Finite Group Databasebeta

← #4309028

Subgroups — #4309028

Order 512.

4432 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#126C211yesmaximal, characteristic, complemented
2C2×#2854C211yesmaximal, characteristic, complemented
3C2×#2618C211yesmaximal, characteristic, complemented
4C2×#2617C211yesmaximal, characteristic
5#16673C214yesmaximal, complemented
6#16658C218yesmaximal, complemented
7#16655C214yesmaximal, complemented
8#15091C212yesmaximal, complemented
9#15069C212yesmaximal, complemented
10#14911C212yesmaximal, complemented
11#14911C212yesmaximal, complemented
12#14685C211yesmaximal, characteristic, complemented
13#14339C211yesmaximal, characteristic, complemented
14#14338C211yesmaximal, characteristic, complemented
15C22×#126—28no—
16C22×#126—28no—
17C22×#126—28no—
18C22×#126C2214yes—
19C22×#126C2214yes—
20C22×#123C2212yes—
21C22×#123C2212yes—
22C22×#121C2211yescharacteristic
23C22×#120C2212yes—
24C23×C4⋊C4C2211yescharacteristic
25C23×C4⋊C4C2211yescharacteristic
26C23×C4⋊C4C2211yescharacteristic
27C23×C22⋊C4C2211yescharacteristic
28C23×C22⋊C4C2211yescharacteristic
29C23×C22⋊C4C2211yescharacteristic
30C23×C42C2211yescharacteristic
31#2854—28no—
32#2618—28no—
33#2617—28no—
34C2×#398C2212yescomplemented
35C2×#397C2212yes—
36C2×#396C2214yescomplemented
37C2×#394C2214yes—
38C2×#394C2212yescomplemented
39C2×#392C2212yes—
40C2×#390C2212yescomplemented
41C2×#389C2212yescomplemented
42C2×#388C2214yescomplemented
43C2×#386C2212yes—
44C2×#386C2212yes—
45C2×#386C2212yes—
46C2×#381C2212yes—
47C2×#381C2212yes—
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×#379C2212yes—
52#1444C2218yescomplemented
53#1442C2218yescomplemented
54#1441C2214yescomplemented
55#1441C2214yescomplemented
56#1439C2214yes—
57#1439C2214yes—
58#1439C2218yes—
59#1439C2218yes—
60#1439C2218yescomplemented
61#1439C2218yescomplemented
62#1438C2218yes—
63#1438C2218yes—
64C4⋊#95C2214yescomplemented
65#1434C2214yescomplemented
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C26C2311yescharacteristic
70C23×D4C2312yes—
71C23×D4C2311yescharacteristic
72C23×D4C2314yes—
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C2×#126—2128no—
80C2×#126—216no—
81C2×#126—216no—
82C2×#126—216no—
83C2×#126—216no—
84C2×#126—216no—
85C2×#126—216no—
86C2×#123—28no—
87C2×#123—28no—
88C2×#123—44no—
89C2×#123—44no—
90C2×#123—44no—
91C2×#123—44no—
92C2×#121—24no—
93C2×#121—24no—
94C2×#121—24no—
95C2×#120—28no—
96C2×#120—44no—
97C2×#120—44no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4D418yescomplemented
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C22⋊C4—24no—
118C22×C22⋊C4—24no—
119C22×C22⋊C4—24no—
120C22×C22⋊C4—24no—
121C22×C22⋊C4—24no—
122C22×C22⋊C4—24no—
123C22×C22⋊C4—24no—
124C22×C22⋊C4—24no—
125C22×C22⋊C4D418yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2314yes—
136C22×C22⋊C4C2314yes—
137C22×C22⋊C4C2314yes—
138C22×C22⋊C4C2314yes—
139C22×C22⋊C4C2314yes—
140C22×C22⋊C4C2318yes—
141C22×C42—24no—
142C22×C42—24no—
143C22×C42D418yescomplemented
144C22×C42C2312yes—
145C22×C42C2312yes—
146#398—28no—
147#397—28no—
148#396—216no—
149#394—28no—
150#394—216no—
151#392—28no—
152#390—28no—
153#389—28no—
154#388—216no—
155#386—28no—
156#386—28no—
157#386—28no—
158#381—44no—
159#381—44no—
160#380—24no—
