Finite Group Databasebeta

← #4930465

Subgroups — #4930465

Order 512.

2926 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2C2×#2423C212yesmaximal, complemented
3C2×#2276C211yesmaximal, characteristic, complemented
4#15122C212yesmaximal, complemented
5#15120C212yesmaximal
6#15106C212yesmaximal, complemented
7#15084C212yesmaximal
8#14805C214yesmaximal, complemented
9#14788C214yesmaximal, complemented
10#14277C211yesmaximal, characteristic, complemented
11#13974C212yesmaximal, complemented
12#13969C212yesmaximal, complemented
13#13942C211yesmaximal, characteristic, complemented
14#13910C211yesmaximal, characteristic, complemented
15#13905C211yesmaximal, characteristic
16#13876C212yesmaximal, complemented
17#13847C211yesmaximal, characteristic
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23#2538—28no—
24#2423—216no—
25C2×#399C2214yes—
26C2×#398C2214yes—
27C2×#396C2214yes—
28C2×#391C2212yes—
29C2×#390C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2214yes—
33C2×#388C2214yes—
34C2×#388C2214yes—
35C2×#388C2214yes—
36C2×#387C2214yes—
37C2×#384C2212yes—
38C2×#384C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2212yes—
43#2276—28no—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C22⋊C4C2212yes—
52C2×C4×C22⋊C4C2212yes—
53C2×#376C2212yes—
54C2×#376C2211yescharacteristic
55#1445C2211yescharacteristic
56#1445C2212yes—
57#1445C2211yescharacteristic
58#1444C2212yes—
59#1444C2212yes—
60#1444C2212yes—
61#1444C2212yes—
62#1443C2214yes—
63#1442C2212yes—
64#1442C2214yes—
65#1442C2212yes—
66#1441C2214yes—
67#1441C2214yes—
68#1440C2214yes—
69#1439C2214yes—
70#1439C2212yes—
71#1439C2212yes—
72#1439C2214yes—
73#1438C2211yescharacteristic
74#1438C2212yes—
75#1438C2211yescharacteristic
76#1437C2214yes—
77#1437C2214yes—
78#1436C2214yes—
79#1436C2214yes—
80#1436C2214yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84#1434C2212yes—
85#1434C2212yes—
86C23×Q8C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×Q8D418yes—
100C2×C4×Q8D418yescomplemented
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2312yes—
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139#399—216no—
140#398—216no—
141#396—216no—
142#391—28no—
143#390—28no—
144#389—28no—
145#389—28no—
146#389—48no—
147#388—216no—
148#388—48no—
149#388—48no—
150#387—216no—
151#384—28no—
152#384—24no—
153#382—24no—
154#380—28no—
155#380—28no—
156#380—28no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4—24no—
159C4×C4⋊C4—28no—
160C4×C4⋊C4—28no—
161C4×C4⋊C4—28no—
162C4×C4⋊C4—24no—
163C4×C4⋊C4—24no—
164C4×C22⋊C4—28no—
165C4×C22⋊C4—28no—
166#376—28no—
167#376—24no—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2311yescharacteristic
175C2×#95C2311yescharacteristic
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2311yescharacteristic
184C2×#95C2312yes—
185C2×#95C2311yescharacteristic
186C2×#95C2312yes—
187C2×#95C2311yescharacteristic
188C2×#95C2311yescharacteristic
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C25C2412yes—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8C2×D414yes—
201C22×Q8C2×D414yes—
202C22×Q8#4114yes—
203C22×D4—24no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—42no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C22⋊Q8—432no—
232C4×Q8—232no—
233C4×Q8—232no—
234C4×Q8—232no—
235C4×D4—416no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—44no—
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⋊C4C2×D414yes—
278C2×C4⋊C4C2×D418yes—
279C2×C4⋊C4C2×D414yes—
280C2×C4⋊C4C2×D418yes—
281C2×C4⋊C4#4114yes—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—22no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—44no—
327C2×C42—44no—
328C2×C42—44no—
329C2×C42C2×D418yes—
330C2×C42C2×D414yes—
331C2×C42C2×D418yes—
332C2×C42C2×D414yes—
333C2×C42#4114yes—
334C2×C42#4114yes—
335C2×C42#4114yes—
336#95—22no—
337#95—24no—
338#95—24no—
339#95—22no—
