Finite Group Databasebeta

← #637038

Subgroups — #637038

Order 512.

5708 conjugacy classes of subgroups.

The subgroup lattice has 767 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#2486C211yesmaximal, characteristic, complemented
3C2×#2409C211yesmaximal, characteristic, complemented
4C2×C4×#120C211yesmaximal, characteristic
5#16632C214yesmaximal, complemented
6#16606C214yesmaximal, complemented
7#15179C214yesmaximal, complemented
8#15177C214yesmaximal, complemented
9#14918C212yesmaximal, complemented
10#14911C212yesmaximal
11#14680C211yesmaximal, characteristic, complemented
12#14263C212yesmaximal, complemented
13#14256C212yesmaximal, complemented
14#13922C211yesmaximal, characteristic, complemented
15#13904C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C25×C4C2211yescharacteristic
18C22×#123C2212yes—
19C22×C22⋊Q8—28no—
20C22×C22⋊Q8—28no—
21C22×C22⋊Q8—28no—
22C22×C22⋊Q8C2214yes—
23C22×C22⋊Q8C2214yes—
24C22×C22⋊Q8C2212yescomplemented
25C22×#120C2211yescharacteristic
26C22×C4×D4C2211yescharacteristic
27C22×C4×D4C2212yes—
28C23×C4⋊C4C2212yes—
29C23×C4⋊C4C2211yescharacteristic
30C23×C22⋊C4C2212yes—
31#2486—28no—
32#2409—28no—
33C2×#398C2212yescomplemented
34C2×#397C2212yescomplemented
35C2×#395C2212yescomplemented
36C2×#393C2212yescomplemented
37C4×#120—28no—
38C2×#391C2212yescomplemented
39C2×#391C2214yescomplemented
40C2×#389C2212yescomplemented
41C2×#386C2212yes—
42C2×#386C2211yescharacteristic
43C2×#386C2212yes—
44C2×#385C2212yes—
45C2×#385C2211yescharacteristic
46C2×#385C2212yes—
47C2×#381C2212yes—
48C2×#381C2212yes—
49C2×#381C2212yes—
50C2×#380C2211yescharacteristic
51C2×#380C2211yescharacteristic
52C2×#380C2214yes—
53C2×#380C2212yescomplemented
54C2×#379C2211yescharacteristic
55C2×#379C2212yes—
56C2×C4×C22⋊C4C2212yes—
57C2×C4×C22⋊C4C2211yescharacteristic
58#1446C2214yescomplemented
59#1445C2214yescomplemented
60#1444C2218yescomplemented
61#1442C2218yescomplemented
62#1441C2218yescomplemented
63#1440C2218yes—
64#1439C2218yes—
65#1439C2218yes—
66#1439C2218yescomplemented
67#1438C2218yes—
68C4⋊#95C2214yescomplemented
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2214yes—
73C26C2311yescharacteristic
74C23×Q8—24no—
75C23×Q8—24no—
76C23×Q8—48no—
77C23×Q8D418yescomplemented
78C23×Q8C2312yes—
79C23×Q8C2312yes—
80C23×D4C2311yescharacteristic
81C23×D4C2312yes—
82C24×C4—24no—
83C24×C4—24no—
84C24×C4—48no—
85C24×C4D418yes—
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2312yes—
92C24×C4C2312yes—
93C2×#123—28no—
94C2×#123—28no—
95C2×#123—44no—
96C2×C22⋊Q8—2128no—
97C2×C22⋊Q8—216no—
98C2×C22⋊Q8—216no—
99C2×C22⋊Q8—216no—
100C2×C22⋊Q8—216no—
101C2×C22⋊Q8—28no—
102C2×C22⋊Q8—216no—
103C2×C22⋊Q8—216no—
104C2×C22⋊Q8—48no—
105C2×#120—24no—
106C2×#120—24no—
107C2×#120—24no—
108C2×C4×D4—24no—
109C2×C4×D4—24no—
110C2×C4×D4—28no—
111C2×C4×D4—24no—
112C2×C4×D4—48no—
113C22×C4⋊C4—28no—
114C22×C4⋊C4—28no—
115C22×C4⋊C4—28no—
116C22×C4⋊C4—24no—
117C22×C4⋊C4—24no—
118C22×C4⋊C4D418yescomplemented
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2318yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2314yes—
127C22×C4⋊C4C2312yes—
128C22×C22⋊C4—28no—
129C22×C22⋊C4—28no—
130C22×C22⋊C4—28no—
131C22×C22⋊C4C2311yescharacteristic
132C22×C22⋊C4C2318yes—
133C22×C22⋊C4C2311yescharacteristic
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2311yescharacteristic
