Finite Group Databasebeta

← #6804727

Subgroups — #6804727

Order 512.

1538 conjugacy classes of subgroups.

The subgroup lattice has 747 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic, complemented
2#36573C211yesmaximal, characteristic, complemented
3#33286C211yesmaximal, characteristic, complemented
4#32554C214yesmaximal, complemented
5#31830C214yesmaximal, complemented
6#30228C211yesmaximal, characteristic, complemented
7#30204C211yesmaximal, characteristic, complemented
8#29220C211yesmaximal, characteristic, complemented
9#29190C211yesmaximal, characteristic, complemented
10#29177C211yesmaximal, characteristic, complemented
11#28716C211yesmaximal, characteristic, complemented
12#28615C211yesmaximal, characteristic
13#28615C211yesmaximal, characteristic
14#27574C214yesmaximal, complemented
15#27438C214yesmaximal, complemented
16#22269C211yesmaximal, characteristic
17#22219C211yesmaximal, characteristic, complemented
18#21646C211yesmaximal, characteristic, complemented
19#21340C211yesmaximal, characteristic, complemented
20C2×#556—21no—
21C2×#493C2211yescharacteristic
22C2×#486C2211yescharacteristic
23C2×#486C2211yescharacteristic
24#2868C2211yescharacteristic
25#2859C2211yescharacteristic
26#2859C2211yescharacteristic
27#2859C2211yescharacteristic
28#2858C2211yescharacteristic
29#2858C2211yescharacteristic
30#2580—21no—
31#2580—21no—
32#2580C2212yes—
33#2547—21no—
34#2545—21no—
35#2545—21no—
36#2538—21no—
37#2538C2212yes—
38#2512—21no—
39#2512—21no—
40#2512C2212yes—
41#2512C2212yes—
42#2454—21no—
43#2454—21no—
44#2454—21no—
45#2454—21no—
46#2454C2212yes—
47#2454C2212yes—
48#2354—21no—
49#2354C2212yes—
50#2349—21no—
51#2349—21no—
52#2349C2212yes—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×#127C2211yescharacteristic
56C4×#127C2211yescharacteristic
57C4×#127C2211yescharacteristic
58C4×#127C2211yescharacteristic
59C42×Q8C2211yescharacteristic
60#2101C2214yes—
61#2099C2214yes—
62#2099C2214yes—
63#2088C2214yes—
64#2073C2214yes—
65#2071C2214yes—
66#2065C2214yes—
67#2054C2214yes—
68#2054C2214yes—
69#2025C2214yes—
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1989C2214yes—
77#1985C2212yes—
78#1957C2211yescharacteristic
79#1957C2211yescharacteristic
80#1957C2211yescharacteristic
81#1957C2211yescharacteristic
82#1940C2212yes—
83#1936C2214yes—
84#1936C2214yes—
85#1936C2214yes—
86#1926C2214yes—
87#1919C2214yes—
88#1917C2212yes—
89#1917C2212yes—
90#1879C2214yes—
91#1879C2214yes—
92#1879C2214yes—
93#1879C2214yes—
94#1864C2214yes—
95#1864C2214yes—
96#1844C2211yescharacteristic
97#1834C2211yescharacteristic
98#1834C2211yescharacteristic
99#1802C2214yes—
100C4×C4⋊C8C2211yescharacteristic
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103#1762C2214yes—
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C4×#97C2211yescharacteristic
107C22×D8C2314yes—
108#556—48no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×#127C2311yescharacteristic
114C2×#127C2311yescharacteristic
115C2×#126—22no—
116C2×#126—22no—
117C2×#125—21no—
118C2×#125—21no—
119C2×#125C2312yes—
120C2×#124—21no—
121C2×#124—21no—
122C2×#124—21no—
123C2×#124—21no—
124C2×C22⋊Q8—22no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8D411yescharacteristic
128C2×C4×Q8D411yescharacteristic
129C2×C4×Q8C2311yescharacteristic
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×C4×D4—21no—
136C2×C4×D4—21no—
137C2×C4×D4—21no—
138C2×C4×D4C2312yes—
