Finite Group Databasebeta

← #412943

Subgroups — #412943

Order 512.

1396 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×#2076C212yesmaximal, complemented
2C2×#1954C211yesmaximal, characteristic
3C2×#1947C211yesmaximal, characteristic, complemented
4#13906C211yesmaximal, characteristic
5#13906C211yesmaximal, characteristic
6#11200C212yesmaximal, complemented
7#11200C212yesmaximal, complemented
8#11096C212yesmaximal, complemented
9#10849C211yesmaximal, characteristic, complemented
10#10841C211yesmaximal, characteristic, complemented
11C4×#340C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#107C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C4×C8C2211yescharacteristic
18C2×#390—21no—
19C2×#390—21no—
20C2×#384C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#380C2212yes—
23C2×#380C2212yes—
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#2076—28no—
29#1954—24no—
30#1947—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2211yescharacteristic
40#1439—22no—
41#1439—22no—
42C4⋊#95—21no—
43C4⋊#95—21no—
44C23×D4D411yescharacteristic
45C24×C4C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4D412yes—
71C22×C42—21no—
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74C2×#107—24no—
75C2×#107—24no—
76C2×#107—42no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8D414yes—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—42no—
85C2×C22⋊C8—42no—
86C2×C4×C8—22no—
87C2×C4×C8—22no—
88C2×C4×C8—22no—
89#390—44no—
90#390—44no—
91#384—22no—
92#384—22no—
93#384—22no—
94#384—22no—
95#380—24no—
96#380—24no—
97#380—24no—
98#380—24no—
99#380—42no—
100#380—42no—
101#380—42no—
102#380—42no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—21no—
113#340—24no—
114#340—24no—
115#340—22no—
116#340—24no—
117#340—22no—
118#340—22no—
119#340—42no—
120#340—42no—
121#340D414yescomplemented
122C25C2×D412yes—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4Q1614yes—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8C2×D412yes—
173C22×C8C2×D414yes—
174C22×C8#4112yes—
175C4×D4—44no—
176C4×D4—44no—
177C4×D4—44no—
178C4×D4—44no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—81no—
223C2×C4⋊C4—81no—
224C2×C4⋊C4—81no—
225C2×C4⋊C4—81no—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4#4112yes—
229C2×C4⋊C4#4112yes—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—82no—
242C2×C22⋊C4—82no—
243C2×C22⋊C4—82no—
244C2×C22⋊C4—82no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42C2×D412yes—
252C2×C42#4112yes—
253C2×C42#4112yes—
254C2×C42#4112yes—
255C2×C42#4112yes—
256C2×C42#4112yes—
257#107—48no—
258C4⋊C8—28no—
259C22⋊C8—48no—
260C22⋊C8—48no—
261C4×C8—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—42no—
267#95—42no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24C2×Q1614yes—
283C24C22×D411yescharacteristic
284C2×D4—28no—
285C2×D4—28no—
286C2×D4—28no—
287C2×D4—44no—
288C2×D4—44no—
289C2×D4—44no—
290C2×D4—44no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4#12112yes—
373C22×C4#12112yes—
374C22×C4#12312yes—
375C22×C4#12312yes—
376C22×C4#12312yes—
377C22×C4#12312yes—
378C22×C4C2×Q1612yes—
379C22×C4#13512yes—
380C22×C4AGL(1,ℤ/8)12yes—
381C22×C4AGL(1,ℤ/8)12yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4H(5,𝔽2)11yescharacteristic
393C22×C4H(5,𝔽2)11yescharacteristic
394C22×C4H(5,𝔽2)11yescharacteristic
395C22×C42−1+411yescharacteristic
396C2×C8—24no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—24no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—42no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—42no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—44no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—84no—
443C4⋊C4—84no—
444C4⋊C4—84no—
445C4⋊C4—84no—
446C22⋊C4—48no—
447C22⋊C4—88no—
448C22⋊C4—88no—
449C22⋊C4—88no—
450C22⋊C4—88no—
451C42—84no—
452C42—84no—
453C42#12314yes—
454C42#12414yes—
455C42#12614yes—
456C42#12614yes—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23C2×#12111yescharacteristic
493C23C2×#12311yescharacteristic
