Finite Group Databasebeta

← #6574544

Subgroups — #6574544

Order 512.

2037 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3078C212yesmaximal, complemented
2#36134C211yesmaximal, characteristic
3C4×D4×Q8C211yesmaximal, characteristic
4#33196C211yesmaximal, characteristic, complemented
5#32954C212yesmaximal, complemented
6#28512C211yesmaximal, characteristic, complemented
7#28284C211yesmaximal, characteristic, complemented
8#26461C212yesmaximal, complemented
9#26360C214yesmaximal, complemented
10#26043C212yesmaximal, complemented
11#25973C214yesmaximal, complemented
12#24845C212yesmaximal, complemented
13#24303C212yesmaximal, complemented
14#22663C212yesmaximal
15#22650C211yesmaximal, characteristic
16#22430C212yesmaximal
17C4×#477C211yesmaximal, characteristic, complemented
18C2×D4×Q8C2211yescharacteristic
19C2×#533C2212yes—
20C22×C4×Q8C2212yes—
21#3078—216no—
22C2×#479C2214yes—
23C2×#479C2214yes—
24C2×#477C2212yes—
25C2×#477C2211yescharacteristic
26C2×#475C2212yes—
27C2×#475C2212yes—
28C2×#475C2212yes—
29C2×#436C2212yes—
30C2×#428C2214yes—
31C2×#423C2212yes—
32C2×#419C2214yes—
33#2868C2211yescharacteristic
34#2857C2212yes—
35#2552C2212yes—
36#2542—21no—
37#2538—21no—
38#2441C2214yes—
39#2429—22no—
40#2423—22no—
41#2365—21no—
42#2365—21no—
43#2365C2211yescharacteristic
44#2360—22no—
45#2360C2212yes—
46#2354C2212yes—
47Q8×C4⋊C4—21no—
48Q8×C4⋊C4C2211yescharacteristic
49D4×C4⋊C4—21no—
50D4×C4⋊C4C2211yescharacteristic
51Q8×C22⋊C4C2212yes—
52C4×C4⋊Q8—21no—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C22⋊Q8—22no—
56C4×C22⋊Q8C2212yes—
57C42×Q8C2211yescharacteristic
58C42×D4—21no—
59C42×D4C2211yescharacteristic
60#2096C2214yes—
61#2088C2214yes—
62#2068C2214yes—
63#2068C2214yes—
64#2059C2214yes—
65#2026C2214yes—
66#1988C2212yes—
67#1987C2211yescharacteristic
68#1981C2212yes—
69#1956C2212yes—
70#1948C2212yes—
71#1948C2212yes—
72#1947C2211yescharacteristic
73#1947C2212yes—
74#1939C2212yes—
75#1921C2212yes—
76#1894C2214yes—
77#1873C2214yes—
78#1865C2214yes—
79#1865C2214yes—
80#1863C2212yes—
81#1854C2214yes—
82#1853C2214yes—
83#1851C2214yes—
84#1844C2212yes—
85#1842C2211yescharacteristic
86#1809C2214yes—
87#1803C2212yes—
88#1803C2211yescharacteristic
89#1803C2212yes—
90#1803C2212yes—
91#1797C2212yes—
92#1787C2214yes—
93C4×#107C2212yes—
94C4×C4⋊C8C2211yescharacteristic
95C4×#103C2212yes—
96C23×Q8C2312yes—
97C22×M16C2314yes—
98D4×Q8—24no—
99D4×Q8Q818yescomplemented
100#533—28no—
101#533—44no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×#125C2314yes—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8C2312yes—
114C2×C4×Q8—22no—
115C2×C4×Q8—21no—
116C2×C4×Q8—22no—
117C2×C4×Q8—22no—
118C2×C4×Q8—22no—
119C2×C4×Q8—28no—
120C2×C4×Q8—44no—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×Q8C2311yescharacteristic
126C2×C4×Q8C2311yescharacteristic
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4D411yescharacteristic
130C2×C4×D4D411yescharacteristic
131C2×C4×D4C2311yescharacteristic
132C2×#117C2314yes—
133C22×C4⋊C4—22no—
134C22×C4⋊C4C2312yes—
135C22×C42—22no—
136C22×C42C2312yes—
137#479—216no—
138#479—216no—
139#477—28no—
140#477D418yescomplemented
141#475—28no—
142#475—28no—
143#475—44no—
144#436—28no—
145#428—216no—
146C2×#107C2314yes—
147C2×#107C2312yes—
148C2×#107C2312yes—
149C2×#106C2314yes—
150C2×#106C2314yes—
151C2×#106C2314yes—
152#423—28no—
153C2×C4⋊C8C2311yescharacteristic
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2314yes—
156C2×C4⋊C8C2312yes—
157C2×C4⋊C8C2312yes—
158C2×C4⋊C8C2311yescharacteristic
159#419—216no—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2312yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#103C2312yes—
167C2×C22⋊C8C2314yes—
168C2×C4×C8C2312yes—
169#399—24no—
170#398—24no—
171#396—24no—
172#391—21no—
