Finite Group Databasebeta

← #366394

Subgroups — #366394

Order 512.

1636 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1853C211yesmaximal, characteristic
2C2×#1823C211yesmaximal, characteristic
3C2×#1823C211yesmaximal, characteristic
4C4×#380C211yesmaximal, characteristic
5C2×C4⋊#95C211yesmaximal, characteristic
6#11316C212yesmaximal, complemented
7#11257C212yesmaximal, complemented
8#10908C212yesmaximal, complemented
9#10807C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10712C211yesmaximal, characteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×#106C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C2×#380C2211yescharacteristic
19C2×#380C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C22⋊C4—21no—
23C22×#95—21no—
24C22×#95—21no—
25#1853—24no—
26#1823—24no—
27#1823C418yescomplemented
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C4⋊#95—21no—
38C4⋊#95—21no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43#1434—21no—
44C22⋊#95—21no—
45C4×#95—21no—
46C24×C4—21no—
47C24×C4—21no—
48C24×C4D411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—24no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—42no—
64C22×C4⋊C4C2×C414yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C42—22no—
74C22×C42—22no—
75C22×C42—24no—
76C22×C42—21no—
77C22×C42—21no—
78C22×C42—42no—
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×#106—24no—
84C2×#106—24no—
85C2×#106—42no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8C2×C414yes—
92#380—22no—
93#380—22no—
94#380C2×C414yes—
95#380D414yes—
96C4×C4⋊C4—24no—
97C4×C4⋊C4—42no—
98C4×C22⋊C4—44no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—44no—
112C2×#95—44no—
113#340—24no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—42no—
118#340C2×C418yescomplemented
119#340C2×C418yescomplemented
120#336—24no—
121#336—42no—
122C25C2×D411yescharacteristic
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—41no—
145C23×C4—41no—
146C23×C4—41no—
147C23×C4—41no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4SD1614yes—
161C23×C4C22×C412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4#4112yes—
169C23×C4#4112yes—
170C23×C4#4112yes—
171C23×C4#4112yes—
172C23×C4C2411yescharacteristic
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8C22×C414yes—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—48no—
214C2×C4⋊C4—48no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4C22×C414yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4#4114yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—48no—
245C2×C42—48no—
246C2×C42—41no—
247C2×C42—41no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42#4114yes—
255#106—48no—
256C22⋊C8—28no—
257C22⋊C8—44no—
258#95—24no—
259#95—24no—
260#95—42no—
261#95—44no—
262#95—44no—
263#95—42no—
264#95—42no—
265#95—42no—
266#95—42no—
267#95—44no—
268#95—44no—
269#95—42no—
270C24—21no—
271C24—21no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24C4×D412yes—
279C24#12112yes—
280C24#12312yes—
281C24C2×SD1612yes—
282C24#13512yes—
283C24AGL(1,ℤ/8)12yes—
284C24AGL(1,ℤ/8)12yes—
285C24C22×D411yescharacteristic
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—28no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—28no—
326C22×C4—28no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—48no—
345C22×C4—48no—
346C22×C4—48no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—48no—
384C22×C4—48no—
385C22×C4—48no—
386C22×C4—84no—
387C22×C4—84no—
388C22×C4—84no—
389C22×C4—84no—
390C22×C4—84no—
391C22×C4—84no—
392C22×C4—84no—
393C22×C4—84no—
394C22×C4#11714yes—
395C22×C4C4×D414yes—
396C22×C4C4×D412yes—
397C22×C4#12112yes—
398C22×C4#12312yes—
399C22×C4#12312yes—
400C22×C4#12312yes—
401C22×C4#12414yes—
402C22×C4#12614yes—
403C22×C4C2×SD1612yes—
404C22×C4C2×SD1612yes—
405C22×C4#13712yes—
406C22×C4#13712yes—
407C22×C4C23×C411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4112yes—
