Finite Group Databasebeta

← #4418778

Subgroups — #4418778

Order 512.

3465 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q82C211yesmaximal, characteristic, complemented
2C2×#2821C211yesmaximal, characteristic
3C2×#2604C211yesmaximal, characteristic
4C2×#2637C211yesmaximal, characteristic
5#16560C212yesmaximal, complemented
6#16511C212yesmaximal
7#16507C212yesmaximal, complemented
8#16506C212yesmaximal
9#15322C214yesmaximal
10#15318C214yesmaximal, complemented
11#14882C212yesmaximal
12#14879C212yesmaximal, complemented
13#14876C212yesmaximal, complemented
14#14875C212yesmaximal
15#14346C211yesmaximal, characteristic, complemented
16#14274C211yesmaximal, characteristic, complemented
17#13961C211yesmaximal, characteristic, complemented
18C2×Q82—28no—
19C2×Q82—28no—
20C2×Q82—28no—
21C22×C4⋊Q8C2212yes—
22C22×C4⋊Q8C2212yes—
23C22×C4⋊Q8C2214yes—
24C22×C4⋊Q8C2211yescharacteristic
25C22×#126C2211yescharacteristic
26C22×C4×Q8C2212yes—
27C22×C4×Q8C2212yes—
28C22×C4×Q8C2211yescharacteristic
29C22×C4×Q8C2211yescharacteristic
30#2821—28no—
31#2637—28no—
32#2604—28no—
33C2×#400C2212yes—
34C2×#400C2211yescharacteristic
35C2×#399C2212yes—
36C2×#399C2211yescharacteristic
37C2×#398C2212yes—
38C2×#398C2211yescharacteristic
39C2×#398C2212yes—
40C2×#397C2212yes—
41C2×#397C2211yescharacteristic
42C2×#396C2212yes—
43C2×#395C2211yescharacteristic
44C2×#395C2211yescharacteristic
45C2×#395C2212yes—
46C2×#395C2214yes—
47C2×#393C2211yescharacteristic
48C2×#393C2211yescharacteristic
49C2×#393C2212yes—
50C2×#393C2214yes—
51C2×#391C2214yes—
52C2×#389C2212yes—
53C2×#388C2212yes—
54C2×#388C2212yes—
55C2×#388C2212yes—
56C2×#387C2212yes—
57C2×#387C2212yes—
58C2×#383C2211yescharacteristic
59C2×#382C2211yescharacteristic
60C2×#376C2211yescharacteristic
61#1446C2212yes—
62#1446C2212yes—
63#1446C2214yes—
64#1446C2214yes—
65#1445C2212yes—
66#1445C2212yes—
67#1445C2212yes—
68#1445C2212yes—
69#1445C2212yes—
70#1445C2212yes—
71#1444C2214yes—
72#1444C2214yes—
73#1442C2214yes—
74#1442C2214yes—
75#1441C2214yes—
76#1441C2214yes—
77#1441C2214yes—
78#1439C2214yes—
79#1439C2214yes—
80#1437C2214yes—
81#1436C2214yes—
82#1436C2214yes—
83#1436C2214yes—
84#1436C2214yes—
85#1434C2212yes—
86#1434C2212yes—
87#1434C2212yes—
88#1434C2212yes—
89C23×Q8C2311yescharacteristic
90C23×Q8C2311yescharacteristic
91C23×Q8C2314yes—
92C23×Q8C2312yes—
93Q82—2128no—
94C2×C4⋊Q8—216no—
95C2×C4⋊Q8—28no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—216no—
98C2×C4⋊Q8—216no—
99C2×C4⋊Q8—28no—
100C2×C4⋊Q8—28no—
101C2×C4⋊Q8—28no—
102C2×C4⋊Q8—28no—
103C2×C4⋊Q8—24no—
104C2×C4⋊Q8—24no—
105C2×C4⋊Q8D418yes—
106C2×#126—24no—
107C2×#126—24no—
108C2×#126—24no—
109C2×C4×Q8—28no—
110C2×C4×Q8—28no—
111C2×C4×Q8—28no—
112C2×C4×Q8—28no—
113C2×C4×Q8—28no—
114C2×C4×Q8—28no—
115C2×C4×Q8—24no—
116C2×C4×Q8—24no—
117C2×C4×Q8—24no—
118C2×C4×Q8—24no—
119C2×C4×Q8D418yes—
120C2×C4×Q8D418yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2314yes—
