Finite Group Databasebeta

← #847685

Subgroups — #847685

Order 512.

4904 conjugacy classes of subgroups.

The subgroup lattice has 770 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#2731C212yesmaximal, complemented
3C2×#2436C211yesmaximal, characteristic
4#16602C214yesmaximal, complemented
5#16093C212yesmaximal, complemented
6#15761C212yesmaximal, complemented
7#15322C214yesmaximal, complemented
8#14873C212yesmaximal, complemented
9#14729C212yesmaximal, complemented
10#14719C212yesmaximal
11#14696C212yesmaximal, complemented
12#14450C212yesmaximal, complemented
13#14039C212yesmaximal, complemented
14#14019C212yesmaximal, complemented
15#13922C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C24×Q8C2211yescharacteristic
18C22×C4⋊Q8—28no—
19C22×C4⋊Q8—216no—
20C22×C4⋊Q8C2212yes—
21C22×C4⋊Q8C2212yes—
22C22×C4⋊Q8C2212yes—
23C22×C4⋊Q8C2212yes—
24C22×#124C2211yescharacteristic
25C22×#124C2211yescharacteristic
26C22×#121C2211yescharacteristic
27C23×C4⋊C4C2212yes—
28C23×C4⋊C4C2212yes—
29C23×C42C2211yescharacteristic
30#2731—216no—
31#2436—28no—
32C2×#396C2214yes—
33C2×#394C2214yes—
34C2×#393C2214yes—
35C2×#393C2212yes—
36C2×#393C2212yes—
37C2×#393C2214yes—
38C2×#391C2212yes—
39C2×#391C2211yescharacteristic
40C2×#391C2211yescharacteristic
41C2×#391C2212yes—
42C2×#388C2214yes—
43C2×#388C2214yes—
44C2×#387C2214yes—
45C2×#386C2212yes—
46C2×#386C2212yes—
47C2×#386C2212yes—
48C2×#386C2212yes—
49C2×#386C2212yes—
50C2×#386C2212yes—
51C2×#385C2212yes—
52C2×#380C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2211yescharacteristic
55#1446C2214yes—
56#1446C2214yes—
57#1446C2212yes—
58#1446C2212yes—
59#1442C2214yes—
60#1442C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1437C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1434C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C23×Q8—28no—
80C23×Q8—28no—
81C23×Q8—24no—
82C23×Q8—42no—
83C23×Q8—42no—
84C23×Q8—42no—
85C23×Q8—42no—
86C23×Q8—44no—
87C23×Q8—44no—
88C23×Q8D418yes—
89C23×Q8C2312yes—
90C23×Q8C2311yescharacteristic
91C23×Q8C2311yescharacteristic
92C23×Q8C2311yescharacteristic
93C23×Q8C2312yes—
94C23×Q8C2311yescharacteristic
95C23×D4C2312yes—
96C23×D4C2311yescharacteristic
97C24×C4C2311yescharacteristic
98C24×C4C2312yes—
99C24×C4C2312yes—
100C24×C4C2311yescharacteristic
101C24×C4C2311yescharacteristic
102C2×C4⋊Q8—2128no—
103C2×C4⋊Q8—28no—
104C2×C4⋊Q8—28no—
105C2×C4⋊Q8—28no—
106C2×C4⋊Q8—28no—
107C2×C4⋊Q8—28no—
108C2×C4⋊Q8—28no—
109C2×C4⋊Q8—216no—
110C2×C4⋊Q8—216no—
111C2×C4⋊Q8—28no—
112C2×C4⋊Q8D4116yescomplemented
113C2×#124—24no—
114C2×#124—24no—
115C2×#124—28no—
116C2×#124—44no—
117C2×#121—24no—
118C2×#121—42no—
119C2×#121—42no—
120C22×C4⋊C4—28no—
121C22×C4⋊C4—28no—
122C22×C4⋊C4—216no—
123C22×C4⋊C4—216no—
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2314yes—
126C22×C4⋊C4C2314yes—
127C22×C4⋊C4C2314yes—
128C22×C4⋊C4C2314yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2314yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2312yes—
139C22×C22⋊C4C2314yes—
140C22×C22⋊C4C2314yes—
141C22×C22⋊C4C2312yes—
142C22×C22⋊C4C2312yes—
143C22×C42—24no—
144C22×C42—28no—
145C22×C42C2312yes—
146C22×C42C2311yescharacteristic
147C22×C42C2312yes—
