Finite Group Databasebeta

← #5956087

Subgroups — #5956087

Order 512.

4106 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic, complemented
2C2×#2316C211yesmaximal, characteristic, complemented
3C2×#2311C211yesmaximal, characteristic
4C2×#2311C211yesmaximal, characteristic
5C2×#2297C211yesmaximal, characteristic
6C22×#389C211yesmaximal, characteristic, complemented
7#15220C214yesmaximal, complemented
8#15189C214yesmaximal, complemented
9#15141C214yesmaximal, complemented
10#15131C214yesmaximal, complemented
11#14921C214yesmaximal, complemented
12#14729C214yesmaximal, complemented
13#13756C211yesmaximal, characteristic, complemented
14C23×#41C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2212yes—
21C23×C42C2211yescharacteristic
22C2×#396C2218yes—
23#2317—28no—
24#2316—28no—
25#2311—28no—
26#2311—28no—
27#2297—28no—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#389—28no—
35C2×#389C2214yes—
36C2×#389C2214yes—
37C2×#389C2214yes—
38C2×#389C2214yes—
39C2×#389C2214yes—
40C2×#386C2212yes—
41C2×#386C2214yes—
42C2×#386C2214yes—
43C2×#386C2212yes—
44C2×#385C2214yes—
45C2×#384C2212yes—
46C2×#382C2212yes—
47C2×#380C2214yes—
48C2×#380C2212yes—
49C2×#380C2214yes—
50C2×#380C2212yes—
51C2×C4×C22⋊C4C2214yes—
52C22×#95C2212yes—
53#1443C2218yes—
54#1441C2218yes—
55#1441C2218yes—
56#1441C2218yes—
57#1439C2218yes—
58#1439C2218yes—
59#1439C2218yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C22⋊#95C2212yes—
63C22⋊#95C2212yes—
64C4×#95C2214yes—
65C22×#41—24no—
66C22×#41—24no—
67C22×#41—24no—
68C22×#41—24no—
69C22×#41—24no—
70C22×#41—24no—
71C22×#41Q818yes—
72C23×Q8C2311yescharacteristic
73C23×D4C2312yes—
74C23×D4C2311yescharacteristic
75C24×C4C2312yes—
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C2×#117—24no—
82C2×#117—24no—
83C2×#117—24no—
84C2×#117—24no—
85C2×#117—24no—
86C2×#117—24no—
87C2×#117—24no—
88C2×#117—24no—
89C2×#117D418yes—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—28no—
95C22×C4⋊C4—48no—
96C22×C4⋊C4D418yes—
97C22×C4⋊C4D418yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2318yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2318yes—
108C22×C4⋊C4C2318yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2318yes—
116C22×C22⋊C4C2314yes—
117C22×C42—24no—
118C22×C42—24no—
119C22×C42—24no—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125#396—416no—
126#391—28no—
127#391—28no—
128#391—28no—
129#390—28no—
130#390—28no—
131#390—28no—
132#389—216no—
133#389—216no—
134#389—216no—
135#389—216no—
136#389—216no—
137#386—216no—
138#386—216no—
139#386—28no—
140#386—28no—
141#385—216no—
142#384—28no—
143#382—28no—
144#380—216no—
145#380—28no—
146#380—28no—
147#380—216no—
148C4×C22⋊C4—216no—
149C2×#95—28no—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2318yes—
153C2×#95C2318yes—
154C2×#95C2318yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2318yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C25C2412yes—
163C25C2411yescharacteristic
164C2×#41—232no—
165C2×#41—232no—
166C2×#41—232no—
167C2×#41—416no—
168C2×#41—416no—
169C2×#41—416no—
170C2×#41—416no—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8C2×Q814yes—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4C2×Q818yes—
191C22×D4C2×Q814yes—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—44no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4C2×D418yes—
217C23×C4C2×D418yes—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×Q818yes—
222C23×C4C2×Q814yes—
223C23×C4#4114yes—
224C23×C4#4114yes—
225C23×C4#4114yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2414yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237#117—232no—
238#117—416no—
239#117—416no—
240C2×C4⋊C4—232no—
241C2×C4⋊C4—232no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—216no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—216no—
263C2×C4⋊C4—216no—
264C2×C4⋊C4—216no—
265C2×C4⋊C4—48no—
266C2×C4⋊C4—432no—
267C2×C4⋊C4—48no—
268C2×C4⋊C4—48no—
269C2×C4⋊C4—48no—
270C2×C4⋊C4—48no—
271C2×C4⋊C4C2×D418yes—
272C2×C4⋊C4C2×D418yes—
273C2×C4⋊C4C2×D418yes—
274C2×C4⋊C4C2×D418yes—
275C2×C4⋊C4C2×D418yes—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—216no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4—48no—
295C2×C22⋊C4—48no—
296C2×C22⋊C4—48no—
297C2×C22⋊C4C2×D418yes—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—416no—
313C2×C42C2×D418yes—
314#95—216no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—216no—
319#95—28no—
320#95—216no—
321#95—216no—
322#95—28no—
323#95—28no—
324#95—44no—
325#95—44no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24C22×D412yes—
341C24C22×D412yes—
342C24C22×D412yes—
343C24C22×Q814yes—
