Finite Group Databasebeta

← #6811856

Subgroups — #6811856

Order 512.

1975 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3238C212yesmaximal, complemented
2D4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#34932C211yesmaximal, characteristic
4#33197C211yesmaximal, characteristic, complemented
5#30801C214yesmaximal, complemented
6#30653C214yesmaximal, complemented
7#29692C212yesmaximal, complemented
8#28816C212yesmaximal, complemented
9#28511C211yesmaximal, characteristic, complemented
10#28284C211yesmaximal, characteristic, complemented
11#27776C212yesmaximal, complemented
12#26234C212yesmaximal, complemented
13#25010C212yesmaximal
14#24711C212yesmaximal, complemented
15#24302C212yesmaximal, complemented
16#22799C211yesmaximal, characteristic, complemented
17C4×#476C211yesmaximal, characteristic, complemented
18C2×D4×Q8—21no—
19C22×C4⋊Q8C2212yes—
20C2×#529C2212yes—
21#3238—216no—
22C2×#501C2214yes—
23C2×#489C2214yes—
24C2×#483C2214yes—
25C2×#480C2214yes—
26C2×#477C2212yes—
27C2×#477C2212yes—
28C2×#477C2212yes—
29C2×#476C2212yes—
30C2×#476C2211yescharacteristic
31C2×#432C2212yes—
32#2868C2211yescharacteristic
33#2857C2212yes—
34#2553—22no—
35#2552C2211yescharacteristic
36#2551—21no—
37#2551C2212yes—
38#2547C2211yescharacteristic
39#2546—21no—
40#2542C2211yescharacteristic
41#2537C2211yescharacteristic
42#2496—21no—
43#2489C2212yes—
44#2440—22no—
45#2440C2214yes—
46#2434—22no—
47#2427C2212yes—
48#2422—21no—
49#2395—22no—
50D4×C4⋊C4—21no—
51D4×C4⋊C4C2212yes—
52C4×C4⋊Q8C2211yescharacteristic
53C4×C4⋊Q8C2211yescharacteristic
54C4×C22⋊Q8C2212yes—
55C4×#121—21no—
56#2285—21no—
57C42×D4C2211yescharacteristic
58C42×D4C2211yescharacteristic
59#2095C2214yes—
60#2075C2214yes—
61#2068C2214yes—
62#2059C2214yes—
63#2057C2214yes—
64#2042C2214yes—
65#2035C2214yes—
66#2035C2214yes—
67#2033C2214yes—
68#2024C2214yes—
69#1987C2211yescharacteristic
70#1987C2212yes—
71#1981C2212yes—
72#1956C2212yes—
73#1954C2212yes—
74#1948C2212yes—
75#1948C2211yescharacteristic
76#1948C2212yes—
77#1947C2212yes—
78#1944C2214yes—
79#1935C2214yes—
80#1924C2214yes—
81#1920C2212yes—
82#1872C2214yes—
83#1864C2212yes—
84#1852C2214yes—
85#1845C2212yes—
86#1842C2211yescharacteristic
87#1803C2212yes—
88#1802C2211yescharacteristic
89#1802C2212yes—
90#1802C2212yes—
91#1797C2212yes—
92C4×#106C2212yes—
93#1770C2212yes—
94C4×C4⋊C8C2211yescharacteristic
95C4×#102C2212yes—
96C23×Q8—22no—
97D4×Q8—48no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—28no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—44no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2314yes—
108C2×C4⋊Q8C2311yescharacteristic
109#529—28no—
110#529—44no—
111C2×#126—22no—
112C2×#125C2312yes—
113C2×#124C2312yes—
114C2×#123—22no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8C2312yes—
118C2×C22⋊Q8C2312yes—
119C2×#121—21no—
120C2×#121C2312yes—
121C2×C4×Q8—21no—
122C2×C4×Q8C2312yes—
123C2×C4×D4—22no—
124C2×C4×D4—22no—
125C2×C4×D4D411yescharacteristic
126C2×C4×D4D411yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2311yescharacteristic
130C2×C4×D4C2311yescharacteristic
131C2×#117—22no—
132C22×C4⋊C4—22no—
133C22×C4⋊C4C2314yes—
134C22×C42—21no—
135C22×C42C2312yes—
136#501—216no—
137#489—216no—
138#483—216no—
139#480—216no—
140#477—28no—
141#477—28no—
142#477—44no—
143#476—28no—
144#476D418yescomplemented
145#432—28no—
146C2×#107C2314yes—
147C2×#107C2314yes—
148C2×#107C2314yes—
149C2×#106C2314yes—
150C2×#106C2312yes—
151C2×#106C2312yes—
152C2×C4⋊C8C2311yescharacteristic
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2311yescharacteristic
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#102C2314yes—
