Finite Group Databasebeta

← #416055

Subgroups — #416055

Order 512.

1666 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2548C211yesmaximal, characteristic
2C2×#1985C211yesmaximal, characteristic, complemented
3C2×#1982C211yesmaximal, characteristic
4C2×#1961C211yesmaximal, characteristic
5C2×#1776C211yesmaximal, characteristic, complemented
6#13925C211yesmaximal, characteristic, complemented
7#11189C212yesmaximal, complemented
8#11189C212yesmaximal, complemented
9#11123C212yesmaximal, complemented
10#11121C212yesmaximal, complemented
11#10850C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C22×#124—21no—
14C22×C4×Q8C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×#102C2212yes—
21C22×#102C2212yes—
22C22×#97C2211yescharacteristic
23#2548—24no—
24#2548—24no—
25C2×#394—22no—
26C2×#390—21no—
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#382C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#1985—24no—
33#1982—24no—
34#1961—24no—
35#1776—24no—
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1446—21no—
43#1445—21no—
44#1436—22no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8D411yescharacteristic
48C23×D4—21no—
49C23×D4—21no—
50C23×D4C2312yes—
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×#127—22no—
55C2×#127—22no—
56C2×#127—22no—
57C2×#127—22no—
58C2×#127—22no—
59C2×#127D414yes—
60C2×#124—28no—
61C2×#124—48no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C22×C4⋊C4D412yes—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C42—21no—
80C22×C42—21no—
81C22×C42C2312yes—
82C22×C42C2311yescharacteristic
83C2×#107—22no—
84C2×#107—22no—
85C2×#107—22no—
86C2×#106—22no—
87C2×#106—22no—
88C2×#106—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—42no—
101C2×#102—42no—
102C2×#102—42no—
103C2×#102—42no—
104C2×#97—22no—
105C2×#97—22no—
106C2×#97D414yes—
107#394—216no—
108#390—28no—
109#390—24no—
110#390—24no—
111#390—24no—
112#390—24no—
113#382—22no—
114#382—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4D414yes—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—22no—
127#340—22no—
128#340—22no—
129#340—42no—
130#340—42no—
131#336—24no—
132#336—24no—
133C25—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—24no—
138C22×Q8—42no—
139C22×D4—24no—
140C22×D4—24no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—44no—
155C22×D4—44no—
156C22×D4—42no—
157C22×D4—42no—
158C22×D4—42no—
159C22×D4—42no—
160C22×D4—82no—
161C22×D4—82no—
162C22×D4C2×D414yes—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—22no—
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2411yescharacteristic
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8C2×D412yes—
191C22×C8C2×D412yes—
192C22×C8#4112yes—
193C22×C8#4112yes—
194#127—28no—
195#127—28no—
196#127—44no—
197#127—44no—
198#124—432no—
199C4×Q8—44no—
200C4×Q8—44no—
201C4×Q8—44no—
202C4×Q8—44no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4#4112yes—
238C2×C4⋊C4#4112yes—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—82no—
248C2×C22⋊C4—82no—
249C2×C22⋊C4—82no—
250C2×C22⋊C4—82no—
251C2×C42—24no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—44no—
259C2×C42—42no—
260C2×C42—82no—
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42#4112yes—
265C2×C42#4112yes—
266C2×C42#4112yes—
267#107—28no—
268#106—28no—
269C4⋊C8—44no—
270#102—48no—
271#102—48no—
272#102—48no—
273#102—48no—
274#97—28no—
275#95—24no—
276#95—44no—
277#95—42no—
278#95—42no—
279#95—42no—
280#95—44no—
281#95—82no—
282#95—82no—
283C24—24no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24—42no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24C22×D411yescharacteristic
299C2×Q8—28no—
300C2×Q8—48no—
301C2×Q8—48no—
302C2×Q8—44no—
303C2×Q8—44no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—24no—
310C2×D4—48no—
311C2×D4—48no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—48no—
335C2×D4—48no—
336C2×D4—84no—
337C2×D4—84no—
338C2×D4—88no—
339C2×D4—88no—
340C2×D4—88no—
341C2×D4—88no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4#12012yes—
387C22×C4#12012yes—
388C22×C4#12112yes—
389C22×C4#12112yes—
390C22×C4#12312yes—
