Finite Group Databasebeta

← #4306713

Subgroups — #4306713

Order 512.

2540 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#2346C211yesmaximal, characteristic
3#15202C211yesmaximal, characteristic, complemented
4#15153C211yesmaximal, characteristic, complemented
5#15137C212yesmaximal, complemented
6#15135C211yesmaximal, characteristic
7#15131C212yesmaximal
8#15121C212yesmaximal, complemented
9#14959C214yesmaximal
10#14938C214yesmaximal, complemented
11#14875C212yesmaximal
12#14870C212yesmaximal
13C4×#393C211yesmaximal, characteristic
14#13850C212yesmaximal, complemented
15#13821C211yesmaximal, characteristic, complemented
16#13818C212yesmaximal, complemented
17#13801C212yesmaximal
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C22×#117C2211yescharacteristic
21C2×#400C2214yes—
22C2×#399C2212yes—
23C2×#398C2214yes—
24C2×#396C2214yes—
25C2×#393C2211yescharacteristic
26Q8×C4⋊C4—28no—
27#2346—28no—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#389C2214yes—
35C2×#388C2212yes—
36C2×#388C2212yes—
37C2×#384C2212yes—
38C2×#384C2211yescharacteristic
39C2×#382C2212yes—
40C2×#380C2211yescharacteristic
41C2×#380C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×#376C2212yes—
50#1446C2211yescharacteristic
51#1446C2212yes—
52#1445C2212yes—
53#1445C2212yes—
54#1445C2212yes—
55#1444C2212yes—
56#1444C2214yes—
57#1444C2212yes—
58#1444C2214yes—
59#1442C2211yescharacteristic
60#1442C2214yes—
61#1442C2212yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1439C2211yescharacteristic
66#1439C2214yes—
67#1439C2212yes—
68#1439C2212yes—
69#1437C2214yes—
70#1437C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74C4⋊#95C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79#1434C2212yes—
80#1434C2214yes—
81C4×#95C2214yes—
82C4×#95C2214yes—
83C4×#95C2212yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×C4×Q8—28no—
88C2×C4×Q8—24no—
89C2×C4×Q8—28no—
90C2×C4×Q8—28no—
91C2×C4×Q8—44no—
92C2×#117—24no—
93C2×#117—24no—
94C2×#117Q818yescomplemented
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2314yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2314yes—
128#400—48no—
129#399—44no—
130#398—48no—
131#396—48no—
132#393—24no—
133#389—24no—
134#389—28no—
135#389—28no—
136#389—24no—
137#389—28no—
138#389—28no—
139#389—216no—
140#388—28no—
141#388—28no—
142#384—28no—
143#384—24no—
144#382—28no—
145#380—24no—
146#380—28no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—44no—
151C4×C4⋊C4D418yes—
152C4×C22⋊C4—24no—
153C4×C22⋊C4—28no—
154#376—28no—
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2311yescharacteristic
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2311yescharacteristic
178C2×#95C2311yescharacteristic
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C25C2411yescharacteristic
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8—22no—
189C22×Q8—24no—
190C22×Q8#4114yes—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4C2×Q814yes—
196C23×C4C2411yescharacteristic
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C4×Q8—232no—
213C4×Q8—432no—
214#117—232no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4C2×D414yes—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4C2×Q814yes—
257C2×C4⋊C4C2×Q814yes—
258C2×C4⋊C4#4118yes—
259C2×C4⋊C4#4114yes—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4C2×Q818yes—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—28no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—28no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—28no—
297C2×C42—44no—
298C2×C42—42no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42C2×D418yes—
302C2×C42C2×D414yes—
303C2×C42C2×Q818yes—
304C2×C42#4118yes—
305#95—22no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—22no—
310#95—24no—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—22no—
320#95—24no—
321#95—22no—
322#95—24no—
323#95—28no—
324#95—24no—
325#95—28no—
326#95—24no—
327#95—24no—
328#95—28no—
329#95—24no—
330#95—42no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95C2×D418yes—