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2312yes—
139C22×C22⋊C4C2314yes—
140C22×C22⋊C4C2311yescharacteristic
141C22×C22⋊C4C2314yes—
142C22×C22⋊C4C2312yes—
143C22×C22⋊C4C2312yes—
144C22×C22⋊C4C2318yes—
145C22×C42C2312yes—
146C22×C42C2311yescharacteristic
147#398—28no—
148#397—28no—
149#395—28no—
150#393—28no—
151#391—28no—
152#391—48no—
153#389—28no—
154#386—28no—
155#386—24no—
156#386—28no—
157#385—28no—
158#385—24no—
159#385—28no—
160#381—28no—
161#381—28no—
162#381—44no—
163#380—24no—
164#380—24no—
165#380—28no—
166#380—48no—
167#379—24no—
168#379—44no—
169C4×C22⋊C4—28no—
170C4×C22⋊C4—24no—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2318yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2318yes—
182C2×#95C2318yes—
183C2×#95C2318yes—
184C2×#95C2318yes—
185C25—22no—
186C25—22no—
187C25—28no—
188C25—22no—
189C25—22no—
190C25—22no—
191C25—22no—
192C25—44no—
193C25C2×D414yes—
194C25C2411yescharacteristic
195C25C2412yes—
196C25C2411yescharacteristic
197C22×Q8—232no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—28no—
203C22×Q8—28no—
204C22×Q8—48no—
205C22×Q8—432no—
206C22×Q8—432no—
207C22×Q8—432no—
208C22×Q8C2×D418yes—
209C22×Q8C2×D418yes—
210C22×D4—24no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—24no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—28no—
219C22×D4—24no—
220C22×D4—44no—
221C22×D4#4114yes—
222C23×C4—28no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—232no—
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—24no—
237C23×C4—24no—
238C23×C4—28no—
239C23×C4—28no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—28no—
244C23×C4—432no—
245C23×C4—44no—
246C23×C4—44no—
247C23×C4—44no—
248C23×C4—48no—
249C23×C4—432no—
250C23×C4—432no—
251C23×C4C2×D418yes—
252C23×C4C2×D418yes—
253C23×C4C2×D414yes—
254C23×C4C2×D414yes—
255C23×C4C2×D414yes—
256C23×C4C2×D418yes—
257C23×C4C2×D414yes—
258C23×C4C2×D418yes—
259C23×C4C2414yes—
260C23×C4C2414yes—
261C23×C4C2411yescharacteristic
262C23×C4C2412yes—
263C23×C4C2411yescharacteristic
264C23×C4C2414yes—
265C23×C4C2414yes—
266C23×C4C2412yes—
267C23×C4C2411yescharacteristic
268C23×C4C2412yes—
269C23×C4C2411yescharacteristic
270C23×C4C2411yescharacteristic
271#123—432no—
272C22⋊Q8—2128no—
273C22⋊Q8—2128no—
274C22⋊Q8—432no—
275#120—232no—
276C4×D4—432no—
277C4×D4—416no—
278C2×C4⋊C4—264no—
279C2×C4⋊C4—232no—
280C2×C4⋊C4—216no—
281C2×C4⋊C4—216no—
282C2×C4⋊C4—216no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—48no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4C2×D418yes—
302C2×C4⋊C4C2×D418yes—
303C2×C22⋊C4—264no—
304C2×C22⋊C4—216no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—216no—
309C2×C22⋊C4—216no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—28no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—28no—
333C2×C22⋊C4—28no—
334C2×C22⋊C4—216no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—48no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—48no—
340C2×C22⋊C4—48no—
341C2×C22⋊C4#4114yes—
342C2×C42—22no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—28no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—28no—
351#95—24no—
352#95—24no—
353#95—28no—
354#95—28no—
355#95—28no—
356#95—216no—
357#95—48no—
358#95—48no—
359#95—48no—
360#95—48no—
361C24—24no—
362C24—24no—
363C24—216no—
364C24—216no—
365C24—28no—
