Finite Group Databasebeta

← #636502

Subgroups — #636502

Order 512.

3322 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×#2857C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic, complemented
3C2×C4×#121C211yesmaximal, characteristic
4C2×#2285C211yesmaximal, characteristic, complemented
5#16566C214yesmaximal, complemented
6#16265C214yesmaximal, complemented
7#15234C212yesmaximal, complemented
8#15181C214yesmaximal, complemented
9#15134C212yesmaximal, complemented
10#15131C214yesmaximal, complemented
11#14340C212yesmaximal, complemented
12#14257C212yesmaximal
13#14161C211yesmaximal, characteristic, complemented
14#13910C211yesmaximal, characteristic, complemented
15#13906C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×#121C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2212yes—
21C22×#117C2212yes—
22C23×C42C2211yescharacteristic
23#2857—28no—
24#2563—28no—
25C2×#396C2218yes—
26C2×#393C2214yes—
27C4×#121—28no—
28C2×#391C2212yes—
29C2×#390C2211yescharacteristic
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2211yescharacteristic
35C2×#389C2214yes—
36C2×#388C2212yes—
37C2×#386C2212yes—
38#2285—28no—
39C2×#384C2211yescharacteristic
40C2×#384C2212yes—
41C2×#384C2212yes—
42C2×#382C2212yes—
43C2×#382C2212yes—
44C2×#380C2211yescharacteristic
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×#380C2214yes—
48C2×#380C2212yes—
49C2×#380C2212yes—
50C2×#380C2214yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2212yes—
56#1446C2214yes—
57#1444C2214yes—
58#1442C2218yes—
59#1441C2218yes—
60#1441C2218yes—
61#1439C2218yes—
62#1439C2218yes—
63#1439C2218yes—
64#1439C2214yes—
65#1438C2218yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C23×Q8C2312yes—
72C23×D4C2311yescharacteristic
73C23×D4C2312yes—
74C24×C4C2312yes—
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—24no—
81C2×C22⋊Q8—28no—
82C2×C22⋊Q8—48no—
83C2×#121—24no—
84C2×#121—24no—
85C2×#121Q818yescomplemented
86C2×C4×D4—28no—
87C2×C4×D4—24no—
88C2×C4×D4—24no—
89C2×C4×D4—24no—
90C2×C4×D4—48no—
91C2×#117—28no—
92C2×#117—28no—
93C2×#117—44no—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2318yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2318yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2318yes—
118C22×C42—24no—
119C22×C42—24no—
120C22×C42D418yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128#396—416no—
129#393—216no—
130#391—28no—
131#390—28no—
132#390—24no—
133#390—44no—
134#389—28no—
135#389—28no—
136#389—24no—
137#389—48no—
138#388—28no—
139#386—28no—
140#384—28no—
141#384—24no—
142#384—44no—
143#382—28no—
144#382—28no—
145#380—28no—
146#380—28no—
147#380—24no—
148#380—216no—
149#380—28no—
150#380—28no—
151#380—48no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4D418yescomplemented
156C4×C22⋊C4—28no—
157C2×#95C2318yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2318yes—
166C2×#95C2318yes—
167C2×#95C2318yes—
168C2×#95C2318yes—
169C2×#95C2312yes—
170C25C2412yes—
171C25C2411yescharacteristic
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—44no—
176C22×Q8—44no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4C2×Q814yes—
188C22×D4C2×Q818yes—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—28no—
204C23×C4—28no—
205C23×C4—42no—
206C23×C4C2×D414yes—
207C23×C4C2×D414yes—
208C23×C4C2×D414yes—
209C23×C4C2×Q814yes—
210C23×C4#4114yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2414yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C4⋊Q8—416no—
225C22⋊Q8—432no—
226#121—232no—
227C4×D4—432no—
228C4×D4—416no—
229#117—432no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—216no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—48no—
267C2×C4⋊C4—48no—
268C2×C4⋊C4—48no—
269C2×C4⋊C4C2×D418yes—
270C2×C4⋊C4C2×Q814yes—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—216no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—48no—
288C2×C22⋊C4—48no—
289C2×C22⋊C4—48no—
290C2×C22⋊C4—48no—
291C2×C22⋊C4—48no—
292C2×C22⋊C4—48no—
293C2×C22⋊C4C2×Q818yes—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—232no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—28no—
308C2×C42C2×D418yes—
309C2×C42C2×D414yes—
310C2×C42C2×D414yes—
311C2×C42C2×D414yes—
312C2×C42C2×D414yes—
313C2×C42C2×D414yes—
314C2×C42#4114yes—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—216no—
321#95—28no—
322#95—28no—
323#95—48no—
324#95—48no—
325#95—48no—
326#95—48no—
327#95C2×D418yes—
328C24—21no—
329C24—21no—
330C24—22no—
331C24—21no—
332C24—22no—
333C24—22no—
334C24—21no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—22no—
342C24—24no—
343C24—24no—
344C24—22no—
345C24—24no—
346C24C22×D412yes—
