Finite Group Databasebeta

← #6809649

Subgroups — #6809649

Order 512.

1965 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3239C212yesmaximal
2#36116C211yesmaximal, characteristic
3C2×#2551C211yesmaximal, characteristic, complemented
4#33109C211yesmaximal, characteristic, complemented
5#30965C214yesmaximal, complemented
6#30821C214yesmaximal, complemented
7#29642C212yesmaximal, complemented
8#28816C212yesmaximal, complemented
9#27910C211yesmaximal, characteristic, complemented
10#27776C212yesmaximal, complemented
11#27645C211yesmaximal, characteristic
12#26462C212yesmaximal, complemented
13#26040C212yesmaximal, complemented
14#24606C212yesmaximal, complemented
15#24067C212yesmaximal
16#22603C211yesmaximal, characteristic, complemented
17C4×#481C211yesmaximal, characteristic
18C22×C4⋊Q8—21no—
19C22×C4⋊Q8C2212yes—
20C2×#529C2212yes—
21#3239—216no—
22C2×#501C2214yes—
23C2×#489C2214yes—
24C2×#484C2212yes—
25C2×#484C2212yes—
26C2×#484C2212yes—
27C2×#481C2212yes—
28C2×#481C2211yescharacteristic
29C2×#475C2214yes—
30C2×#474C2214yes—
31C2×#432C2212yes—
32#2857C2211yescharacteristic
33#2855C2212yes—
34#2552C2211yescharacteristic
35#2551—21no—
36#2551—22no—
37#2551—21no—
38#2551C2212yes—
39#2551C2214yes—
40#2551C2212yes—
41#2547C2211yescharacteristic
42#2496—21no—
43#2486—21no—
44#2478—22no—
45#2453C2212yes—
46#2447C2212yes—
47#2419—22no—
48C2×#395—22no—
49#2365C2211yescharacteristic
50D4×C4⋊C4—21no—
51D4×C4⋊C4C2211yescharacteristic
52#2344—21no—
53C4×#121C2211yescharacteristic
54C2×#389—21no—
55C2×#389C2212yes—
56C2×#384C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59#2095C2214yes—
60#2086C2214yes—
61#2084C2212yes—
62#2071C2212yes—
63#2066C2214yes—
64#2059C2214yes—
65#2057C2214yes—
66#2057C2214yes—
67#2042C2214yes—
68#2024C2214yes—
69#2024C2214yes—
70#1981C2212yes—
71#1956C2212yes—
72#1948C2212yes—
73#1947C2212yes—
74#1944C2214yes—
75#1939C2212yes—
76#1935C2214yes—
77#1928C2214yes—
78#1926C2212yes—
79#1919C2212yes—
80#1917C2214yes—
81#1874C2211yescharacteristic
82#1872C2212yes—
83#1865C2214yes—
84#1862C2214yes—
85#1853C2211yescharacteristic
86#1845C2212yes—
87#1824C2212yes—
88#1824C2211yescharacteristic
89#1817C2212yes—
90#1802C2212yes—
91#1797C2211yescharacteristic
92C4×#106C2212yes—
93#1770C2212yes—
94C4×#102C2212yes—
95C4×C22⋊C8C2211yescharacteristic
96C23×Q8—22no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—28no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8—48no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2314yes—
107C2×C4⋊Q8C2314yes—
108#529—28no—
109#529—44no—
110C2×#126—22no—
111C2×#125C2312yes—
112C2×#124C2312yes—
113C2×#123—22no—
114C2×C22⋊Q8C2312yes—
115C2×#121C2311yescharacteristic
116C2×C4×Q8C2312yes—
117C2×C4×D4—22no—
118C2×C4×D4—21no—
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2312yes—
121C2×#117—22no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C42—21no—
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135#501—216no—
136#489—216no—
137#484—28no—
138#484—28no—
139#484—44no—
140#481—28no—
141#481D418yes—
142#475—216no—
143#474—216no—
144#432—28no—
145C2×#107C2314yes—
146C2×#107C2314yes—
147C2×#107C2314yes—
148C2×#106C2314yes—
149C2×#106C2312yes—
150C2×#106C2312yes—
151C2×C4⋊C8C2314yes—
152C2×C4⋊C8C2312yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#102C2314yes—
158C2×#102C2312yes—
