Finite Group Databasebeta

← #923043

Subgroups — #923043

Order 512.

3316 conjugacy classes of subgroups.

The subgroup lattice has 745 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2429C212yesmaximal, complemented
2C2×#2616C212yesmaximal
3#17132C212yesmaximal, complemented
4#17109C212yesmaximal, complemented
5#16549C214yesmaximal, complemented
6#16541C214yesmaximal, complemented
7#15782C212yesmaximal, complemented
8#15779C212yesmaximal, complemented
9#15182C212yesmaximal, complemented
10#15178C211yesmaximal, characteristic, complemented
11#15135C211yesmaximal, characteristic, complemented
12#15132C212yesmaximal, complemented
13#13969C212yesmaximal, complemented
14#13969C212yesmaximal, complemented
15#13851C211yesmaximal, characteristic
16C23×#41C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C22⋊Q8C2212yes—
20C22×C22⋊Q8C2212yes—
21C22×C4×Q8C2212yes—
22#2616—216no—
23#2429—216no—
24C2×#399C2212yes—
25C2×#398C2214yes—
26C2×#397C2214yes—
27C2×#396C2214yes—
28C2×#395C2214yes—
29C2×#394C2214yes—
30C2×#394C2214yes—
31C2×#394C2214yes—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#389C2212yes—
35C2×#389C2214yes—
36C2×#388C2214yes—
37C2×#384C2211yescharacteristic
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×C4×C22⋊C4C2212yes—
42C2×C4×C22⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2211yescharacteristic
45#1446C2211yescharacteristic
46#1446C2211yescharacteristic
47#1445C2212yes—
48#1444C2212yes—
49#1444C2212yes—
50#1444C2212yes—
51#1443C2214yes—
52#1442C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1440C2214yes—
58#1439C2212yes—
59#1439C2212yes—
60#1439C2212yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1438C2212yes—
66#1438C2212yes—
67#1438C2211yescharacteristic
68#1438C2211yescharacteristic
69#1437C2214yes—
70#1437C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73C4⋊#95C2211yescharacteristic
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2211yescharacteristic
77C22×#41—28no—
78C22×#41—24no—
79C22×#41—44no—
80C22×#41—42no—
81C22×#41—42no—
82C23×Q8C2312yes—
83C23×Q8C2311yescharacteristic
84C23×D4C2312yes—
85C23×D4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C2×C4⋊Q8—28no—
89C2×C4⋊Q8—42no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—44no—
95C2×C22⋊Q8—44no—
96C2×C22⋊Q8—44no—
97C2×C22⋊Q8—44no—
98C2×C22⋊Q8D4116yes—
99C2×C4×Q8—28no—
100C2×C4×Q8—44no—
101C2×C4×Q8—44no—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2314yes—
131#399—44no—
132#398—216no—
133#397—48no—
134#396—216no—
135#395—216no—
136#394—216no—
137#394—216no—
138#394—216no—
139#391—28no—
140#391—44no—
141#389—216no—
142#389—28no—
143#388—216no—
144#384—42no—
145#380—28no—
146#380—28no—
147#380—28no—
148C4×C22⋊C4—28no—
149C4×C22⋊C4—42no—
150C4×C22⋊C4—44no—
151C4×C22⋊C4—42no—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2311yescharacteristic
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2311yescharacteristic
173C2×#95C2312yes—
174C2×#95C2311yescharacteristic
175C2×#95C2311yescharacteristic
176C25C2412yes—
177C25C2411yescharacteristic
178C2×#41—416no—
179C2×#41—432no—
180C2×#41—432no—
181C2×#41—416no—
182C2×#41—416no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—41no—
192C22×Q8—42no—
193C22×Q8—41no—
194C22×Q8C2×D418yes—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—22no—
200C22×D4—42no—
201C22×D4—41no—
202C22×D4—41no—
203C22×D4—42no—
204C22×D4—42no—
205C22×D4—42no—
206C22×D4—42no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4C2×D418yes—
218C23×C4#4114yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C4⋊Q8—416no—
233C22⋊Q8—432no—
234C22⋊Q8—432no—
235C22⋊Q8—432no—
236C4×Q8—432no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4C2×D418yes—
269C2×C4⋊C4C2×D4116yes—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—41no—
299C2×C22⋊C4—41no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4C2×D4116yes—
309C2×C42—24no—
310C2×C42—28no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—44no—
318C2×C42—44no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—24no—
331#95—44no—
332#95—42no—
333#95—42no—
334#95—41no—
335#95—41no—
336#95—41no—
337#95—41no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—42no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—21no—
348C24—22no—
349C24—22no—
350C24—21no—
