Finite Group Databasebeta

← #6802678

Subgroups — #6802678

Order 512.

1731 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36580C211yesmaximal, characteristic
2C2×#2582C211yesmaximal, characteristic, complemented
3#32903C211yesmaximal, characteristic
4#32657C214yesmaximal, complemented
5#32611C214yesmaximal, complemented
6#29760C212yesmaximal, complemented
7#28306C212yesmaximal, complemented
8#28265C212yesmaximal, complemented
9#28118C212yesmaximal, complemented
10#27909C211yesmaximal, characteristic
11#27647C211yesmaximal, characteristic, complemented
12#25295C212yesmaximal, complemented
13#25063C212yesmaximal, complemented
14#24190C212yesmaximal, complemented
15Q8×C22⋊C8C211yesmaximal, characteristic, complemented
16#22075C212yesmaximal, complemented
17C4×#451C211yesmaximal, characteristic, complemented
18C2×#559C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#483C2212yes—
22C2×#480C2212yes—
23C2×#480C2212yes—
24C2×#460C2212yes—
25C2×#451C2211yescharacteristic
26C2×#448C2212yes—
27C2×C8×Q8C2212yes—
28C2×C4×Q16C2212yes—
29#2856C2211yescharacteristic
30#2850C2212yes—
31#2592C2212yes—
32#2582—21no—
33#2582—21no—
34#2582—22no—
35#2582C2214yes—
36#2582C2211yescharacteristic
37#2582C2212yes—
38#2582C2212yes—
39#2580C2212yes—
40#2548C2211yescharacteristic
41#2458—22no—
42#2448—21no—
43#2443—22no—
44#2436—21no—
45C2×#398—22no—
46#2367—21no—
47#2361—21no—
48#2355C2211yescharacteristic
49C4×C22⋊Q8C2211yescharacteristic
50C4×#121C2212yes—
51C2×#391—21no—
52C2×#391C2212yes—
53C2×#389—21no—
54C2×#389C2212yes—
55C2×#384C2211yescharacteristic
56#2097C2214yes—
57#2093C2214yes—
58#2085C2214yes—
59#2084C2212yes—
60#2072C2212yes—
61#2068C2214yes—
62#2068C2214yes—
63#2065C2214yes—
64#2055C2214yes—
65#2042C2214yes—
66#2015C2214yes—
67#1984C2214yes—
68#1984C2212yes—
69#1969C2214yes—
70#1968C2214yes—
71#1957C2212yes—
72#1948C2212yes—
73#1942C2214yes—
74#1934C2214yes—
75#1928C2214yes—
76#1918C2214yes—
77#1880C2214yes—
78#1878C2212yes—
79#1877C2214yes—
80#1874C2212yes—
81#1872C2211yescharacteristic
82#1863C2212yes—
83#1862C2214yes—
84#1852C2211yescharacteristic
85#1845C2212yes—
86#1823C2212yes—
87#1803C2212yes—
88#1803C2212yes—
89#1797C2212yes—
90#1797C2211yescharacteristic
91#1787C2211yescharacteristic
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C23×Q8D411yescharacteristic
96C23×Q8C2311yescharacteristic
97C22×Q16C2312yes—
98C22×SD16C2314yes—
99#559—44no—
100C2×#127C2311yescharacteristic
101C2×#124—21no—
102C2×#124—21no—
103C2×#124C2312yes—
104C2×#123—22no—
105C2×C22⋊Q8C2311yescharacteristic
106C2×#121—21no—
107C2×#121—21no—
108C2×#121C2312yes—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8—44no—
112C2×C4×Q8—44no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D412yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2312yes—
122C2×C4×D4—22no—
123C2×C4×D4—21no—
124C2×C4×D4C2312yes—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4D412yes—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136#483—44no—
137#480—28no—
138#480—44no—
139#460—44no—
140#451—24no—
141#448—44no—
142C8×Q8—44no—
143C4×Q16—44no—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#107C2312yes—
147C2×#106C2314yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2314yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×C22⋊C8C2311yescharacteristic
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2311yescharacteristic
162C2×C4×C8C2312yes—
163C2×C4×C8C2314yes—
164#399—22no—
165#398—24no—
166#398—24no—
167#398—24no—
168#398—24no—
169#398—22no—
