Finite Group Databasebeta

← #405347

Subgroups — #405347

Order 512.

1652 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×#2503C211yesmaximal, characteristic
2C2×#1892C211yesmaximal, characteristic
3C2×#1872C211yesmaximal, characteristic
4C2×#1823C211yesmaximal, characteristic
5C22×#340C211yesmaximal, characteristic
6#14130C211yesmaximal, characteristic
7#11268C212yesmaximal, complemented
8#11208C212yesmaximal, complemented
9#11190C212yesmaximal, complemented
10#11067C212yesmaximal, complemented
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19#2503—24no—
20#2503—24no—
21C2×#400—21no—
22C2×#400—21no—
23C2×#391C2211yescharacteristic
24C2×#391C2211yescharacteristic
25C2×#389C2212yes—
26C2×#385—21no—
27C2×#382C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×#381—21no—
30C2×#381—21no—
31C2×#380C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33#1892—24no—
34#1872—24no—
35#1823—24no—
36C2×#340—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1444—21no—
46#1442—21no—
47C4⋊#95C2212yes—
48C22⋊#95—21no—
49C23×Q8C2311yescharacteristic
50C24×C4—21no—
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C23×C8—22no—
54C23×C8—24no—
55C23×C8—22no—
56C23×C8—22no—
57C23×C8—42no—
58C23×C8C2312yes—
59C23×C8C2312yes—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8D414yes—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—24no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—42no—
79C22×C4⋊C4D414yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4D411yescharacteristic
88C22×C22⋊C4D411yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—42no—
96C2×#103—42no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100#400—28no—
101#400—28no—
102#391—22no—
103#391—22no—
104#391—22no—
105#391—22no—
106#389—24no—
107#389—42no—
108#385—44no—
109#382—22no—
110#382—22no—
111#382—22no—
112#382—22no—
113#381—28no—
114#381—28no—
115#380—22no—
116#380D414yes—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—22no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95D411yescharacteristic
132C2×#95D411yescharacteristic
133#340—24no—
134#340—24no—
135#340—24no—
136#340—24no—
137#340—24no—
138#340—24no—
139#336—24no—
140#336—42no—
141C25C2×D411yescharacteristic
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8C2×D412yes—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—24no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—44no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—82no—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D412yes—
180C23×C4C2×D412yes—
181C23×C4C2×D412yes—
182C23×C4#4112yes—
183C23×C4#4112yes—
184C23×C4#4112yes—
185C23×C4C2411yescharacteristic
186C22×C8—28no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—48no—
201C22×C8—48no—
202C22×C8—44no—
203C22×C8—44no—
204C22×C8#4114yes—
205C22⋊Q8—28no—
206C22⋊Q8—28no—
207C22⋊Q8—44no—
208C22⋊Q8—44no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—48no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—82no—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4#4114yes—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277#103—48no—
278#103—48no—
279C22⋊C8—44no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—22no—
289#95—22no—
290#95—24no—
291#95—44no—
292#95—44no—
293#95—82no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24#12012yes—
299C24#12112yes—
300C24#12412yes—
301C24#13512yes—
302C24#13512yes—
303C24AGL(1,ℤ/8)12yes—
304C24#13712yes—
305C24C22×D411yescharacteristic
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—48no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—88no—
403C22×C4—88no—
404C22×C4—84no—
405C22×C4—84no—
406C22×C4—84no—
407C22×C4—84no—
408C22×C4—84no—
409C22×C4—84no—
410C22×C4#12012yes—
411C22×C4#12114yes—
412C22×C4#12114yes—
413C22×C4#12112yes—
414C22×C4#12314yes—
415C22×C4#12314yes—
416C22×C4#12412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4H(5,𝔽2)12yes—
426C22×C42−1+412yes—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—416no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—48no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—88no—
466C22⋊C4—48no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—48no—
470C42—41no—
471C42—41no—
472C42—41no—
473C42—41no—
474C42—42no—
475C42—42no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—24no—
485C23—22no—
486C23—22no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23#39414yes—
493C23#44914yes—
494C23#45214yes—
495C23#46314yes—
496C23#46414yes—
497C23#48814yes—
498C23#48914yes—
499C23C2×#12011yescharacteristic
500C23C2×#12111yescharacteristic
501C23C2×#12411yescharacteristic
502C23#54612yes—
503C23#54712yes—
504C23C2×#13511yescharacteristic
505C23C2×#13511yescharacteristic
506C23C2×AGL(1,ℤ/8)11yescharacteristic
507C23C2×#13711yescharacteristic
508C23#57712yes—
509C23#57812yes—
510Q8—44no—
511Q8—44no—
512Q8—81no—
513Q8—81no—
514Q8—81no—
515Q8—81no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—816no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—816no—
586C2×C4#39414yes—
587C2×C4C2×#12111yescharacteristic
588C2×C4C2×#12111yescharacteristic
589C2×C4C2×#12311yescharacteristic
590C2×C4C2×#12311yescharacteristic
591C2×C4#52511yescharacteristic
592C2×C4#52511yescharacteristic
593C2×C4#52912yes—
594C2×C4#52912yes—
595C2×C4#53511yescharacteristic
596C2×C4#53611yescharacteristic
597C2×C4#53912yes—
598C2×C4#53912yes—
599C2×C4#54612yes—
600C2×C4#54612yes—
601C2×C4#54712yes—
602C2×C4#54712yes—
603C2×C4#55112yes—
604C2×C4#55212yes—
605C8—44no—
606C8—82no—
607C22—24no—
608C22—24no—
609C22—28no—
610C22—24no—
611C22—28no—
612C22—28no—
613C22—28no—
614C22—24no—
615C22—24no—
616C22—24no—
617C22#204218yes—
618C22C2×#39411yescharacteristic
619C22#240411yescharacteristic
620C22#240611yescharacteristic
621C22#242512yes—
622C22#248812yes—
623C22C2×#44911yescharacteristic
624C22C2×#45211yescharacteristic
625C22#301011yescharacteristic
626C22#301111yescharacteristic
627C22C2×#46311yescharacteristic
628C22C2×#46411yescharacteristic
629C22#304612yes—
630C22C2×#48811yescharacteristic
631C22C2×#48911yescharacteristic
632C22#313712yes—
633C22#329612yes—
634C22#330012yes—
635C22#331012yes—
636C22#331212yes—
637C22#333612yes—
638C22#333712yes—
639C22#334312yes—
640C22#334412yes—
641C4—24no—
642C4—81no—
643C4—81no—
644C4—82no—
645C4—84no—
646C4—81no—
647C4—84no—
648C4—81no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4#241111yescharacteristic
653C4#241311yescharacteristic
654C4#241912yes—
655C4#242512yes—
656C4#262311yescharacteristic
657C4#285311yescharacteristic
658C2—28no—
659C2#1491211yescharacteristic
660C2C2×#204211yescharacteristic
661C2#2351011yescharacteristic
662C2#2396611yescharacteristic
663C2#2396711yescharacteristic
664C2#2409411yescharacteristic
665C2#2409511yescharacteristic
666C2#2520812yes—
667C2#2521212yes—
668C2#2769912yes—
669C2#2771312yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #11268C2semidirect
8. #11208C2semidirect
9. #11190C2semidirect
10. #11067C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
C2×#18922
C2×#18722
C2×#18232
C22×#3402
#141302
#112682
#112082
#111902
#110672
#107142