366C24—216no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—24no—
373C24—24no—
374C24—24no—
375C24—22no—
376C24—28no—
377C24—28no—
378C24—21no—
379C24—21no—
380C24—21no—
381C24—21no—
382C24—21no—
383C24—24no—
384C24—21no—
385C24—24no—
386C24—48no—
387C24—416no—
388C24—416no—
389C24—416no—
390C24—416no—
391C24—48no—
392C24—48no—
393C24—44no—
394C24—44no—
395C24—44no—
396C24—44no—
397C24C22×D414yes—
398C24C22×D412yes—
399C24C22×D412yes—
400C24C22×D412yes—
401C24C2×#4112yes—
402C24C2×#4112yes—
403C24C2×#4112yes—
404C24H(5,𝔽2)12yes—
405C242−1+418yes—
406C242−1+412yes—
407C24C2511yescharacteristic
408C2×Q8—216no—
409C2×Q8—216no—
410C2×Q8—216no—
411C2×Q8—216no—
412C2×Q8—416no—
413C2×Q8—416no—
414C2×Q8—416no—
415C2×Q8—4128no—
416C2×Q8—416no—
417C2×Q8—416no—
418C2×D4—216no—
419C2×D4—28no—
420C2×D4—28no—
421C2×D4—28no—
422C2×D4—416no—
423C2×D4—416no—
424C2×D4—48no—
425C2×D4—44no—
426C2×D4—48no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—44no—
430C22×C4—24no—
431C22×C4—216no—
432C22×C4—216no—
433C22×C4—216no—
434C22×C4—216no—
435C22×C4—216no—
436C22×C4—28no—
437C22×C4—216no—
438C22×C4—216no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—28no—
457C22×C4—24no—
458C22×C4—28no—
459C22×C4—21no—
460C22×C4—24no—
461C22×C4—21no—
462C22×C4—24no—
463C22×C4—216no—
464C22×C4—216no—
465C22×C4—216no—
466C22×C4—216no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—28no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—28no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—22no—
488C22×C4—28no—
489C22×C4—28no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—416no—
494C22×C4—416no—
495C22×C4—416no—
496C22×C4—4128no—
497C22×C4—48no—
498C22×C4—48no—
499C22×C4—416no—
500C22×C4—416no—
501C22×C4—48no—
502C22×C4—48no—
503C22×C4—48no—
504C22×C4—48no—
505C22×C4—416no—
506C22×C4—416no—
507C22×C4—44no—
508C22×C4—48no—
509C22×C4—48no—
510C22×C4—48no—
511C22×C4#120116yes—
512C22×C4C22×D418yes—
513C22×C4C22×D418yes—
514C22×C4C22×D414yes—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D412yes—
519C22×C4C22×D412yes—
520C22×C4C22×D412yes—
521C22×C4C2×#4114yes—
522C22×C4C2×#4112yes—
523C22×C4H(5,𝔽2)18yes—
524C4⋊C4—264no—
525C4⋊C4—216no—
526C4⋊C4—216no—
527C4⋊C4—416no—
528C4⋊C4—48no—
529C4⋊C4—416no—
530C4⋊C4—44no—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C22⋊C4—264no—
534C22⋊C4—216no—
535C22⋊C4—28no—
536C22⋊C4—416no—
537C22⋊C4—48no—
538C22⋊C4—416no—
539C22⋊C4—48no—
540C22⋊C4—44no—
541C22⋊C4—48no—
542C22⋊C4—44no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—44no—
546C22⋊C4—816no—
547C42—44no—
548C42—48no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—28no—
556C23—28no—
557C23—28no—
558C23—28no—
559C23—232no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—28no—
567C23—28no—
568C23—28no—
569C23—28no—
570C23—28no—
571C23—28no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—28no—
576C23—432no—
577C23—432no—
578C23—416no—
579C23—432no—
580C23—432no—
581C23—432no—
582C23—432no—
583C23—416no—
584C23—44no—
585C23—44no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—41no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—41no—
600C23—41no—
601C23—41no—
602C23—42no—
603C23—42no—
604C23—44no—
605C23—44no—
606C23—44no—
607C23C2×#12014yes—
608C23#52514yes—
609C23#53618yes—
610C23#53714yes—
611C23#54614yes—
612C23#54714yes—
613C23#54814yes—
614C23#55014yes—
