Finite Group Databasebeta

← #4462725

Subgroups — #4462725

Order 512.

2732 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2619C211yesmaximal, characteristic
2C2×#2396C211yesmaximal, characteristic, complemented
3#17141C212yesmaximal, complemented
4#17105C212yesmaximal, complemented
5#17101C214yesmaximal, complemented
6#17092C214yesmaximal, complemented
7#17088C212yesmaximal, complemented
8#17084C212yesmaximal, complemented
9#15234C212yesmaximal, complemented
10#15157C212yesmaximal, complemented
11#15148C212yesmaximal, complemented
12#15127C212yesmaximal, complemented
13#15126C212yesmaximal, complemented
14#15126C212yesmaximal, complemented
15#14343C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#121C2212yes—
18C22×C4×Q8C2211yescharacteristic
19C22×#117C2211yescharacteristic
20#2619—28no—
21#2396—28no—
22C2×#400C2211yescharacteristic
23C2×#399C2211yescharacteristic
24C2×#399C2214yes—
25C2×#398C2212yes—
26C2×#396C2212yes—
27C2×#395C2211yescharacteristic
28C2×#394C2212yes—
29C2×#394C2212yes—
30C2×#394C2214yes—
31C2×#394C2214yes—
32C2×#394C2214yes—
33C2×#394C2214yes—
34C2×#393C2211yescharacteristic
35C2×#393C2214yes—
36C2×#392C2212yes—
37C2×#392C2212yes—
38C2×#390C2212yes—
39C2×#390C2212yes—
40C2×#390C2212yes—
41C2×#390C2212yes—
42C2×#390C2212yes—
43C2×#389C2212yes—
44C2×#389C2212yes—
45C2×#382C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51#1445C2212yes—
52#1445C2212yes—
53#1442C2214yes—
54#1442C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1440C2212yes—
58#1440C2212yes—
59#1440C2214yes—
60#1440C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1438C2214yes—
70#1438C2214yes—
71#1438C2214yes—
72#1438C2214yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C23×Q8C2311yescharacteristic
78C23×D4C2312yes—
79C23×D4C2314yes—
80C23×D4C2312yes—
81C23×D4C2312yes—
82C24×C4C2311yescharacteristic
83C2×#127—24no—
84C2×#127—42no—
85C2×#127—42no—
86C2×#121—28no—
87C2×#121—44no—
88C2×#121—44no—
89C2×C4×Q8—24no—
90C2×C4×Q8—42no—
91C2×C4×Q8—42no—
92C2×#117—42no—
93C2×#117—42no—
94C2×#117D418yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#400—24no—
129#399—24no—
130#399—48no—
131#398—28no—
132#396—28no—
133#395—24no—
134#394—28no—
135#394—28no—
136#394—48no—
137#394—48no—
138#394—48no—
139#394—48no—
140#393—24no—
141#393—48no—
142#392—28no—
143#392—28no—
144#390—28no—
145#390—44no—
146#390—44no—
147#390—44no—
148#390—44no—
149#389—44no—
150#389—44no—
151#382—24no—
152#380—42no—
153#380—42no—
154C4×C4⋊C4—28no—
155C4×C4⋊C4—24no—
156C4×C22⋊C4—28no—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C25C2414yes—
178C25C2411yescharacteristic
179C22×Q8—22no—
180C22×Q8—24no—
181C22×Q8—41no—
182C22×Q8—41no—
183C22×Q8—42no—
184C22×D4—28no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—42no—
197C22×D4—42no—
198C22×D4—42no—
199C22×D4—42no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4—44no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4C2×D414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226#127—416no—
227#121—432no—
228C4×Q8—416no—
229#117—416no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4C2×D414yes—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—41no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
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—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
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⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4C2×D414yes—
314C2×C22⋊C4C2×D414yes—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42C2×D414yes—
326C2×C42C2×D414yes—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—22no—
331#95—22no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—28no—
336#95—44no—
337#95—42no—
338#95—42no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347C24—22no—
348C24—22no—
349C24—21no—
