Finite Group Databasebeta

← #412598

Subgroups — #412598

Order 512.

1378 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2076C212yesmaximal
2C2×#1947C211yesmaximal, characteristic, complemented
3C2×#1797C211yesmaximal, characteristic
4#14686C211yesmaximal, characteristic
5C4×#380C211yesmaximal, characteristic
6#11316C212yesmaximal, complemented
7#11256C212yesmaximal, complemented
8#11200C212yesmaximal
9#10849C211yesmaximal, characteristic, complemented
10#10849C211yesmaximal, characteristic, complemented
11#10849C211yesmaximal, characteristic, complemented
12C23×C42C2211yescharacteristic
13C22×#107C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C22⋊C8C2212yes—
17C2×#384C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#380—21no—
20C2×#380—21no—
21C2×#380C2212yes—
22C2×#380C2211yescharacteristic
23C2×#380C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C22⋊C4—21no—
27#2076—28no—
28#1947—24no—
29#1797—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38C4⋊#95—21no—
39C4⋊#95—21no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43#1434—21no—
44C22⋊#95—21no—
45C4×#95—21no—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—22no—
64C22×C42—22no—
65C22×C42—22no—
66C22×C42—22no—
67C22×C42—22no—
68C22×C42—22no—
69C22×C42—22no—
70C22×C42—21no—
71C22×C42—21no—
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×#107—24no—
77C2×#107—24no—
78C2×#107—42no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8D414yes—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—42no—
88#384—22no—
89#384—22no—
90#384—22no—
91#384—22no—
92#380—22no—
93#380—22no—
94#380—22no—
95#380—24no—
96#380—24no—
97#380—44no—
98#380—44no—
99#380D414yes—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C22⋊C4—44no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—24no—
122#340—42no—
123#336—24no—
124C25C2×D411yescharacteristic
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—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—41no—
155C23×C4—41no—
156C23×C4Q1614yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4#4112yes—
164C23×C4C2411yescharacteristic
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8C2×D414yes—
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—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
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—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
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—42no—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4#4112yes—
226C2×C4⋊C4#4112yes—
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×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—44no—
249C2×C42—44no—
250C2×C42—44no—
251C2×C42—44no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42C2×D412yes—
259C2×C42#4112yes—
260C2×C42#4112yes—
261C2×C42#4112yes—
262C2×C42#4112yes—
263C2×C42#4112yes—
264#107—48no—
265C4⋊C8—28no—
266C4⋊C8—44no—
267C22⋊C8—48no—
268#95—24no—
269#95—24no—
270#95—42no—
271#95—42no—
272#95—42no—
273#95—44no—
274#95—42no—
275#95—44no—
276#95—42no—
277#95—42no—
278#95—44no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24—21no—
288C24—21no—
289C24—21no—
290C24—21no—
291C24#12112yes—
292C24C2×Q1612yes—
293C24#13712yes—
294C24C22×D411yescharacteristic
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—28no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
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—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—28no—
331C22×C4—28no—
332C22×C4—28no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
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—41no—
372C22×C4—41no—
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—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4#12112yes—
402C22×C4#12112yes—
403C22×C4#12112yes—
404C22×C4#12312yes—
405C22×C4#12312yes—
406C22×C4#12312yes—
407C22×C4#12312yes—
408C22×C4C2×Q1612yes—
409C22×C4C2×Q1612yes—
410C22×C4#13512yes—
411C22×C4AGL(1,ℤ/8)12yes—
412C22×C4AGL(1,ℤ/8)12yes—
413C22×C4AGL(1,ℤ/8)12yes—
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C2×#4111yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4H(5,𝔽2)11yescharacteristic
425C22×C42−1+411yescharacteristic
426C22×C42−1+411yescharacteristic
427C22×C42−1+411yescharacteristic
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
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—22no—
449C4⋊C4—22no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—84no—
