Finite Group Databasebeta

← #366821

Subgroups — #366821

Order 512.

1952 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2501C211yesmaximal, characteristic
2C2×#1895C211yesmaximal, characteristic
3C2×#1875C211yesmaximal, characteristic
4C2×#1851C211yesmaximal, characteristic, complemented
5C2×#1824C211yesmaximal, characteristic, complemented
6C2×#1736C211yesmaximal, characteristic, complemented
7#13801C211yesmaximal, characteristic, complemented
8#11357C212yesmaximal, complemented
9#11076C212yesmaximal, complemented
10#10931C212yesmaximal, complemented
11#10849C212yesmaximal, complemented
12C23×M16C2211yescharacteristic
13C22×#123—22no—
14C22×#121C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C22×C4⋊C8C2212yes—
18C22×#102C2211yescharacteristic
19C22×#102C2212yes—
20C22×#102C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22#2501—24no—
23#2501C418yescomplemented
24C2×#399—21no—
25C2×#399—21no—
26C2×#390C2212yes—
27C2×#384C2212yes—
28C2×#380C2211yescharacteristic
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C22⋊C4C2211yescharacteristic
31#1895—24no—
32#1875—24no—
33#1851—24no—
34#1824—24no—
35#1736—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2211yescharacteristic
42C2×#340C2212yes—
43C2×#336C2212yes—
44#1437—22no—
45C4⋊#95C2212yes—
46#1434—21no—
47C4×#95—21no—
48C23×D4—21no—
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C24×C4C2311yescharacteristic
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—22no—
55C22×M16—24no—
56C22×M16—42no—
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C23×C8C2312yes—
60C2×#123—216no—
61C2×#123—48no—
62C2×#123—48no—
63C2×#121—22no—
64C2×#121—22no—
65C2×#121—22no—
66C2×#121—22no—
67C2×#121—22no—
68C2×#121C2×C414yes—
69C2×#117—22no—
70C2×#117—24no—
71C2×#117—42no—
72C2×#117D414yes—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4C2×C414yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C22⋊C4—21no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—22no—
90C22×C22⋊C4—22no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4D412yes—
93C22×C42—21no—
94C22×C42—22no—
95C22×C42C2312yes—
96C2×C4⋊C8—24no—
97C2×C4⋊C8—24no—
98C2×C4⋊C8—42no—
99C2×#102—24no—
100C2×#102—22no—
101C2×#102—22no—
102C2×#102—24no—
103C2×#102—24no—
104C2×#102—42no—
105C2×#102—42no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109#399—28no—
110#399—28no—
111#390—24no—
112#390C2×C418yes—
113#384—24no—
114#384C2×C418yes—
115#380—22no—
116#380D414yes—
117C4×C4⋊C4—24no—
118C4×C4⋊C4—42no—
119C4×C22⋊C4C2×C414yes—
120C4×C22⋊C4D414yes—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95D411yescharacteristic
126C2×#95D411yescharacteristic
127#340—24no—
128#340—24no—
129#340—22no—
130#340—22no—
131#340—24no—
132#340—24no—
133#340—24no—
134#336—24no—
135C25—21no—
136C25C2×D411yescharacteristic
137C22×D4—24no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—44no—
148C22×D4—44no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4C22×C412yes—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—24no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4C22×C412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4#4112yes—
174C23×C4#4112yes—
175C23×C4C2411yescharacteristic
176C2×M16—28no—
177C2×M16—48no—
178C2×M16—48no—
179C2×M16—44no—
180C2×M16—44no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8#4114yes—
198#123—464no—
199#121—28no—
200#121—28no—
201#121—44no—
202#121—44no—
203#117—28no—
204#117—44no—
205#117—48no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4C22×C414yes—
240C2×C4⋊C4C22×C412yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D414yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4#4114yes—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—81no—
260C2×C22⋊C4—81no—
261C2×C22⋊C4C22⋊C418yes—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—24no—
267C2×C42—44no—
268C2×C42—44no—
269C2×C42—44no—
270C2×C42—82no—
271C2×C42C22×C414yes—
272C2×C42C2×D414yes—
273C4⋊C8—48no—
274#102—48no—
275#102—44no—
276#102—44no—
277C22⋊C8—44no—
278#95—24no—
279#95—44no—
280#95—44no—
281#95—82no—
282#95C22⋊C414yes—
283#95C22⋊C414yes—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—21no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—22no—
293C24—41no—
294C24—41no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24—41no—
303C24C2×C22⋊C412yes—
304C24#12112yes—
305C24#12112yes—
306C24C22×D411yescharacteristic
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—24no—
310C2×D4—24no—
311C2×D4—416no—
312C2×D4—48no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—42no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—84no—
328C2×D4—84no—