615C23C23×D411yescharacteristic
616C23C23×D411yescharacteristic
617C23C23×D411yescharacteristic
618C23C22×#4111yescharacteristic
619C23C22×#4111yescharacteristic
620C23C2×H(5,𝔽2)14yes—
621C23C2×H(5,𝔽2)11yescharacteristic
622C23C2×2−1+414yes—
623C23C2×2−1+411yescharacteristic
624Q8—432no—
625Q8—432no—
626D4—432no—
627D4—416no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—232no—
632C2×C4—28no—
633C2×C4—28no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—22no—
637C2×C4—24no—
638C2×C4—28no—
639C2×C4—24no—
640C2×C4—28no—
641C2×C4—28no—
642C2×C4—22no—
643C2×C4—28no—
644C2×C4—22no—
645C2×C4—28no—
646C2×C4—432no—
647C2×C4—416no—
648C2×C4—432no—
649C2×C4—432no—
650C2×C4—41no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—48no—
655C2×C4—44no—
656C2×C4—48no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—41no—
664C2×C4—44no—
665C2×C4—432no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—48no—
684C2×C4—48no—
685C2×C4—48no—
686C2×C4—48no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—88no—
690C2×C4C2×#12014yes—
691C2×C4C2×#12014yes—
692C2×C4#53418yes—
693C2×C4#53418yes—
694C2×C4#53518yes—
695C22—28no—
696C22—28no—
697C22—28no—
698C22—24no—
699C22—28no—
700C22—28no—
701C22—24no—
702C22—28no—
703C22—28no—
704C22—28no—
705C22—28no—
706C22—464no—
707C22—48no—
708C22—44no—
709C22—44no—
710C22—44no—
711C22—48no—
712C22—48no—
713C22—44no—
714C22—48no—
715C22—44no—
716C22—42no—
717C22—44no—
718C22—44no—
719C22—82no—
720C22—82no—
721C22—81no—
722C22—82no—
723C22—81no—
724C22#240718yes—
725C22C22×#12011yescharacteristic
726C22C2×#52511yescharacteristic
727C22C2×#53412yes—
728C22C2×#53412yes—
729C22C2×#53512yes—
730C22C2×#53612yes—
731C22C2×#53711yescharacteristic
732C22C2×#54611yescharacteristic
733C22C2×#54711yescharacteristic
734C22C2×#54811yescharacteristic
735C22C2×#55011yescharacteristic
736C22#348612yes—
737C22#348612yes—
738C22#352014yes—
739C22#352014yes—
740C22#353614yes—
741C22#353814yes—
742C4—24no—
743C4—44no—
744C4—48no—
745C4—42no—
746C4—44no—
747C4—48no—
748C4—82no—
749C4—81no—
750C4—84no—
751C4—81no—
752C4—84no—
753C4#261318yes—
754C2—28no—
755C2—48no—
756C2—82no—
757C2C2×#261311yescharacteristic
758C2C2×#240711yescharacteristic
759C2#3508412yes—
760C2#3524211yescharacteristic
761C2#3744212yes—
762C2#3890512yes—
763C2#3891712yes—
764C2#3912112yes—
765C2#3913312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
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×C22⋊Q8C2semidirect
2. C2×#2486C2semidirect
3. C2×#2409C2semidirect
5. #16632C2semidirect
6. #16606C2semidirect
7. #15179C2semidirect
8. #15177C2semidirect
9. #14918C2semidirect
11. #14680C2semidirect
12. #14263C2semidirect
13. #14256C2semidirect
14. #13922C2semidirect
15. #13904C2semidirect
24. C22×C22⋊Q8C22semidirect
33. C2×#398C22semidirect
34. C2×#397C22semidirect
35. C2×#395C22semidirect
36. C2×#393C22semidirect
38. C2×#391C22semidirect
39. C2×#391C22semidirect
40. C2×#389C22semidirect
53. C2×#380C22semidirect
58. #1446C22semidirect
59. #1445C22semidirect
60. #1444C22semidirect
61. #1442C22semidirect
62. #1441C22semidirect
66. #1439C22semidirect
68. C4⋊#95C22semidirect
77. C23×Q8D4semidirect
118. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C2×#24862
C2×#24092
C2×C4×#1202
#166322
#166062
#151792
#151772
#149182
#149112
#146802
#142632
#142562
#139222
#139042