413C22×C4C2×#4112yes—
414C22×C4C2×#4111yescharacteristic
415C22×C4H(5,𝔽2)12yes—
416C22×C42−1+412yes—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—48no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4—48no—
451C42—24no—
452C42—24no—
453C42—44no—
454C42—44no—
455C42—42no—
456C42—42no—
457C42—84no—
458C42—84no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23#38814yes—
488C23C4×SD1614yes—
489C23#44214yes—
490C23#46014yes—
491C23#46314yes—
492C23#48114yes—
493C23#48214yes—
494C23C2×C4×D411yescharacteristic
495C23C2×#12111yescharacteristic
496C23C2×#12311yescharacteristic
497C23C2×#12311yescharacteristic
498C23#52511yescharacteristic
499C23#53411yescharacteristic
500C23#53611yescharacteristic
501C23C22×SD1611yescharacteristic
502C23C2×#13511yescharacteristic
503C23C2×AGL(1,ℤ/8)11yescharacteristic
504C23C2×AGL(1,ℤ/8)11yescharacteristic
505C23C2×#13711yescharacteristic
506C23C2×#13711yescharacteristic
507C23#57511yescharacteristic
508C23#57511yescharacteristic
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—416no—
535C2×C4—416no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—816no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—82no—
583C2×C4—816no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4#38814yes—
593C2×C4#48314yes—
594C2×C4C2×#11711yescharacteristic
595C2×C4C2×C4×D411yescharacteristic
596C2×C4C4×#4112yes—
597C2×C4#52012yes—
598C2×C4C2×#12411yescharacteristic
599C2×C4C2×#12611yescharacteristic
600C2×C4#52912yes—
601C2×C4#53912yes—
602C2×C4#54612yes—
603C2×C4#54712yes—
604C2×C4#55112yes—
605C2×C4#55112yes—
606C2×C4#55112yes—
607C2×C4#55212yes—
608C2×C4#55212yes—
609C2×C4#55212yes—
610C8—44no—
611C8—82no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22#192618yes—
627C22C2×#38811yescharacteristic
628C22C4×#12311yescharacteristic
629C22#232211yescharacteristic
630C22#248311yescharacteristic
631C22#248411yescharacteristic
632C22#248611yescharacteristic
633C22#253411yescharacteristic
634C22C2×C4×SD1611yescharacteristic
635C22C2×#44211yescharacteristic
636C22C4×#13711yescharacteristic
637C22C4×#13711yescharacteristic
638C22C2×#46311yescharacteristic
639C22C2×#46011yescharacteristic
640C22#303911yescharacteristic
641C22#304011yescharacteristic
642C22C2×#48311yescharacteristic
643C22C2×#48211yescharacteristic
644C22C2×#48111yescharacteristic
645C22#309111yescharacteristic
646C22#309211yescharacteristic
647C22#309411yescharacteristic
648C22#309511yescharacteristic
649C22#309511yescharacteristic
650C22#310111yescharacteristic
651C22#310811yescharacteristic
652C22#311211yescharacteristic
653C22#311411yescharacteristic
654C22#318711yescharacteristic
655C22#319011yescharacteristic
656C22#319311yescharacteristic
657C22#319411yescharacteristic
658C22#322911yescharacteristic
659C22#323211yescharacteristic
660C22#323411yescharacteristic
661C22#323811yescharacteristic
662C4—24no—
663C4—24no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—82no—
668C4—84no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—84no—
673C4#234312yes—
674C4#235912yes—
675C4#248812yes—
676C4#249312yes—
677C2—28no—
678C2#1399211yescharacteristic
679C2C2×#192611yescharacteristic
680C2#2106111yescharacteristic
681C2C4×#48311yescharacteristic
682C2#2157811yescharacteristic
683C2#2207811yescharacteristic
684C2#2208111yescharacteristic
685C2#2760111yescharacteristic
686C2#2760211yescharacteristic
687C2#2761311yescharacteristic
688C2#2761611yescharacteristic
689C2#2764711yescharacteristic
690C2#2765111yescharacteristic
691C2#2811811yescharacteristic
692C2#2812011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11316C2semidirect
7. #11257C2semidirect
8. #10908C2semidirect
9. #10807C2semidirect
27. #1823C4semidirect
118. #340C2×C4semidirect
119. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18532
C2×#18232
C4×#3802
C2×C4⋊#952
#113162
#112572
#109082
#108072
#107302
#107122