335C24—21no—
336C24—21no—
337C24—21no—
338C24—22no—
339C24—22no—
340C24—21no—
341C24—22no—
342C24—22no—
343C24—21no—
344C24—21no—
345C24C22×Q812yes—
346C24C2511yescharacteristic
347C2×Q8—28no—
348C2×Q8—28no—
349C2×Q8—216no—
350C2×Q8—28no—
351C2×Q8—216no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—28no—
401C22×C4—216no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—44no—
427C22×C4—48no—
428C22×C4—44no—
429C22×C4C22×D414yes—
430C22×C4C22×D412yes—
431C22×C4C22×D414yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×Q812yes—
435C22×C4C22×Q814yes—
436C22×C4C22×Q814yes—
437C22×C4C22×Q812yes—
438C22×C4C2×#4118yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4114yes—
443C22×C4C2×#4114yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4114yes—
446C22×C4C2×#4114yes—
447C22×C4H(5,𝔽2)14yes—
448C22×C4H(5,𝔽2)12yes—
449C22×C42−1+412yes—
450C22×C42−1+414yes—
451C4⋊C4—28no—
452C4⋊C4—216no—
453C4⋊C4—28no—
454C4⋊C4—216no—
455C4⋊C4—28no—
456C4⋊C4—416no—
457C4⋊C4—416no—
458C4⋊C4—44no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—88no—
462C4⋊C4—84no—
463C4⋊C4—88no—
464C4⋊C4—84no—
465C4⋊C4—84no—
466C4⋊C4—88no—
467C4⋊C4C22⋊Q8116yes—
468C22⋊C4—216no—
469C22⋊C4—44no—
470C22⋊C4—48no—
471C22⋊C4—44no—
472C22⋊C4—48no—
473C22⋊C4—84no—
474C22⋊C4—88no—
475C42—216no—
476C42—28no—
477C42—216no—
478C42—216no—
479C42—48no—
480C42—416no—
481C42—48no—
482C42—44no—
483C42—88no—
484C23—22no—
485C23—24no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—41no—
490C23—42no—
491C23—41no—
492C23—41no—
493C23—42no—
494C23—41no—
495C23—42no—
496C23—41no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—41no—
501C23—42no—
502C23—41no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23C23×D411yescharacteristic
508C23C23×Q811yescharacteristic
509C23C22×#4111yescharacteristic
510C23C22×#4112yes—
511C23C22×#4112yes—
512C23C22×#4111yescharacteristic
513C23C2×H(5,𝔽2)11yescharacteristic
514C23C2×H(5,𝔽2)12yes—
515C23C2×2−1+412yes—
516C23C2×2−1+411yescharacteristic
517C23#58511yescharacteristic
518Q8—232no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—416no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4C2×C22⋊Q818yes—
607C2×C4C2×C22⋊Q814yes—
608C2×C4#52514yes—
609C2×C4#52918yes—
610C2×C4#53214yes—
611C2×C4#53318yes—
612C2×C4#54614yes—
613C2×C4#55218yes—
614C22—44no—
615C22—42no—
616C22—42no—
617C22—44no—
618C22—42no—
619C22—81no—
620C22—81no—
621C22—82no—
622C22—81no—
623C22—82no—
624C22—81no—
625C22C22×C22⋊Q811yescharacteristic
626C22C2×#52511yescharacteristic
627C22C2×#52912yes—
628C22C2×#53211yescharacteristic
629C22C2×#53312yes—
630C22C2×#54611yescharacteristic
631C22C2×#55212yes—
632C22Q8×#4111yescharacteristic
633C22#348211yescharacteristic
634C22#348312yes—
635C22#348611yescharacteristic
636C22#348911yescharacteristic
637C22#349212yes—
638C22#349412yes—
639C22#349811yescharacteristic
640C22#350311yescharacteristic
641C22#350312yes—
642C22#351112yes—
643C22#351411yescharacteristic
644C22#351512yes—
645C22#351912yes—
646C22#352212yes—
647C22#352312yes—
648C4—24no—
649C4—28no—
650C4—28no—
651C4—44no—
652C4—48no—
653C4—44no—
654C4—42no—
655C4—44no—
656C4—42no—
657C4—42no—
658C4—44no—
659C4—81no—
660C4—84no—
661C4—82no—
662C4—82no—
663C4#239618yes—
664C2—82no—
665C2C2×#239611yescharacteristic
666C2#3574712yes—
667C2#3574811yescharacteristic
668C2#3594712yes—
669C2#3594911yescharacteristic
670C2#3608711yescharacteristic
671C2#3616311yescharacteristic
672C2#3627411yescharacteristic
673C2#3628712yes—
674C2#3631212yes—
675C2#3641211yescharacteristic
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. C2×Q8×C4⋊C4C2semidirect
3. #15202C2semidirect
4. #15153C2semidirect
5. #15137C2semidirect
8. #15121C2semidirect
10. #14938C2semidirect
14. #13850C2semidirect
15. #13821C2semidirect
16. #13818C2semidirect
94. C2×#117Q8semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C4⋊C42
C2×#23462
#152022
#151532
#151372
#151352
#151312
#151212
#149592
#149382
#148752
#148702
C4×#3932
#138502
#138212
#138182
#138012