Finite Group Databasebeta

← #674680

Subgroups — #674680

Order 512.

4246 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2537C211yesmaximal, characteristic
2C2×#2501C212yesmaximal, complemented
3C2×#2422C214yesmaximal, complemented
4C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
5#15384C218yesmaximal, complemented
6#15150C212yesmaximal, complemented
7#15135C214yesmaximal, complemented
8#15130C212yesmaximal, complemented
9#14692C211yesmaximal, characteristic, complemented
10#13910C211yesmaximal, characteristic, complemented
11#13906C211yesmaximal, characteristic, complemented
12#13880C212yesmaximal, complemented
13C4×#390C211yesmaximal, characteristic, complemented
14C4×#390C211yesmaximal, characteristic, complemented
15C22×#123C2218yes—
16C22×#121C2212yes—
17C22×#121C2212yes—
18C22×C4×D4C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22#2537—28no—
23#2501—216no—
24#2422—232no—
25C2×#399C2218yes—
26C2×#397C2218yes—
27C2×#394C2214yes—
28C2×#394C2218yes—
29D4×C4⋊C4—28no—
30C2×#390C2211yescharacteristic
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#390C2211yescharacteristic
34C2×#389C2212yes—
35C2×#389C2214yes—
36C2×#385C2214yes—
37C2×#385C2218yes—
38C2×#384C2214yes—
39C2×#384C2212yes—
40C2×#384C2212yes—
41C2×#384C2211yescharacteristic
42C2×#384C2211yescharacteristic
43C2×#384C2214yes—
44C2×#384C2212yes—
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×#380C2214yes—
48C2×#380C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2212yes—
52C2×C4×C22⋊C4C2214yes—
53C2×C4×C22⋊C4C2212yes—
54C2×C43C2211yescharacteristic
55#1445C2218yes—
56#1444C2214yes—
57#1444C2218yes—
58#1439C2214yes—
59#1439C2218yes—
60#1437C2214yes—
61#1437C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2212yes—
66C4⋊#95C2211yescharacteristic
67C23×D4C2311yescharacteristic
68C23×D4C2314yes—
69C23×D4C2311yescharacteristic
70C24×C4C2312yes—
71C24×C4C2312yes—
72C24×C4C2314yes—
73C2×#123—232no—
74C2×#123—232no—
75C2×#123—416no—
76C2×#121—216no—
77C2×#121—28no—
78C2×#121—28no—
79C2×#121—28no—
80C2×#121—44no—
81C2×C4×D4—28no—
82C2×C4×D4—216no—
83C2×C4×D4—24no—
84C2×C4×D4—24no—
85C2×C4×D4—24no—
86C2×C4×D4—24no—
87C2×C4×D4—24no—
88C2×C4×D4D418yes—
89C22×C4⋊C4—28no—
90C22×C4⋊C4—28no—
91C22×C4⋊C4—44no—
92C22×C4⋊C4C2318yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2318yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2318yes—
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2318yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#399—232no—
125#397—232no—
126#394—232no—
127#394—216no—
128#390—28no—
129#390—24no—
130#390D4116yescomplemented
131#390D418yescomplemented
132#389—28no—
133#389—216no—
134#385—216no—
135#385—232no—
136#384—28no—
137#384—24no—
138#384—216no—
139#384—28no—
140#384—24no—
141#384—44no—
142#384—48no—
143#380—28no—
144#380—28no—
145#380—216no—
146#380—28no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—24no—
149C4×C22⋊C4—216no—
150C4×C22⋊C4—28no—
151C4×C22⋊C4—28no—
152C2×#95C2318yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2318yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C43—24no—
163C25C2412yes—
164C25C2412yes—
165C22×D4—28no—
166C22×D4—28no—
167C22×D4—216no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—22no—
173C22×D4—22no—
174C22×D4—22no—
175C22×D4—24no—
176C22×D4—22no—
177C22×D4—44no—
178C22×D4—48no—
179C22×D4C2×D414yes—
180C22×D4C2×D414yes—
181C22×D4C2×D418yes—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—28no—
189C23×C4—216no—
190C23×C4—28no—
191C23×C4—24no—
192C23×C4—28no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—28no—
