Finite Group Databasebeta

← #394827

Subgroups — #394827

Order 512.

1904 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#2296C211yesmaximal, characteristic
3C2×#2058C211yesmaximal, characteristic, complemented
4C2×#2057C211yesmaximal, characteristic, complemented
5C2×#1854C212yesmaximal
6#11328C212yesmaximal, complemented
7#11327C212yesmaximal, complemented
8#11207C211yesmaximal, characteristic, complemented
9#11207C211yesmaximal, characteristic, complemented
10#10731C212yesmaximal
11#10712C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8—21no—
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C22×#107C2212yes—
18C22×#106C2212yes—
19C22×C22⋊C8C2212yes—
20#2857—28no—
21#2296—24no—
22#2296—24no—
23C2×#391—21no—
24C2×#389—21no—
25C2×#389C2212yes—
26C2×#389C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#385—21no—
31C2×#385—21no—
32C2×#384C2211yescharacteristic
33C2×#380—21no—
34C2×#380C2212yes—
35C2×C4×C4⋊C4—21no—
36C2×C4×C4⋊C4—21no—
37#2058—24no—
38#2057—24no—
39#1854—28no—
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×C22⋊Q8—28no—
60C2×C22⋊Q8—44no—
61C2×C22⋊Q8—44no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—24no—
70C2×#117—42no—
71C2×#117Q814yescomplemented
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4—22no—
89C22×C42—21no—
90C22×C42—24no—
91C22×C42—21no—
92C22×C42—22no—
93C22×C42—22no—
94C22×C42—42no—
95C22×C42D411yescharacteristic
96C22×C42D411yescharacteristic
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99C2×#107—24no—
100C2×#107—24no—
101C2×#107—42no—
102C2×#106—24no—
103C2×#106—24no—
104C2×#106—42no—
105C2×C22⋊C8—24no—
106C2×C22⋊C8—24no—
107C2×C22⋊C8Q818yescomplemented
108#391—28no—
109#389—28no—
110#389—24no—
111#389—24no—
112#389—22no—
113#389—22no—
114#389—22no—
115#389—22no—
116#389—22no—
117#389—22no—
118#389D414yes—
119#389D414yes—
120#385—28no—
121#385—28no—
122#384—24no—
123#380—28no—
124#380—24no—
125#380D418yes—
126C4×C4⋊C4—28no—
127C4×C4⋊C4—28no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—22no—
133C2×#95—22no—
134#340—24no—
135#340—24no—
136#340—24no—
137#340—24no—
138#340—24no—
139#340—24no—
140#336—22no—
141#336—22no—
142#336—42no—
143C25C2×D411yescharacteristic
144C22×Q8—24no—
145C22×Q8—42no—
146C22×Q8—42no—
147C22×Q8—42no—
148C22×Q8—42no—
149C22×Q8—42no—
150C22×Q8—42no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—41no—
174C23×C4—41no—
175C23×C4—41no—
176C23×C4—41no—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D414yes—
183C23×C4C2×Q814yes—
184C23×C4C2×Q812yes—
185C23×C4#4112yes—
186C23×C4C2411yescharacteristic
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×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8C2×Q814yes—
200C22×C8C2×Q814yes—
201C4⋊Q8—48no—
202C4⋊Q8—48no—
203C22⋊Q8—432no—
204#117—28no—
205#117—28no—
206#117—28no—
207#117—28no—
208#117—48no—
209#117—48no—
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—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
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⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×Q812yes—
263C2×C4⋊C4C2×Q812yes—
264C2×C4⋊C4#4112yes—
265C2×C4⋊C4#4112yes—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—42no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—22no—
289C2×C42—48no—
290C2×C42—48no—
291C2×C42—41no—
292C2×C42—41no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42C2×D412yes—
296C2×C42C2×D412yes—
297#107—48no—
298#106—48no—
299C22⋊C8—216no—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—24no—
304#95—24no—
305#95—28no—
306C24—21no—
307C24—21no—
308C24—21no—
309C24—21no—
310C24—21no—
311C24—21no—
312C24—21no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24#12012yes—
319C24C22⋊Q812yes—
320C24C4⋊Q812yes—
321C24C22×D411yescharacteristic
322C2×Q8—48no—
323C2×Q8—48no—
324C2×Q8—48no—
325C2×Q8—84no—
326C2×Q8—84no—
327C2×Q8—84no—
328C2×Q8—84no—
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—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—42no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—42no—
423C22×C4#12012yes—
424C22×C4#12012yes—
425C22×C4#12012yes—
426C22×C4C22⋊Q818yes—
427C22×C4C22⋊Q812yes—
428C22×C4C4⋊Q818yes—
429C22×C4C4⋊Q812yes—
430C22×C4#13512yes—
431C22×C4#13512yes—