350C24—21no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—24no—
361C24—24no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—24no—
366C24—24no—
367C24—24no—
368C24—24no—
369C24—24no—
370C24C22×D412yes—
371C24C2511yescharacteristic
372C2×Q8—48no—
373C2×Q8—44no—
374C2×Q8—44no—
375C2×Q8—48no—
376C2×Q8—44no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—416no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—48no—
385C2×D4—48no—
386C2×D4—48no—
387C2×D4—88no—
388C2×D4—88no—
389C2×D4—88no—
390C2×D4—88no—
391C2×D4—84no—
392C2×D4—84no—
393C2×D4—88no—
394C2×D4—88no—
395C2×D4—84no—
396C2×D4—84no—
397C2×D4—88no—
398C2×D4—88no—
399C2×D4—88no—
400C2×D4—88no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—28no—
434C22×C4—28no—
435C22×C4—28no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4H(5,𝔽2)14yes—
480C22×C4H(5,𝔽2)14yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C4H(5,𝔽2)12yes—
483C22×C4H(5,𝔽2)14yes—
484C22×C42−1+414yes—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—88no—
496C4⋊C4—88no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—416no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C22⋊C4—88no—
506C22⋊C4—88no—
507C22⋊C4—88no—
508C22⋊C4—84no—
509C22⋊C4—88no—
510C22⋊C4—88no—
511C22⋊C4—84no—
512C22⋊C4—88no—
513C22⋊C4—88no—
514C42—28no—
515C42—28no—
516C42—48no—
517C42—48no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—24no—
522C23—24no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
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—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23C23×D411yescharacteristic
575C23C2×H(5,𝔽2)11yescharacteristic
576C23C2×H(5,𝔽2)11yescharacteristic
577C23C2×H(5,𝔽2)12yes—
578C23C2×H(5,𝔽2)12yes—
579C23C2×H(5,𝔽2)12yes—
580C23C2×2−1+412yes—
581C23#58512yes—
582C23#58512yes—
583Q8—416no—
584D4—816no—
585D4—816no—
586D4—832no—
587D4—816no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—416no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4#53414yes—
664C2×C4#53514yes—
665C2×C4#53614yes—
666C2×C4#56218yes—
667C2×C4#56218yes—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—44no—
672C22—44no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—84no—
678C22—84no—
679C22—84no—
680C22—84no—
681C22—84no—
682C22—84no—
683C22—84no—
684C22—84no—
685C22—84no—
686C22—84no—
687C22—84no—
688C22—84no—
689C22—84no—
690C22—84no—
691C22—84no—
692C22C2×#53411yescharacteristic
693C22C2×#53511yescharacteristic
694C22C2×#53611yescharacteristic
695C22#348812yes—
696C22#348812yes—
697C22C2×#56212yes—
698C22C2×#56212yes—
699C22#353711yescharacteristic
700C22#354011yescharacteristic
701C22#354212yes—
702C22#354312yes—
703C22#354612yes—
704C22#354612yes—
705C22#354812yes—
706C22#354911yescharacteristic
707C22#355111yescharacteristic
708C22#355412yes—
709C22#355412yes—
710C22#355712yes—
711C22#355812yes—
712C22#356212yes—
713C4—42no—
714C4—42no—
715C4—42no—
716C4—44no—
717C4—42no—
718C4—42no—
719C4—44no—
720C4—84no—
721C4—82no—
722C4—82no—
723C4—84no—
724C4—84no—
725C4#281618yes—
726C2—82no—
727C2—164no—
728C2C2×#281611yescharacteristic
729C2#3726911yescharacteristic
730C2#3727311yescharacteristic
731C2#3727412yes—
732C2#3727812yes—
733C2#5297312yes—
734C2#5298412yes—
735C2#5310512yes—
736C2#5310612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2619C2
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×#2396C2semidirect
3. #17141C2semidirect
4. #17105C2semidirect
5. #17101C2semidirect
6. #17092C2semidirect
7. #17088C2semidirect
8. #17084C2semidirect
9. #15234C2semidirect
10. #15157C2semidirect
11. #15148C2semidirect
12. #15127C2semidirect
13. #15126C2semidirect
14. #15126C2semidirect
15. #14343C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26192
C2×#23962
#171412
#171052
#171012
#170922
#170882
#170842
#152342
#151572
#151482
#151272
#151262
#143432