196C23×C4—44no—
197C23×C4—48no—
198C23×C4C2×D418yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211#123—4128no—
212#121—432no—
213#121—432no—
214C4×D4—232no—
215C4×D4—432no—
216C4×D4—416no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—216no—
235C2×C4⋊C4—216no—
236C2×C4⋊C4—216no—
237C2×C4⋊C4—216no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—216no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—216no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—432no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—48no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—48no—
254C2×C4⋊C4—48no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2×D418yes—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—216no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—216no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—216no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—216no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—48no—
283C2×C22⋊C4C2×D418yes—
284C2×C22⋊C4C2×D4116yes—
285C2×C22⋊C4C2×D4116yes—
286C2×C22⋊C4C2×D4116yes—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—22no—
296C2×C42—28no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42C2×D414yes—
302C2×C42C2×D418yes—
303C2×C42C2×D414yes—
304C2×C42#4114yes—
305C2×C42#4114yes—
306C2×C42#4114yes—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—24no—
311#95—28no—
312#95—28no—
313#95—216no—
314#95—216no—
315#95—28no—
316#95—24no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—24no—
322C24—22no—
323C24—24no—
324C24—22no—
325C24—24no—
326C24—24no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—24no—
332C24—22no—
333C24—24no—
334C24—24no—
335C24—22no—
336C24C22×D414yes—
337C24C22×D418yes—
338C24C22×D414yes—
339C24C2511yescharacteristic
340C2×D4—28no—
341C2×D4—28no—
342C2×D4—28no—
343C2×D4—28no—
344C2×D4—216no—
345C2×D4—216no—
346C2×D4—28no—
347C2×D4—416no—
348C2×D4—432no—
349C2×D4—416no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—88no—
355C2×D4—816no—
356C2×D4#127116yes—
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—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—216no—
384C22×C4—216no—
385C22×C4—216no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—28no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—28no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—28no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—28no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—416no—
407C22×C4—432no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—416no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—416no—
417C22×C4—416no—
418C22×C4—88no—
419C22×C4—816no—
420C22×C4C22×D418yes—
421C22×C4C22×D414yes—
422C22×C4C22×D414yes—
423C22×C4C22×D412yes—
424C22×C4C22×D414yes—
425C22×C4C22×D414yes—
426C22×C4C22×D414yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D414yes—
430C22×C4C22×D412yes—
431C22×C4C2×#4112yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4114yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4112yes—
439C22×C4H(5,𝔽2)14yes—
440C22×C4H(5,𝔽2)12yes—
441C22×C42−1+414yes—
442C22×C42−1+414yes—
443C22×C42−1+412yes—
444C4⋊C4—28no—
445C4⋊C4—28no—
446C4⋊C4—216no—
447C4⋊C4—416no—
448C4⋊C4—432no—
449C4⋊C4—44no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—432no—
455C4⋊C4—44no—
456C4⋊C4—84no—
457C4⋊C4—816no—
458C4⋊C4—84no—
459C22⋊C4—216no—
460C22⋊C4—232no—
461C22⋊C4—232no—
462C22⋊C4—232no—
463C22⋊C4—416no—
464C22⋊C4—432no—
465C22⋊C4—416no—
466C22⋊C4—416no—
467C22⋊C4—48no—
468C22⋊C4—416no—
469C42—216no—
470C42—28no—