162C2×#102C2312yes—
163C2×#102C2312yes—
164C2×C22⋊C8C2314yes—
165C2×C22⋊C8C2314yes—
166C2×#97C2314yes—
167C2×C4×C8C2312yes—
168#400—22no—
169#399—22no—
170#398—22no—
171#398—24no—
172#397—22no—
173#397—22no—
174#397—22no—
175#395—24no—
176#395—24no—
177#394—22no—
178#394—22no—
179#391—21no—
180#391—22no—
181#391C2312yes—
182#391C2312yes—
183#390—21no—
184#389—22no—
185#389—22no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—22no—
191#389C2312yes—
192#389C2312yes—
193#389C2314yes—
194#388—22no—
195#387—22no—
196#386—22no—
197#385—22no—
198#385—24no—
199#385—24no—
200#384—21no—
201#384—21no—
202#384—21no—
203#384—21no—
204#384C2311yescharacteristic
205#384C2311yescharacteristic
206#384C2312yes—
207#384C2312yes—
208#382—22no—
209#380—22no—
210#380—21no—
211#380—21no—
212#380—21no—
213#380—22no—
214#380C2312yes—
215#380C2314yes—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2311yescharacteristic
227C4×C22⋊C4—22no—
228C4×C22⋊C4—21no—
229C4×C22⋊C4—21no—
230C4×C22⋊C4—21no—
231C4×C22⋊C4D412yes—
232C4×C22⋊C4D412yes—
233C4×C22⋊C4C2312yes—
234C43C2311yescharacteristic
235#340C2312yes—
236#340C2314yes—
237#340C2312yes—
238#340C2314yes—
239#340C2314yes—
240#340C2312yes—
241#340C2312yes—
242#336C2314yes—
243#336C2312yes—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8—24no—
247C22×Q8—48no—
248C22×Q8—48no—
249C22×Q8—84no—
250C22×Q8C2412yes—
251C22×D4—22no—
252C22×D4C2×D411yescharacteristic
253C22×D4C2411yescharacteristic
254C23×C4—21no—
255C23×C4—24no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2412yes—
261C22×C8C2412yes—
262C22×C8C2414yes—
263C22×C8C2412yes—
264C4⋊Q8—28no—
265C4⋊Q8—24no—
266C4⋊Q8—22no—
267C4⋊Q8—24no—
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270C4⋊Q8—44no—
271C4⋊Q8—48no—
272C4⋊Q8—48no—
273C4⋊Q8—44no—
274C4⋊Q8—48no—
275C4⋊Q8—82no—
276C4⋊Q8C2×D414yes—
277#126—48no—
278#125—24no—
279#125—42no—
280#124—24no—
281#124—42no—
282#123—48no—
283C22⋊Q8—24no—
284C22⋊Q8—24no—
285C22⋊Q8—416no—
286C22⋊Q8—42no—
287C22⋊Q8—42no—
288C22⋊Q8—48no—
289#121—24no—
290#121—42no—
291#121—44no—
292C4×Q8—24no—
293C4×Q8—24no—
294C4×Q8—44no—
295C4×D4—24no—
296C4×D4—22no—
297C4×D4—24no—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—41no—
304C4×D4—48no—
305C4×D4—48no—
306C4×D4C2×D414yes—
307#117—48no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4—48no—
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2414yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2411yescharacteristic
343C2×C22⋊C4—21no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C42—24no—
357C2×C42—24no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—44no—
364C2×C42C2×D412yes—
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D412yes—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2411yescharacteristic
371C2×C42C2412yes—
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374#107—28no—
375#107—28no—
376#107—44no—
377#106—28no—
378#106—24no—
379#106C2×D418yes—
380C4⋊C8—24no—
381C4⋊C8—22no—
382C4⋊C8—24no—
383C4⋊C8—42no—
384C4⋊C8C2×D414yes—
385#103—28no—
386#103—28no—
387#103—44no—
388#103—44no—
389#102—28no—
390#102—42no—
391#102C2×D418yes—
392C22⋊C8—28no—
393C22⋊C8—44no—
394#97—28no—
395C4×C8—24no—
396#95—21no—
397#95—21no—
398#95—21no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—24no—
408#95—24no—
409#95C2×D412yes—
410#95C2×D412yes—
411#95C2×D412yes—
412C24—21no—
413C24C22×D412yes—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—48no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8—44no—
425C2×Q8—48no—
426C2×Q8—44no—
427C2×Q8—44no—
428C2×Q8—84no—
429C2×Q8—832no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—82no—
