Finite Group Databasebeta

← #1394007

Subgroups — #1394007

Order 512.

2830 conjugacy classes of subgroups.

The subgroup lattice has 754 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2491C211yesmaximal, characteristic
2C2×#2632C212yesmaximal, complemented
3C2×#2365C211yesmaximal, characteristic, complemented
4#16665C212yesmaximal, complemented
5#16559C212yesmaximal, complemented
6#16353C214yesmaximal, complemented
7#16346C214yesmaximal, complemented
8#15148C211yesmaximal, characteristic, complemented
9#15144C211yesmaximal, characteristic, complemented
10#15128C212yesmaximal, complemented
11#15126C212yesmaximal
12#15125C211yesmaximal, characteristic, complemented
13#14875C212yesmaximal, complemented
14#14022C212yesmaximal, complemented
15#13974C212yesmaximal, complemented
16#13921C211yesmaximal, characteristic, complemented
17C4×#399C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×#124C2212yes—
21C22×#121C2211yescharacteristic
22C22×C4×Q8C2212yes—
23#2632—216no—
24#2491—28no—
25C2×#399C2212yes—
26C2×#399C2211yescharacteristic
27C2×#395C2214yes—
28C2×#394C2214yes—
29C2×#394C2212yes—
30C2×#393C2212yes—
31C2×#393C2212yes—
32C2×#393C2212yes—
33C2×#393C2212yes—
34#2365—28no—
35C2×#391C2212yes—
36C2×#390C2212yes—
37C2×#390C2211yescharacteristic
38C2×#390C2211yescharacteristic
39C2×#390C2212yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#388C2212yes—
43C2×#388C2214yes—
44C2×#388C2214yes—
45C2×#387C2214yes—
46C2×#383C2212yes—
47C2×#380C2212yes—
48C2×#380C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2212yes—
55#1446C2212yes—
56#1446C2214yes—
57#1445C2211yescharacteristic
58#1445C2212yes—
59#1444C2214yes—
60#1444C2211yescharacteristic
61#1443C2214yes—
62#1443C2214yes—
63#1442C2212yes—
64#1441C2214yes—
65#1440C2214yes—
66#1440C2214yes—
67#1440C2214yes—
68#1439C2212yes—
69#1439C2214yes—
70#1439C2212yes—
71#1439C2214yes—
72#1439C2211yescharacteristic
73#1439C2212yes—
74#1439C2214yes—
75#1439C2214yes—
76#1438C2214yes—
77#1437C2214yes—
78#1436C2214yes—
79C4⋊#95C2212yes—
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84#1434C2212yes—
85C4×#95C2211yescharacteristic
86C23×Q8C2312yes—
87C23×Q8C2312yes—
88C23×D4C2311yescharacteristic
89C23×D4C2312yes—
90C24×C4C2311yescharacteristic
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—44no—
95C2×#124—28no—
96C2×#124—28no—
97C2×#124—44no—
98C2×#121—24no—
99C2×#121—44no—
100C2×C4×Q8—28no—
101C2×C4×Q8—28no—
102C2×C4×Q8—44no—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2314yes—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137#399—28no—
138#399D418yescomplemented
139#395—216no—
140#394—216no—
141#394—28no—
142#393—28no—
143#393—28no—
144#393—28no—
145#393—44no—
146#391—28no—
147#390—24no—
148#390—28no—
149#390—24no—
150#390—44no—
151#389—28no—
152#389—44no—
153#388—216no—
154#388—216no—
155#388—28no—
156#387—216no—
157#383—28no—
158#380—24no—
159#380—28no—
160C4×C4⋊C4—28no—
161C4×C4⋊C4—24no—
162C4×C4⋊C4—44no—
163C4×C4⋊C4D418yes—
164C4×C22⋊C4—24no—
165C4×C22⋊C4—44no—
166C2×#95C2311yescharacteristic
167C2×#95C2311yescharacteristic
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2311yescharacteristic
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2311yescharacteristic
191C2×#95C2312yes—
192C25C2412yes—
193C25C2411yescharacteristic
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—28no—
199C22×Q8—24no—
200C22×Q8—44no—
201C22×Q8—42no—
202C22×Q8—44no—
203C22×Q8—44no—
204C22×D4—24no—
205C22×D4—22no—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—44no—
212C22×D4—42no—
213C22×D4—44no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C4⋊Q8—232no—
235C4⋊Q8—416no—
236#124—432no—
237#121—416no—
238C4×Q8—432no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4C2×D414yes—
269C2×C4⋊C4C2×D414yes—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2×D418yes—
272C2×C4⋊C4#4118yes—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
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—24no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—41no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4C2×D414yes—
315C2×C22⋊C4C2×D418yes—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—28no—
320C2×C42—28no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—44no—
330C2×C42C2×D418yes—
331C2×C42C2×D414yes—
332#95—22no—
333#95—22no—
334#95—24no—
335#95—28no—
336#95—24no—
337#95—28no—
338#95—24no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—22no—
345#95—44no—
346#95—42no—
347#95—44no—
348#95—42no—
349#95—44no—
350#95—42no—
351#95—42no—
352#95—42no—
353#95—42no—
354#95—44no—
355#95—44no—
356#95C2×D418yes—
357#95C2×D414yes—
358C24—21no—
359C24—22no—
