Finite Group Databasebeta

← #6793066

Subgroups — #6793066

Order 512.

1759 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36127C211yesmaximal, characteristic
2#34941C211yesmaximal, characteristic, complemented
3#33198C211yesmaximal, characteristic, complemented
4#31626C214yesmaximal, complemented
5#31566C214yesmaximal, complemented
6#29938C212yesmaximal, complemented
7#29218C211yesmaximal, characteristic, complemented
8#28280C211yesmaximal, characteristic, complemented
9#28150C212yesmaximal, complemented
10#26106C212yesmaximal, complemented
11#25346C212yesmaximal, complemented
12#24846C212yesmaximal, complemented
13#24815C212yesmaximal, complemented
14#24156C212yesmaximal, complemented
15#22650C211yesmaximal, characteristic, complemented
16#21893C212yesmaximal, complemented
17C4×#460C211yesmaximal, characteristic, complemented
18C2×#547—21no—
19C2×#532C2212yes—
20C2×#477C2212yes—
21C2×#476C2212yes—
22C2×#460C2211yescharacteristic
23C2×C4×SD16C2212yes—
24C2×#436C2212yes—
25C2×#423C2212yes—
26#2868C2211yescharacteristic
27#2857C2212yes—
28#2574—22no—
29#2567C2211yescharacteristic
30#2563C2211yescharacteristic
31#2552C2211yescharacteristic
32#2547—21no—
33#2538C2211yescharacteristic
34#2537—21no—
35#2537C2212yes—
36#2508—21no—
37#2497C2212yes—
38#2454—21no—
39#2432—22no—
40#2423C2212yes—
41#2422—22no—
42#2422C2214yes—
43#2396—22no—
44D4×C4⋊C4—21no—
45D4×C4⋊C4C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127C2211yescharacteristic
48C4×#125—21no—
49C4×C22⋊Q8C2212yes—
50#2293—21no—
51#2285C2212yes—
52C42×Q8C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2100C2214yes—
55#2099C2214yes—
56#2086C2214yes—
57#2084C2214yes—
58#2077C2214yes—
59#2077C2214yes—
60#2066C2214yes—
61#2065C2214yes—
62#2033C2214yes—
63#2000C2214yes—
64#1990C2212yes—
65#1987C2212yes—
66#1987C2212yes—
67#1987C2212yes—
68#1982C2212yes—
69#1982C2211yescharacteristic
70#1948C2212yes—
71#1948C2211yescharacteristic
72#1947C2212yes—
73#1947C2212yes—
74#1939C2212yes—
75#1926C2212yes—
76#1895C2214yes—
77#1895C2214yes—
78#1894C2214yes—
79#1880C2212yes—
80#1877C2214yes—
81#1862C2214yes—
82#1851C2214yes—
83#1842C2211yescharacteristic
84#1834C2212yes—
85#1809C2214yes—
86#1803C2211yescharacteristic
87#1803C2212yes—
88#1803C2212yes—
89#1802C2214yes—
90#1802C2212yes—
91#1802C2214yes—
92#1797C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C22×#41—22no—
96C22×SD16C2312yes—
97C22×M16C2314yes—
98#547—48no—
99#532—28no—
100#532—44no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×#127C2311yescharacteristic
106C2×#125—21no—
107C2×#124—22no—
108C2×#123—24no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2312yes—
111C2×#121—22no—
112C2×#121—22no—
113C2×#121C2314yes—
114C2×C4×Q8—21no—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—22no—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2312yes—
124C2×C4×D4C2314yes—
125C2×C4×D4C2311yescharacteristic
126C2×#117—22no—
127C22×C4⋊C4—22no—
128C22×C42C2312yes—
129#477—44no—
130#476—44no—
131#460—24no—
132C4×SD16—44no—
133#436—44no—
134C2×#107C2314yes—
135C2×#106C2314yes—
136C2×#106C2312yes—
137#423—44no—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2314yes—
143C2×C4⋊C8C2312yes—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×C22⋊C8C2314yes—
152C2×C4×C8C2312yes—
153#400—22no—
154#399—22no—
155#398—22no—
156#398—22no—
157#397—24no—
158#396—22no—
159#396—22no—
160#395—22no—
161#394—24no—
162#394—24no—
163#393—22no—
164#391—21no—
165#391—22no—
166#391C2312yes—
167#390—21no—
168#390—21no—
169#390—21no—
170#390—21no—
171#390C2312yes—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—22no—
178#389C2312yes—
179#389C2314yes—
180#389C2312yes—
181#388—22no—
182#388—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#385—22no—
187#385—24no—
188#385—24no—
189#384—21no—
190#384—21no—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2312yes—
195#384C2312yes—
196#384C2312yes—
197#382—21no—
198#382—22no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380—22no—
203#380C2312yes—
204#380C2312yes—
205#380C2314yes—