159C2×#102C2312yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2311yescharacteristic
162C2×C22⋊C8C2311yescharacteristic
163C2×#97C2314yes—
164C2×C4×C8C2312yes—
165#400—22no—
166#398—22no—
167#398—22no—
168#397—22no—
169#397—22no—
170#397—22no—
171#395—24no—
172#395—24no—
173#395—24no—
174#395—24no—
175#391C2312yes—
176#391C2312yes—
177#391C2312yes—
178#390C2311yescharacteristic
179#389—21no—
180#389—22no—
181#389—21no—
182#389—22no—
183#389—22no—
184#389—24no—
185#389—21no—
186#389—21no—
187#389—22no—
188#389—22no—
189#389C2312yes—
190#389C2312yes—
191#389C2314yes—
192#389C2314yes—
193#389C2311yescharacteristic
194#388—21no—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#386—21no—
200#385—22no—
201#385—22no—
202#385—22no—
203#384—21no—
204#384—21no—
205#384—21no—
206#384C2312yes—
207#384C2312yes—
208#384C2311yescharacteristic
209#381—22no—
210#381—22no—
211#380—21no—
212#380—21no—
213#380—21no—
214#380—21no—
215#380—21no—
216#380C2311yescharacteristic
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4D412yes—
222C4×C4⋊C4D411yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2311yescharacteristic
229C4×C22⋊C4—21no—
230C2×#95—21no—
231C2×#95—22no—
232C2×#95D412yes—
233#340C2312yes—
234#340C2312yes—
235#340C2314yes—
236#340C2314yes—
237#340C2312yes—
238#340C2314yes—
239#340C2312yes—
240#336C2312yes—
241#336C2314yes—
242#336C2314yes—
243#336C2312yes—
244C22×Q8—24no—
245C22×Q8—24no—
246C22×Q8—48no—
247C22×Q8—48no—
248C22×Q8—84no—
249C22×Q8C2412yes—
250C22×D4—21no—
251C22×D4C2411yescharacteristic
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—22no—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4C2×D412yes—
258C23×C4C2×D412yes—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2411yescharacteristic
262C22×C8C2412yes—
263C22×C8C2414yes—
264C22×C8C2412yes—
265C4⋊Q8—28no—
266C4⋊Q8—28no—
267C4⋊Q8—24no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8—48no—
271C4⋊Q8—48no—
272C4⋊Q8—416no—
273C4⋊Q8—48no—
274C4⋊Q8—48no—
275#126—48no—
276#125—24no—
277#125—42no—
278#124—24no—
279#124—42no—
280#123—48no—
281C22⋊Q8—24no—
282C22⋊Q8—24no—
283#121—22no—
284#121C2×D414yes—
285C4×Q8—24no—
286C4×Q8—42no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—42no—
290C4×D4—48no—
291C4×D4—44no—
292#117—48no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—48no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—48no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—82no—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D414yes—
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2414yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2412yes—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42—22no—
353C2×C42—21no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—44no—
357C2×C42C2×D412yes—
358C2×C42C2×D412yes—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2412yes—
362C2×C42C2412yes—
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365#107—28no—
366#107—28no—
367#107—44no—
368#106—28no—
369#106—24no—
370#106C2×D418yes—
371C4⋊C8—28no—
372C4⋊C8—24no—
373#103—28no—
374#103—28no—
375#103—28no—
376#103—44no—
377#102—28no—
378#102—24no—
379#102C2×D418yes—
380C22⋊C8—24no—
381C22⋊C8—22no—
382C22⋊C8C2×D414yes—
383#97—28no—
384C4×C8—24no—
385#95—22no—
386#95—22no—
387#95—24no—
388#95—24no—
389#95—21no—
390#95—21no—
391#95—22no—
392#95—21no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95C2×D412yes—