360C24—21no—
361C24—22no—
362C24—21no—
363C24—21no—
364C24—21no—
365C24—22no—
366C24—21no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—24no—
371C24—22no—
372C24—24no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—24no—
377C24—22no—
378C24—24no—
379C24C22×D412yes—
380C24C2511yescharacteristic
381C2×Q8—216no—
382C2×Q8—416no—
383C2×Q8—416no—
384C2×Q8—48no—
385C2×Q8—48no—
386C2×Q8—48no—
387C2×Q8—48no—
388C2×Q8—88no—
389C2×Q8—84no—
390C2×Q8—88no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—84no—
399C2×D4—88no—
400C2×D4—88no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4C22×D414yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D414yes—
483C22×C4C22×D414yes—
484C22×C4C22×D414yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C2×#4118yes—
489C22×C4C2×#4114yes—
490C22×C4H(5,𝔽2)12yes—
491C22×C4H(5,𝔽2)14yes—
492C22×C42−1+412yes—
493C22×C42−1+412yes—
494C22×C42−1+412yes—
495C4⋊C4—216no—
496C4⋊C4—216no—
497C4⋊C4—28no—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—48no—
501C4⋊C4—416no—
502C4⋊C4—44no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—48no—
507C4⋊C4—88no—
508C22⋊C4—28no—
509C22⋊C4—48no—
510C22⋊C4—416no—
511C22⋊C4—44no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—416no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—88no—
520C22⋊C4—84no—
521C22⋊C4—88no—
522C22⋊C4—88no—
523C42—28no—
524C42—216no—
525C42—416no—
526C42—48no—
527C42—44no—
528C42—48no—
529C23—22no—
530C23—24no—
531C23—24no—
532C23—22no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—42no—
538C23—42no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—41no—
549C23—41no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—44no—
556C23—44no—
557C23—42no—
558C23—44no—
559C23—42no—
560C23—42no—
561C23—44no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—42no—
572C23—42no—
573C23—44no—
574C23—44no—
575C23C23×D411yescharacteristic
576C23C23×D411yescharacteristic
577C23C22×#4112yes—
578C23C2×H(5,𝔽2)12yes—
579C23C2×H(5,𝔽2)11yescharacteristic
580C23C2×2−1+411yescharacteristic
581C23C2×2−1+411yescharacteristic
582C23C2×2−1+411yescharacteristic
583C23#58512yes—
584C23#58511yescharacteristic
585C23#58512yes—
586Q8—432no—
587Q8—816no—
588D4—816no—
589D4—88no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—28no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—28no—
615C2×C4—24no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—41no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—88no—
674C2×C4#52514yes—
675C2×C4#53514yes—
676C2×C4#53614yes—
677C2×C4#54218yes—
678C2×C4#54218yes—
679C2×C4D4214yes—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—44no—
684C22—44no—
685C22—81no—
686C22—82no—
687C22—82no—
688C22—81no—
689C22—81no—
690C22—81no—
691C22—82no—
692C22—82no—
693C22—84no—
694C22—84no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—84no—
700C22—82no—
701C22—84no—
702C22C2×#52511yescharacteristic
703C22C2×#53511yescharacteristic
704C22C2×#53611yescharacteristic
705C22C2×#54212yes—
706C22C2×#54212yes—
707C22C2×D4211yescharacteristic
708C22D4×#4112yes—
709C22#348711yescharacteristic
710C22#348911yescharacteristic
711C22#349612yes—
712C22#349711yescharacteristic
713C22#349911yescharacteristic
714C22#350011yescharacteristic
715C22#351312yes—
716C22#351812yes—
717C22#352412yes—
718C22#354012yes—
719C22#354112yes—
720C22#354612yes—
721C22#354711yescharacteristic
722C22#355011yescharacteristic
723C22#355712yes—
724C22#356512yes—
725C4—24no—
726C4—48no—
727C4—42no—
728C4—44no—
729C4—44no—
730C4—44no—
731C4—42no—
732C4—44no—
733C4—42no—
734C4—81no—
735C4—82no—
736C4—82no—
737C4—82no—
738C4—84no—
739C4#263218yescomplemented
740C2—82no—
741C2—162no—
742C2C2×#263211yescharacteristic
743C2#3607711yescharacteristic
744C2#3610211yescharacteristic
745C2#3934311yescharacteristic
746C2#3936211yescharacteristic
747C2#3940312yes—
748C2#3941112yes—
749C2#4254212yes—
750C2#4260412yes—
751C2#4287511yescharacteristic
752C2#4288711yescharacteristic
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
2. C2×#2632C2semidirect
3. C2×#2365C2semidirect
4. #16665C2semidirect
5. #16559C2semidirect
6. #16353C2semidirect
7. #16346C2semidirect
8. #15148C2semidirect
9. #15144C2semidirect
10. #15128C2semidirect
12. #15125C2semidirect
13. #14875C2semidirect
14. #14022C2semidirect
15. #13974C2semidirect
16. #13921C2semidirect
17. C4×#399C2semidirect
138. #399D4semidirect
739. C4#2632semidirect

Maximal subgroups

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

subgroupindex
C2×#24912
C2×#26322
C2×#23652
#166652
#165592
#163532
#163462
#151482
#151442
#151282
#151262
#151252
#148752
#140222
#139742
#139212
C4×#3992