471C42—216no—
472C42—28no—
473C42—48no—
474C42—48no—
475C42#123116yescomplemented
476C23—24no—
477C23—24no—
478C23—28no—
479C23—24no—
480C23—28no—
481C23—24no—
482C23—28no—
483C23—28no—
484C23—24no—
485C23—28no—
486C23—28no—
487C23—24no—
488C23—24no—
489C23—28no—
490C23—24no—
491C23—24no—
492C23—28no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—42no—
511C23—42no—
512C23—44no—
513C23—42no—
514C23—44no—
515C23—44no—
516C23—42no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23C2×#12718yes—
522C23C23×D411yescharacteristic
523C23C23×D412yes—
524C23C23×D411yescharacteristic
525C23C22×#4111yescharacteristic
526C23C22×#4111yescharacteristic
527C23C22×#4111yescharacteristic
528C23C2×H(5,𝔽2)11yescharacteristic
529C23C2×H(5,𝔽2)12yes—
530C23C2×2−1+412yes—
531C23C2×2−1+411yescharacteristic
532C23C2×2−1+412yes—
533D4—232no—
534D4—416no—
535D4—832no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—432no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—48no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—48no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—44no—
589C2×C4—48no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—832no—
593C2×C4—816no—
594C2×C4C2×#12314yes—
595C2×C4C2×#12314yes—
596C2×C4C2×#12314yes—
597C2×C4C2×#12714yes—
598C2×C4#53214yes—
599C2×C4#53614yes—
600C2×C4#54214yes—
601C2×C4D4218yes—
602C2×C4D4214yes—
603C2×C4#54718yes—
604C2×C4#54714yes—
605C2×C4#54714yes—
606C2×C4#54718yes—
607C2×C4#54718yes—
608C2×C4#55314yes—
609C22—28no—
610C22—28no—
611C22—28no—
612C22—48no—
613C22—44no—
614C22—44no—
615C22—48no—
616C22—48no—
617C22—44no—
618C22—44no—
619C22—48no—
620C22—44no—
621C22—48no—
622C22—44no—
623C22—48no—
624C22—44no—
625C22—82no—
626C22—82no—
627C22—84no—
628C22—82no—
629C22—82no—
630C22—84no—
631C22C22×#12311yescharacteristic
632C22C22×#12711yescharacteristic
633C22C2×#53211yescharacteristic
634C22C2×#53611yescharacteristic
635C22C2×#54211yescharacteristic
636C22C2×D4212yes—
637C22C2×D4211yescharacteristic
638C22C2×#54712yes—
639C22C2×#54712yes—
640C22C2×#54711yescharacteristic
641C22C2×#54711yescharacteristic
642C22C2×#54712yes—
643C22C2×#55311yescharacteristic
644C22#349212yes—
645C22#349811yescharacteristic
646C22#349812yes—
647C22#349912yes—
648C22#349912yes—
649C22#349911yescharacteristic
650C22#351112yes—
651C22#351112yes—
652C22#354412yes—
653C22#355212yes—
654C4—28no—
655C4—28no—
656C4—24no—
657C4—44no—
658C4—48no—
659C4—44no—
660C4—84no—
661C4—84no—
662C4—82no—
663C4—82no—
664C4#242218yescomplemented
665C4#250118yescomplemented
666C2—48no—
667C2—84no—
668C2C2×#242211yescharacteristic
669C2C2×#250111yescharacteristic
670C2#3553011yescharacteristic
671C2D4×#12711yescharacteristic
672C2#3612712yes—
673C2#3617911yescharacteristic
674C2#3936011yescharacteristic
675C2#3937712yes—
676C2#3948911yescharacteristic
677C2#5930412yes—
678C2#5931312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2537C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2501C2semidirect
3. C2×#2422C2semidirect
4. C2×D4×C4⋊C4C2semidirect
5. #15384C2semidirect
6. #15150C2semidirect
7. #15135C2semidirect
8. #15130C2semidirect
9. #14692C2semidirect
10. #13910C2semidirect
11. #13906C2semidirect
12. #13880C2semidirect
13. C4×#390C2semidirect
14. C4×#390C2semidirect
130. #390D4semidirect
131. #390D4semidirect
475. C42#123semidirect
664. C4#2422semidirect
665. C4#2501semidirect

Maximal subgroups

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

subgroupindex
C2×#25372
C2×#25012
C2×#24222
C2×D4×C4⋊C42
#153842
#151502
#151352
#151302
#146922
#139102
#139062
#138802
C4×#3902