351C24—21no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—21no—
362C24—21no—
363C24—21no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24C22×D414yes—
368C24C2×#4112yes—
369C24C2511yescharacteristic
370#41—4128no—
371C2×Q8—216no—
372C2×Q8—216no—
373C2×Q8—216no—
374C2×Q8—44no—
375C2×Q8—48no—
376C2×Q8—48no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×D4—48no—
384C2×D4—48no—
385C2×D4—48no—
386C2×D4—48no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—44no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—216no—
440C22×C4—28no—
441C22×C4—216no—
442C22×C4—216no—
443C22×C4—216no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—44no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—44no—
472C22×C4C22×D414yes—
473C22×C4C22×D418yes—
474C22×C4C22×D418yes—
475C22×C4C22×D414yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C4H(5,𝔽2)14yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C42−1+412yes—
483C22×C42−1+414yes—
484C22×C42−1+412yes—
485C22×C42−1+412yes—
486C4⋊C4—216no—
487C4⋊C4—416no—
488C4⋊C4—416no—
489C4⋊C4—416no—
490C4⋊C4—416no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—416no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C22⋊C4—216no—
499C22⋊C4—416no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—416no—
505C22⋊C4—416no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—88no—
509C22⋊C4—88no—
510C22⋊C4—88no—
511C42—416no—
512C42—48no—
513C42—44no—
514C42—44no—
515C23—24no—
516C23—22no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—22no—
526C23—22no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—41no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—41no—
546C23—41no—
547C23—42no—
548C23—42no—
549C23—41no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—41no—
567C23—42no—
568C23—42no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—42no—
573C23C23×D412yes—
574C23C22×#4111yescharacteristic
575C23C2×H(5,𝔽2)11yescharacteristic
576C23C2×H(5,𝔽2)12yes—
577C23C2×H(5,𝔽2)11yescharacteristic
578C23C2×H(5,𝔽2)11yescharacteristic
579C23C2×2−1+412yes—
580C23C2×2−1+411yescharacteristic
581C23C2×2−1+411yescharacteristic
582C23C2×2−1+411yescharacteristic
583C23#58512yes—
584Q8—416no—
585Q8—432no—
586D4—432no—
587D4—416no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—28no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—432no—
642C2×C4—416no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4#53418yes—
665C2×C4#53814yes—
666C2×C4D4214yes—
667C2×C4#54618yes—
668C2×C4#56014yes—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—42no—
675C22—42no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—42no—
681C22—81no—
682C22—82no—
683C22—81no—
684C22—82no—
685C22—81no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—81no—
694C22—82no—
695C22C2×#53412yes—
696C22C2×#53811yescharacteristic
697C22#346112yes—
698C22C2×D4211yescharacteristic
699C22C2×#54612yes—
700C22#348612yes—
701C22#348812yes—
702C22#349712yes—
703C22#349912yes—
704C22#350911yescharacteristic
705C22#351111yescharacteristic
706C22C2×#56011yescharacteristic
707C22#353611yescharacteristic
708C22#353811yescharacteristic
709C22#353812yes—
710C22#354112yes—
711C22#354112yes—
712C22#354712yes—
713C22#354812yes—
714C22#354811yescharacteristic
715C22#355011yescharacteristic
716C22#355011yescharacteristic
717C22#355111yescharacteristic
718C4—42no—
719C4—42no—
720C4—42no—
721C4—44no—
722C4—44no—
723C4—42no—
724C4—48no—
725C4—48no—
726C4—42no—
727C4—84no—
728C4—82no—
729C4—84no—
730C4#266818yes—
731C2—84no—
732C2—82no—
733C2#3732411yescharacteristic
734C2#3751411yescharacteristic
735C2#3778112yes—
736C2#3943512yes—
737C2#4278612yes—
738C2#4291612yes—
739C2#4524411yescharacteristic
740C2#4532111yescharacteristic
741C2C2×#266811yescharacteristic
742C2#5300611yescharacteristic
743C2#5300711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2429C2semidirect
3. #17132C2semidirect
4. #17109C2semidirect
5. #16549C2semidirect
6. #16541C2semidirect
7. #15782C2semidirect
8. #15779C2semidirect
9. #15182C2semidirect
10. #15178C2semidirect
11. #15135C2semidirect
12. #15132C2semidirect
13. #13969C2semidirect
14. #13969C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24292
C2×#26162
#171322
#171092
#165492
#165412
#157822
#157792
#151822
#151782
#151352
#151322
#139692
#138512