Finite Group Databasebeta

← #627731

Subgroups — #627731

Order 512.

3330 conjugacy classes of subgroups.

The subgroup lattice has 746 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2856C211yesmaximal, characteristic
2C2×#2851C211yesmaximal, characteristic, complemented
3C2×#2641C211yesmaximal, characteristic, complemented
4C2×#2330C211yesmaximal, characteristic, complemented
5#15902C214yesmaximal, complemented
6#15458C214yesmaximal, complemented
7#15338C214yesmaximal, complemented
8#15316C214yesmaximal, complemented
9#15141C212yesmaximal, complemented
10#15126C212yesmaximal, complemented
11#14276C211yesmaximal, characteristic, complemented
12#14272C211yesmaximal, characteristic, complemented
13#14141C212yesmaximal, complemented
14#13998C212yesmaximal, complemented
15#13753C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×#121C2212yes—
19C22×#121C2214yes—
20C23×C42C2211yescharacteristic
21#2856—28no—
22#2851—28no—
23#2641—28no—
24C2×#396C2214yes—
25C2×#394C2214yes—
26C2×#394C2214yes—
27C2×#394C2214yes—
28#2330—28no—
29C2×#390C2214yes—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#390C2214yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#389C2214yes—
39C2×#389C2212yes—
40C2×#388C2214yes—
41C2×#387C2214yes—
42C2×#387C2214yes—
43C2×#385C2214yes—
44C2×#384C2211yescharacteristic
45C2×#382C2211yescharacteristic
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×#376C2211yescharacteristic
51C2×#376C2211yescharacteristic
52#1441C2214yes—
53#1441C2214yes—
54#1440C2214yes—
55#1440C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1438C2214yes—
63#1438C2214yes—
64#1438C2214yes—
65#1438C2214yes—
66#1437C2214yes—
67#1437C2214yes—
68C4⋊#95C2212yes—
69#1434C2212yes—
70#1434C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C4×#95C2211yescharacteristic
74C4×#95C2211yescharacteristic
75C23×D4C2312yes—
76C23×D4C2314yes—
77C23×D4C2314yes—
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C2×#127—24no—
81C2×#127—24no—
82C2×#127—24no—
83C2×#125—24no—
84C2×#125—24no—
85C2×#125D418yescomplemented
86C2×#121—28no—
87C2×#121—28no—
88C2×#121—216no—
89C2×#121—48no—
90C2×#121—44no—
91C2×#121—48no—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C22⋊C4C2311yescharacteristic
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C42—24no—
116C22×C42—24no—
117C22×C42—24no—
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2314yes—
125C22×C42C2311yescharacteristic
126#396—48no—
127#394—216no—
128#394—216no—
129#394—48no—
130#390—28no—
131#390—216no—
132#390—28no—
133#390—28no—
134#390—28no—
135#390—44no—
136#390—48no—
137#389—216no—
138#389—28no—
139#389—28no—
140#389—44no—
141#388—48no—
142#387—216no—
143#387—216no—
144#385—216no—
145#384D418yes—
146#382—24no—
147#380—28no—
148C4×C4⋊C4—28no—
149C4×C22⋊C4—24no—
150C4×C22⋊C4—24no—
151#376—24no—
152#376—24no—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2311yescharacteristic
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C25C2414yes—
173C25C2411yescharacteristic
174C22×D4—28no—
175C22×D4—28no—
176C22×D4—28no—
177C22×D4—28no—
178C22×D4—28no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—28no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—28no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—44no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4#4114yes—
209C23×C4C2411yescharacteristic
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222#127—416no—
223#125—232no—
224#121—432no—
225#121—464no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4C2×D418yes—
247C2×C4⋊C4C2×D418yes—
248C2×C4⋊C4C2×D4116yes—
249C2×C4⋊C4C2×D4116yes—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
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—44no—
292C2×C22⋊C4—44no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—28no—
311C2×C42—416no—
312C2×C42—44no—
313C2×C42—44no—
314C2×C42C2×D414yes—
315C2×C42C2×D414yes—
316C2×C42#4114yes—
317C2×C42#4114yes—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—24no—
328#95—22no—
329#95—22no—
330#95—24no—
331#95—24no—
332#95—42no—
333#95—42no—
334#95—44no—
335#95—44no—
336#95—44no—
337C24—21no—
338C24—21no—
339C24—21no—
340C24—21no—
341C24—21no—
342C24—22no—
343C24—21no—
344C24—22no—