173#391—22no—
174#391—22no—
175#391—22no—
176#391C2312yes—
177#391C2312yes—
178#390—22no—
179#389—21no—
180#389—21no—
181#389—22no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—22no—
186#389—22no—
187#389—24no—
188#389—22no—
189#389C2312yes—
190#389C2312yes—
191#389C2314yes—
192#388—24no—
193#387—22no—
194#387—22no—
195#387—22no—
196#387—24no—
197#385—22no—
198#385—22no—
199#385—22no—
200#384—21no—
201#384—22no—
202#384—21no—
203#384—21no—
204#384C2311yescharacteristic
205#384C2311yescharacteristic
206#384C2312yes—
207#384C2312yes—
208#380—22no—
209#380—22no—
210#380C2312yes—
211#380C2314yes—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4D411yescharacteristic
223C4×C4⋊C4D411yescharacteristic
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2311yescharacteristic
228C4×C4⋊C4C2311yescharacteristic
229C4×C22⋊C4—22no—
230C4×C22⋊C4—22no—
231C4×C22⋊C4—22no—
232C4×C22⋊C4D412yes—
233C4×C22⋊C4D412yes—
234C4×C22⋊C4C2312yes—
235C43—21no—
236C43C2311yescharacteristic
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#340C2312yes—
241#340C2312yes—
242#340C2312yes—
243#340C2312yes—
244#336C2314yes—
245#336C2312yes—
246C22×Q8—21no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—24no—
250C22×Q8—24no—
251C22×Q8—24no—
252C22×Q8—42no—
253C22×Q8—42no—
254C22×Q8C2×Q818yes—
255C22×Q8C2412yes—
256C22×Q8C2411yescharacteristic
257C22×D4C2×D411yescharacteristic
258C23×C4—22no—
259C23×C4—22no—
260C23×C4C2×D412yes—
261C23×C4C2×D412yes—
262C23×C4C2412yes—
263C2×M16—28no—
264C2×M16—28no—
265C2×M16—44no—
266C22×C8C2414yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C4⋊Q8—28no—
270C4⋊Q8—24no—
271C4⋊Q8—22no—
272C4⋊Q8—22no—
273C4⋊Q8—24no—
274C4⋊Q8—44no—
275C4⋊Q8—82no—
276C4⋊Q8C2×D414yes—
277C4⋊Q8C2×Q814yes—
278#125—28no—
279#125—44no—
280C22⋊Q8—24no—
281C22⋊Q8—216no—
282C22⋊Q8—48no—
283C22⋊Q8—48no—
284C22⋊Q8—84no—
285C22⋊Q8C2×Q818yes—
286C4×Q8—22no—
287C4×Q8—22no—
288C4×Q8—24no—
289C4×Q8—22no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×Q8—48no—
294C4×Q8—42no—
295C4×Q8—42no—
296C4×Q8—41no—
297C4×Q8—48no—
298C4×Q8—48no—
299C4×Q8—48no—
300C4×Q8—44no—
301C4×Q8C2×D414yes—
302C4×Q8C2×Q814yes—
303C4×D4—28no—
304C4×D4—24no—
305C4×D4—42no—
306C4×D4—44no—
307C4×D4C2×Q814yes—
308C4×D4#4114yes—
309#117—28no—
310#117—44no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—42no—
338C2×C4⋊C4—48no—
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2414yes—
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2411yescharacteristic
349C2×C22⋊C4—22no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C42—21no—
357C2×C42—21no—
358C2×C42—24no—
359C2×C42—24no—
360C2×C42—24no—
361C2×C42—21no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—21no—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42—48no—
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D412yes—
372C2×C42C2×D412yes—
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2412yes—
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380#107—28no—
381#107—42no—
382#107C2×D418yes—
383#106—28no—
384#106—28no—
385#106—44no—
386C4⋊C8—28no—
387C4⋊C8—24no—
388C4⋊C8—24no—
389C4⋊C8—22no—
390C4⋊C8—42no—
391C4⋊C8C2×D414yes—
392#103—28no—
393#103—24no—
394#103—28no—
395#103—28no—
396#103—44no—
397#103—44no—
398#103C2×D418yes—
399C22⋊C8—28no—
400C4×C8—24no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—24no—
411#95—22no—
412#95—22no—
413#95—24no—
414#95C2×D412yes—
415#95C2×D412yes—
416#95C2×D412yes—
417C24C22×D412yes—
418C2×Q8—28no—
419C2×Q8—24no—
420C2×Q8—21no—
421C2×Q8—22no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—216no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—44no—
430C2×Q8—44no—
431C2×Q8—48no—
432C2×Q8—48no—
433C2×Q8—48no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8C22×D412yes—