127C22×C4⋊C4C2314yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2314yes—
137C22×C4⋊C4C2312yes—
138C22×C22⋊C4C2311yescharacteristic
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2311yescharacteristic
141C22×C22⋊C4C2311yescharacteristic
142C22×C22⋊C4C2312yes—
143C22×C22⋊C4C2312yes—
144C22×C22⋊C4C2312yes—
145C22×C22⋊C4C2314yes—
146C22×C42C2314yes—
147C22×C42C2312yes—
148C22×C42C2312yes—
149C22×C42C2312yes—
150C22×C42C2311yescharacteristic
151#400—28no—
152#400—24no—
153#399—28no—
154#399—24no—
155#398—28no—
156#398—24no—
157#398—28no—
158#397—28no—
159#397—24no—
160#396—28no—
161#395—28no—
162#395—24no—
163#395—24no—
164#395—48no—
165#393—28no—
166#393—24no—
167#393—24no—
168#393—48no—
169#391—48no—
170#389—28no—
171#388—28no—
172#388—28no—
173#388—28no—
174#387—28no—
175#387—28no—
176#383—24no—
177#382—24no—
178#376—24no—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2311yescharacteristic
183C2×#95C2312yes—
184C2×#95C2311yescharacteristic
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2311yescharacteristic
189C2×#95C2312yes—
190C2×#95C2311yescharacteristic
191C2×#95C2312yes—
192C2×#95C2314yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C2×#95C2312yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C2×#95C2314yes—
203C2×#95C2312yes—
204C2×#95C2312yes—
205C2×#95C2314yes—
206C2×#95C2314yes—
207C2×#95C2314yes—
208C2×#95C2314yes—
209C2×#95C2314yes—
210C25C2411yescharacteristic
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—28no—
216C22×Q8—28no—
217C22×Q8—28no—
218C22×Q8—24no—
219C22×Q8—24no—
220C22×Q8—24no—
221C22×Q8—28no—
222C22×Q8—24no—
223C22×Q8—44no—
224C22×Q8—44no—
225C22×Q8—44no—
226C22×Q8—44no—
227C22×Q8—44no—
228C22×Q8—44no—
229C22×Q8C2×D414yes—
230C22×Q8C2×D414yes—
231C22×Q8C2×D418yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2414yes—
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2412yes—
247C23×C4C2412yes—
248C4⋊Q8—2128no—
249C4⋊Q8—264no—
250C4⋊Q8—264no—
251C4⋊Q8—232no—
252#126—416no—
253C4×Q8—264no—
254C4×Q8—264no—
255C4×Q8—232no—
256C4×Q8—232no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4C2×D418yes—
300C2×C4⋊C4C2×D418yes—
301C2×C4⋊C4C2×D418yes—
302C2×C4⋊C4C2×D418yes—
303C2×C4⋊C4C2×D414yes—
304C2×C4⋊C4C2×D414yes—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—28no—
323C2×C22⋊C4—42no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C42—28no—
328C2×C42—28no—
329C2×C42—28no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—42no—
337C2×C42C2×D418yes—
338C2×C42C2×D418yes—
339C2×C42C2×D414yes—
340C2×C42C2×D414yes—
341C2×C42C2×D414yes—
342#95—24no—
343#95—22no—
344#95—22no—
345#95—24no—
346#95—22no—
347#95—22no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—28no—
360#95—28no—
361#95—28no—
362#95—28no—
363#95—28no—
364#95—28no—
365#95—28no—
366#95—28no—
367#95—28no—
368#95—44no—
369#95—44no—
370#95—44no—
371#95—44no—
372#95—44no—
373C24—21no—
374C24—21no—
375C24—21no—
376C24—22no—
377C24—22no—