345C24—22no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—24no—
361C24C2×#4112yes—
362C24H(5,𝔽2)14yes—
363C24C2511yescharacteristic
364C2×D4—416no—
365C2×D4—416no—
366C2×D4—416no—
367C2×D4—416no—
368C2×D4—416no—
369C2×D4—416no—
370C2×D4—416no—
371C2×D4—48no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—88no—
379C2×D4—88no—
380C2×D4—88no—
381C2×D4—88no—
382C2×D4—88no—
383C2×D4—88no—
384C2×D4—88no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—216no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4C22×D418yes—
468C22×C4C22×D418yes—
469C22×C4C22×D414yes—
470C22×C4C22×D414yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4114yes—
475C22×C4C2×#4114yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4H(5,𝔽2)14yes—
480C22×C4H(5,𝔽2)12yes—
481C22×C42−1+412yes—
482C4⋊C4—232no—
483C4⋊C4—216no—
484C4⋊C4—216no—
485C4⋊C4—416no—
486C4⋊C4—416no—
487C4⋊C4—48no—
488C4⋊C4—88no—
489C4⋊C4—88no—
490C22⋊C4—416no—
491C22⋊C4—416no—
492C22⋊C4—416no—
493C22⋊C4—416no—
494C22⋊C4—48no—
495C22⋊C4—416no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—88no—
499C22⋊C4—88no—
500C22⋊C4—88no—
501C22⋊C4—88no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C42—28no—
506C42—48no—
507C42—44no—
508C42—44no—
509C42—44no—
510C42—44no—
511C42—88no—
512C42#121116yes—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—41no—
526C23—41no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
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#54014yes—
571C23C23×D411yescharacteristic
572C23C23×D411yescharacteristic
573C23C22×#4111yescharacteristic
574C23C22×#4111yescharacteristic
575C23C22×#4111yescharacteristic
576C23C2×H(5,𝔽2)12yes—
577C23C2×H(5,𝔽2)11yescharacteristic
578C23C2×H(5,𝔽2)12yes—
579C23C2×2−1+411yescharacteristic
580C23#58512yes—
581C23#58512yes—
582D4—832no—
583D4—832no—
584D4—816no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—28no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—432no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—416no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4C2×#12118yes—
663C2×C4C2×#12114yes—
664C2×C4#52914yes—
665C2×C4#54214yes—
666C2×C4#54614yes—
667C2×C4#54814yes—
668C2×C4#54814yes—
669C2×C4#55014yes—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—42no—
674C22—42no—
675C22—44no—
676C22—44no—
677C22—42no—
678C22—44no—
679C22—42no—
680C22—44no—
681C22—84no—
682C22—84no—
683C22—84no—
684C22—84no—
685C22—84no—
686C22—84no—
687C22—84no—
688C22—84no—
689C22—84no—
690C22—84no—
691C22—84no—
692C22—84no—
693C22—84no—
694C22—84no—
695C22—84no—
696C22C22×#12111yescharacteristic
697C22C2×#52911yescharacteristic
698C22C2×#54011yescharacteristic
699C22C2×#54211yescharacteristic
700C22#345511yescharacteristic
701C22#346012yes—
702C22C2×#54611yescharacteristic
703C22C2×#54811yescharacteristic
704C22C2×#54811yescharacteristic
705C22C2×#55011yescharacteristic
706C22#348412yes—
707C22#348712yes—
708C22#348812yes—
709C22#348812yes—
710C22#349412yes—
711C22#349412yes—
712C22#349412yes—
713C22#350811yescharacteristic
714C22#351211yescharacteristic
715C22#354312yes—
716C22#354712yes—
717C22#355912yes—
718C22#356212yes—
719C4—28no—
720C4—48no—
721C4—44no—
722C4—42no—
723C4—48no—
724C4—44no—
725C4—81no—
726C4—81no—
727C4—81no—
728C4—81no—
729C4—84no—
730C4—82no—
731C4#242618yes—
732C2—44no—
733C2—164no—
734C2C2×#242611yescharacteristic
735C2#3527111yescharacteristic
736C2#3529511yescharacteristic
737C2#3556312yes—
738C2#3729212yes—
739C2#3769411yescharacteristic
740C2#3770111yescharacteristic
741C2#3770611yescharacteristic
742C2#3771011yescharacteristic
743C2#3775612yes—
744C2#3777512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2856C2
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×#2851C2semidirect
3. C2×#2641C2semidirect
4. C2×#2330C2semidirect
5. #15902C2semidirect
6. #15458C2semidirect
7. #15338C2semidirect
8. #15316C2semidirect
9. #15141C2semidirect
10. #15126C2semidirect
11. #14276C2semidirect
12. #14272C2semidirect
13. #14141C2semidirect
14. #13998C2semidirect
15. #13753C2semidirect
85. C2×#125D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28562
C2×#28512
C2×#26412
C2×#23302
#159022
#154582
#153382
#153162
#151412
#151262
#142762
#142722
#141412
#139982
#137532