139C2×C4×D4C2312yes—
140C2×C4×D4C2312yes—
141C2×C4×D4C2312yes—
142C2×#117—22no—
143#493—24no—
144#486—24no—
145#486—24no—
146C2×#106C2314yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#97C2312yes—
157C2×#97C2312yes—
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162#399—22no—
163#399—22no—
164#399—22no—
165#397—22no—
166#397—22no—
167#396—22no—
168#396—22no—
169#396—22no—
170#396—22no—
171#396—22no—
172#396—22no—
173#393—22no—
174#391—21no—
175#391—21no—
176#390—21no—
177#390—21no—
178#390—21no—
179#390—21no—
180#390—21no—
181#390—21no—
182#390—21no—
183#390—21no—
184#390—21no—
185#390—21no—
186#390—21no—
187#390—21no—
188#390C2312yes—
189#390C2312yes—
190#390C2312yes—
191#390C2312yes—
192#390C2312yes—
193#390C2312yes—
194#390C2312yes—
195#390C2312yes—
196#389—21no—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389—21no—
202#389—21no—
203#389—21no—
204#389—21no—
205#389—21no—
206#389C2312yes—
207#389C2312yes—
208#389C2312yes—
209#389C2312yes—
210#389C2312yes—
211#389C2312yes—
212#388—22no—
213#388—22no—
214#388—22no—
215#388—22no—
216#385—22no—
217#385—22no—
218#385—22no—
219#385—22no—
220#385—22no—
221#385—22no—
222#384—21no—
223#384—21no—
224#384C2311yescharacteristic
225#384C2311yescharacteristic
226#384C2311yescharacteristic
227#384C2311yescharacteristic
228#384C2312yes—
229#384C2312yes—
230#382—21no—
231#382—21no—
232#382C2312yes—
233#380—22no—
234#380—22no—
235#380—22no—
236#380—22no—
237#380—22no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4D411yescharacteristic
241C4×C4⋊C4D411yescharacteristic
242C4×C4⋊C4D411yescharacteristic
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4D411yescharacteristic
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2311yescharacteristic
249C4×C4⋊C4C2311yescharacteristic
250C4×C4⋊C4C2311yescharacteristic
251C2×#95—22no—
252C43C2311yescharacteristic
253C43C2311yescharacteristic
254C43C2311yescharacteristic
255#340C2314yes—
256#340C2314yes—
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#336C2312yes—
264#336C2312yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270C22×Q8—21no—
271C22×Q8—21no—
272C22×Q8C2×D411yescharacteristic
273C22×D4—21no—
274C22×D4—21no—
275C22×D4—21no—
276C22×D4—21no—
277C22×D4C2412yes—
278C22×D4C2412yes—
279C23×C4—22no—
280C23×C4—22no—
281C23×C4—22no—
282C23×C4—22no—
283C2×D8—28no—
284C2×D8—44no—
285C2×D8—44no—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—44no—
295C4⋊Q8—82no—
296#127—22no—
297#127—22no—
298#127—22no—
299#127#4114yes—
300#126—48no—
301#126—48no—
302#125—44no—
303#125—82no—
304#125—82no—
305#124—44no—
306#124—44no—
307#124—82no—
308#124—82no—
309C22⋊Q8—48no—
310C4×Q8—24no—
311C4×Q8—42no—
312C4×Q8—42no—
313C4×Q8—44no—
314C4×Q8—44no—
315C4×D4—44no—
316C4×D4—44no—
317C4×D4—44no—
318C4×D4—44no—
319C4×D4—44no—
320C4×D4—44no—
321C4×D4—44no—
322C4×D4—44no—
323C4×D4—82no—
324C4×D4—82no—
325C4×D4—82no—
326C4×D4—82no—
327#117—48no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406#106—44no—
407C4⋊C8—42no—
408C4⋊C8—42no—
409C4⋊C8—42no—
410#102—28no—
411#102—28no—
412#102—44no—
413#102—44no—
414#102—44no—
415#102—44no—
416#97—42no—
417#97—42no—
418C4×C8—22no—
419C4×C8—22no—
420C4×C8—22no—
421C4×C8—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95C2×D411yescharacteristic
433#95C2×D411yescharacteristic
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437#95C2×D411yescharacteristic
438C24—22no—
