Finite Group Databasebeta

← #405294

Subgroups — #405294

Order 512.

1852 conjugacy classes of subgroups.

The subgroup lattice has 739 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2C2×#1869C211yesmaximal, characteristic
3C2×#1853C211yesmaximal, characteristic, complemented
4C2×#1823C211yesmaximal, characteristic, complemented
5C22×#340C211yesmaximal, characteristic, complemented
6#13872C211yesmaximal, characteristic
7#11293C212yesmaximal, complemented
8#11220C212yesmaximal, complemented
9#11200C212yesmaximal, complemented
10#11095C212yesmaximal, complemented
11#10721C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×#123—21no—
14C22×C4×D4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C22×#106C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20#2486—24no—
21#2486—24no—
22C2×#398—21no—
23C2×#397—21no—
24C2×#389C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#386—21no—
27C2×#381—21no—
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×#380C2211yescharacteristic
31C2×#380C2212yes—
32C2×#380C2211yescharacteristic
33#1869—24no—
34#1853—24no—
35#1823—24no—
36C2×#340—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45#1441—21no—
46#1439—21no—
47C4⋊#95C2212yes—
48C22⋊#95—21no—
49C23×D4D411yescharacteristic
50C24×C4—21no—
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C24×C4C2311yescharacteristic
54C23×C8—22no—
55C23×C8—24no—
56C23×C8—22no—
57C23×C8—22no—
58C23×C8—42no—
59C23×C8C2312yes—
60C23×C8C2312yes—
61C2×#123—28no—
62C2×#123—44no—
63C2×#123—44no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—24no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—42no—
83C22×C4⋊C4D411yescharacteristic
84C22×C4⋊C4D414yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4—21no—
94C22×C22⋊C4—21no—
95C22×C22⋊C4—22no—
96C22×C22⋊C4D411yescharacteristic
97C22×C22⋊C4D411yescharacteristic
98C22×C42C2311yescharacteristic
99C2×#106—24no—
100C2×#106—24no—
101C2×#106—42no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—24no—
105C2×C22⋊C8—24no—
106C2×C22⋊C8—42no—
107C2×C22⋊C8D414yes—
108#398—28no—
109#397—28no—
110#389—22no—
111#389—22no—
112#389—22no—
113#389—22no—
114#386—44no—
115#381—28no—
116#380—22no—
117#380—22no—
118#380—22no—
119#380—22no—
120#380—22no—
121#380—22no—
122#380—22no—
123#380—24no—
124#380—42no—
125#380D414yes—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—21no—
134C2×#95—22no—
135#340—24no—
136#340—24no—
137#340—24no—
138#340—24no—
139#340—24no—
140#340—24no—
141#340—42no—
142#336—24no—
143C25C2×D411yescharacteristic
144C25C2×D411yescharacteristic
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4SD1614yes—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—24no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—44no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—82no—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D412yes—
189C23×C4C2×D412yes—
190C23×C4C2×D412yes—
191C23×C4#4112yes—
192C23×C4#4112yes—
193C23×C4#4112yes—
194C23×C4C2411yescharacteristic
195C22×C8—28no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—48no—
209C22×C8—48no—
210C22×C8—44no—
211C22×C8—44no—
212C22×C8C2×D414yes—
213C22×C8#4114yes—
214#123—432no—
215C4×D4—44no—
216C4×D4—44no—
217C4×D4—44no—
218C4×D4—44no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—48no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
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⋊C4C2×D412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4#4114yes—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—82no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299#106—48no—
300C22⋊C8—28no—
301C22⋊C8—48no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—44no—
308#95—44no—
309#95—42no—
310#95—82no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—22no—
319C24—22no—
320C24—21no—
321C24—21no—
322C24—21no—
323C24—22no—
324C24—22no—
325C24#12012yes—
326C24#12112yes—
327C24#12312yes—
328C24C2×SD1612yes—
329C24C2×SD1612yes—
330C24#13512yes—
331C24AGL(1,ℤ/8)12yes—
332C24AGL(1,ℤ/8)12yes—
333C24C22×D411yescharacteristic
334C2×D4—28no—
335C2×D4—28no—
336C2×D4—28no—
337C2×D4—44no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—44no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—48no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—88no—