148C22×C42C2311yescharacteristic
149C22×C42C2312yes—
150#396—216no—
151#394—216no—
152#393—28no—
153#393—28no—
154#393—216no—
155#393—48no—
156#391—24no—
157#391—24no—
158#391—28no—
159#391—44no—
160#388—216no—
161#388—216no—
162#387—216no—
163#386—28no—
164#386—28no—
165#386—28no—
166#386—28no—
167#386—44no—
168#386—44no—
169#385—28no—
170#380—42no—
171C4×C22⋊C4—24no—
172C4×C22⋊C4—42no—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2314yes—
195C25C2412yes—
196C25C2411yescharacteristic
197C22×Q8—232no—
198C22×Q8—232no—
199C22×Q8—28no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8—24no—
207C22×Q8—24no—
208C22×Q8—28no—
209C22×Q8—416no—
210C22×Q8—41no—
211C22×Q8—41no—
212C22×Q8—42no—
213C22×Q8—41no—
214C22×Q8—41no—
215C22×Q8—41no—
216C22×Q8—41no—
217C22×Q8—41no—
218C22×Q8—41no—
219C22×Q8—42no—
220C22×Q8—42no—
221C22×Q8—42no—
222C22×Q8—42no—
223C22×Q8—42no—
224C22×Q8—42no—
225C22×Q8—42no—
226C22×Q8—416no—
227C22×Q8—432no—
228C22×Q8—432no—
229C22×Q8—432no—
230C22×Q8—432no—
231C22×Q8—416no—
232C22×Q8—416no—
233C22×Q8C2×D414yes—
234C22×Q8C2×D414yes—
235C22×Q8C2×D414yes—
236C22×Q8C2×D418yes—
237C22×Q8C2×D414yes—
238C22×Q8C2×D418yes—
239C22×D4—24no—
240C22×D4—24no—
241C22×D4—24no—
242C22×D4—24no—
243C22×D4—24no—
244C22×D4—24no—
245C22×D4—24no—
246C22×D4—22no—
247C22×D4—24no—
248C22×D4—41no—
249C22×D4—41no—
250C22×D4—42no—
251C23×C4—24no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—24no—
255C23×C4—28no—
256C23×C4—24no—
257C23×C4—24no—
258C23×C4—28no—
259C23×C4—24no—
260C23×C4—22no—
261C23×C4—22no—
262C23×C4—22no—
263C23×C4—22no—
264C23×C4—28no—
265C23×C4—24no—
266C23×C4—24no—
267C23×C4—24no—
268C23×C4—24no—
269C23×C4—42no—
270C23×C4—42no—
271C23×C4—44no—
272C23×C4C2×D414yes—
273C23×C4C2×D414yes—
274C23×C4C2×D414yes—
275C23×C4#4118yes—
276C23×C4C2414yes—
277C23×C4C2412yes—
278C23×C4C2414yes—
279C23×C4C2412yes—
280C23×C4C2412yes—
281C23×C4C2411yescharacteristic
282C23×C4C2411yescharacteristic
283C23×C4C2414yes—
284C23×C4C2411yescharacteristic
285C23×C4C2411yescharacteristic
286C23×C4C2411yescharacteristic
287C23×C4C2414yes—
288C23×C4C2411yescharacteristic
289C4⋊Q8—264no—
290C4⋊Q8—264no—
291C4⋊Q8—264no—
292C4⋊Q8—264no—
293#124—416no—
294#124—416no—
295#121—416no—
296C2×C4⋊C4—264no—
297C2×C4⋊C4—264no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—28no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—28no—
302C2×C4⋊C4—28no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—28no—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—41no—
314C2×C4⋊C4—41no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4C2×D4116yes—
318C2×C4⋊C4C2×D4116yes—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—28no—
334C2×C22⋊C4—28no—
335C2×C22⋊C4—28no—
336C2×C22⋊C4—28no—
337C2×C22⋊C4—28no—
338C2×C22⋊C4—28no—
339C2×C22⋊C4—28no—
340C2×C22⋊C4—28no—
341C2×C22⋊C4—28no—
342C2×C22⋊C4—28no—
343C2×C22⋊C4—28no—
344C2×C22⋊C4—42no—
345C2×C22⋊C4—42no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4—44no—
349C2×C22⋊C4—44no—
350C2×C22⋊C4—44no—
351C2×C42—232no—
352C2×C42—24no—
353C2×C42—28no—
354C2×C42—24no—
355C2×C42—24no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—24no—
359C2×C42—24no—
360C2×C42—42no—
361C2×C42—42no—