344C24C22×Q812yes—
345C24C2×#4112yes—
346C24C2×#4112yes—
347C24C2×#4112yes—
348C24C2511yescharacteristic
349#41—4128no—
350C2×Q8—28no—
351C2×Q8—28no—
352C2×Q8—28no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×D4—216no—
358C2×D4—216no—
359C2×D4—216no—
360C2×D4—28no—
361C2×D4—28no—
362C2×D4—28no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—44no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—216no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—216no—
407C22×C4—216no—
408C22×C4—216no—
409C22×C4—216no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—416no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—416no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—88no—
446C22×C4—88no—
447C22×C4#121116yes—
448C22×C4#121116yes—
449C22×C4#121116yes—
450C22×C4C22⋊Q8116yes—
451C22×C4C22⋊Q8116yes—
452C22×C4C22⋊Q8116yes—
453C22×C4#126116yes—
454C22×C4C22×D418yes—
455C22×C4C22×D418yes—
456C22×C4C22×D418yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×Q814yes—
464C22×C4C22×Q812yes—
465C22×C4C22×Q812yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4H(5,𝔽2)14yes—
473C22×C42−1+414yes—
474C4⋊C4—216no—
475C4⋊C4—216no—
476C4⋊C4—216no—
477C4⋊C4—216no—
478C4⋊C4—216no—
479C4⋊C4—432no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—816no—
484C4⋊C4—816no—
485C22⋊C4—216no—
486C22⋊C4—416no—
487C22⋊C4—416no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C22⋊C4—816no—
493C42—216no—
494C42—48no—
495C42—48no—
496C42—48no—
497C42—48no—
498C23—24no—
499C23—24no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—28no—
518C23—28no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23C2×#12114yes—
538C23C2×#12114yes—
539C23C2×#12114yes—
540C23C2×C22⋊Q814yes—
541C23C2×C22⋊Q814yes—
542C23C2×C22⋊Q814yes—
543C23C2×#12614yes—
544C23C23×D411yescharacteristic
545C23C23×D411yescharacteristic
546C23C23×D411yescharacteristic
547C23C23×Q811yescharacteristic
548C23C22×#4111yescharacteristic
549C23C22×#4111yescharacteristic
550C23C22×#4111yescharacteristic
551C23C2×H(5,𝔽2)12yes—
552C23C2×2−1+412yes—
553C23#58512yes—
554C23#58512yes—
555Q8—416no—
556D4—432no—
557D4—416no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—216no—
575C2×C4—416no—
576C2×C4—432no—
577C2×C4—416no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—48no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—48no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—48no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—432no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—816no—
618C2×C4#396164yescomplemented
619C2×C4C2×#12114yes—
620C2×C4C2×#12114yes—
621C2×C4C2×#12114yes—
622C2×C4C2×#12114yes—
623C2×C4C2×#12114yes—
624C2×C4C2×#12114yes—
625C2×C4C2×C22⋊Q814yes—
626C2×C4C2×C22⋊Q814yes—
627C2×C4C2×C22⋊Q814yes—
628C2×C4C2×C22⋊Q814yes—
629C2×C4C2×C22⋊Q814yes—
630C2×C4C2×C22⋊Q814yes—
631C2×C4C2×#12614yes—
632C2×C4C2×#12614yes—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—28no—
638C22—24no—
639C22—28no—
640C22—28no—
641C22—24no—
642C22—28no—
643C22—44no—
644C22—48no—
645C22—44no—
646C22—44no—
647C22—48no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—84no—
653C22—84no—
654C22C2×#39618yes—
655C22C22×#12111yescharacteristic
656C22C22×#12111yescharacteristic
657C22C22×#12111yescharacteristic
658C22C22×C22⋊Q811yescharacteristic
659C22C22×C22⋊Q811yescharacteristic
660C22C22×C22⋊Q811yescharacteristic
661C22C22×#12611yescharacteristic
662C22#346012yes—
663C22#346112yes—
664C22#348411yescharacteristic
665C22#348511yescharacteristic
666C22#348611yescharacteristic
667C22#348711yescharacteristic
668C22#348812yes—
669C22#348912yes—
670C22#349412yes—
671C22#349512yes—
672C22#349612yes—
673C22#349712yes—
674C22#350312yes—
675C22#350412yes—
676C22#351112yes—
677C22#351212yes—
678C4—44no—
679C4—48no—
680C4—44no—
681C4—48no—
682C4—44no—
683C4—48no—
684C4—82no—
685C4—82no—
686C4—84no—
687C4C2×#39618yes—
688C4C2×#39618yes—
689C2—28no—
690C2—84no—
691C2C22×#39611yescharacteristic
692C2#3515311yescharacteristic
693C2#3516811yescharacteristic
694C2#3525712yes—
695C2#3526712yes—
696C2#3551411yescharacteristic
697C2#3556412yes—
698C2#3557411yescharacteristic
699C2#3557512yes—
700C2#3563911yescharacteristic
701C2#3566111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2317C2semidirect
2. C2×#2316C2semidirect
6. C22×#389C2semidirect
7. #15220C2semidirect
8. #15189C2semidirect
9. #15141C2semidirect
10. #15131C2semidirect
11. #14921C2semidirect
12. #14729C2semidirect
13. #13756C2semidirect
618. C2×C4#396semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C2×#23162
C2×#23112
C2×#22972
C22×#3892
#152202
#151892
#151412
#151312
#149212
#147292
#137562