Finite Group Databasebeta

← #1409996

Subgroups — #1409996

Order 512.

3142 conjugacy classes of subgroups.

The subgroup lattice has 757 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic, complemented
2C2×#2550C212yesmaximal, complemented
3C2×#2490C211yesmaximal, characteristic
4C2×#2491C212yesmaximal
5#16812C214yesmaximal, complemented
6#16811C212yesmaximal, complemented
7#16511C212yesmaximal, complemented
8#16510C214yesmaximal, complemented
9#15244C211yesmaximal, characteristic, complemented
10#15243C212yesmaximal, complemented
11#15223C212yesmaximal, complemented
12#15148C211yesmaximal, characteristic, complemented
13#15148C212yesmaximal, complemented
14#15135C211yesmaximal, characteristic, complemented
15#14322C212yesmaximal, complemented
16C4×#399C211yesmaximal, characteristic, complemented
17#13812C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2212yes—
19C22×C4⋊Q8C2212yes—
20C22×#125C2211yescharacteristic
21C22×#124C2214yes—
22C22×#121C2211yescharacteristic
23#2551—28no—
24#2550—216no—
25#2491—216no—
26#2490—28no—
27C2×#399C2211yescharacteristic
28C2×#399C2212yes—
29C2×#399C2212yes—
30C2×#399C2214yes—
31C2×#398C2214yes—
32C2×#397C2212yes—
33C2×#395C2214yes—
34C2×#395C2214yes—
35C2×#394C2212yes—
36C2×#394C2214yes—
37C2×#393C2214yes—
38C2×#391C2214yes—
39C2×#390C2212yes—
40C2×#390C2211yescharacteristic
41C2×#390C2212yes—
42C2×#390C2211yescharacteristic
43C2×#390C2212yes—
44C2×#389C2212yes—
45C2×#389C2212yes—
46C2×#389C2212yes—
47C2×#389C2214yes—
48C2×#389C2212yes—
49C2×#388C2212yes—
50C2×#388C2214yes—
51C2×#384C2211yescharacteristic
52C2×#382C2212yes—
53C2×#380C2212yes—
54C2×#380C2211yescharacteristic
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2212yes—
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C22⋊C4C2211yescharacteristic
60C2×C4×C22⋊C4C2212yes—
61#1445C2212yes—
62#1445C2212yes—
63#1445C2211yescharacteristic
64#1445C2214yes—
65#1445C2214yes—
66#1444C2212yes—
67#1444C2211yescharacteristic
68#1439C2212yes—
69#1439C2212yes—
70#1439C2211yescharacteristic
71#1439C2214yes—
72#1439C2214yes—
73#1439C2212yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2214yes—
77#1439C2214yes—
78#1439C2214yes—
79#1439C2212yes—
80#1439C2212yes—
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2212yes—
83#1434C2214yes—
84#1434C2212yes—
85C4×#95C2211yescharacteristic
86C4×#95C2212yes—
87C23×Q8C2314yes—
88C23×D4C2311yescharacteristic
89C23×D4C2312yes—
90C24×C4C2311yescharacteristic
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8—28no—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—48no—
96C2×#125—24no—
97C2×#125—28no—
98C2×#124—216no—
99C2×#124—48no—
100C2×#124—48no—
101C2×#121—24no—
102C2×#121—28no—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2314yes—
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139C22×C42C2314yes—
140C22×C42C2312yes—
141#399—24no—
142#399—28no—
143#399—44no—
144#399—48no—
145#398—216no—
146#397—28no—
147#395—216no—
148#395—216no—
149#394—28no—
150#394—216no—
151#393—216no—
152#391—216no—
153#390—24no—
154#390—28no—
155#390—24no—
156#390—28no—
157#390—28no—
158#389—28no—
159#389—28no—
160#389—28no—
161#389—216no—
162#389—28no—
163#388—28no—
164#388—216no—
165#384—24no—
166#382—28no—
167#380—24no—
168#380—28no—
169C4×C4⋊C4—28no—
170C4×C4⋊C4—24no—
171C4×C4⋊C4D418yes—
172C4×C4⋊C4D4116yes—
173C4×C22⋊C4—24no—
174C4×C22⋊C4—28no—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2311yescharacteristic