391C22×C4#12312yes—
392C22×C4AGL(1,ℤ/8)14yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4H(5,𝔽2)11yescharacteristic
404C22×C4H(5,𝔽2)11yescharacteristic
405C22×C42−1+411yescharacteristic
406C22×C42−1+411yescharacteristic
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8#12414yescomplemented
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—48no—
445C4⋊C4—44no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C22⋊C4—416no—
451C22⋊C4—416no—
452C22⋊C4—88no—
453C22⋊C4—88no—
454C42—48no—
455C42—42no—
456C42—42no—
457C42—42no—
458C42—42no—
459C42—84no—
460C42#12114yes—
461C42#12114yes—
462C42#12314yes—
463C42#12414yes—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23—82no—
494C23—82no—
495C23—82no—
496C23—82no—
497C23—82no—
498C23—82no—
499C23C2×#12011yescharacteristic
500C23C2×#12111yescharacteristic
501C23C2×#12311yescharacteristic
502C23#54811yescharacteristic
503C23#54811yescharacteristic
504C23#55011yescharacteristic
505C23#55011yescharacteristic
506C23C2×AGL(1,ℤ/8)12yes—
507C23#57512yes—
508C23#57612yes—
509C23#57712yes—
510Q8—416no—
511D4—44no—
512D4—44no—
513D4—44no—
514D4—44no—
515D4—816no—
516D4—816no—
517D4—82no—
518D4—82no—
519D4—82no—
520D4—82no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4#39414yes—
571C2×C4#39414yes—
572C2×C4C2×#12111yescharacteristic
573C2×C4C2×#12111yescharacteristic
574C2×C4C2×#12311yescharacteristic
575C2×C4#52511yescharacteristic
576C2×C4#52511yescharacteristic
577C2×C4C2×#12411yescharacteristic
578C2×C4C2×#12411yescharacteristic
579C2×C4#52911yescharacteristic
580C2×C4#52911yescharacteristic
581C2×C4#53211yescharacteristic
582C2×C4#53511yescharacteristic
583C2×C4#53611yescharacteristic
584C2×C4#53711yescharacteristic
585C2×C4#53711yescharacteristic
586C2×C4#54211yescharacteristic
587C2×C4#54211yescharacteristic
588C2×C4#54611yescharacteristic
589C2×C4#54611yescharacteristic
590C2×C4#54711yescharacteristic
591C2×C4#54711yescharacteristic
592C2×C4#55611yescharacteristic
593C2×C4#55611yescharacteristic
594C2×C4#55611yescharacteristic
595C2×C4#55611yescharacteristic
596C2×C4#55911yescharacteristic
597C2×C4#55911yescharacteristic
598C2×C4#55911yescharacteristic
599C2×C4#55911yescharacteristic
600C8—24no—
601C8—42no—
602C8—82no—
603C22—84no—
604C22—84no—
605C22—84no—
606C22—84no—
607C22—84no—
608C22—84no—
609C22—84no—
610C22—162no—
611C22—162no—
612C22—162no—
613C22—162no—
614C22—162no—
615C22—162no—
616C22—162no—
617C22—162no—
618C22C2×#39411yescharacteristic
619C22#240711yescharacteristic
620C22#240711yescharacteristic
621C22#244811yescharacteristic
622C22#244811yescharacteristic
623C22#254511yescharacteristic
624C22#254811yescharacteristic
625C22#300311yescharacteristic
626C22#300511yescharacteristic
627C22#300811yescharacteristic
628C22#300911yescharacteristic
629C22#305512yes—
630C22#306212yes—
631C22#309412yes—
632C22#309612yes—
633C22#336612yes—
634C22#336812yes—
635C22#337012yes—
636C22#337112yes—
637C22#339712yes—
638C22#339912yes—
639C22#340112yes—
640C22#340312yes—
641C4—24no—
642C4—24no—
643C4—44no—
644C4—82no—
645C4—82no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4—162no—
650C4#244711yescharacteristic
651C4#244711yescharacteristic
652C4#254411yescharacteristic
653C4#254711yescharacteristic
654C4#259211yescharacteristic
655C4#259211yescharacteristic
656C4#262011yescharacteristic
657C4#285011yescharacteristic
658C2—164no—
659C2#1521711yescharacteristic
660C2#2348011yescharacteristic
661C2#2348311yescharacteristic
662C2#2419611yescharacteristic
663C2#2419811yescharacteristic
664C2#2420011yescharacteristic
665C2#2420211yescharacteristic
666C2#2633212yes—
667C2#2633412yes—
668C2#2865012yes—
669C2#2874612yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1985C2semidirect
5. C2×#1776C2semidirect
6. #13925C2semidirect
7. #11189C2semidirect
8. #11189C2semidirect
9. #11123C2semidirect
10. #11121C2semidirect
11. #10850C2semidirect
427. C2×C8#124semidirect

Maximal subgroups

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

subgroupindex
C2×#25482
C2×#19852
C2×#19822
C2×#19612
C2×#17762
#139252
#111892
#111232
#111212
#108502