440C2×D4AGL(1,ℤ/8)14yes—
441C2×D4C22×D412yes—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—48no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—48no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—81no—
473C22×C4—84no—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C2511yescharacteristic
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—24no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8C22×D414yes—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—48no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—81no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4C22×D414yes—
530C4⋊C4C22×D412yes—
531C4⋊C42−1+412yes—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—42no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—48no—
543C22⋊C4—44no—
544C22⋊C4—42no—
545C42—24no—
546C42—24no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—21no—
552C42—21no—
553C42—41no—
554C42—44no—
555C42—44no—
556C42—42no—
557C42—42no—
558C42—42no—
559C42#13714yes—
560C42C22×D412yes—
561C42H(5,𝔽2)12yes—
562C422−1+412yes—
563C422−1+412yes—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23C2×AGL(1,ℤ/8)14yes—
578C23C23×D411yescharacteristic
579Q8—48no—
580Q8—42no—
581Q8—42no—
582Q8—42no—
583Q8—42no—
584Q8—88no—
585Q8—816no—
586D4—28no—
587D4—22no—
588D4—22no—
589D4—41no—
590D4—41no—
591D4—88no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—48no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—48no—
623C2×C4—48no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—84no—
642C2×C4—816no—
643C2×C4—84no—
644C2×C4#53612yes—
645C2×C4#53612yes—
646C2×C4D4212yes—
647C2×C4D4212yes—
648C2×C4C2×AGL(1,ℤ/8)12yes—
649C2×C4C2×#13712yes—
650C2×C4C2×#13712yes—
651C2×C4C2×#13712yes—
652C2×C4#57812yes—
653C2×C4C23×D411yescharacteristic
654C2×C4C2×H(5,𝔽2)11yescharacteristic
655C2×C4C2×2−1+411yescharacteristic
656C2×C4C2×2−1+411yescharacteristic
657C2×C4C2×2−1+411yescharacteristic
658C2×C4#58511yescharacteristic
659C8—24no—
660C8—44no—
661C8—42no—
662C22—24no—
663C22—24no—
664C22—24no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22C2×#53611yescharacteristic
677C22C2×D4211yescharacteristic
678C22#348911yescharacteristic
679C22C22×AGL(1,ℤ/8)11yescharacteristic
680C22C22×#13711yescharacteristic
681C22C2×#57811yescharacteristic
682C22#358511yescharacteristic
683C4—22no—
684C4—22no—
685C4—22no—
686C4—42no—
687C4—42no—
688C4—42no—
689C4—44no—
690C4—42no—
691C4—42no—
692C4—41no—
693C4—84no—
694C4—82no—
695C4—82no—
696C4—81no—
697C4—82no—
698C4#323814yescomplemented
699C4#327614yes—
700C4#349711yescharacteristic
701C4#349911yescharacteristic
702C4#354711yescharacteristic
703C4#355011yescharacteristic
704C2—44no—
705C2—82no—
706C2#3934311yescharacteristic
707C2C2×#323811yescharacteristic
708C2#6089311yescharacteristic
709C2C2×#327611yescharacteristic
710C2D4×AGL(1,ℤ/8)11yescharacteristic
711C2#6210811yescharacteristic
712C2#6213211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2537C22
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×#3238C2semidirect
2. D4×C4⋊Q8C2semidirect
4. #33197C2semidirect
5. #30801C2semidirect
6. #30653C2semidirect
7. #29692C2semidirect
8. #28816C2semidirect
9. #28511C2semidirect
10. #28284C2semidirect
11. #27776C2semidirect
12. #26234C2semidirect
14. #24711C2semidirect
15. #24302C2semidirect
16. #22799C2semidirect
17. C4×#476C2semidirect
144. #476D4semidirect
698. C4#3238semidirect

Maximal subgroups

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

subgroupindex
C2×#32382
D4×C4⋊Q82
#349322
#331972
#308012
#306532
#296922
#288162
#285112
#282842
#277762
#262342
#250102
#247112
#243022
#227992
C4×#4762