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2311yescharacteristic
185C2×#95C2312yes—
186C2×#95C2311yescharacteristic
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2311yescharacteristic
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2314yes—
194C2×#95C2312yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2312yes—
198C25C2412yes—
199C25C2411yescharacteristic
200C22×Q8—28no—
201C22×Q8—28no—
202C22×Q8—28no—
203C22×Q8—28no—
204C22×Q8—44no—
205C22×Q8—44no—
206C22×Q8—44no—
207C22×D4—24no—
208C22×D4—22no—
209C22×D4—24no—
210C22×D4—28no—
211C22×D4—24no—
212C22×D4—24no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—44no—
216C22×D4—42no—
217C23×C4—22no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4C2411yescharacteristic
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C4⋊Q8—432no—
238C4⋊Q8—432no—
239#125—416no—
240#124—464no—
241#121—416no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D418yes—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2×D418yes—
286C2×C4⋊C4#4114yes—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—41no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—22no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—44no—
336C2×C42—44no—
337C2×C42—44no—
338C2×C42C2×D418yes—
339C2×C42C2×D414yes—
340C2×C42C2×D4116yes—
341C2×C42C2×D418yes—
342#95—24no—
343#95—22no—
344#95—24no—
345#95—22no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—28no—
350#95—24no—
351#95—28no—
352#95—28no—
353#95—28no—
354#95—28no—
355#95—28no—
356#95—22no—
357#95—24no—
358#95—22no—
359#95—42no—
360#95—44no—
361#95—42no—
362#95—44no—
363#95C2×D4116yes—
364#95C2×D418yes—
365C24—22no—
366C24—21no—
367C24—21no—
368C24—22no—
369C24—21no—
370C24—21no—
371C24—21no—
372C24—22no—
373C24—22no—
374C24—21no—
375C24—22no—
376C24—24no—
377C24—22no—
378C24—24no—
379C24—22no—
380C24—22no—
381C24—24no—
382C24—22no—
383C24—22no—
384C24—24no—
385C24—22no—
386C24H(5,𝔽2)12yes—
387C24C2511yescharacteristic
388C2×Q8—416no—
389C2×Q8—416no—
390C2×Q8—416no—
391C2×Q8—416no—
392C2×Q8—88no—
393C2×Q8—88no—
394C2×Q8—88no—
395C2×D4—416no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—48no—
401C2×D4—48no—
402C2×D4—44no—
403C2×D4—84no—
404C2×D4—88no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
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—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—24no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—24no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—44no—
480C22×C4C22×D414yes—
481C22×C4C22×D414yes—
482C22×C4C22×D412yes—
483C22×C4C22×D418yes—
484C22×C4C22×D414yes—
485C22×C4C22×D418yes—
486C22×C4C22×D414yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D414yes—
490C22×C4C2×#4114yes—
491C22×C4C2×#4112yes—
492C22×C4H(5,𝔽2)14yes—
493C22×C42−1+412yes—
494C22×C42−1+414yes—
495C22×C42−1+412yes—
496C22×C42−1+412yes—
497C4⋊C4—28no—
498C4⋊C4—28no—
499C4⋊C4—216no—
500C4⋊C4—28no—
501C4⋊C4—48no—
502C4⋊C4—416no—
503C4⋊C4—48no—
504C4⋊C4—416no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—44no—
509C4⋊C4—416no—
510C4⋊C4—48no—
511C4⋊C4—88no—
512C4⋊C4—84no—
513C4⋊C4#127116yes—
514C22⋊C4—216no—
515C22⋊C4—48no—
516C22⋊C4—416no—
517C22⋊C4—416no—
518C22⋊C4—44no—
519C22⋊C4—416no—
520C22⋊C4—416no—
521C22⋊C4—48no—
522C22⋊C4—44no—
523C22⋊C4—48no—
524C22⋊C4—84no—
525C22⋊C4—88no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C22⋊C4—84no—
529C42—28no—
530C42—216no—
531C42—416no—