440C2×Q8C22×Q814yes—
441C2×Q8C22×Q812yes—
442C2×Q82−1+412yes—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4C22⋊Q814yes—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—28no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—82no—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×Q814yes—
481C22×C4C2×#4114yes—
482C22×C4C2511yescharacteristic
483M16—416no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—24no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8C22×D414yes—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—28no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—48no—
517C4⋊C4—48no—
518C4⋊C4—42no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—81no—
527C4⋊C4—82no—
528C4⋊C4—84no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4C22⋊Q814yes—
532C4⋊C4C22×D414yes—
533C4⋊C4C22×D412yes—
534C4⋊C4C22×Q812yes—
535C22⋊C4—28no—
536C22⋊C4—24no—
537C22⋊C4—42no—
538C22⋊C4—48no—
539C22⋊C4—44no—
540C22⋊C4—84no—
541C22⋊C4C22⋊Q818yes—
542C42—24no—
543C42—24no—
544C42—24no—
545C42—24no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—21no—
551C42—21no—
552C42—41no—
553C42—44no—
554C42—44no—
555C42—48no—
556C42—42no—
557C42—44no—
558C42—42no—
559C42—44no—
560C42—42no—
561C42—81no—
562C42#13714yes—
563C42C22×D412yes—
564C42C22×Q812yes—
565C42C2×#4112yes—
566C42H(5,𝔽2)12yes—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23C2×C22⋊Q814yes—
574C23C23×D411yescharacteristic
575Q8—22no—
576Q8—22no—
577Q8—22no—
578Q8—24no—
579Q8—24no—
580Q8—416no—
581Q8—48no—
582Q8—42no—
583Q8—42no—
584Q8—88no—
585Q8D4×Q814yes—
586D4—28no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—416no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—48no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—84no—
638C2×C4—88no—
639C2×C4C2×C22⋊Q812yes—
640C2×C4C2×C22⋊Q814yes—
641C2×C4C2×C22⋊Q812yes—
642C2×C4D4212yes—
643C2×C4D4212yes—
644C2×C4C2×#13712yes—
645C2×C4C2×#13712yes—
646C2×C4C2×#13712yes—
647C2×C4#57512yes—
648C2×C4#57712yes—
649C2×C4C23×D411yescharacteristic
650C2×C4C23×Q811yescharacteristic
651C2×C4C22×#4111yescharacteristic
652C2×C4C2×H(5,𝔽2)11yescharacteristic
653C2×C4C2×2−1+411yescharacteristic
654C2×C4#58511yescharacteristic
655C8—24no—
656C8—44no—
657C8—42no—
658C22—24no—
659C22—24no—
660C22—24no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22C22×C22⋊Q811yescharacteristic
666C22C2×D4211yescharacteristic
667C22#348911yescharacteristic
668C22C22×#13711yescharacteristic
669C22C2×#57511yescharacteristic
670C22C2×#57711yescharacteristic
671C22#358611yescharacteristic
672C4—22no—
673C4—22no—
674C4—22no—
675C4—24no—
676C4—24no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—44no—
682C4—41no—
683C4—44no—
684C4—84no—
685C4—82no—
686C4—81no—
687C4—82no—
688C4#307814yescomplemented
689C4#327814yes—
690C4C2×D4×Q811yescharacteristic
691C4D4×#4111yescharacteristic
692C4#350111yescharacteristic
693C4#352011yescharacteristic
694C2—44no—
695C2D4×C22⋊Q811yescharacteristic
696C2C2×#307811yescharacteristic
697C2#6049711yescharacteristic
698C2C2×#327811yescharacteristic
699C2#6106711yescharacteristic
700C2#6211311yescharacteristic
701C2#6214511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3078C2semidirect
4. #33196C2semidirect
5. #32954C2semidirect
6. #28512C2semidirect
7. #28284C2semidirect
8. #26461C2semidirect
9. #26360C2semidirect
10. #26043C2semidirect
11. #25973C2semidirect
12. #24845C2semidirect
13. #24303C2semidirect
17. C4×#477C2semidirect
99. D4×Q8Q8semidirect
140. #477D4semidirect
688. C4#3078semidirect

Maximal subgroups

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

subgroupindex
C2×#30782
#361342
C4×D4×Q82
#331962
#329542
#285122
#282842
#264612
#263602
#260432
#259732
#248452
#243032
#226632
#226502
#224302
C4×#4772