Finite Group Databasebeta

← #4309126

Subgroups — #4309126

Order 512.

3248 conjugacy classes of subgroups.

The subgroup lattice has 776 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2619C211yesmaximal, characteristic, complemented
2C2×#2290C211yesmaximal, characteristic, complemented
3#16648C214yesmaximal, complemented
4#16537C214yesmaximal, complemented
5#16284C214yesmaximal, complemented
6#16266C214yesmaximal, complemented
7#15235C211yesmaximal, characteristic, complemented
8#15223C211yesmaximal, characteristic, complemented
9#15127C211yesmaximal, characteristic, complemented
10#15126C212yesmaximal, complemented
11#15124C211yesmaximal, characteristic, complemented
12#15123C212yesmaximal, complemented
13#14698C212yesmaximal, complemented
14#14343C211yesmaximal, characteristic, complemented
15#14260C212yesmaximal, complemented
16C22×#127C2211yescharacteristic
17C22×#121C2212yes—
18C22×#121C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2619—28no—
23C2×#394C2214yes—
24C2×#394C2214yes—
25C2×#394C2214yes—
26C2×#392C2214yes—
27C2×#392C2214yes—
28C2×#392C2214yes—
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#389C2212yes—
34#2290—28no—
35C2×#388C2212yes—
36C2×#387C2212yes—
37C2×#386C2212yes—
38C2×#386C2214yes—
39C2×#386C2214yes—
40C2×#385C2212yes—
41C2×#382C2211yescharacteristic
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×#376C2211yescharacteristic
48#1445C2214yes—
49#1445C2214yes—
50#1443C2214yes—
51#1442C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1440C2214yes—
55#1440C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1438C2214yes—
62#1438C2214yes—
63#1438C2214yes—
64#1438C2214yes—
65#1438C2214yes—
66#1437C2214yes—
67#1437C2214yes—
68C4⋊#95C2211yescharacteristic
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C23×D4C2312yes—
74C23×D4C2311yescharacteristic
75C23×D4C2314yes—
76C23×D4C2314yes—
77C23×D4C2314yes—
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C2×#127—24no—
82C2×#127—44no—
83C2×#121—28no—
84C2×#121—28no—
85C2×#121—44no—
86C2×#121—44no—
87C2×#121—44no—
88C2×#121—44no—
89C2×C4×D4—24no—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C2×#117—44no—
93C2×#117D418yescomplemented
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C42—24no—
119C22×C42—44no—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126#394—216no—
127#394—48no—
128#394—48no—
129#392—216no—
130#392—48no—
131#392—48no—
132#390—28no—
133#390—28no—
134#390—44no—
135#390—44no—
136#389—44no—
137#388—28no—
138#387—28no—
139#386—28no—
140#386—48no—
141#386—48no—
142#385—28no—
143#382—24no—
144#380—44no—
145#380—44no—
146C4×C4⋊C4—24no—
147C4×C22⋊C4—24no—
148C4×C22⋊C4—24no—
149#376—24no—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C25C2414yes—
168C25C2411yescharacteristic
169C25C2411yescharacteristic
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—28no—
173C22×D4—24no—
174C22×D4—28no—
175C22×D4—28no—
176C22×D4—28no—
177C22×D4—24no—
178C22×D4—28no—
179C22×D4—22no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—42no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—44no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4C2×D414yes—
219C23×C4#4114yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2414yes—
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232#127—416no—
233#121—432no—
234#121—432no—
235C4×D4—232no—
236#117—416no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
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—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4C2×D418yes—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
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—42no—
280C2×C22⋊C4—42no—
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×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⋊C4C2×D418yes—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—416no—
319C2×C42—42no—
320C2×C42—42no—
321C2×C42C2×D414yes—
322C2×C42C2×D414yes—
323C2×C42#4114yes—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—28no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341C24—21no—
342C24—21no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—21no—
349C24—21no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
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—
370C24—24no—
371C24C22×D412yes—
372C24C2×#4112yes—
373C24H(5,𝔽2)14yes—
374C24C2511yescharacteristic
375C2×D4—28no—
376C2×D4—48no—
377C2×D4—416no—