467C4⋊C4—84no—
468C4⋊C4—84no—
469C4⋊C4—84no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—88no—
473C42—41no—
474C42—41no—
475C42—41no—
476C42—41no—
477C42—44no—
478C42—44no—
479C42—84no—
480C42—84no—
481C42#12314yes—
482C42#12414yes—
483C42#12614yes—
484C42#12614yes—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23#46214yes—
520C23C2×#12111yescharacteristic
521C23C2×#12111yescharacteristic
522C23C2×#12311yescharacteristic
523C23C2×#12311yescharacteristic
524C23#52511yescharacteristic
525C23#53611yescharacteristic
526C23#53711yescharacteristic
527C23C22×Q1611yescharacteristic
528C23C2×#13511yescharacteristic
529C23C2×AGL(1,ℤ/8)11yescharacteristic
530C23C2×AGL(1,ℤ/8)11yescharacteristic
531C23C2×AGL(1,ℤ/8)11yescharacteristic
532C23C2×#13711yescharacteristic
533C23#57511yescharacteristic
534C23#57511yescharacteristic
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—416no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—416no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4#39714yes—
615C2×C4#39714yes—
616C2×C4#46714yes—
617C2×C4#46914yes—
618C2×C4#48214yes—
619C2×C4#48214yes—
620C2×C4#48414yes—
621C2×C4#48414yes—
622C2×C4C2×#12311yescharacteristic
623C2×C4C2×#12411yescharacteristic
624C2×C4C2×#12611yescharacteristic
625C2×C4C2×#12611yescharacteristic
626C2×C4#52911yescharacteristic
627C2×C4#52911yescharacteristic
628C2×C4#53211yescharacteristic
629C2×C4#53211yescharacteristic
630C2×C4#54111yescharacteristic
631C2×C4#54111yescharacteristic
632C2×C4#54211yescharacteristic
633C2×C4#54211yescharacteristic
634C2×C4D4211yescharacteristic
635C2×C4#54711yescharacteristic
636C2×C4#54711yescharacteristic
637C2×C4#54711yescharacteristic
638C2×C4#54711yescharacteristic
639C2×C4#54811yescharacteristic
640C2×C4#54811yescharacteristic
641C2×C4#55211yescharacteristic
642C2×C4#55211yescharacteristic
643C2×C4#55211yescharacteristic
644C2×C4#55211yescharacteristic
645C2×C4#55311yescharacteristic
646C2×C4#55311yescharacteristic
647C2×C4#55511yescharacteristic
648C2×C4#55611yescharacteristic
649C2×C4#55611yescharacteristic
650C8—44no—
651C8—82no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22C2×#39711yescharacteristic
668C22#241511yescharacteristic
669C22#241611yescharacteristic
670C22#248511yescharacteristic
671C22#248611yescharacteristic
672C22#262511yescharacteristic
673C22#285411yescharacteristic
674C22C2×#46211yescharacteristic
675C22#303511yescharacteristic
676C22#303611yescharacteristic
677C22#303811yescharacteristic
678C22C2×#46711yescharacteristic
679C22C2×#46911yescharacteristic
680C22#305511yescharacteristic
681C22#305611yescharacteristic
682C22C2×#48211yescharacteristic
683C22C2×#48211yescharacteristic
684C22C2×#48411yescharacteristic
685C22C2×#48411yescharacteristic
686C22#309211yescharacteristic
687C22#309311yescharacteristic
688C22#309411yescharacteristic
689C22#309511yescharacteristic
690C22#310411yescharacteristic
691C22#310611yescharacteristic
692C22#311011yescharacteristic
693C22#311511yescharacteristic
694C22#323011yescharacteristic
695C22#323311yescharacteristic
696C22#323511yescharacteristic
697C22#323911yescharacteristic
698C22#324811yescharacteristic
699C22#324911yescharacteristic
700C22#325111yescharacteristic
701C22#325311yescharacteristic
702C4—24no—
703C4—42no—
704C4—42no—
705C4—42no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—162no—
711C4—162no—
712C4#207618yescomplemented
713C4#242111yescharacteristic
714C4#242211yescharacteristic
715C4#248911yescharacteristic
716C4#249111yescharacteristic
717C4#249411yescharacteristic
718C4#249511yescharacteristic
719C4#249611yescharacteristic
720C4#249611yescharacteristic
721C2—44no—
722C2#1513211yescharacteristic
723C2C2×#207611yescharacteristic
724C2#2364211yescharacteristic
725C2#2470211yescharacteristic
726C2#2470911yescharacteristic
727C2#2475211yescharacteristic
728C2#2475811yescharacteristic
729C2#2762611yescharacteristic
730C2#2762911yescharacteristic
731C2#2764611yescharacteristic
732C2#2764811yescharacteristic
733C2#2993911yescharacteristic
734C2#2994111yescharacteristic
735C2#3310911yescharacteristic
736C2#3311311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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
2. C2×#1947C2semidirect
6. #11316C2semidirect
7. #11256C2semidirect
9. #10849C2semidirect
10. #10849C2semidirect
11. #10849C2semidirect
712. 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×#19472
C2×#17972
#146862
C4×#3802
#113162
#112562
#112002
#108492