532C42—48no—
533C42—44no—
534C42—48no—
535C23—24no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—24no—
540C23—41no—
541C23—42no—
542C23—41no—
543C23—42no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—41no—
548C23—42no—
549C23—42no—
550C23—41no—
551C23—42no—
552C23—41no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—41no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—44no—
561C23—44no—
562C23—42no—
563C23—42no—
564C23—44no—
565C23—44no—
566C23—42no—
567C23—44no—
568C23—42no—
569C23—44no—
570C23—42no—
571C23—44no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—42no—
579C23—44no—
580C23—44no—
581C23C23×D411yescharacteristic
582C23C23×D412yes—
583C23C22×#4111yescharacteristic
584C23C2×H(5,𝔽2)12yes—
585C23C2×H(5,𝔽2)11yescharacteristic
586C23C2×2−1+412yes—
587C23C2×2−1+411yescharacteristic
588C23C2×2−1+411yescharacteristic
589C23C2×2−1+411yescharacteristic
590C23#58512yes—
591C23#58511yescharacteristic
592Q8—832no—
593D4—816no—
594D4—88no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—28no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—28no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—28no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—41no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—41no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—41no—
674C2×C4—42no—
675C2×C4—88no—
676C2×C4C2×#12718yes—
677C2×C4C2×#12714yes—
678C2×C4#53614yes—
679C2×C4#53618yes—
680C2×C4#53618yes—
681C2×C4#53614yes—
682C2×C4#54814yes—
683C22—42no—
684C22—44no—
685C22—42no—
686C22—44no—
687C22—42no—
688C22—81no—
689C22—82no—
690C22—81no—
691C22—81no—
692C22—81no—
693C22—82no—
694C22—82no—
695C22—84no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—84no—
700C22—82no—
701C22—82no—
702C22—84no—
703C22—84no—
704C22—82no—
705C22C22×#12711yescharacteristic
706C22C2×#53611yescharacteristic
707C22C2×#53612yes—
708C22C2×#53612yes—
709C22C2×#53611yescharacteristic
710C22C2×#54811yescharacteristic
711C22D4×#4112yes—
712C22#348411yescharacteristic
713C22#348412yes—
714C22#348812yes—
715C22#348811yescharacteristic
716C22#349712yes—
717C22#350011yescharacteristic
718C22#351112yes—
719C22#351811yescharacteristic
720C22#352211yescharacteristic
721C22#354411yescharacteristic
722C22#354712yes—
723C22#355212yes—
724C22#355412yes—
725C22#355511yescharacteristic
726C22#355912yes—
727C22#356512yes—
728C4—24no—
729C4—28no—
730C4—44no—
731C4—44no—
732C4—44no—
733C4—44no—
734C4—42no—
735C4—48no—
736C4—44no—
737C4—81no—
738C4—81no—
739C4—84no—
740C4—81no—
741C4—82no—
742C4#265818yes—
743C2—82no—
744C2—162no—
745C2C2×#265811yescharacteristic
746C2#3592111yescharacteristic
747C2#3646511yescharacteristic
748C2#3912511yescharacteristic
749C2#3972011yescharacteristic
750C2#3987712yes—
751C2#3989112yes—
752C2#4242312yes—
753C2#4243111yescharacteristic
754C2#4280412yes—
755C2#4294411yescharacteristic
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
1. C2×#2551C2semidirect
2. C2×#2550C2semidirect
5. #16812C2semidirect
6. #16811C2semidirect
7. #16511C2semidirect
8. #16510C2semidirect
9. #15244C2semidirect
10. #15243C2semidirect
11. #15223C2semidirect
12. #15148C2semidirect
13. #15148C2semidirect
14. #15135C2semidirect
15. #14322C2semidirect
16. C4×#399C2semidirect
17. #13812C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
C2×#25502
C2×#24902
C2×#24912
#168122
#168112
#165112
#165102
#152442
#152432
#152232
#151482
#151352
#143222
C4×#3992
#138122