Finite Group Databasebeta

← #368313

Subgroups — #368313

Order 512.

1790 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C8×D4C211yesmaximal, characteristic
2C2×#1955C211yesmaximal, characteristic
3C2×#1940C211yesmaximal, characteristic, complemented
4C2×#1850C212yesmaximal, complemented
5C2×#1768C211yesmaximal, characteristic, complemented
6#13847C211yesmaximal, characteristic
7#11042C212yesmaximal, complemented
8#11042C212yesmaximal, complemented
9#10834C211yesmaximal, characteristic, complemented
10#10833C211yesmaximal, characteristic, complemented
11#10714C212yesmaximal, complemented
12C24×C8C2212yes—
13C22×C4×D4C2211yescharacteristic
14C2×C8×D4—24no—
15C2×C8×D4—24no—
16C2×C8×D4—24no—
17C22×C4⋊C8C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×C4⋊C8C2212yes—
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×C22⋊C8C2212yes—
23C22×C22⋊C8C2212yes—
24C22×C22⋊C8C2212yes—
25C22×#97C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2212yes—
29C2×C4×C22⋊C4C2212yes—
30C2×#376C2211yescharacteristic
31#1955—24no—
32#1940—24no—
33#1850—28no—
34#1768—24no—
35C2×#336C2211yescharacteristic
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1445C411yescharacteristic
43#1445C411yescharacteristic
44#1439C412yes—
45#1439C412yes—
46#1438C411yescharacteristic
47#1438C411yescharacteristic
48C23×D4C2×C411yescharacteristic
49C24×C4C2312yes—
50C23×C8—24no—
51C23×C8—24no—
52C23×C8—24no—
53C23×C8—24no—
54C23×C8—24no—
55C23×C8—42no—
56C23×C8—42no—
57C23×C8C2312yes—
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C23×C8C2312yes—
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4D414yes—
68C2×C4×D4Q814yes—
69C22×C4⋊C4C2×C412yes—
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C22⋊C4C2×C412yes—
72C22×C22⋊C4C2×C412yes—
73C22×C22⋊C4C2×C412yes—
74C22×C22⋊C4C2×C412yes—
75C22×C22⋊C4C2×C412yes—
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C8×D4—232no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—24no—
89C2×C4⋊C8—24no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—42no—
94C2×C4⋊C8D414yes—
95C2×C4⋊C8D414yescomplemented
96C2×C22⋊C8—24no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—24no—
101C2×C22⋊C8—24no—
102C2×C22⋊C8—24no—
103C2×C22⋊C8—42no—
104C2×C22⋊C8—42no—
105C2×#97—22no—
106C2×#97—22no—
107C2×#97—22no—
108C2×C4×C8—22no—
109C2×C4×C8—22no—
110C2×C4×C8D414yes—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C22⋊C4—24no—
114C4×C22⋊C4—24no—
115C4×C22⋊C4—42no—
116C4×C22⋊C4—42no—
117#376—22no—
118#376—22no—
119C2×#95C2×C411yescharacteristic
120C2×#95C2×C411yescharacteristic
121C2×#95C2×C411yescharacteristic
122C2×#95C2×C411yescharacteristic
123C2×#95C2×C411yescharacteristic
124C2×#95C2×C411yescharacteristic
125C2×#95C2×C412yes—
126C2×#95C2×C412yes—
127C2×#95C2×C412yes—
128C2×#95C2×C412yes—
129#336—24no—
130#336—24no—
131#336—22no—
132#336—22no—
133#336—24no—
134#336—42no—
135#336—42no—
136C25C22×C412yes—
137C22×D4—24no—
138C22×D4—24no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4C4⋊C414yes—
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×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4C22×C412yes—
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C412yes—
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C412yes—
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C2×D414yes—
164C23×C4C2×Q814yes—
165C23×C4C2411yescharacteristic
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—216no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—48no—
203C22×C8—48no—
204C22×C8—48no—
205C22×C8—48no—
206C22×C8—48no—
207C22×C8—48no—
208C22×C8C2×D414yes—
209C22×C8C2×D412yes—
210C22×C8C2×D412yes—
211C22×C8C2×D412yes—
212C22×C8C2×D412yes—
213C22×C8#4112yes—
214C22×C8#4112yes—
215C4×D4—28no—
216C4×D4—28no—
217C4×D4—28no—
218C4×D4—28no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4C4⋊C414yes—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4C4⋊C418yes—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—21no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—21no—
251C2×C42—22no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—21no—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42C2×Q812yes—
261C2×C42#4112yes—
262C2×C42#4112yes—
263C2×C42#4112yes—
264C4⋊C8—28no—
265C4⋊C8—28no—
266C4⋊C8—28no—
267C4⋊C8—48no—
268C4⋊C8—44no—
269C22⋊C8—216no—
270C22⋊C8—48no—
271C22⋊C8—48no—
272#97—44no—
273C4×C8—28no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—24no—
279#95—24no—
280#95—22no—
281#95—22no—
282#95—42no—
283#95—42no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24C2×C4⋊C414yes—
297C24#13114yes—
298C24#13114yes—
299C24C23×C411yescharacteristic
