Finite Group Databasebeta

← #637488

Subgroups — #637488

Order 512.

3192 conjugacy classes of subgroups.

The subgroup lattice has 751 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2545C211yesmaximal, characteristic, complemented
2C2×#2536C211yesmaximal, characteristic, complemented
3C2×#2540C211yesmaximal, characteristic
4C2×#2295C211yesmaximal, characteristic, complemented
5#16463C214yesmaximal, complemented
6#16289C214yesmaximal, complemented
7#15215C212yesmaximal, complemented
8#15179C212yesmaximal, complemented
9#15178C212yesmaximal, complemented
10#15126C212yesmaximal, complemented
11#15120C214yesmaximal, complemented
12#14708C211yesmaximal, characteristic, complemented
13#14312C214yesmaximal, complemented
14#13955C211yesmaximal, characteristic, complemented
15C4×#390C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C22×C22⋊Q8C2212yes—
21C22×#121C2212yes—
22C22×C4×Q8C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24#2545—28no—
25#2540—28no—
26#2536—28no—
27C2×#399C2212yes—
28C2×#398C2212yes—
29C2×#396C2214yes—
30C2×#394C2214yes—
31C2×#394C2212yes—
32C2×#392C2212yes—
33#2295—28no—
34C2×#391C2212yes—
35C2×#391C2211yescharacteristic
36C2×#391C2212yes—
37C2×#390C2214yes—
38C2×#390C2211yescharacteristic
39C2×#390C2212yes—
40C2×#390C2212yes—
41C2×#389C2212yes—
42C2×#388C2214yes—
43C2×#388C2214yes—
44C2×#387C2214yes—
45C2×#382C2211yescharacteristic
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2212yes—
52C2×#376C2212yes—
53C2×C43C2211yescharacteristic
54#1444C2214yes—
55#1444C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2214yes—
59#1440C2214yes—
60#1440C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1438C2214yes—
68#1438C2214yes—
69#1438C2214yes—
70#1437C2212yes—
71#1437C2212yes—
72#1437C2214yes—
73#1437C2212yes—
74#1437C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78#1434C2214yes—
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C23×D4C2314yes—
84C24×C4C2312yes—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—44no—
87C2×#124—24no—
88C2×#124—24no—
89C2×#124—24no—
90C2×#124—24no—
91C2×#124—44no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—44no—
95C2×C22⋊Q8—44no—
96C2×C22⋊Q8—44no—
97C2×C22⋊Q8—44no—
98C2×#121—28no—
99C2×#121—44no—
100C2×#121—44no—
101C2×C4×Q8—24no—
102C2×C4×Q8—24no—
103C2×C4×Q8D418yescomplemented
104C2×C4×D4—24no—
105C2×C4×D4—24no—
106C2×C4×D4—24no—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2312yes—
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139C22×C42C2312yes—
140C22×C42C2311yescharacteristic
141#399—28no—
142#398—28no—
143#396—216no—
144#394—28no—
145#394—216no—
146#392—28no—
147#391—24no—
148#391—28no—
149#391—44no—
150#390—24no—
151#390—28no—
152#390—216no—
153#390—44no—
154#389—28no—
155#388—216no—
156#388—216no—
157#387—216no—
158#382—24no—
159#380—28no—
160C4×C4⋊C4—24no—
161C4×C4⋊C4—28no—
162C4×C4⋊C4D418yes—
163C4×C22⋊C4—44no—
164C4×C22⋊C4—44no—
165#376—28no—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C43—24no—
186C25C2412yes—
187C25C2412yes—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—22no—
193C22×Q8—24no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8C2×D414yes—
198C22×D4—24no—
199C22×D4—28no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—28no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—42no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4—44no—
213C22×D4—44no—
214C22×D4—44no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C4⋊Q8—416no—
236#124—416no—
237#124—416no—
238C22⋊Q8—432no—
239C22⋊Q8—432no—
240#121—432no—
241C4×Q8—232no—
242C4×D4—416no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2×D418yes—
272C2×C4⋊C4C2×D418yes—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4#4118yes—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—44no—
333C2×C42—42no—
334C2×C42—42no—
335C2×C42C2×D418yes—
336C2×C42C2×D418yes—
337C2×C42C2×D414yes—
338C2×C42C2×D414yes—
339C2×C42#4114yes—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—24no—
344#95—28no—
345#95—24no—
346#95—24no—
347#95—28no—
348#95—28no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—42no—
353#95—42no—
354#95—42no—