362C2×C42—42no—
363C2×C42C2×D418yes—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—28no—
370#95—28no—
371#95—28no—
372#95—28no—
373#95—28no—
374#95—28no—
375#95—28no—
376#95—28no—
377#95—28no—
378#95—28no—
379#95—42no—
380#95—42no—
381#95—44no—
382#95—44no—
383#95—44no—
384#95—44no—
385#95—44no—
386C24—22no—
387C24—22no—
388C24—24no—
389C24—24no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24—24no—
394C24—22no—
395C24—22no—
396C24—24no—
397C24—24no—
398C24—22no—
399C24—22no—
400C24C22×D412yes—
401C24C2×#4114yes—
402C24H(5,𝔽2)12yes—
403C24H(5,𝔽2)12yes—
404C242−1+414yes—
405C24C2511yescharacteristic
406C2×Q8—216no—
407C2×Q8—28no—
408C2×Q8—28no—
409C2×Q8—28no—
410C2×Q8—216no—
411C2×Q8—28no—
412C2×Q8—28no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—48no—
420C2×Q8—48no—
421C2×Q8—48no—
422C2×Q8—48no—
423C2×Q8—48no—
424C2×Q8—48no—
425C2×Q8—48no—
426C2×Q8—48no—
427C2×Q8—416no—
428C2×Q8—416no—
429C2×Q8—4128no—
430C2×Q8—4128no—
431C2×Q8—48no—
432C2×Q8—48no—
433C2×Q8—48no—
434C2×Q8—48no—
435C2×Q8—48no—
436C2×Q8—48no—
437C2×Q8—48no—
438C2×Q8—48no—
439C2×Q8#120116yes—
440C2×D4—44no—
441C2×D4—44no—
442C2×D4—48no—
443C2×D4—48no—
444C2×D4—48no—
445C2×D4—48no—
446C2×D4—48no—
447C2×D4—48no—
448C2×D4—48no—
449C2×D4—44no—
450C2×D4—84no—
451C2×D4—84no—
452C22×C4—216no—
453C22×C4—232no—
454C22×C4—216no—
455C22×C4—216no—
456C22×C4—216no—
457C22×C4—28no—
458C22×C4—216no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—28no—
476C22×C4—28no—
477C22×C4—232no—
478C22×C4—216no—
479C22×C4—28no—
480C22×C4—28no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—24no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—22no—
495C22×C4—28no—
496C22×C4—28no—
497C22×C4—24no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—24no—
502C22×C4—24no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—24no—
507C22×C4—24no—
508C22×C4—24no—
509C22×C4—22no—
510C22×C4—24no—
511C22×C4—24no—
512C22×C4—24no—
513C22×C4—24no—
514C22×C4—24no—
515C22×C4—22no—
516C22×C4—22no—
517C22×C4—24no—
518C22×C4—24no—
519C22×C4—24no—
520C22×C4—22no—
521C22×C4—24no—
522C22×C4—24no—
523C22×C4—22no—
524C22×C4—24no—
525C22×C4—24no—
526C22×C4—22no—
527C22×C4—22no—
528C22×C4—44no—
529C22×C4—44no—
530C22×C4—48no—
531C22×C4—48no—
532C22×C4—48no—
533C22×C4—48no—
534C22×C4—416no—
535C22×C4—416no—
536C22×C4—48no—
537C22×C4—48no—
538C22×C4C22×D412yes—
539C22×C4C22×D412yes—
540C22×C4C22×D418yes—
541C22×C4C22×D412yes—
542C22×C4C22×D414yes—
543C22×C4C22×D414yes—
544C22×C4C22×D412yes—
545C22×C4C22×D412yes—
546C22×C4C22×D414yes—
547C22×C4C22×D412yes—
548C22×C4C22×D418yes—
549C22×C4C2×#4118yes—
550C22×C4H(5,𝔽2)14yes—
551C22×C42−1+418yes—
552C4⋊C4—232no—
553C4⋊C4—232no—
554C4⋊C4—232no—
555C4⋊C4—232no—
556C4⋊C4—44no—
557C4⋊C4—416no—
558C4⋊C4—84no—
559C22⋊C4—48no—
560C22⋊C4—48no—
561C22⋊C4—48no—
562C22⋊C4—48no—
563C22⋊C4—48no—
564C22⋊C4—48no—
565C22⋊C4—48no—
566C22⋊C4—48no—
567C22⋊C4—416no—
568C22⋊C4—416no—
569C22⋊C4—88no—
570C22⋊C4—88no—
571C42—216no—
572C42—216no—
573C42—44no—
574C42—44no—
575C42—84no—
576C23—22no—
577C23—22no—
578C23—28no—
579C23—24no—
580C23—24no—
581C23—24no—
582C23—28no—
583C23—28no—
584C23—24no—
585C23—24no—