340#95—22no—
341#95—24no—
342#95—22no—
343#95—24no—
344#95—24no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—24no—
351#95—24no—
352#95—22no—
353#95—22no—
354#95—28no—
355#95—24no—
356#95—24no—
357#95—28no—
358#95—28no—
359#95—28no—
360#95—44no—
361#95—44no—
362#95—44no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C242−1+414yes—
378C24C2511yescharacteristic
379C2×Q8—28no—
380C2×Q8—28no—
381C2×Q8—28no—
382C2×Q8—28no—
383C2×Q8—28no—
384C2×Q8—28no—
385C2×Q8#121116yes—
386C2×D4—28no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—44no—
390C2×D4—44no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4C22×D412yes—
468C22×C4C22×D414yes—
469C22×C4C22×D412yes—
470C22×C4C22×D414yes—
471C22×C4C22×D414yes—
472C22×C4C22×D414yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4114yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4114yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4H(5,𝔽2)14yes—
491C22×C42−1+414yes—
492C4⋊C4—216no—
493C4⋊C4—216no—
494C4⋊C4—216no—
495C4⋊C4—28no—
496C4⋊C4—216no—
497C4⋊C4—28no—
498C4⋊C4—28no—
499C4⋊C4—416no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—416no—
503C4⋊C4—44no—
504C4⋊C4—48no—
505C4⋊C4—84no—
506C4⋊C4—88no—
507C4⋊C4—88no—
508C22⋊C4—216no—
509C22⋊C4—216no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—416no—
513C22⋊C4—416no—
514C22⋊C4—48no—
515C22⋊C4—88no—
516C22⋊C4—88no—
517C42—216no—
518C42—216no—
519C42—28no—
520C42—216no—
521C42—28no—
522C42—48no—
523C42—84no—
524C42—84no—
525C42—84no—
526C42#123116yes—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23C23×D411yescharacteristic
563C23C23×D411yescharacteristic
564C23C22×#4111yescharacteristic
565C23C22×#4111yescharacteristic
566C23C22×#4111yescharacteristic
567C23C22×#4111yescharacteristic
568C23C22×#4111yescharacteristic
569C23C2×H(5,𝔽2)12yes—
570C23C2×2−1+412yes—
571C23C2×2−1+412yes—
572C23#58512yes—
573Q8—232no—
574D4—416no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—28no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—816no—
660C2×C4C2×#12118yes—
661C2×C4C2×#12114yes—
662C2×C4C2×#12314yes—
663C2×C4C2×#12314yes—
664C2×C4C2×#12314yes—
665C2×C4#52514yes—
666C2×C4#52914yes—
667C2×C4#52914yes—
668C2×C4#53214yes—
669C2×C4#53214yes—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22C22×#12111yescharacteristic
686C22C22×#12311yescharacteristic
687C22C2×#52511yescharacteristic
688C22C2×#52911yescharacteristic
689C22C2×#52911yescharacteristic
690C22C2×#53211yescharacteristic
691C22C2×#53211yescharacteristic
692C22#348612yes—
693C22#348611yescharacteristic
694C22#348711yescharacteristic
695C22#348812yes—
696C22#349212yes—
697C22#349212yes—
698C22#349512yes—
699C22#349512yes—
700C22#349512yes—
701C22#349811yescharacteristic
702C22#349911yescharacteristic
703C22#351112yes—
704C22#351712yes—
705C22#351812yes—
706C22#352012yes—
707C22#352312yes—
708C4—28no—
709C4—24no—
710C4—24no—
711C4—44no—
712C4—44no—
713C4—42no—
714C4—42no—
715C4—44no—
716C4—42no—
717C4—44no—
718C4—42no—
719C4—44no—
720C4—82no—
721C4—84no—
722C4—84no—
723C4#285418yes—
724C2—84no—
725C2#3553011yescharacteristic
726C2#3557312yes—
727C2#3560912yes—
728C2#3567211yescharacteristic
729C2#3570011yescharacteristic
730C2#3570111yescharacteristic
731C2#3592612yes—
732C2#3594512yes—
733C2#3602911yescharacteristic
734C2#3603011yescharacteristic
735C2C2×#285411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2423C2semidirect
3. C2×#2276C2semidirect
4. #15122C2semidirect
6. #15106C2semidirect
8. #14805C2semidirect
9. #14788C2semidirect
10. #14277C2semidirect
11. #13974C2semidirect
12. #13969C2semidirect
13. #13942C2semidirect
14. #13910C2semidirect
16. #13876C2semidirect
100. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#24232
C2×#22762
#151222
#151202
#151062
#150842
#148052
#147882
#142772
#139742
#139692
#139422
#139102
#139052
#138762
#138472