Finite Group Databasebeta

← #642476

Subgroups — #642476

Order 512.

3480 conjugacy classes of subgroups.

The subgroup lattice has 747 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2641C211yesmaximal, characteristic, complemented
2C2×#2629C211yesmaximal, characteristic, complemented
3C2×#2619C211yesmaximal, characteristic
4C2×C4×#127C211yesmaximal, characteristic, complemented
5#16596C214yesmaximal, complemented
6#16552C214yesmaximal, complemented
7#15138C212yesmaximal, complemented
8#15126C212yesmaximal, complemented
9#14824C214yesmaximal, complemented
10#14732C214yesmaximal, complemented
11#14709C211yesmaximal, characteristic, complemented
12#14276C211yesmaximal, characteristic, complemented
13#13970C212yesmaximal, complemented
14#13920C212yesmaximal, complemented
15#13761C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×#121C2212yes—
19C22×C4×D4C2212yes—
20C22×C4×D4C2214yes—
21C22×#117C2211yescharacteristic
22#2641—28no—
23#2629—28no—
24#2619—28no—
25C2×#398C2214yes—
26C2×#397C2214yes—
27C2×#396C2214yes—
28C2×#394C2214yes—
29C2×#394C2214yes—
30C2×#394C2214yes—
31C2×#394C2214yes—
32C2×#392C2214yes—
33C4×#127—28no—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#390C2212yes—
38C2×#390C2214yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#386C2214yes—
42C2×#384C2211yescharacteristic
43C2×#382C2211yescharacteristic
44C2×#380C2212yes—
45C2×#380C2214yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×#376C2211yescharacteristic
49C2×C43C2211yescharacteristic
50#1441C2214yes—
51#1441C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1440C2214yes—
55#1440C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1438C2214yes—
61#1438C2214yes—
62#1437C2212yes—
63#1437C2212yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2212yes—
67#1436C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C22⋊#95C2214yes—
72C22⋊#95C2214yes—
73C4×#95C2211yescharacteristic
74C4×#95C2211yescharacteristic
75C23×D4C2312yes—
76C23×D4C2314yes—
77C23×D4C2314yes—
78C24×C4C2311yescharacteristic
79C24×C4C2314yes—
80C2×#127—24no—
81C2×#127—24no—
82C2×#127D418yes—
83C2×#125—24no—
84C2×#125—24no—
85C2×#125—24no—
86C2×#121—28no—
87C2×#121—28no—
88C2×#121—44no—
89C2×C4×D4—28no—
90C2×C4×D4—216no—
91C2×C4×D4—28no—
92C2×C4×D4—48no—
93C2×C4×D4—44no—
94C2×C4×D4—48no—
95C2×#117—24no—
96C2×#117—24no—
97C2×#117—24no—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129#398—216no—
130#397—48no—
131#396—216no—
132#394—216no—
133#394—216no—
134#394—216no—
135#394—48no—
136#392—216no—
137#390—28no—
138#390—28no—
139#390—28no—
140#390—28no—
141#390—216no—
142#389—28no—
143#389—44no—
144#386—48no—
145#384D418yescomplemented
146#382—24no—
147#380—28no—
148#380—48no—
149C4×C22⋊C4—24no—
150C4×C22⋊C4—24no—
151#376—24no—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C43—24no—
172C25C2414yes—
173C25C2411yescharacteristic
174C22×D4—28no—
175C22×D4—28no—
176C22×D4—28no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—28no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—44no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4C2×D418yes—
194C22×D4C2×D4116yes—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—28no—
202C23×C4—28no—
203C23×C4—28no—
204C23×C4—28no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4#4114yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222#127—232no—
223#125—416no—
224#121—432no—
225C4×D4—432no—
226C4×D4—464no—
227#117—416no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4C2×D418yes—
251C2×C4⋊C4C2×D4116yes—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
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×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—42no—
314C2×C42—42no—
315C2×C42C2×D414yes—
316C2×C42C2×D414yes—
317C2×C42#4114yes—
318C2×C42#4114yes—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—22no—
325#95—22no—
326#95—24no—
327#95—24no—
328#95—28no—
329#95—28no—
330#95—24no—
331#95—44no—
332#95—44no—
333#95—42no—
334#95—42no—
335#95—44no—
336#95—44no—
337#95—44no—
338C24—21no—
339C24—24no—
340C24—21no—
341C24—21no—
342C24—21no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—22no—
348C24—22no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—21no—
354C24—21no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—22no—
359C24—22no—
360C24—24no—
361C24—24no—
362C24C22×D418yes—
363C24C2×#4112yes—
364C24C2511yescharacteristic
365C2×D4—232no—
366C2×D4—216no—
367C2×D4—216no—
368C2×D4—216no—
369C2×D4—416no—
370C2×D4—416no—
371C2×D4—416no—
372C2×D4—416no—
373C2×D4—416no—
374C2×D4—416no—
375C2×D4—416no—
