Finite Group Databasebeta

← #636393

Subgroups — #636393

Order 512.

3538 conjugacy classes of subgroups.

The subgroup lattice has 721 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic, complemented
2C2×#2547C211yesmaximal, characteristic, complemented
3C2×#2620C212yesmaximal
4C2×C4×#121C211yesmaximal, characteristic
5C2×#2285C211yesmaximal, characteristic, complemented
6#16518C214yesmaximal, complemented
7#16401C214yesmaximal, complemented
8#15217C212yesmaximal, complemented
9#15184C212yesmaximal, complemented
10#15131C214yesmaximal, complemented
11#15129C214yesmaximal, complemented
12#14258C212yesmaximal, complemented
13#14155C211yesmaximal, characteristic, complemented
14#13910C211yesmaximal, characteristic, complemented
15#13906C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#124C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×#121C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2212yes—
22C22×#117C2212yes—
23C23×C42C2211yescharacteristic
24#2857—28no—
25#2620—216no—
26#2547—28no—
27C2×#397C2214yes—
28C2×#396C2218yes—
29C2×#394C2218yes—
30C4×#121—28no—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#390C2212yes—
36C2×#390C2211yescharacteristic
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2211yescharacteristic
40C2×#389C2214yes—
41C2×#388C2212yes—
42C2×#386C2212yes—
43#2285—28no—
44C2×#384C2211yescharacteristic
45C2×#384C2212yes—
46C2×#384C2212yes—
47C2×#382C2212yes—
48C2×#380C2212yes—
49C2×#380C2212yes—
50C2×#380C2212yes—
51C2×#380C2212yes—
52C2×#380C2214yes—
53C2×#380C2211yescharacteristic
54C2×#380C2214yes—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2212yes—
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C22⋊C4C2212yes—
60#1445C2214yes—
61#1444C2214yes—
62#1441C2218yes—
63#1441C2218yes—
64#1441C2218yes—
65#1440C2218yes—
66#1440C2218yes—
67#1439C2218yes—
68#1439C2214yes—
69C4⋊#95C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C23×Q8C2312yes—
73C23×D4C2311yescharacteristic
74C23×D4C2312yes—
75C24×C4C2312yes—
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—24no—
82C2×#124—28no—
83C2×#124—28no—
84C2×#124—44no—
85C2×C22⋊Q8—28no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—28no—
88C2×#121—24no—
89C2×#121—24no—
90C2×#121D418yes—
91C2×C4×D4—28no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C2×C4×D4—48no—
96C2×#117—28no—
97C2×#117—28no—
98C2×#117—44no—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2318yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2318yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2318yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C42—24no—
124C22×C42—24no—
125C22×C42D418yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133#397—216no—
134#396—232no—
135#394—232no—
136#391—28no—
137#391—28no—
138#391—28no—
139#391—44no—
140#390—28no—
141#390—24no—
142#389—28no—
143#389—28no—
144#389—24no—
145#389—48no—
146#388—28no—
147#386—28no—
148#384—28no—
149#384—28no—
150#384—24no—
151#382—28no—
152#380—28no—
153#380—28no—
154#380—24no—
155#380—28no—
156#380—216no—
157#380—28no—
158#380—48no—
159C4×C4⋊C4—28no—
160C4×C4⋊C4—24no—
161C4×C4⋊C4—24no—
162C4×C4⋊C4D418yescomplemented
163C4×C22⋊C4—28no—
164C2×#95C2318yes—
165C2×#95C2318yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2318yes—
174C2×#95C2318yes—
175C2×#95C2318yes—
176C2×#95C2312yes—
177C25C2412yes—
178C25C2411yescharacteristic
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4C2×D414yes—
197C22×D4C2×D418yes—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—28no—
214C23×C4—42no—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4#4114yes—
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C4⋊Q8—416no—
234#124—432no—
235C22⋊Q8—432no—
236#121—232no—
237C4×D4—432no—
238C4×D4—416no—
239#117—432no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—216no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—48no—
279C2×C4⋊C4—48no—
280C2×C4⋊C4C2×D414yes—
281C2×C4⋊C4C2×D418yes—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—216no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—216no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—216no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—48no—
303C2×C22⋊C4—48no—
304C2×C22⋊C4—48no—
305C2×C22⋊C4C2×D418yes—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—232no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—28no—
320C2×C42C2×D418yes—
321C2×C42C2×D414yes—
322C2×C42C2×D414yes—
323C2×C42C2×D414yes—
324C2×C42C2×D414yes—
325C2×C42C2×D414yes—
326C2×C42#4114yes—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—24no—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—216no—
335#95—216no—
336#95—48no—
337#95—48no—
338#95—48no—