329C2×D4—88no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—416no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
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—41no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—48no—
387C22×C4—84no—
388C22×C4—84no—
389C22×C4—88no—
390C22×C4C2×C22⋊C414yes—
391C22×C4C2×C22⋊C412yes—
392C22×C4C2×C22⋊C414yes—
393C22×C4C4×D414yes—
394C22×C4C4×D414yes—
395C22×C4#12112yes—
396C22×C4#12112yes—
397C22×C4#12414yes—
398C22×C4#12714yes—
399C22×C4AGL(1,ℤ/8)14yes—
400C22×C4AGL(1,ℤ/8)14yes—
401C22×C4C23×C411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4112yes—
408C22×C4C2×#4112yes—
409C22×C4H(5,𝔽2)12yes—
410C22×C42−1+412yes—
411M16—416no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—416no—
440C4⋊C4—42no—
441C4⋊C4—41no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—41no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—48no—
448C4⋊C4—416no—
449C4⋊C4#12314yescomplemented
450C22⋊C4—216no—
451C22⋊C4—48no—
452C22⋊C4—416no—
453C22⋊C4—48no—
454C22⋊C4—416no—
455C22⋊C4—84no—
456C42—24no—
457C42—24no—
458C42—24no—
459C42—24no—
460C42—48no—
461C42—88no—
462C23—24no—
463C23—24no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—24no—
469C23—22no—
470C23—22no—
471C23—24no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—44no—
485C23—44no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—82no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23—82no—
494C23—82no—
495C23—81no—
496C23—81no—
497C23—82no—
498C23—81no—
499C23—81no—
500C23—82no—
501C23—81no—
502C23—81no—
503C23—81no—
504C23—81no—
505C23—82no—
506C23—82no—
507C23#39014yes—
508C23C22×C22⋊C411yescharacteristic
509C23C2×#12111yescharacteristic
510C23C2×#12111yescharacteristic
511C23#54612yes—
512C23#54712yes—
513C23C2×AGL(1,ℤ/8)12yes—
514C23C2×AGL(1,ℤ/8)12yes—
515C23#57711yescharacteristic
516C23#57711yescharacteristic
517C23#57811yescharacteristic
518C23#57811yescharacteristic
519D4—42no—
520D4—44no—
521D4—42no—
522D4—816no—
523D4—82no—
524D4—81no—
525D4—81no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—816no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4#39014yes—
580C2×C4#41818yes—
581C2×C4C2×C4×D411yescharacteristic
582C2×C4C2×C4×D411yescharacteristic
583C2×C4C4×#4112yes—
584C2×C4#52012yes—
585C2×C4C2×#12311yescharacteristic
586C2×C4#52511yescharacteristic
587C2×C4C2×#12411yescharacteristic
588C2×C4#52912yes—
589C2×C4C2×#12711yescharacteristic
590C2×C4#53411yescharacteristic
591C2×C4#53611yescharacteristic
592C2×C4#53912yes—
593C2×C4#54612yes—
594C2×C4#54612yes—
595C2×C4#54712yes—
596C2×C4#54712yes—
597C2×C4#55112yes—
598C2×C4#55212yes—
599C8—44no—
600C8—81no—
601C8—81no—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—42no—
606C22—42no—
607C22—44no—
608C22—44no—
609C22—42no—
610C22—44no—
611C22—42no—
612C22—44no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—84no—
617C22—84no—
618C22—161no—
619C22—161no—
620C22—162no—
621C22—162no—
622C22—161no—
623C22—161no—
624C22C2×#39011yescharacteristic
625C22#233912yes—
626C22#241211yescharacteristic
627C22#241311yescharacteristic
628C22#242512yes—
629C22C2×#41812yes—
630C22#289811yescharacteristic
631C22#289811yescharacteristic
632C22#303811yescharacteristic
633C22#303811yescharacteristic
634C22#303812yes—
635C22#304211yescharacteristic
636C22#304411yescharacteristic
637C22#304611yescharacteristic
638C22#304611yescharacteristic
639C22#330612yes—
640C22#330812yes—
641C22#333012yes—
642C22#333212yes—
643C22#334312yes—
644C22#334512yes—
645C22#335312yes—
646C22#335412yes—
647C4—24no—
648C4—28no—
649C4—44no—
650C4—44no—
651C4—84no—
652C4—82no—
653C4—84no—
654C4—82no—
655C4C4×#12311yescharacteristic
656C4#232211yescharacteristic
657C4#235912yes—
658C4#248812yes—
659C4#250111yescharacteristic
660C4#253911yescharacteristic
661C2—44no—
662C2—162no—
663C2#1399611yescharacteristic
664C2#2126011yescharacteristic
665C2#2126211yescharacteristic
666C2#2249512yes—
667C2#2250012yes—
668C2#2454211yescharacteristic
669C2#2454411yescharacteristic
670C2#2459211yescharacteristic
671C2#2459411yescharacteristic
672C2#2521712yes—
673C2#2522112yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×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
4. C2×#1851C2semidirect
5. C2×#1824C2semidirect
6. C2×#1736C2semidirect
7. #13801C2semidirect
8. #11357C2semidirect
9. #11076C2semidirect
10. #10931C2semidirect
11. #10849C2semidirect
23. #2501C4semidirect
449. C4⋊C4#123semidirect

Maximal subgroups

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

subgroupindex
C2×#25012
C2×#18952
C2×#18752
C2×#18512
C2×#18242
C2×#17362
#138012
#113572
#110762
#109312
#108492