378C24—24no—
379C24—22no—
380C24—22no—
381C24C2511yescharacteristic
382C2×Q8—216no—
383C2×Q8—28no—
384C2×Q8—28no—
385C2×Q8—216no—
386C2×Q8—216no—
387C2×Q8—232no—
388C2×Q8—28no—
389C2×Q8—28no—
390C2×Q8—216no—
391C2×Q8—216no—
392C2×Q8—216no—
393C2×Q8—416no—
394C2×Q8—88no—
395C2×Q8—84no—
396C2×Q8—84no—
397C2×Q8—88no—
398C2×Q8—88no—
399C2×Q8—88no—
400C2×Q8—88no—
401C2×Q8—88no—
402C2×Q8—88no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4C22×D418yes—
472C22×C4C22×D414yes—
473C22×C4C22×D414yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D414yes—
484C22×C4C22×D414yes—
485C22×C4H(5,𝔽2)14yes—
486C22×C42−1+414yes—
487C22×C42−1+418yes—
488C22×C42−1+414yes—
489C4⋊C4—232no—
490C4⋊C4—232no—
491C4⋊C4—216no—
492C4⋊C4—216no—
493C4⋊C4—216no—
494C4⋊C4—216no—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—48no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—84no—
504C4⋊C4—88no—
505C4⋊C4—84no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—88no—
513C22⋊C4—84no—
514C42—232no—
515C42—216no—
516C42—216no—
517C42—48no—
518C42#120116yes—
519C23—41no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23—41no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23C23×D411yescharacteristic
536C23C23×D411yescharacteristic
537C23C23×D411yescharacteristic
538C23C2×H(5,𝔽2)12yes—
539C23C2×2−1+412yes—
540C23C2×2−1+414yes—
541C23C2×2−1+412yes—
542C23#58512yes—
543Q8—232no—
544Q8—232no—
545Q8—816no—
546Q8—832no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4C2×#12014yes—
629C2×C4C2×#12014yes—
630C2×C4C2×#12014yes—
631C2×C4#53618yes—
632C2×C4#53618yes—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—84no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22C22×#12011yescharacteristic
642C22C2×#53612yes—
643C22C2×#53612yes—
644C22#348612yes—
645C22#348611yescharacteristic
646C22#348611yescharacteristic
647C22#348612yes—
648C22#348612yes—
649C22#348812yes—
650C22#349612yes—
651C22#349811yescharacteristic
652C22#350011yescharacteristic
653C22#354414yes—
654C22#355014yes—
655C22#355514yes—
656C22#355514yes—
657C4—24no—
658C4—24no—
659C4—28no—
660C4—28no—
661C4—24no—
662C4—44no—
663C4—44no—
664C4—48no—
665C4—44no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—84no—
670C4—81no—
671C4—81no—
672C4—81no—
673C2—161no—
674C2#3524211yescharacteristic
675C2#3524211yescharacteristic
676C2#3551311yescharacteristic
677C2#4278912yes—
678C2#4279312yes—
679C2#4280912yes—
680C2#4281312yes—
681C2#5207914yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×Q82C2semidirect
5. #16560C2semidirect
7. #16507C2semidirect
10. #15318C2semidirect
12. #14879C2semidirect
13. #14876C2semidirect
15. #14346C2semidirect
16. #14274C2semidirect
17. #13961C2semidirect

Maximal subgroups

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

subgroupindex
C22×Q822
C2×#28212
C2×#26042
C2×#26372
#165602
#165112
#165072
#165062
#153222
#153182
#148822
#148792
#148762
#148752
#143462
#142742
#139612