216no—
481C42—416no—
482C42—416no—
483C42—48no—
484C42—44no—
485C42—44no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23C23×D411yescharacteristic
522C23C23×D412yes—
523C23C22×#4112yes—
524C23C22×#4111yescharacteristic
525C23C2×H(5,𝔽2)11yescharacteristic
526C23C2×2−1+411yescharacteristic
527C23C2×2−1+411yescharacteristic
528C23C2×2−1+412yes—
529C23C2×2−1+411yescharacteristic
530C23#58512yes—
531C23#58511yescharacteristic
532Q8—432no—
533D4—416no—
534C2×C4—22no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—22no—
540C2×C4—28no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—28no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—432no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—44no—
613C2×C4—41no—
614C2×C4C2×#121116yes—
615C2×C4C2×#12118yes—
616C2×C4#53714yes—
617C2×C4D4214yes—
618C2×C4#54714yes—
619C2×C4#54718yes—
620C2×C4#55514yes—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22C22×#12112yes—
637C22C2×#53711yescharacteristic
638C22C2×D4211yescharacteristic
639C22C2×#54711yescharacteristic
640C22C2×#54712yes—
641C22D4×#4112yes—
642C22C2×#55511yescharacteristic
643C22#348212yes—
644C22#348612yes—
645C22#348712yes—
646C22#348912yes—
647C22#349711yescharacteristic
648C22#349912yes—
649C22#350511yescharacteristic
650C22#350911yescharacteristic
651C22#351512yes—
652C22#351612yes—
653C22#351812yes—
654C22#353811yescharacteristic
655C22#354011yescharacteristic
656C22#355011yescharacteristic
657C22#355512yes—
658C22#356311yescharacteristic
659C4—24no—
660C4—216no—
661C4—42no—
662C4—44no—
663C4—48no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—44no—
668C4—81no—
669C4—84no—
670C4—82no—
671C4—84no—
672C4#253718yes—
673C2—84no—
674C2C2×#253711yescharacteristic
675C2#3577112yes—
676C2#3625912yes—
677C2#3948711yescharacteristic
678C2#4292911yescharacteristic
679C2#4523812yes—
680C2#4532111yescharacteristic
681C2#4541011yescharacteristic
682C2#5900111yescharacteristic
683C2#5911412yes—
684C2#5995111yescharacteristic
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×#2429C2semidirect
3. C2×#2362C2semidirect
6. #15778C2semidirect
7. #15774C2semidirect
8. #15182C2semidirect
9. #15180C2semidirect
10. #15081C2semidirect
11. #14151C2semidirect
12. #13951C2semidirect
13. #13935C2semidirect
14. #13910C2semidirect
152. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24152
C2×#24292
C2×#23622
#165142
#157802
#157782
#157742
#151822
#151802
#150812
#141512
#139512
#139352
#139102
C4×#3942