494C23C2×#12311yescharacteristic
495C23D4211yescharacteristic
496C23#54611yescharacteristic
497C23#54611yescharacteristic
498C23#54711yescharacteristic
499C23C22×Q1611yescharacteristic
500C23C2×#13511yescharacteristic
501C23C2×AGL(1,ℤ/8)11yescharacteristic
502C23C2×AGL(1,ℤ/8)11yescharacteristic
503C23#57611yescharacteristic
504C23#57711yescharacteristic
505C23#57711yescharacteristic
506C23#57811yescharacteristic
507D4—416no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4#39714yes—
572C2×C4#39714yes—
573C2×C4#46714yes—
574C2×C4#46914yes—
575C2×C4#48214yes—
576C2×C4#48214yes—
577C2×C4#48414yes—
578C2×C4#48414yes—
579C2×C4C2×#12311yescharacteristic
580C2×C4C2×#12411yescharacteristic
581C2×C4C2×#12611yescharacteristic
582C2×C4C2×#12611yescharacteristic
583C2×C4#52911yescharacteristic
584C2×C4#52911yescharacteristic
585C2×C4#52911yescharacteristic
586C2×C4#53211yescharacteristic
587C2×C4#53211yescharacteristic
588C2×C4#53211yescharacteristic
589C2×C4#54011yescharacteristic
590C2×C4#54011yescharacteristic
591C2×C4#54011yescharacteristic
592C2×C4#54211yescharacteristic
593C2×C4#54211yescharacteristic
594C2×C4#54211yescharacteristic
595C2×C4D4211yescharacteristic
596C2×C4#54611yescharacteristic
597C2×C4#54711yescharacteristic
598C2×C4#54811yescharacteristic
599C2×C4#55011yescharacteristic
600C2×C4#55011yescharacteristic
601C2×C4#55311yescharacteristic
602C2×C4#55311yescharacteristic
603C2×C4#55511yescharacteristic
604C2×C4#55611yescharacteristic
605C2×C4#55911yescharacteristic
606C2×C4#55911yescharacteristic
607C8—44no—
608C8—42no—
609C8—42no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22C2×#39711yescharacteristic
626C22#242011yescharacteristic
627C22#242211yescharacteristic
628C22#242711yescharacteristic
629C22#242811yescharacteristic
630C22#244511yescharacteristic
631C22#244711yescharacteristic
632C22C2×#46711yescharacteristic
633C22C2×#46911yescharacteristic
634C22#306211yescharacteristic
635C22#306311yescharacteristic
636C22C2×#48211yescharacteristic
637C22C2×#48211yescharacteristic
638C22C2×#48411yescharacteristic
639C22C2×#48411yescharacteristic
640C22#309611yescharacteristic
641C22#309711yescharacteristic
642C22#309811yescharacteristic
643C22#309911yescharacteristic
644C22#327211yescharacteristic
645C22#327411yescharacteristic
646C22#328411yescharacteristic
647C22D4×Q1611yescharacteristic
648C22#329911yescharacteristic
649C22#329911yescharacteristic
650C22#330211yescharacteristic
651C22#330511yescharacteristic
652C22#332411yescharacteristic
653C22#332511yescharacteristic
654C22#333011yescharacteristic
655C22#333211yescharacteristic
656C22#333811yescharacteristic
657C22#334011yescharacteristic
658C22#335311yescharacteristic
659C22#335411yescharacteristic
660C4—24no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—162no—
667C4—162no—
668C4#207618yescomplemented
669C4#244511yescharacteristic
670C4#244811yescharacteristic
671C4#249011yescharacteristic
672C4#249111yescharacteristic
673C4#251211yescharacteristic
674C4#251311yescharacteristic
675C4#264411yescharacteristic
676C4#286011yescharacteristic
677C2—84no—
678C2#1513811yescharacteristic
679C2C2×#207611yescharacteristic
680C2#2364311yescharacteristic
681C2#2496811yescharacteristic
682C2#2497111yescharacteristic
683C2#2501311yescharacteristic
684C2#2501611yescharacteristic
685C2#2534011yescharacteristic
686C2#2534711yescharacteristic
687C2#2538611yescharacteristic
688C2#2539211yescharacteristic
689C2#2623511yescharacteristic
690C2#2623611yescharacteristic
691C2#2630111yescharacteristic
692C2#2630211yescharacteristic
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
1. C2×#2076C2semidirect
3. C2×#1947C2semidirect
6. #11200C2semidirect
7. #11200C2semidirect
8. #11096C2semidirect
9. #10849C2semidirect
10. #10841C2semidirect
11. C4×#340C2semidirect
121. #340D4semidirect
668. C4#2076semidirect

Maximal subgroups

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

subgroupindex
C2×#20762
C2×#19542
C2×#19472
#139062
#112002
#110962
#108492
#108412
C4×#3402