355#95—42no—
356#95—44no—
357#95—44no—
358#95C2×D418yes—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
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—24no—
377C24—24no—
378C24—24no—
379C24—24no—
380C24—24no—
381C24—24no—
382C24C2×#4114yes—
383C24C2511yescharacteristic
384C2×Q8—28no—
385C2×Q8—28no—
386C2×Q8—28no—
387C2×Q8—44no—
388C2×Q8—48no—
389C2×Q8—48no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—48no—
393C2×Q8—48no—
394C2×D4—28no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—44no—
398C2×D4—416no—
399C2×D4—416no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—48no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—48no—
406C2×D4—88no—
407C2×D4—88no—
408C2×D4—88no—
409C2×D4—88no—
410C2×D4—88no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4C22×D414yes—
481C22×C4C22×D418yes—
482C22×C4C22×D414yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C2×#4114yes—
490C22×C4C2×#4114yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4112yes—
494C22×C4H(5,𝔽2)14yes—
495C22×C4H(5,𝔽2)14yes—
496C22×C42−1+414yes—
497C4⋊C4—28no—
498C4⋊C4—216no—
499C4⋊C4—216no—
500C4⋊C4—48no—
501C4⋊C4—416no—
502C4⋊C4—416no—
503C4⋊C4—416no—
504C4⋊C4—416no—
505C4⋊C4—44no—
506C4⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—416no—
509C22⋊C4—48no—
510C22⋊C4—416no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—416no—
514C22⋊C4—416no—
515C22⋊C4—416no—
516C22⋊C4—416no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—88no—
521C22⋊C4—88no—
522C42—28no—
523C42—216no—
524C42—48no—
525C42—48no—
526C42—44no—
527C42—48no—
528C42—44no—
529C42—44no—
530C42#121116yes—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
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—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—44no—
570C23—42no—
571C23—44no—
572C23—42no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—42no—
577C23—42no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23C23×D411yescharacteristic
588C23C23×D411yescharacteristic
589C23C22×#4111yescharacteristic
590C23C22×#4112yes—
591C23C2×H(5,𝔽2)12yes—
592C23C2×H(5,𝔽2)12yes—
593C23C2×2−1+412yes—
594C23#58512yes—
595C23#58512yes—
596Q8—416no—
597Q8—88no—
598D4—416no—
599D4—832no—
600D4—88no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—816no—
674C2×C4C2×#12114yes—
675C2×C4C2×#12114yes—
676C2×C4C2×#12114yes—
677C2×C4#53514yes—
678C2×C4#53518yes—
679C2×C4#53514yes—
680C2×C4#54018yes—
681C22—44no—
682C22—44no—
683C22—44no—
684C22—44no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—84no—
697C22—84no—
698C22—84no—
699C22—82no—
700C22—82no—
701C22—84no—
702C22—82no—
703C22—84no—
704C22—84no—
705C22C22×#12111yescharacteristic
706C22#344411yescharacteristic
707C22C2×#53511yescharacteristic
708C22C2×#53511yescharacteristic
709C22C2×#53512yes—
710C22C2×#54012yes—
711C22#345511yescharacteristic
712C22#346112yes—
713C22D4×#4112yes—
714C22D4×#4112yes—
715C22#348511yescharacteristic
716C22#348611yescharacteristic
717C22#349712yes—
718C22#350512yes—
719C22#351112yes—
720C22#351412yes—
721C22#352012yes—
722C22#354712yes—
723C22#355112yes—
724C22#356112yes—
725C22#356212yes—
726C4—24no—
727C4—24no—
728C4—44no—
729C4—44no—
730C4—44no—
731C4—44no—
732C4—44no—
733C4—48no—
734C4—82no—
735C4—84no—
736C4—82no—
737C4—82no—
738C4#261918yes—
739C2—84no—
740C2—162no—
741C2C2×#261911yescharacteristic
742C2#3474311yescharacteristic
743C2#3510812yes—
744C2#3515311yescharacteristic
745C2#3757912yes—
746C2#3985412yes—
747C2#3988112yes—
748C2#3995012yes—
749C2#3995412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2540C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#2545C2semidirect
2. C2×#2536C2semidirect
4. C2×#2295C2semidirect
5. #16463C2semidirect
6. #16289C2semidirect
7. #15215C2semidirect
8. #15179C2semidirect
9. #15178C2semidirect
10. #15126C2semidirect
11. #15120C2semidirect
12. #14708C2semidirect
13. #14312C2semidirect
14. #13955C2semidirect
15. C4×#390C2semidirect
103. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25452
C2×#25362
C2×#25402
C2×#22952
#164632
#162892
#152152
#151792
#151782
#151262
#151202
#147082
#143122
#139552
C4×#3902