339#95C2×D418yes—
340C24—21no—
341C24—21no—
342C24—21no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—21no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—24no—
354C24—24no—
355C24—24no—
356C24—22no—
357C24—24no—
358C24C22×D414yes—
359C24C22×D412yes—
360C24C22×D412yes—
361C24C2×#4112yes—
362C24C2511yescharacteristic
363C2×Q8—48no—
364C2×Q8—48no—
365C2×Q8—48no—
366C2×Q8—48no—
367C2×Q8—48no—
368C2×Q8—48no—
369C2×Q8—48no—
370C2×D4—216no—
371C2×D4—28no—
372C2×D4—28no—
373C2×D4—28no—
374C2×D4—416no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—416no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—28no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—28no—
445C22×C4—44no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—416no—
454C22×C4—416no—
455C22×C4#121116yes—
456C22×C4C22×D412yes—
457C22×C4C22×D414yes—
458C22×C4C22×D418yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4H(5,𝔽2)14yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C42−1+414yes—
477C22×C42−1+414yes—
478C22×C42−1+412yes—
479C4⋊C4—28no—
480C4⋊C4—216no—
481C4⋊C4—416no—
482C4⋊C4—416no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—44no—
486C4⋊C4—48no—
487C4⋊C4—416no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—816no—
494C22⋊C4—216no—
495C22⋊C4—416no—
496C22⋊C4—432no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—416no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—816no—
503C42—28no—
504C42—28no—
505C42—28no—
506C42—216no—
507C42—48no—
508C42—48no—
509C42#121116yes—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—28no—
528C23—28no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—44no—
542C23—42no—
543C23—42no—
544C23—44no—
545C23—42no—
546C23—42no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23C2×#12114yes—
556C23C23×D411yescharacteristic
557C23C23×D411yescharacteristic
558C23C23×D411yescharacteristic
559C23C22×#4111yescharacteristic
560C23C22×#4111yescharacteristic
561C23C2×H(5,𝔽2)12yes—
562C23C2×H(5,𝔽2)11yescharacteristic
563C23C2×2−1+412yes—
564C23C2×2−1+412yes—
565C23C2×2−1+411yescharacteristic
566C23#58512yes—
567Q8—816no—
568D4—432no—
569D4—416no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—232no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—416no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—416no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—48no—
627C2×C4—44no—
628C2×C4—48no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—816no—
634C2×C4C2×#12114yes—
635C2×C4C2×#12114yes—
636C2×C4C2×#12114yes—
637C2×C4C2×#12114yes—
638C2×C4C2×#12114yes—
639C2×C4#53514yes—
640C2×C4#53614yes—
641C2×C4#53614yes—
642C2×C4#53718yes—
643C2×C4#53714yes—
644C2×C4#54218yes—
645C2×C4D4214yes—
646C2×C4#54714yes—
647C2×C4#54814yes—
648C2×C4#55314yes—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—42no—
653C22—42no—
654C22—44no—
655C22—42no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—42no—
660C22—44no—
661C22—44no—
662C22—48no—
663C22—48no—
664C22—44no—
665C22—82no—
666C22—82no—
667C22—84no—
668C22—82no—
669C22—84no—
670C22—82no—
671C22C22×#12111yescharacteristic
672C22C22×#12111yescharacteristic
673C22C2×#53511yescharacteristic
674C22C2×#53611yescharacteristic
675C22C2×#53611yescharacteristic
676C22C2×#53711yescharacteristic
677C22C2×#53712yes—
678C22#344911yescharacteristic
679C22C2×#54212yes—
680C22C2×D4211yescharacteristic
681C22C2×#54711yescharacteristic
682C22C2×#54811yescharacteristic
683C22C2×#55311yescharacteristic
684C22#348912yes—
685C22#349512yes—
686C22#349612yes—
687C22#349912yes—
688C22#350612yes—
689C22#350912yes—
690C22#354712yes—
691C22#354812yes—
692C22#355012yes—
693C22#355512yes—
694C4—24no—
695C4—24no—
696C4—24no—
697C4—42no—
698C4—44no—
699C4—48no—
700C4—82no—
701C4—82no—
702C4—84no—
703C4—84no—
704C4—84no—
705C4#241618yes—
706C4#262018yescomplemented
707C2—44no—
708C2—84no—
709C2C2×#262011yescharacteristic
710C2C2×#241611yescharacteristic
711C2#3516511yescharacteristic
712C2#3530611yescharacteristic
713C2#3531611yescharacteristic
714C2#3763311yescharacteristic
715C2#3764011yescharacteristic
716C2#3934012yes—
717C2#3935012yes—
718C2#3972712yes—
719C2#3973712yes—
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×#2547C2semidirect
5. C2×#2285C2semidirect
6. #16518C2semidirect
7. #16401C2semidirect
8. #15217C2semidirect
9. #15184C2semidirect
10. #15131C2semidirect
11. #15129C2semidirect
12. #14258C2semidirect
13. #14155C2semidirect
14. #13910C2semidirect
15. #13906C2semidirect
162. C4×C4⋊C4D4semidirect
706. C4#2620semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#25472
C2×#26202
C2×C4×#1212
C2×#22852
#165182
#164012
#152172
#151842
#151312
#151292
#142582
#141552
#139102
#139062