586C23—28no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—44no—
593C23—44no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—44no—
598C23—44no—
599C23—42no—
600C23—42no—
601C23—44no—
602C23—44no—
603C23—44no—
604C23—44no—
605C23—42no—
606C23—44no—
607C23—44no—
608C23—44no—
609C23—44no—
610C23#53414yes—
611C23#54218yes—
612C23#54818yes—
613C23#55514yes—
614C23#56314yes—
615C23C23×D411yescharacteristic
616C23C23×D412yes—
617C23C22×#4112yes—
618C23C2×H(5,𝔽2)11yescharacteristic
619C23C2×H(5,𝔽2)11yescharacteristic
620C23C2×H(5,𝔽2)12yes—
621C23C2×2−1+412yes—
622C23C2×2−1+414yes—
623Q8—416no—
624Q8—416no—
625Q8—416no—
626Q8—416no—
627Q8—432no—
628Q8—432no—
629D4—88no—
630D4—816no—
631C2×C4—264no—
632C2×C4—232no—
633C2×C4—28no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—28no—
643C2×C4—28no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—28no—
653C2×C4—28no—
654C2×C4—28no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—28no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—22no—
662C2×C4—28no—
663C2×C4—28no—
664C2×C4—416no—
665C2×C4—416no—
666C2×C4—432no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—44no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—44no—
683C2×C4—42no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—44no—
690C2×C4—42no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—42no—
694C2×C4—44no—
695C2×C4—42no—
696C2×C4—44no—
697C2×C4—44no—
698C2×C4—44no—
699C2×C4—44no—
700C2×C4—44no—
701C2×C4—42no—
702C2×C4—44no—
703C2×C4—44no—
704C2×C4—44no—
705C2×C4—44no—
706C2×C4C2×#12014yes—
707C2×C4C2×#12014yes—
708C2×C4C2×#12014yes—
709C2×C4#53414yes—
710C2×C4#548116yes—
711C22—28no—
712C22—216no—
713C22—24no—
714C22—24no—
715C22—28no—
716C22—28no—
717C22—28no—
718C22—82no—
719C22—82no—
720C22—84no—
721C22—84no—
722C22—82no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22—84no—
728C22—84no—
729C22#273118yescomplemented
730C22C22×#12011yescharacteristic
731C22C2×#53411yescharacteristic
732C22C2×#53411yescharacteristic
733C22C2×#54212yes—
734C22C2×#54812yes—
735C22C2×#54814yes—
736C22C2×#55511yescharacteristic
737C22#348412yes—
738C22#348412yes—
739C22#348612yes—
740C22#348614yes—
741C22#351114yes—
742C22C2×#56311yescharacteristic
743C22#354412yes—
744C22#354412yes—
745C22#355214yes—
746C4—24no—
747C4—24no—
748C4—24no—
749C4—24no—
750C4—216no—
751C4—42no—
752C4—42no—
753C4—48no—
754C4—82no—
755C4—84no—
756C4—82no—
757C4—82no—
758C4—84no—
759C2—28no—
760C2—162no—
761C2#3498011yescharacteristic
762C2#3657512yes—
763C2#3725811yescharacteristic
764C2#3770414yes—
765C2#3972112yes—
766C2#4632512yes—
767C2C2×#273111yescharacteristic
768C2#5913512yes—
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. C23×C4⋊Q8C2semidirect
2. C2×#2731C2semidirect
4. #16602C2semidirect
5. #16093C2semidirect
6. #15761C2semidirect
7. #15322C2semidirect
8. #14873C2semidirect
9. #14729C2semidirect
11. #14696C2semidirect
12. #14450C2semidirect
13. #14039C2semidirect
14. #14019C2semidirect
15. #13922C2semidirect
112. C2×C4⋊Q8D4semidirect
729. C22#2731semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊Q82
C2×#27312
C2×#24362
#166022
#160932
#157612
#153222
#148732
#147292
#147192
#146962
#144502
#140392
#140192
#139222