439C24—22no—
440C2×Q8—22no—
441C2×Q8—41no—
442C2×Q8—41no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×D4—24no—
450C2×D4—24no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—24no—
455C2×D4—24no—
456C2×D4—24no—
457C2×D4—24no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4C2×#4114yes—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—82no—
482C22×C4—82no—
483C22×C4—82no—
484C22×C4—82no—
485C22×C4—82no—
486C22×C4—82no—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C2511yescharacteristic
502D8—88no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—41no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4—41no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—44no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—44no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4—82no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—82no—
565C22⋊C4—82no—
566C22⋊C4—82no—
567C22⋊C4—82no—
568C22⋊C4—82no—
569C22⋊C4—82no—
570C22⋊C4—82no—
571C22⋊C4—82no—
572C22⋊C4—82no—
573C22⋊C4—82no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—44no—
592C42—44no—
593C42AGL(1,ℤ/8)14yes—
594C42C2×#4112yes—
595C422−1+412yes—
596C23—24no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23—42no—
610C23C23×D411yescharacteristic
611Q8—44no—
612Q8—84no—
613Q8—84no—
614D4—28no—
615D4—48no—
616D4—48no—
617D4—44no—
618D4—44no—
619D4—44no—
620D4—84no—
621D4—84no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—88no—
675C2×C4—164no—
676C2×C4—164no—
677C2×C4—164no—
678C2×C4#54812yes—
679C2×C4#54812yes—
680C2×C4C2×AGL(1,ℤ/8)12yes—
681C2×C4C2×AGL(1,ℤ/8)12yes—
682C2×C4C2×AGL(1,ℤ/8)12yes—
683C2×C4#57512yes—
684C2×C4#57612yes—
685C2×C4#57712yes—
686C2×C4C22×#4111yescharacteristic
687C2×C4C2×2−1+411yescharacteristic
688C2×C4#58512yes—
689C2×C4#58512yes—
690C8—42no—
691C8—42no—
692C8—42no—
693C8—42no—
694C22—44no—
695C22—44no—
696C22—44no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22#344911yescharacteristic
709C22C2×#54811yescharacteristic
710C22#350011yescharacteristic
711C22C22×AGL(1,ℤ/8)11yescharacteristic
712C22C2×#57511yescharacteristic
713C22C2×#57611yescharacteristic
714C22C2×#57711yescharacteristic
715C4—22no—
716C4—22no—
717C4—41no—
718C4—41no—
719C4—41no—
720C4—41no—
721C4—41no—
722C4—41no—
723C4—41no—
724C4—41no—
725C4—41no—
726C4—41no—
727C4—82no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4—82no—
732C4—82no—
733C4#336614yes—
734C4#337614yes—
735C4#352112yes—
736C4#355712yes—
737C2—84no—
738C2—162no—
739C2#3916311yescharacteristic
740C2#6096511yescharacteristic
741C2#6100611yescharacteristic
742C2C2×#336611yescharacteristic
743C2C2×#337611yescharacteristic
744C2#6254111yescharacteristic
745C2#6254611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#493C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36594C2semidirect
2. #36573C2semidirect
3. #33286C2semidirect
4. #32554C2semidirect
5. #31830C2semidirect
6. #30228C2semidirect
7. #30204C2semidirect
8. #29220C2semidirect
9. #29190C2semidirect
10. #29177C2semidirect
11. #28716C2semidirect
14. #27574C2semidirect
15. #27438C2semidirect
17. #22219C2semidirect
18. #21646C2semidirect
19. #21340C2semidirect

Maximal subgroups

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

subgroupindex
#365942
#365732
#332862
#325542
#318302
#302282
#302042
#292202
#291902
#291772
#287162
#286152
#275742
#274382
#222692
#222192
#216462
#213402