Finite Group Databasebeta

← #454519

Subgroups — #454519

Order 512.

1720 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#2851C211yesmaximal, characteristic
3C2×#2059C212yesmaximal, complemented
4C2×#2058C211yesmaximal, characteristic, complemented
5C2×#2057C211yesmaximal, characteristic, complemented
6#11208C212yesmaximal, complemented
7#11207C211yesmaximal, characteristic, complemented
8#11207C211yesmaximal, characteristic, complemented
9#11096C212yesmaximal
10#11095C212yesmaximal
11#10712C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×C22⋊Q8—21no—
15C23×C42C2211yescharacteristic
16C22×#107C2212yes—
17C22×#106C2212yes—
18C22×C22⋊C8C2212yes—
19#2857—28no—
20#2851—24no—
21#2851—24no—
22C2×#391—21no—
23C2×#389—21no—
24C2×#389C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#389C2212yes—
27C2×#387—21no—
28C2×#387—21no—
29C2×#385—21no—
30C2×#385—21no—
31C2×#384C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#380—21no—
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×#380C2212yes—
37#2059—28no—
38#2058—24no—
39#2057—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#336C2211yescharacteristic
47C2×#336C2211yescharacteristic
48C2×#336C2212yes—
49C23×Q8—21no—
50C24×C4D411yescharacteristic
51C24×C4D411yescharacteristic
52C24×C4C2311yescharacteristic
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—42no—
59C2×#125—22no—
60C2×#125—22no—
61C2×#125—24no—
62C2×#125—42no—
63C2×C22⋊Q8—28no—
64C2×C22⋊Q8—44no—
65C2×C22⋊Q8—44no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C42—22no—
84C22×C42—22no—
85C22×C42—24no—
86C22×C42—42no—
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C2×#107—24no—
92C2×#107—24no—
93C2×#107—42no—
94C2×#106—24no—
95C2×#106—24no—
96C2×#106—42no—
97C2×C22⋊C8—42no—
98C2×C22⋊C8—42no—
99C2×C22⋊C8Q818yescomplemented
100#391—28no—
101#389—28no—
102#389—22no—
103#389—22no—
104#389—24no—
105#389D414yes—
106#389D414yes—
107#389D418yes—
108#387—28no—
109#387—28no—
110#385—28no—
111#385—28no—
112#384—24no—
113#382—22no—
114#382—22no—
115#380—28no—
116#380—22no—
117#380—22no—
118#380—22no—
119#380—22no—
120#380—24no—
121#380—24no—
122C2×#95—22no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130#340—24no—
131#340—24no—
132#340—24no—
133#340—24no—
134#340—42no—
135#340—42no—
136#336—22no—
137#336—22no—
138#336—24no—
139C25C2×D411yescharacteristic
140C22×Q8—24no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×Q8—42no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—41no—
173C23×C4—41no—
174C23×C4—41no—
175C23×C4—41no—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×Q814yes—
182C23×C4C2411yescharacteristic
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8C2×Q814yes—
196C22×C8C2×Q814yes—
197C4⋊Q8—48no—
198C4⋊Q8—48no—
199#125—48no—
200#125—48no—
201C22⋊Q8—432no—
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—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2×D414yes—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D414yes—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—42no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—48no—
277C2×C42—48no—
278C2×C42—41no—
279C2×C42—41no—
280C2×C42C2×D414yes—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283#107—48no—
284#106—48no—
285C22⋊C8—48no—
286#95—24no—
287#95—24no—
288#95—28no—
289#95—24no—
290#95—24no—
291#95—28no—
292#95—24no—
293#95—24no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—21no—
302C24—21no—
303C24—21no—
304C24—21no—
305C24—21no—
306C24C4⋊Q812yes—
307C24AGL(1,ℤ/8)12yes—
308C24#13712yes—
309C24C22×D411yescharacteristic
310C2×Q8—48no—
311C2×Q8—48no—
312C2×Q8—48no—
313C2×Q8—84no—
314C2×Q8—84no—
315C2×Q8—84no—
316C2×Q8—84no—
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—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4#12012yes—
416C22×C4#12012yes—
417C22×C4#12012yes—
418C22×C4#12012yes—
419C22×C4C4⋊Q812yes—
420C22×C4#13512yes—
421C22×C4#13512yes—
422C22×C4AGL(1,ℤ/8)12yes—
423C22×C4#13712yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D412yes—
427C22×C4C22×Q812yes—
428C22×C4H(5,𝔽2)11yescharacteristic
429C22×C4H(5,𝔽2)12yes—
430C22×C42−1+411yescharacteristic