432C22×C4AGL(1,ℤ/8)12yes—
433C22×C4AGL(1,ℤ/8)12yes—
434C22×C4#13712yes—
435C22×C4#13712yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×Q811yescharacteristic
440C22×C4C22×Q812yes—
441C22×C4C2×#4111yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4C2×#4111yescharacteristic
444C22×C4H(5,𝔽2)12yes—
445C22×C42−1+412yes—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—416no—
477C4⋊C4—48no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4C22⋊Q814yescomplemented
487C4⋊C4C22⋊Q814yescomplemented
488C22⋊C4—216no—
489C22⋊C4—48no—
490C22⋊C4—416no—
491C42—216no—
492C42—24no—
493C42—24no—
494C42—44no—
495C42—44no—
496C42—48no—
497C42—42no—
498C42—42no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23#39514yes—
534C23C2×#12011yescharacteristic
535C23C2×#12011yescharacteristic
536C23C2×C22⋊Q811yescharacteristic
537C23#52511yescharacteristic
538C23C2×C4⋊Q811yescharacteristic
539C23#53411yescharacteristic
540C23#53611yescharacteristic
541C23C2×#13511yescharacteristic
542C23C2×#13511yescharacteristic
543C23C2×AGL(1,ℤ/8)11yescharacteristic
544C23C2×AGL(1,ℤ/8)11yescharacteristic
545C23C2×#13711yescharacteristic
546C23C2×#13711yescharacteristic
547C23#57511yescharacteristic
548C23#57511yescharacteristic
549Q8—816no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—416no—
581C2×C4—416no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—81no—
617C2×C4—82no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—82no—
621C2×C4—81no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4—81no—
629C2×C4—82no—
630C2×C4#39514yes—
631C2×C4#44914yes—
632C2×C4#45214yes—
633C2×C4C2×C22⋊Q811yescharacteristic
634C2×C4C2×C22⋊Q811yescharacteristic
635C2×C4C2×C22⋊Q812yes—
636C2×C4#52511yescharacteristic
637C2×C4#52511yescharacteristic
638C2×C4C2×C4⋊Q812yes—
639C2×C4#53912yes—
640C2×C4#54412yes—
641C2×C4D4212yes—
642C2×C4D4212yes—
643C2×C4#54712yes—
644C2×C4#54712yes—
645C2×C4D4×Q812yes—
646C2×C4D4×Q812yes—
647C2×C4#55412yes—
648C2×C4#55412yes—
649C8—44no—
650C8—44no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22C2×#39511yescharacteristic
667C22#240311yescharacteristic
668C22#240611yescharacteristic
669C22#243411yescharacteristic
670C22#258311yescharacteristic
671C22#262711yescharacteristic
672C22#284811yescharacteristic
673C22C2×#44911yescharacteristic
674C22C2×#45211yescharacteristic
675C22#300211yescharacteristic
676C22#300211yescharacteristic
677C22#300311yescharacteristic
678C22#300411yescharacteristic
679C22#300611yescharacteristic
680C22#300711yescharacteristic
681C22#307611yescharacteristic
682C22#307611yescharacteristic
683C22#307911yescharacteristic
684C22#307911yescharacteristic
685C22#310611yescharacteristic
686C22#310711yescharacteristic
687C22#311511yescharacteristic
688C22#311611yescharacteristic
689C22#316011yescharacteristic
690C22#316211yescharacteristic
691C22#316511yescharacteristic
692C22#316511yescharacteristic
693C22#318811yescharacteristic
694C22#319011yescharacteristic
695C22#319411yescharacteristic
696C22#319611yescharacteristic
697C22#322811yescharacteristic
698C22#323111yescharacteristic
699C22#323511yescharacteristic
700C22#323711yescharacteristic
701C4—24no—
702C4—24no—
703C4—24no—
704C4—42no—
705C4—42no—
706C4—44no—
707C4—44no—
708C4—82no—
709C4—84no—
710C4—82no—
711C4—84no—
712C4#244012yes—
713C4#244012yes—
714C4#244912yes—
715C4#244912yes—
716C2—44no—
717C2#1472011yescharacteristic
718C2#2355111yescharacteristic
719C2#2355211yescharacteristic
720C2#2392011yescharacteristic
721C2#2392211yescharacteristic
722C2#2406211yescharacteristic
723C2#2406511yescharacteristic
724C2#2581211yescharacteristic
725C2#2581811yescharacteristic
726C2#2929911yescharacteristic
727C2#2930411yescharacteristic
728C2#2997011yescharacteristic
729C2#2997111yescharacteristic
730C2#3295511yescharacteristic
731C2#3295811yescharacteristic
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×#2058C2semidirect
4. C2×#2057C2semidirect
6. #11328C2semidirect
7. #11327C2semidirect
8. #11207C2semidirect
9. #11207C2semidirect
71. C2×#117Q8semidirect
107. C2×C22⋊C8Q8semidirect
486. C4⋊C4C22⋊Q8semidirect
487. C4⋊C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#22962
C2×#20582
C2×#20572
C2×#18542
#113282
#113272
#112072
#107312
#107122