347C24C22×Q814yes—
348C24C22×Q812yes—
349C24C2×#4112yes—
350C24C2511yescharacteristic
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—88no—
355C2×Q8—88no—
356C2×D4—216no—
357C2×D4—28no—
358C2×D4—28no—
359C2×D4—28no—
360C2×D4—416no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—416no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—44no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—28no—
416C22×C4—24no—
417C22×C4—28no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—28no—
429C22×C4—44no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—416no—
438C22×C4—416no—
439C22×C4C22⋊Q8116yes—
440C22×C4C22×D412yes—
441C22×C4C22×D418yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×Q814yes—
451C22×C4C22×Q812yes—
452C22×C4C22×Q812yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4H(5,𝔽2)14yes—
459C22×C4H(5,𝔽2)14yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C42−1+414yes—
462C22×C42−1+412yes—
463C4⋊C4—28no—
464C4⋊C4—216no—
465C4⋊C4—416no—
466C4⋊C4—416no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—416no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—816no—
478C22⋊C4—216no—
479C22⋊C4—416no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—416no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—816no—
486C22⋊C4—816no—
487C42—28no—
488C42—28no—
489C42—28no—
490C42—216no—
491C42—48no—
492C42—48no—
493C42#121116yes—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—28no—
512C23—28no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—44no—
526C23—42no—
527C23—42no—
528C23—44no—
529C23—42no—
530C23—42no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23C2×C22⋊Q814yes—
540C23C23×D411yescharacteristic
541C23C23×D411yescharacteristic
542C23C23×Q811yescharacteristic
543C23C22×#4111yescharacteristic
544C23C22×#4111yescharacteristic
545C23C2×H(5,𝔽2)12yes—
546C23C2×H(5,𝔽2)12yes—
547C23C2×H(5,𝔽2)11yescharacteristic
548C23C2×2−1+412yes—
549C23C2×2−1+411yescharacteristic
550C23#58512yes—
551Q8—816no—
552D4—432no—
553D4—416no—
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—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—232no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—416no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—416no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—48no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—48no—
614C2×C4—44no—
615C2×C4—816no—
616C2×C4C2×#12114yes—
617C2×C4C2×#12114yes—
618C2×C4C2×#12114yes—
619C2×C4C2×C22⋊Q814yes—
620C2×C4C2×C22⋊Q814yes—
621C2×C4#53514yes—
622C2×C4#53514yes—
623C2×C4#53614yes—
624C2×C4#53718yes—
625C2×C4#53714yes—
626C2×C4#54218yes—
627C2×C4#54614yes—
628C2×C4D4×Q814yes—
629C2×C4#55214yes—
630C2×C4#55414yes—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—42no—
635C22—42no—
636C22—44no—
637C22—42no—
638C22—44no—
639C22—44no—
640C22—42no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—48no—
645C22—48no—
646C22—44no—
647C22—82no—
648C22—82no—
649C22—84no—
650C22—82no—
651C22—84no—
652C22—82no—
653C22C22×#12111yescharacteristic
654C22C22×C22⋊Q811yescharacteristic
655C22C2×#53511yescharacteristic
656C22C2×#53511yescharacteristic
657C22C2×#53611yescharacteristic
658C22C2×#53711yescharacteristic
659C22C2×#53712yes—
660C22#344911yescharacteristic
661C22C2×#54212yes—
662C22C2×#54611yescharacteristic
663C22C2×D4×Q811yescharacteristic
664C22C2×#55211yescharacteristic
665C22C2×#55411yescharacteristic
666C22#348812yes—
667C22#349412yes—
668C22#350112yes—
669C22#350312yes—
670C22#350612yes—
671C22#351112yes—
672C22#354712yes—
673C22#354812yes—
674C22#354912yes—
675C22#355512yes—
676C4—24no—
677C4—24no—
678C4—24no—
679C4—48no—
680C4—42no—
681C4—44no—
682C4—82no—
683C4—84no—
684C4—84no—
685C4—82no—
686C4—84no—
687C4#241718yes—
688C4#262018yes—
689C2—44no—
690C2—84no—
691C2C2×#262011yescharacteristic
692C2C2×#241711yescharacteristic
693C2#3517211yescharacteristic
694C2#3525211yescharacteristic
695C2#3527911yescharacteristic
696C2#3763311yescharacteristic
697C2#3763911yescharacteristic
698C2#3946912yes—
699C2#3947912yes—
700C2#3959812yes—
701C2#3960812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2857C2semidirect
2. C2×#2563C2semidirect
4. C2×#2285C2semidirect
5. #16566C2semidirect
6. #16265C2semidirect
7. #15234C2semidirect
8. #15181C2semidirect
9. #15134C2semidirect
10. #15131C2semidirect
11. #14340C2semidirect
13. #14161C2semidirect
14. #13910C2semidirect
15. #13906C2semidirect
85. C2×#121Q8semidirect
155. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#25632
C2×C4×#1212
C2×#22852
#165662
#162652
#152342
#151812
#151342
#151312
#143402
#142572
#141612
#139102
#139062