Finite Group Databasebeta

← #410469

Subgroups — #410469

Order 512.

1180 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1934C211yesmaximal, characteristic
2C2×#1926C211yesmaximal, characteristic
3C2×#1918C211yesmaximal, characteristic, complemented
4C2×C4×#106C211yesmaximal, characteristic, complemented
5#14050C211yesmaximal, characteristic, complemented
6#14001C211yesmaximal, characteristic
7#11378C211yesmaximal, characteristic, complemented
8#11371C211yesmaximal, characteristic, complemented
9#11325C211yesmaximal, characteristic, complemented
10#11323C211yesmaximal, characteristic, complemented
11#11123C211yesmaximal, characteristic
12#11121C211yesmaximal, characteristic
13#11121C211yesmaximal, characteristic
14#11051C211yesmaximal, characteristic
15#10797C211yesmaximal, characteristic, complemented
16C22×#107C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C22×#106C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C22×#102C2211yescharacteristic
21C22×#102C2211yescharacteristic
22C22×#102C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#388—22no—
30C2×#384C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1934—24no—
35#1926—24no—
36#1918—24no—
37C4×#106—24no—
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#340C2211yescharacteristic
46C2×#336C2211yescharacteristic
47C2×#336C2211yescharacteristic
48C2×#336C2211yescharacteristic
49C2×#336C2211yescharacteristic
50C2×#336C2211yescharacteristic
51#1443—22no—
52#1439—22no—
53#1436—22no—
54C4⋊#95C2211yescharacteristic
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2211yescharacteristic
58C4×#95C2211yescharacteristic
59C23×D4C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4—22no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—21no—
77C22×C42—21no—
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×#107—22no—
83C2×#107—22no—
84C2×#107—22no—
85C2×#106—22no—
86C2×#106—22no—
87C2×#106—22no—
88C2×#106—22no—
89C2×#106—22no—
90C2×#106—22no—
91C2×#106—22no—
92C2×#106—22no—
93C2×#106—41no—
94C2×#102—22no—
95C2×#102—22no—
96C2×#102—22no—
97C2×#102—22no—
98C2×#102—22no—
99C2×#102—22no—
100C2×#102—22no—
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—22no—
105C2×#102—41no—
106C2×C4×C8—22no—
107C2×C4×C8—22no—
108C2×C4×C8D414yes—
109#390—22no—
110#390—22no—
111#390—22no—
112#390—22no—
113#390—22no—
114#390—41no—
115#389—22no—
116#389—22no—
117#388—48no—
118#384—22no—
119#384—41no—
120#382—22no—
121#382—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95D412yes—
135C2×#95D412yes—
136#340—22no—
137#340—22no—
138#340—22no—
139#340—22no—
140#340—22no—
141#340—22no—
142#340—22no—
143#340—22no—
144#336—22no—
145#336—22no—
146#336—41no—
147#336—41no—
148#336D414yes—
149C25—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4#4112yes—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4C2×D412yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D412yes—
174C23×C4C2×D411yescharacteristic
175C23×C4C2411yescharacteristic
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8C2×D412yes—
199C22×C8C2×D412yes—
200C22×C8C2×D412yes—
201C22×C8C2×D412yes—
202C22×C8#4112yes—
203C22×C8#4112yes—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
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—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—82no—
246C2×C4⋊C4#4112yes—
247C2×C4⋊C4#4112yes—
248C2×C4⋊C4#4112yes—
249C2×C4⋊C4#4112yes—
250C2×C4⋊C4#4112yes—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—82no—
260C2×C22⋊C4—82no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—42no—
278C2×C42—42no—
279C2×C42—44no—
280C2×C42—82no—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283#107—28no—
284#106—28no—
285#106—28no—
286#106—44no—
287#102—28no—
288#102—28no—
289#102—44no—
290#102—44no—
291C4×C8—28no—
292#95—28no—
293#95—24no—
294#95—24no—
295#95—44no—
296#95—42no—
297#95—42no—
298#95—42no—
299#95—44no—
300#95—82no—
301#95—82no—
302C24—22no—
303C24—41no—
304C24—41no—
305C24—41no—
306C24—41no—
307C24—41no—
308C24—41no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24C22×D411yescharacteristic
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
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—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—41no—