431C22×C42−1+412yes—
432C22×C42−1+411yescharacteristic
433C22×C42−1+411yescharacteristic
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—42no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—416no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—42no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—416no—
479C42—24no—
480C42—24no—
481C42—24no—
482C42—24no—
483C42—44no—
484C42—44no—
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#50114yes—
520C23C2×#12011yescharacteristic
521C23C2×#12011yescharacteristic
522C23C2×C4⋊Q811yescharacteristic
523C23#53411yescharacteristic
524C23#53611yescharacteristic
525C23#53611yescharacteristic
526C23#53711yescharacteristic
527C23C2×#13511yescharacteristic
528C23C2×#13511yescharacteristic
529C23C2×AGL(1,ℤ/8)11yescharacteristic
530C23C2×AGL(1,ℤ/8)11yescharacteristic
531C23C2×#13711yescharacteristic
532C23C2×#13711yescharacteristic
533C23#57511yescharacteristic
534C23#57511yescharacteristic
535Q8—816no—
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—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—416no—
558C2×C4—416no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—81no—
594C2×C4—82no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—82no—
598C2×C4—81no—
599C2×C4—82no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—82no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—82no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—82no—
610C2×C4—81no—
611C2×C4—82no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—82no—
617C2×C4#44914yes—
618C2×C4#45214yes—
619C2×C4#53611yescharacteristic
620C2×C4#53611yescharacteristic
621C2×C4#53612yes—
622C2×C4#53711yescharacteristic
623C2×C4#53711yescharacteristic
624C2×C4#53712yes—
625C2×C4#54312yes—
626C2×C4#54412yes—
627C2×C4D4212yes—
628C2×C4D4212yes—
629C2×C4D4×Q812yes—
630C2×C4D4×Q812yes—
631C2×C4D4×Q812yes—
632C2×C4D4×Q812yes—
633C2×C4#56312yes—
634C2×C4#56312yes—
635C8—44no—
636C8—44no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22#240311yescharacteristic
653C22#243411yescharacteristic
654C22#243411yescharacteristic
655C22#261611yescharacteristic
656C22#261611yescharacteristic
657C22#265911yescharacteristic
658C22#281711yescharacteristic
659C22C2×#44911yescharacteristic
660C22C2×#45211yescharacteristic
661C22#300211yescharacteristic
662C22#300211yescharacteristic
663C22#300311yescharacteristic
664C22#300411yescharacteristic
665C22#300611yescharacteristic
666C22#300711yescharacteristic
667C22#316011yescharacteristic
668C22C2×#50111yescharacteristic
669C22#316511yescharacteristic
670C22#316611yescharacteristic
671C22#318811yescharacteristic
672C22#319011yescharacteristic
673C22#319411yescharacteristic
674C22#319611yescharacteristic
675C22#322811yescharacteristic
676C22#323111yescharacteristic
677C22#323311yescharacteristic
678C22#323311yescharacteristic
679C22#323411yescharacteristic
680C22#323611yescharacteristic
681C22#323811yescharacteristic
682C22#323911yescharacteristic
683C22#324711yescharacteristic
684C22#324811yescharacteristic
685C22#325111yescharacteristic
686C22#325211yescharacteristic
687C4—24no—
688C4—24no—
689C4—42no—
690C4—42no—
691C4—42no—
692C4—82no—
693C4—84no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4—84no—
698C4#267712yes—
699C4#267712yes—
700C4#268312yes—
701C4#268312yes—
702C2—44no—
703C2#1625411yescharacteristic
704C2#2392011yescharacteristic
705C2#2392211yescharacteristic
706C2#2581211yescharacteristic
707C2#2581411yescharacteristic
708C2#2581611yescharacteristic
709C2#2581911yescharacteristic
710C2#2968911yescharacteristic
711C2#2969211yescharacteristic
712C2#2969411yescharacteristic
713C2#2969511yescharacteristic
714C2#3029411yescharacteristic
715C2#3029511yescharacteristic
716C2#3278711yescharacteristic
717C2#3278911yescharacteristic
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
3. C2×#2059C2semidirect
4. C2×#2058C2semidirect
5. C2×#2057C2semidirect
6. #11208C2semidirect
7. #11207C2semidirect
8. #11207C2semidirect
99. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#28512
C2×#20592
C2×#20582
C2×#20572
#112082
#112072
#110962
#110952
#107122