300C2×D4—28no—
301C2×D4—28no—
302C2×D4—216no—
303C2×D4—216no—
304C2×D4—28no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
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—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
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—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—42no—
373C22×C4—42no—
374C22×C4C2×C4⋊C414yes—
375C22×C4C2×C4⋊C412yes—
376C22×C4C2×C4⋊C412yes—
377C22×C4C4×D412yes—
378C22×C4C4×D412yes—
379C22×C4C4×D412yes—
380C22×C4C4×D412yes—
381C22×C4C4×D412yes—
382C22×C4C4×D412yes—
383C22×C4#13114yes—
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×Q811yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4H(5,𝔽2)11yescharacteristic
395C22×C4H(5,𝔽2)11yescharacteristic
396C22×C4H(5,𝔽2)11yescharacteristic
397C22×C42−1+411yescharacteristic
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—24no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—24no—
413C2×C8—22no—
414C2×C8—24no—
415C2×C8—22no—
416C2×C8—432no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—42no—
420C2×C8—41no—
421C2×C8—42no—
422C2×C8—41no—
423C2×C8—42no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—42no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8#12114yes—
441C4⋊C4—28no—
442C4⋊C4—48no—
443C22⋊C4—216no—
444C22⋊C4—216no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—42no—
457C42—42no—
458C42—42no—
459C42—42no—
460C42#12014yes—
461C42C22⋊Q814yes—
462C42C22⋊Q814yes—
463C42#12314yes—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23#42418yes—
499C23C22×C4⋊C411yescharacteristic
500C23C2×C4×D411yescharacteristic
501C23C2×C4×D411yescharacteristic
502C23C2×C4×D411yescharacteristic
503C23C4×#4111yescharacteristic
504C23#51811yescharacteristic
505C23#52011yescharacteristic
506C23C2×#13112yes—
507C23C2×#13112yes—
508C23C2×#13112yes—
509C23#56812yes—
510D4—232no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4#38514yes—
569C2×C4#38514yes—
570C2×C4C2×#12011yescharacteristic
571C2×C4C2×#12111yescharacteristic
572C2×C4C2×C22⋊Q811yescharacteristic
573C2×C4C2×C22⋊Q811yescharacteristic
574C2×C4C2×#12311yescharacteristic
575C2×C4#52511yescharacteristic
576C2×C4#52911yescharacteristic
577C2×C4#53211yescharacteristic
578C2×C4#53511yescharacteristic
579C2×C4#53711yescharacteristic
580C2×C4#54011yescharacteristic
581C2×C4#54211yescharacteristic
582C2×C4D4211yescharacteristic
583C2×C4D4211yescharacteristic
584C2×C4#54611yescharacteristic
585C2×C4#54611yescharacteristic
586C2×C4#54611yescharacteristic
587C2×C4#54711yescharacteristic
588C2×C4#54811yescharacteristic
589C2×C4D4×Q811yescharacteristic
590C2×C4#55011yescharacteristic
591C2×C4#55011yescharacteristic
592C2×C4#55411yescharacteristic
593C2×C4#55411yescharacteristic
594C2×C4#55411yescharacteristic
595C2×C4#55611yescharacteristic
596C2×C4#55911yescharacteristic
597C2×C4#55911yescharacteristic
598C8—24no—
599C8—42no—
600C8—42no—
601C8—42no—
602C8—44no—
603C8—82no—
604C22—28no—
605C22—28no—
606C22—28no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22C2×#38511yescharacteristic
620C22C4×#12111yescharacteristic
621C22#232411yescharacteristic
622C22D4×C4⋊C411yescharacteristic
623C22#234911yescharacteristic
624C22#236611yescharacteristic
625C22#236711yescharacteristic
626C22C2×#42412yes—
627C22#290612yes—
628C22#292911yescharacteristic
629C22#292911yescharacteristic
630C22#292912yes—
631C22#292912yes—
632C22#293111yescharacteristic
633C22#293211yescharacteristic
634C22#293312yes—
635C22#293312yes—
636C22#296612yes—
637C22#296812yes—
638C22#297612yes—
639C22#297712yes—
640C22#299212yes—
641C22#299412yes—
642C4—24no—
643C4—24no—
644C4—44no—
645C4—44no—
646C4—42no—
647C4—42no—
648C4—81no—
649C4—81no—
650C4—82no—
651C4#240711yescharacteristic
652C4#240811yescharacteristic
653C4#244511yescharacteristic
654C4#244711yescharacteristic
655C4#245211yescharacteristic
656C4#245411yescharacteristic
657C4#253811yescharacteristic
658C4#258011yescharacteristic
659C2—48no—
660C2#1401811yescharacteristic
661C2#2083411yescharacteristic
662C2#2083911yescharacteristic
663C2#2149111yescharacteristic
664C2#2149211yescharacteristic
665C2#2213411yescharacteristic
666C2#2213611yescharacteristic
667C2#2264712yes—
668C2#2269712yes—
669C2#2317712yes—
670C2#2321112yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1940C2semidirect
4. C2×#1850C2semidirect
5. C2×#1768C2semidirect
7. #11042C2semidirect
8. #11042C2semidirect
9. #10834C2semidirect
10. #10833C2semidirect
11. #10714C2semidirect
95. C2×C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C22×C8×D42
C2×#19552
C2×#19402
C2×#18502
C2×#17682
#138472
#110422
#108342
#108332
#107142