161#380—24no—
162#380—24no—
163#379—44no—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2318yes—
171C2×#95C2318yes—
172C2×#95C2318yes—
173C2×#95C2318yes—
174C2×#95C2318yes—
175C25—22no—
176C25—22no—
177C25—22no—
178C25—22no—
179C25—22no—
180C25—22no—
181C25—44no—
182C25—44no—
183C25C2×D414yes—
184C25C2412yes—
185C25C2412yes—
186C25C2411yescharacteristic
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—28no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4—48no—
202C22×D4—48no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4C2414yes—
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2414yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244#126—2128no—
245#126—2128no—
246#123—432no—
247#123—432no—
248#121—416no—
249#120—432no—
250C2×C4⋊C4—232no—
251C2×C4⋊C4—232no—
252C2×C4⋊C4—232no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4C2×D418yes—
279C2×C4⋊C4C2×D418yes—
280C2×C22⋊C4—232no—
281C2×C22⋊C4—232no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—232no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—48no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—48no—
319C2×C22⋊C4—48no—
320C2×C22⋊C4—48no—
321C2×C22⋊C4—48no—
322C2×C22⋊C4—48no—
323C2×C22⋊C4C2×D418yes—
324C2×C22⋊C4C2×D418yes—
325C2×C42—232no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42C2×D418yes—
331C2×C42C2×D418yes—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—28no—
336#95—28no—
337#95—28no—
338#95—48no—
339#95—48no—
340#95—48no—
341#95—48no—
342#95—48no—
343C24—24no—
344C24—24no—
345C24—28no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—28no—
353C24—28no—
354C24—28no—
355C24—28no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—28no—
364C24—28no—
365C24—48no—
366C24—48no—
367C24—48no—
368C24—48no—
369C24—48no—
370C24—48no—
371C24—44no—
372C24—44no—
373C24—44no—
374C24—44no—
375C24—88no—
376C24—88no—
377C24—88no—
378C24—88no—
379C24C22×D414yes—
380C24C22×D412yes—
381C24C22×D412yes—
382C24C22×D412yes—
383C24H(5,𝔽2)12yes—
384C24H(5,𝔽2)12yes—
385C24H(5,𝔽2)12yes—
386C242−1+412yes—
387C24C2511yescharacteristic
388C2×D4—48no—
389C2×D4—416no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—44no—
398C2×D4—84no—
399C2×D4—84no—
400C2×D4—88no—
401C2×D4—88no—
402C2×D4—816no—
403C2×D4—816no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—28no—
435C22×C4—28no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—28no—
454C22×C4—28no—
455C22×C4—28no—
456C22×C4—28no—
457C22×C4—28no—
458C22×C4—28no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4#120116yes—
480C22×C4C22×D414yes—
481C22×C4C22×D414yes—
482C22×C4C22×D414yes—
483C22×C4C22×D414yes—
484C22×C4C22×D414yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C4⋊C4—232no—
492C4⋊C4—232no—
493C4⋊C4—216no—
494C4⋊C4—216no—
495C4⋊C4—48no—
496C4⋊C4—416no—
497C4⋊C4—416no—
498C4⋊C4—44no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C22⋊C4—232no—
502C22⋊C4—232no—
503C22⋊C4—216no—
504C22⋊C4—216no—
505C22⋊C4—416no—
506C22⋊C4—416no—
507C22⋊C4—416no—
508C22⋊C4—416no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—88no—
514C22⋊C4—88no—
515C22⋊C4—816no—
516C42—216no—
517C42—216no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—28no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—28no—