170#397—22no—
171#394—22no—
172#393—22no—
173#393—22no—
174#391—22no—
175#391—24no—
176#391—22no—
177#391—21no—
178#391—21no—
179#391—22no—
180#391—22no—
181#391C2314yes—
182#391C2312yes—
183#391C2314yes—
184#391C2311yescharacteristic
185#390—21no—
186#390—21no—
187#390—21no—
188#390C2312yes—
189#390C2312yes—
190#390C2312yes—
191#389—22no—
192#389—22no—
193#389—24no—
194#389—21no—
195#389—22no—
196#389—22no—
197#389C2314yes—
198#389C2312yes—
199#389C2311yescharacteristic
200#389C2314yes—
201#388—22no—
202#388—22no—
203#388—22no—
204#388—22no—
205#388—22no—
206#387—22no—
207#387—21no—
208#386—22no—
209#386—22no—
210#386—22no—
211#385—21no—
212#385—22no—
213#385—22no—
214#384—21no—
215#384—21no—
216#384C2311yescharacteristic
217#384C2312yes—
218#384C2312yes—
219#382C2312yes—
220#380C2311yescharacteristic
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2311yescharacteristic
225C4×C22⋊C4—21no—
226C4×C22⋊C4—22no—
227C2×#95—22no—
228C2×#95—22no—
229C2×#95—22no—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#336C2312yes—
236#336C2314yes—
237#336C2312yes—
238#336C2314yes—
239C22×Q8—24no—
240C22×Q8—24no—
241C22×Q8—24no—
242C22×Q8—21no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—42no—
246C22×Q8—42no—
247C22×Q8C2×D411yescharacteristic
248C22×Q8C2×D412yes—
249C22×Q8C2×D411yescharacteristic
250C22×Q8C2411yescharacteristic
251C22×Q8C2412yes—
252C22×D4—22no—
253C22×D4—21no—
254C22×D4—21no—
255C22×D4C2412yes—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4—21no—
259C23×C4—22no—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D412yes—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D412yes—
264C23×C4C2411yescharacteristic
265C2×Q16—24no—
266C2×Q16—24no—
267C2×Q16—42no—
268C2×SD16—28no—
269C2×SD16—44no—
270C2×SD16—44no—
271C22×C8C2414yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274#127—24no—
275#124—44no—
276#124—44no—
277#124—44no—
278#123—48no—
279C22⋊Q8—22no—
280C22⋊Q8—22no—
281#121—24no—
282#121—24no—
283#121—44no—
284#121—82no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—44no—
291C4×Q8—44no—
292C4×Q8—42no—
293C4×Q8—42no—
294C4×Q8—42no—
295C4×Q8—42no—
296C4×Q8—44no—
297C4×Q8—44no—
298C4×D4—44no—
299C4×D4—44no—
300C4×D4—84no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—42no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—82no—
325C2×C4⋊C4—84no—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2414yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2411yescharacteristic
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
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×C42—24no—
352C2×C42—21no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—21no—
356C2×C42—22no—
357C2×C42—44no—
358C2×C42—44no—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D412yes—
362C2×C42C2×D412yes—
363C2×C42C2412yes—
364C2×C42C2411yescharacteristic
365C2×C42C2412yes—
366C2×C42C2411yescharacteristic
367#107—28no—
368#107—42no—
369#107—44no—
370#106—44no—
371C4⋊C8—24no—
372C4⋊C8—44no—
373#103—24no—
374#103—24no—
375#103—42no—
376#103—42no—
377#103—44no—
378#103—44no—
379#102—28no—
380#102—44no—
381#102—44no—
382C22⋊C8—24no—
383C22⋊C8—22no—
384C22⋊C8#4114yes—
385C4×C8—24no—
386C4×C8—44no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95—24no—
393#95—21no—
394#95—21no—
395#95—22no—
396#95—21no—
397#95—22no—
398#95—21no—
399#95—21no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404C24—22no—
405C24C22×D411yescharacteristic
406C2×Q8—24no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—24no—