401#95C2×D412yes—
402#95C2×D412yes—
403#95C2×D412yes—
404C24—21no—
405C24C22×D411yescharacteristic
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—48no—
411C2×Q8—48no—
412C2×Q8—48no—
413C2×Q8—48no—
414C2×Q8—84no—
415C2×Q8—84no—
416C2×Q8—832no—
417C2×D4—24no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4C22×D412yes—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—41no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—82no—
460C22×C4#13714yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4H(5,𝔽2)12yes—
472C22×C4C2511yescharacteristic
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8C22×D414yes—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—48no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—48no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—81no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4AGL(1,ℤ/8)14yes—
519C4⋊C4C22×D414yes—
520C4⋊C4C22×D412yes—
521C4⋊C42−1+414yes—
522C22⋊C4—24no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—81no—
531C42—22no—
532C42—22no—
533C42—24no—
534C42—24no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—44no—
541C42—44no—
542C42—44no—
543C42—42no—
544C422−1+414yes—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—22no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23D4212yes—
557C23C2×#13712yes—
558C23#57812yes—
559C23C23×D411yescharacteristic
560Q8—44no—
561Q8—44no—
562Q8—816no—
563Q8—816no—
564D4—24no—
565D4—48no—
566D4—42no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—44no—
594C2×C4—48no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—42no—
598C2×C4—48no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—88no—
620C2×C4—88no—
621C2×C4—84no—
622C2×C4#53612yes—
623C2×C4#53612yes—
624C2×C4D4212yes—
625C2×C4C2×AGL(1,ℤ/8)14yes—
626C2×C4C2×AGL(1,ℤ/8)12yes—
627C2×C4C2×#13712yes—
628C2×C4C2×#13712yes—
629C2×C4C23×D411yescharacteristic
630C2×C4C2×H(5,𝔽2)11yescharacteristic
631C2×C4C2×2−1+412yes—
632C2×C4C2×2−1+412yes—
633C8—24no—
634C8—44no—
635C8—42no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22#327614yes—
650C22C2×#53611yescharacteristic
651C22C2×D4211yescharacteristic
652C22#348611yescharacteristic
653C22C22×AGL(1,ℤ/8)11yescharacteristic
654C22C22×#13711yescharacteristic
655C22C2×#57811yescharacteristic
656C22#358511yescharacteristic
657C4—24no—
658C4—22no—
659C4—22no—
660C4—22no—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—44no—
666C4—44no—
667C4—41no—
668C4—81no—
669C4—82no—
670C4—84no—
671C4—82no—
672C4—81no—
673C4#323914yescomplemented
674C4#349912yes—
675C4#354412yes—
676C2—24no—
677C2—82no—
678C2#3933711yescharacteristic
679C2C2×#323911yescharacteristic
680C2#6087411yescharacteristic
681C2C2×#327611yescharacteristic
682C2D4×AGL(1,ℤ/8)11yescharacteristic
683C2#6183011yescharacteristic
684C2#6183711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2551C2semidirect
4. #33109C2semidirect
5. #30965C2semidirect
6. #30821C2semidirect
7. #29642C2semidirect
8. #28816C2semidirect
9. #27910C2semidirect
10. #27776C2semidirect
12. #26462C2semidirect
13. #26040C2semidirect
14. #24606C2semidirect
16. #22603C2semidirect
673. C4#3239semidirect

Maximal subgroups

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

subgroupindex
C2×#32392
#361162
C2×#25512
#331092
#309652
#308212
#296422
#288162
#279102
#277762
#276452
#264622
#260402
#246062
#240672
#226032
C4×#4812