434C22×C4—88no—
435C22×C4—84no—
436C22×C4—84no—
437C22×C4—84no—
438C22×C4—84no—
439C22×C4—84no—
440C22×C4—84no—
441C22×C4#12012yes—
442C22×C4#12114yes—
443C22×C4#12114yes—
444C22×C4#12112yes—
445C22×C4#12314yes—
446C22×C4#12312yes—
447C22×C4#12414yes—
448C22×C4C2×SD1612yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C2×#4111yescharacteristic
453C22×C4C2×#4111yescharacteristic
454C22×C4C2×#4111yescharacteristic
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C42−1+412yes—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—416no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—42no—
473C2×C8—42no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—42no—
493C4⋊C4—41no—
494C4⋊C4—42no—
495C4⋊C4—48no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—44no—
506C22⋊C4—88no—
507C22⋊C4—84no—
508C42—41no—
509C42—41no—
510C42—41no—
511C42—41no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—24no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—22no—
533C23—42no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23—42no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—41no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23#39414yes—
552C23#44914yes—
553C23#45014yes—
554C23#46514yes—
555C23#46914yes—
556C23#48114yes—
557C23#48214yes—
558C23C2×#12011yescharacteristic
559C23C2×#12111yescharacteristic
560C23C2×#12311yescharacteristic
561C23#54612yes—
562C23#54712yes—
563C23C22×SD1611yescharacteristic
564C23C2×#13511yescharacteristic
565C23C2×AGL(1,ℤ/8)11yescharacteristic
566C23C2×AGL(1,ℤ/8)11yescharacteristic
567C23#57712yes—
568C23#57812yes—
569D4—416no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—816no—
613C2×C4—82no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4—82no—
628C2×C4—82no—
629C2×C4—816no—
630C2×C4—81no—
631C2×C4—81no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4—82no—
637C2×C4—82no—
638C2×C4—82no—
639C2×C4—82no—
640C2×C4—82no—
641C2×C4—82no—
642C2×C4—82no—
643C2×C4#39414yes—
644C2×C4C2×#12111yescharacteristic
645C2×C4C2×#12111yescharacteristic
646C2×C4C2×#12311yescharacteristic
647C2×C4#52511yescharacteristic
648C2×C4#52511yescharacteristic
649C2×C4C2×#12411yescharacteristic
650C2×C4#52912yes—
651C2×C4#52912yes—
652C2×C4#53511yescharacteristic
653C2×C4#53611yescharacteristic
654C2×C4#53912yes—
655C2×C4#53912yes—
656C2×C4#54612yes—
657C2×C4#54612yes—
658C2×C4#54712yes—
659C2×C4#54712yes—
660C2×C4#55112yes—
661C2×C4#55212yes—
662C8—44no—
663C8—44no—
664C22—24no—
665C22—24no—
666C22—24no—
667C22—28no—
668C22—28no—
669C22—28no—
670C22—24no—
671C22—24no—
672C22—28no—
673C22—24no—
674C22—42no—
675C22—44no—
676C22—44no—
677C22—42no—
678C22—42no—
679C22—81no—
680C22—81no—
681C22—82no—
682C22—81no—
683C22—82no—
684C22—81no—
685C22#203418yes—
686C22C2×#39411yescharacteristic
687C22#240411yescharacteristic
688C22#240611yescharacteristic
689C22#241912yes—
690C22#242512yes—
691C22C2×#45011yescharacteristic
692C22C2×#44911yescharacteristic
693C22#301011yescharacteristic
694C22#301111yescharacteristic
695C22C2×#46511yescharacteristic
696C22C2×#46911yescharacteristic
697C22#306312yes—
698C22C2×#48211yescharacteristic
699C22C2×#48111yescharacteristic
700C22#309712yes—
701C22#329612yes—
702C22#329812yes—
703C22#330812yes—
704C22#331012yes—
705C22#333512yes—
706C22#333612yes—
707C22#334312yes—
708C22#334512yes—
709C4—24no—
710C4—42no—
711C4—82no—
712C4—81no—
713C4—82no—
714C4—84no—
715C4—82no—
716C4—82no—
717C4—81no—
718C4—162no—
719C4#241111yescharacteristic
720C4#241311yescharacteristic
721C4#242512yes—
722C4#248812yes—
723C4#262311yescharacteristic
724C4#285311yescharacteristic
725C2—28no—
726C2—82no—
727C2#1491211yescharacteristic
728C2C2×#203411yescharacteristic
729C2#2348411yescharacteristic
730C2#2396111yescharacteristic
731C2#2396611yescharacteristic
732C2#2408911yescharacteristic
733C2#2409411yescharacteristic
734C2#2493712yes—
735C2#2494112yes—
736C2#2523612yes—
737C2#2525012yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#1853C2semidirect
4. C2×#1823C2semidirect
5. C22×#340C2semidirect
7. #11293C2semidirect
8. #11220C2semidirect
9. #11200C2semidirect
10. #11095C2semidirect
11. #10721C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
C2×#18692
C2×#18532
C2×#18232
C22×#3402
#138722
#112932
#112202
#112002
#110952
#107212