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#12712yes—
398C22×C4#12712yes—
399C22×C4#13514yes—
400C22×C4AGL(1,ℤ/8)14yes—
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4H(5,𝔽2)11yescharacteristic
412C22×C42−1+411yescharacteristic
413C22×C42−1+411yescharacteristic
414C22×C42−1+411yescharacteristic
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8#12314yes—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—88no—
472C4⋊C4—84no—
473C22⋊C4—416no—
474C22⋊C4—84no—
475C22⋊C4—88no—
476C22⋊C4—84no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—41no—
482C42—41no—
483C42—41no—
484C42—41no—
485C42—41no—
486C42—41no—
487C42—41no—
488C42—41no—
489C42—84no—
490C42—84no—
491C23—44no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—82no—
497C23—82no—
498C23—82no—
499C23—82no—
500C23—82no—
501C23—82no—
502C23—82no—
503C23—82no—
504C23—81no—
505C23—81no—
506C23—81no—
507C23—81no—
508C23—81no—
509C23—81no—
510C23—81no—
511C23—82no—
512C23—81no—
513C23—82no—
514C23#52511yescharacteristic
515C23#52511yescharacteristic
516C23C2×#12711yescharacteristic
517C23#54611yescharacteristic
518C23#54711yescharacteristic
519C23#54711yescharacteristic
520C23#54811yescharacteristic
521C23C2×#13512yes—
522C23C2×AGL(1,ℤ/8)12yes—
523C23#57712yes—
524C23#57712yes—
525D4—42no—
526D4—42no—
527D4—42no—
528D4—42no—
529D4—81no—
530D4—81no—
531D4—81no—
532D4—81no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
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—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—82no—
591C2×C4#49614yes—
592C2×C4C2×#12311yescharacteristic
593C2×C4#52511yescharacteristic
594C2×C4#52511yescharacteristic
595C2×C4#52511yescharacteristic
596C2×C4#52511yescharacteristic
597C2×C4#52511yescharacteristic
598C2×C4#52911yescharacteristic
599C2×C4#52911yescharacteristic
600C2×C4#52911yescharacteristic
601C2×C4#52911yescharacteristic
602C2×C4#53611yescharacteristic
603C2×C4#53711yescharacteristic
604C2×C4#54111yescharacteristic
605C2×C4#54111yescharacteristic
606C2×C4#54211yescharacteristic
607C2×C4#54211yescharacteristic
608C2×C4#54611yescharacteristic
609C2×C4#54711yescharacteristic
610C2×C4#54711yescharacteristic
611C2×C4#54811yescharacteristic
612C2×C4#55211yescharacteristic
613C2×C4#55211yescharacteristic
614C2×C4#55211yescharacteristic
615C2×C4#55311yescharacteristic
616C2×C4#55511yescharacteristic
617C2×C4#55611yescharacteristic
618C2×C4#55611yescharacteristic
619C2×C4#55611yescharacteristic
620C8—24no—
621C8—42no—
622C8—42no—
623C8—42no—
624C22—82no—
625C22—84no—
626C22—84no—
627C22—82no—
628C22—82no—
629C22—162no—
630C22—161no—
631C22—161no—
632C22—161no—
633C22—162no—
634C22—161no—
635C22#239711yescharacteristic
636C22#246811yescharacteristic
637C22#246911yescharacteristic
638C22#247411yescharacteristic
639C22#247511yescharacteristic
640C22#250111yescharacteristic
641C22#254011yescharacteristic
642C22#310512yes—
643C22#310612yes—
644C22#312512yes—
645C22#312512yes—
646C22#314711yescharacteristic
647C22C2×#49611yescharacteristic
648C22#315411yescharacteristic
649C22#315411yescharacteristic
650C22#330112yes—
651C22#332412yes—
652C22#333712yes—
653C22#334012yes—
654C22#334512yes—
655C22#334512yes—
656C22#336012yes—
657C22#336812yes—
658C4—24no—
659C4—42no—
660C4—44no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—161no—
668C4—161no—
669C4#246811yescharacteristic
670C4#246911yescharacteristic
671C4#247411yescharacteristic
672C4#247511yescharacteristic
673C4#248511yescharacteristic
674C4#259911yescharacteristic
675C4#263711yescharacteristic
676C4#285311yescharacteristic
677C2—162no—
678C2#1507911yescharacteristic
679C2#2379611yescharacteristic
680C2#2381311yescharacteristic
681C2#2709712yes—
682C2#2713912yes—
683C2#2733512yes—
684C2#2736512yes—
685C2#2792511yescharacteristic
686C2#2793411yescharacteristic
687C2#2840611yescharacteristic
688C2#2842011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1918C2semidirect
4. C2×C4×#106C2semidirect
5. #14050C2semidirect
7. #11378C2semidirect
8. #11371C2semidirect
9. #11325C2semidirect
10. #11323C2semidirect
15. #10797C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19342
C2×#19262
C2×#19182
C2×C4×#1062
#140502
#140012
#113782
#113712
#113252
#113232
#111232
#111212
#110512
#107972