532C23—28no—
533C23—416no—
534C23—416no—
535C23—416no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—48no—
545C23—48no—
546C23—48no—
547C23—48no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—48no—
562C23—48no—
563C23—48no—
564C23—48no—
565C23—816no—
566C23—816no—
567C23—816no—
568C23—816no—
569C23—816no—
570C23—816no—
571C23C2×#12014yes—
572C23#53514yes—
573C23#53514yes—
574C23#53514yes—
575C23#53714yes—
576C23#53714yes—
577C23#53714yes—
578C23C23×D411yescharacteristic
579C23C23×D411yescharacteristic
580C23C23×D411yescharacteristic
581C23C2×H(5,𝔽2)11yescharacteristic
582C23C2×H(5,𝔽2)11yescharacteristic
583C23C2×H(5,𝔽2)11yescharacteristic
584C23C2×2−1+411yescharacteristic
585C23#58514yes—
586C23#58514yes—
587D4—88no—
588D4—832no—
589D4—816no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—416no—
610C2×C4—416no—
611C2×C4—416no—
612C2×C4—416no—
613C2×C4—416no—
614C2×C4—416no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—48no—
646C2×C4—48no—
647C2×C4—48no—
648C2×C4—48no—
649C2×C4—48no—
650C2×C4—48no—
651C2×C4C2×#12014yes—
652C2×C4C2×#12014yes—
653C22—24no—
654C22—24no—
655C22—28no—
656C22—28no—
657C22—28no—
658C22—28no—
659C22—28no—
660C22—28no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—48no—
665C22—48no—
666C22—832no—
667C22—84no—
668C22—84no—
669C22—84no—
670C22—84no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—84no—
675C22—84no—
676C22—84no—
677C22—84no—
678C22—84no—
679C22—84no—
680C22#261518yes—
681C22C22×#12011yescharacteristic
682C22C2×#53511yescharacteristic
683C22C2×#53511yescharacteristic
684C22C2×#53511yescharacteristic
685C22C2×#53711yescharacteristic
686C22C2×#53711yescharacteristic
687C22C2×#53711yescharacteristic
688C22#344912yes—
689C22#344912yes—
690C22#348512yes—
691C22#348512yes—
692C22#348512yes—
693C22#348712yes—
694C22#356014yes—
695C22#356014yes—
696C22#356014yes—
697C22#356414yes—
698C4—24no—
699C4—24no—
700C4—44no—
701C4—44no—
702C4—44no—
703C4—44no—
704C4—44no—
705C4—82no—
706C4—84no—
707C4—84no—
708C4—84no—
709C2—28no—
710C2—84no—
711C2—162no—
712C2C2×#261511yescharacteristic
713C2#3508811yescharacteristic
714C2#3514911yescharacteristic
715C2#3696012yes—
716C2#3696112yes—
717C2#3696912yes—
718C2#3697112yes—
719C2#3753812yes—
720C2#3754712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2617C2
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×#126C2semidirect
2. C2×#2854C2semidirect
3. C2×#2618C2semidirect
5. #16673C2semidirect
6. #16658C2semidirect
7. #16655C2semidirect
8. #15091C2semidirect
9. #15069C2semidirect
10. #14911C2semidirect
11. #14911C2semidirect
12. #14685C2semidirect
13. #14339C2semidirect
14. #14338C2semidirect
34. C2×#398C22semidirect
36. C2×#396C22semidirect
38. C2×#394C22semidirect
40. C2×#390C22semidirect
41. C2×#389C22semidirect
42. C2×#388C22semidirect
52. #1444C22semidirect
53. #1442C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
60. #1439C22semidirect
61. #1439C22semidirect
64. C4⋊#95C22semidirect
65. #1434C22semidirect
106. C22×C4⋊C4D4semidirect
143. C22×C42D4semidirect

Maximal subgroups

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

subgroupindex
C23×#1262
C2×#28542
C2×#26182
C2×#26172
#166732
#166582
#166552
#150912
#150692
#149112
#146852
#143392
#143382