378C2×D4—48no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—416no—
382C2×D4—416no—
383C2×D4—416no—
384C2×D4—416no—
385C2×D4—48no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—88no—
393C2×D4—88no—
394C2×D4—88no—
395C2×D4—88no—
396C2×D4—88no—
397C2×D4—88no—
398C2×D4—88no—
399C2×D4—88no—
400C2×D4—88no—
401C2×D4—88no—
402C2×D4—88no—
403C2×D4—88no—
404C2×D4—88no—
405C2×D4—88no—
406C2×D4—88no—
407C2×D4—88no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
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—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—28no—
443C22×C4—28no—
444C22×C4—28no—
445C22×C4—28no—
446C22×C4—216no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4C22×D414yes—
488C22×C4C22×D414yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C4C2×#4112yes—
496C22×C4H(5,𝔽2)14yes—
497C22×C4H(5,𝔽2)14yes—
498C22×C4H(5,𝔽2)14yes—
499C4⋊C4—28no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—88no—
506C22⋊C4—28no—
507C22⋊C4—28no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—416no—
512C22⋊C4—416no—
513C22⋊C4—88no—
514C22⋊C4—88no—
515C22⋊C4—88no—
516C22⋊C4—88no—
517C22⋊C4—88no—
518C22⋊C4—88no—
519C22⋊C4—88no—
520C22⋊C4—88no—
521C22⋊C4—88no—
522C22⋊C4—88no—
523C42—28no—
524C42—28no—
525C42—28no—
526C42—48no—
527C42—44no—
528C42—44no—
529C23—22no—
530C23—24no—
531C23—22no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—24no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—42no—
548C23—41no—
549C23—41no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—41no—
557C23—41no—
558C23—42no—
559C23—42no—
560C23—41no—
561C23—41no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23—44no—
603C23—44no—
604C23—44no—
605C23#54014yes—
606C23C23×D411yescharacteristic
607C23C22×#4111yescharacteristic
608C23C22×#4111yescharacteristic
609C23C2×H(5,𝔽2)12yes—
610C23C2×H(5,𝔽2)12yes—
611C23C2×H(5,𝔽2)12yes—
612C23C2×H(5,𝔽2)12yes—
613C23#58512yes—
614C23#58512yes—
615D4—416no—
616D4—832no—
617D4—832no—
618D4—832no—
619D4—816no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—432no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—416no—
663C2×C4—416no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—42no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—42no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—42no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—42no—
691C2×C4—44no—
692C2×C4#52514yes—
693C2×C4#53414yes—
694C2×C4#53514yes—
695C2×C4#53818yes—
696C2×C4#54018yes—
697C22—28no—
698C22—24no—
699C22—42no—
700C22—44no—
701C22—44no—
702C22—44no—
703C22—42no—
704C22—44no—
705C22—44no—
706C22—42no—
707C22—44no—
708C22—44no—
709C22—42no—
710C22—42no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22—84no—
716C22—84no—
717C22—84no—
718C22—84no—
719C22—84no—
720C22—84no—
721C22—84no—
722C22—84no—
723C22—84no—
724C22—84no—
725C22—84no—
726C22—84no—
727C22—84no—
728C22—84no—
729C22—84no—
730C22C2×#52511yescharacteristic
731C22C2×#53411yescharacteristic
732C22C2×#53511yescharacteristic
733C22C2×#53812yes—
734C22C2×#54012yes—
735C22C2×#54011yescharacteristic
736C22#345511yescharacteristic
737C22#346012yes—
738C22#348812yes—
739C22#348812yes—
740C22#349412yes—
741C22#350811yescharacteristic
742C22#351211yescharacteristic
743C22#353712yes—
744C22#354012yes—
745C22#354212yes—
746C22#354212yes—
747C22#354312yes—
748C22#354612yes—
749C22#355812yes—
750C22#356212yes—
751C4—42no—
752C4—42no—
753C4—44no—
754C4—44no—
755C4—44no—
756C4—42no—
757C4—42no—
758C4—84no—
759C4—82no—
760C4—84no—
761C4—84no—
762C4#263518yes—
763C2—44no—
764C2—82no—
765C2—164no—
766C2C2×#263511yescharacteristic
767C2#3525911yescharacteristic
768C2#3526111yescharacteristic
769C2#3726912yes—
770C2#3727812yes—
771C2#3733212yes—
772C2#3737212yes—
773C2#3749312yes—
774C2#3750112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
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×#2619C2semidirect
2. C2×#2290C2semidirect
3. #16648C2semidirect
4. #16537C2semidirect
5. #16284C2semidirect
6. #16266C2semidirect
7. #15235C2semidirect
8. #15223C2semidirect
9. #15127C2semidirect
10. #15126C2semidirect
11. #15124C2semidirect
12. #15123C2semidirect
13. #14698C2semidirect
14. #14343C2semidirect
15. #14260C2semidirect
93. C2×#117D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26192
C2×#22902
#166482
#165372
#162842
#162662
#152352
#152232
#151272
#151262
#151242
#151232
#146982
#143432
#142602