376C2×D4—416no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—88no—
382C2×D4—88no—
383C2×D4—88no—
384C2×D4—88no—
385C2×D4—88no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—416no—
457C22×C4—416no—
458C22×C4—416no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—416no—
464C22×C4—88no—
465C22×C4—88no—
466C22×C4—88no—
467C22×C4C22×D418yes—
468C22×C4C22×D414yes—
469C22×C4C22×D414yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4114yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4H(5,𝔽2)14yes—
480C22×C4H(5,𝔽2)14yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C42−1+412yes—
483C4⋊C4—216no—
484C4⋊C4—416no—
485C4⋊C4—48no—
486C4⋊C4—416no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—88no—
490C4⋊C4—88no—
491C22⋊C4—416no—
492C22⋊C4—416no—
493C22⋊C4—416no—
494C22⋊C4—48no—
495C22⋊C4—416no—
496C22⋊C4—44no—
497C22⋊C4—48no—
498C22⋊C4—44no—
499C22⋊C4—88no—
500C22⋊C4—88no—
501C22⋊C4—88no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C22⋊C4—88no—
506C42—216no—
507C42—28no—
508C42—48no—
509C42—44no—
510C42—44no—
511C42—44no—
512C42—44no—
513C42#121116yes—
514C23—28no—
515C23—22no—
516C23—24no—
517C23—28no—
518C23—22no—
519C23—28no—
520C23—22no—
521C23—24no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—41no—
527C23—44no—
528C23—41no—
529C23—44no—
530C23—41no—
531C23—44no—
532C23—41no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—42no—
540C23—42no—
541C23—44no—
542C23—44no—
543C23—41no—
544C23—41no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—42no—
549C23—44no—
550C23—44no—
551C23—42no—
552C23—44no—
553C23—44no—
554C23—42no—
555C23—42no—
556C23—44no—
557C23—41no—
558C23—44no—
559C23—41no—
560C23—44no—
561C23—44no—
562C23—42no—
563C23—42no—
564C23—44no—
565C23—44no—
566C23—42no—
567C23—44no—
568C23—44no—
569C23—42no—
570C23—44no—
571C23—44no—
572C23C23×D411yescharacteristic
573C23C23×D411yescharacteristic
574C23C22×#4111yescharacteristic
575C23C22×#4111yescharacteristic
576C23C22×#4111yescharacteristic
577C23C2×H(5,𝔽2)12yes—
578C23C2×H(5,𝔽2)11yescharacteristic
579C23C2×H(5,𝔽2)12yes—
580C23C2×2−1+411yescharacteristic
581C23#58512yes—
582C23#58512yes—
583D4—464no—
584D4—432no—
585D4—832no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—28no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—41no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—416no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—832no—
662C2×C4C2×#12118yes—
663C2×C4C2×#12114yes—
664C2×C4#52914yes—
665C2×C4#54014yes—
666C2×C4#54214yes—
667C2×C4#54614yes—
668C2×C4#54614yes—
669C2×C4#54714yes—
670C2×C4#54814yes—
671C22—42no—
672C22—48no—
673C22—48no—
674C22—44no—
675C22—42no—
676C22—48no—
677C22—48no—
678C22—48no—
679C22—42no—
680C22—44no—
681C22—48no—
682C22—48no—
683C22—84no—
684C22—84no—
685C22—81no—
686C22—81no—
687C22—84no—
688C22—82no—
689C22—84no—
690C22—84no—
691C22—81no—
692C22—81no—
693C22—84no—
694C22—82no—
695C22—84no—
696C22—84no—
697C22C22×#12111yescharacteristic
698C22C2×#52911yescharacteristic
699C22C2×#54011yescharacteristic
700C22C2×#54211yescharacteristic
701C22#345511yescharacteristic
702C22#346012yes—
703C22#346012yes—
704C22#346012yes—
705C22C2×#54611yescharacteristic
706C22C2×#54611yescharacteristic
707C22C2×#54711yescharacteristic
708C22C2×#54811yescharacteristic
709C22#348412yes—
710C22#348712yes—
711C22#349412yes—
712C22#349612yes—
713C22#349612yes—
714C22#350811yescharacteristic
715C22#351211yescharacteristic
716C22#354312yes—
717C22#354712yes—
718C22#355912yes—
719C22#356212yes—
720C4—24no—
721C4—28no—
722C4—48no—
723C4—42no—
724C4—44no—
725C4—42no—
726C4—82no—
727C4—84no—
728C4—81no—
729C4—81no—
730C4—82no—
731C4—84no—
732C4#242718yes—
733C2—82no—
734C2—88no—
735C2#3462111yescharacteristic
736C2C2×#242711yescharacteristic
737C2#3516812yes—
738C2#3528711yescharacteristic
739C2#3759112yes—
740C2#3769411yescharacteristic
741C2#3770111yescharacteristic
742C2#3775612yes—
743C2#3937211yescharacteristic
744C2#3937811yescharacteristic
745C2#3941612yes—
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
1. C2×#2641C2semidirect
2. C2×#2629C2semidirect
4. C2×C4×#127C2semidirect
5. #16596C2semidirect
6. #16552C2semidirect
7. #15138C2semidirect
8. #15126C2semidirect
9. #14824C2semidirect
10. #14732C2semidirect
11. #14709C2semidirect
12. #14276C2semidirect
13. #13970C2semidirect
14. #13920C2semidirect
15. #13761C2semidirect
145. #384D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26412
C2×#26292
C2×#26192
C2×C4×#1272
#165962
#165522
#151382
#151262
#148242
#147322
#147092
#142762
#139702
#139202
#137612