411C2×Q8—22no—
412C2×Q8—24no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—21no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—42no—
424C2×Q8—416no—
425C2×Q8—48no—
426C2×Q8—48no—
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8#13714yes—
430C2×Q82−1+414yes—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—81no—
443C2×D4—81no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—42no—
463C22×C4—41no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—44no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4#13514yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C2×#4112yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C4C2511yescharacteristic
490Q16—48no—
491SD16—88no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—22no—
499C2×C8—24no—
500C2×C8C2×#4114yes—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—42no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—81no—
528C4⋊C4—82no—
529C4⋊C4—84no—
530C4⋊C4—82no—
531C4⋊C4—84no—
532C4⋊C4—82no—
533C4⋊C4—81no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—82no—
544C22⋊C4—82no—
545C22⋊C4—82no—
546C42—22no—
547C42—22no—
548C42—24no—
549C42—22no—
550C42—24no—
551C42—42no—
552C42—42no—
553C42—44no—
554C42—44no—
555C42—42no—
556C42—42no—
557C42—82no—
558C42H(5,𝔽2)14yes—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23#55012yes—
571C23C2×#13512yes—
572C23#57612yes—
573C23C23×D411yescharacteristic
574Q8—28no—
575Q8—24no—
576Q8—24no—
577Q8—48no—
578Q8—48no—
579Q8—44no—
580Q8—42no—
581Q8—42no—
582Q8—42no—
583Q8—42no—
584D4—44no—
585D4—44no—
586D4—88no—
587D4—84no—
588D4—162no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—44no—
614C2×C4—48no—
615C2×C4—48no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—44no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4—88no—
642C2×C4—164no—
643C2×C4#53412yes—
644C2×C4#53412yes—
645C2×C4#55012yes—
646C2×C4C2×#13512yes—
647C2×C4C2×#13512yes—
648C2×C4C2×#13714yes—
649C2×C4C2×#13712yes—
650C2×C4C22×#4111yescharacteristic
651C2×C4C2×H(5,𝔽2)12yes—
652C2×C4C2×H(5,𝔽2)11yescharacteristic
653C2×C4C2×2−1+412yes—
654C8—24no—
655C8—42no—
656C8—44no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22#339614yes—
671C22C2×#53411yescharacteristic
672C22C2×#55011yescharacteristic
673C22#348611yescharacteristic
674C22C22×#13511yescharacteristic
675C22C22×#13711yescharacteristic
676C22C2×#57611yescharacteristic
677C22#358611yescharacteristic
678C4—22no—
679C4—24no—
680C4—22no—
681C4—22no—
682C4—42no—
683C4—41no—
684C4—42no—
685C4—42no—
686C4—44no—
687C4—82no—
688C4—81no—
689C4—81no—
690C4—84no—
691C4—82no—
692C4—81no—
693C4—81no—
694C4#319014yes—
695C4#352012yes—
696C4#353612yes—
697C2—24no—
698C2—162no—
699C2#3913311yescharacteristic
700C2C2×#319011yescharacteristic
701C2#6076211yescharacteristic
702C2C2×#339611yescharacteristic
703C2#6140411yescharacteristic
704C2#6182211yescharacteristic
705C2#6184411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2582C2semidirect
4. #32657C2semidirect
5. #32611C2semidirect
6. #29760C2semidirect
7. #28306C2semidirect
8. #28265C2semidirect
9. #28118C2semidirect
11. #27647C2semidirect
12. #25295C2semidirect
13. #25063C2semidirect
14. #24190C2semidirect
15. Q8×C22⋊C8C2semidirect
16. #22075C2semidirect
17. C4×#451C2semidirect

Maximal subgroups

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

subgroupindex
#365802
C2×#25822
#329032
#326572
#326112
#297602
#283062
#282652
#281182
#279092
#276472
#252952
#250632
#241902
Q8×C22⋊C82
#220752
C4×#4512