206#380C2312yes—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—22no—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4C2312yes—
224C43—21no—
225C43C2311yescharacteristic
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#336C2314yes—
236#336C2312yes—
237C2×#41—48no—
238C2×#41—48no—
239C2×#41—84no—
240C22×Q8—22no—
241C22×Q8—21no—
242C22×Q8C2411yescharacteristic
243C22×D4—24no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4C2×D411yescharacteristic
247C22×D4C2412yes—
248C23×C4—22no—
249C23×C4—24no—
250C23×C4C2×D412yes—
251C23×C4C2×D412yes—
252C23×C4C2412yes—
253C2×SD16—24no—
254C2×SD16—24no—
255C2×SD16—42no—
256C2×M16—44no—
257C2×M16—44no—
258C2×M16—44no—
259C22×C8C2412yes—
260C22×C8C2414yes—
261C22×C8C2412yes—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266C4⋊Q8—44no—
267C4⋊Q8—82no—
268#127—22no—
269#127#4114yes—
270#125—44no—
271#124—48no—
272#123—416no—
273C22⋊Q8—24no—
274C22⋊Q8—42no—
275C22⋊Q8—48no—
276#121—28no—
277#121—44no—
278#121—48no—
279#121—48no—
280C4×Q8—22no—
281C4×Q8—22no—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8—41no—
286C4×Q8—82no—
287C4×D4—24no—
288C4×D4—28no—
289C4×D4—24no—
290C4×D4—22no—
291C4×D4—24no—
292C4×D4—22no—
293C4×D4—24no—
294C4×D4—44no—
295C4×D4—48no—
296C4×D4—44no—
297C4×D4—44no—
298#117—84no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—48no—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D412yes—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363#107—44no—
364#106—42no—
365#106—44no—
366C4⋊C8—22no—
367C4⋊C8—22no—
368C4⋊C8—24no—
369C4⋊C8—24no—
370C4⋊C8—42no—
371C4⋊C8—44no—
372#103—42no—
373#103—44no—
374#102—24no—
375#102—42no—
376#102—44no—
377#102—42no—
378#102—44no—
379C22⋊C8—44no—
380C4×C8—24no—
381#95—22no—
382#95—22no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95—24no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24—22no—
398C24C22×D412yes—
399#41—832no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8—44no—
409C2×Q8—81no—
410C2×Q8—81no—
411C2×Q82−1+412yes—
412C2×D4—22no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—48no—
425C2×D4—48no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—84no—
429C2×D4C2×#4114yes—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—42no—
450C22×C4—48no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—48no—
454C22×C4—44no—
455C22×C4—84no—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C2511yescharacteristic
466SD16—48no—
467M16—88no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—24no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—48no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4—81no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4AGL(1,ℤ/8)14yes—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—48no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C42—24no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—21no—
535C42—21no—
536C42—41no—
537C42—42no—
538C42—42no—
539C42—42no—
540C42—42no—
541C42—42no—
542C42—44no—
543C42—81no—
544C42#13714yes—
545C42C2×#4112yes—
546C42H(5,𝔽2)12yes—
547C42H(5,𝔽2)12yes—
548C422−1+412yes—
549C23—22no—
550C23—22no—
551C23—24no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23C23×D411yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—41no—
567Q8—41no—
568Q8—88no—
569Q8—84no—
570D4—24no—
571D4—24no—
572D4—48no—
573D4—48no—
574D4—44no—
575D4—42no—
576D4—42no—
577D4—816no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—48no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—48no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—44no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—816no—
626C2×C4—88no—
627C2×C4#53712yes—
628C2×C4#53712yes—
629C2×C4#54612yes—
630C2×C4#54612yes—
631C2×C4C2×AGL(1,ℤ/8)14yes—
632C2×C4C2×AGL(1,ℤ/8)12yes—
633C2×C4C2×#13712yes—
634C2×C4C2×#13712yes—
635C2×C4C2×#13712yes—
636C2×C4#57712yes—
637C2×C4C22×#4111yescharacteristic
638C2×C4C2×H(5,𝔽2)11yescharacteristic
639C2×C4C2×H(5,𝔽2)11yescharacteristic
640C2×C4C2×2−1+411yescharacteristic
641C2×C4C2×2−1+411yescharacteristic
642C2×C4#58511yescharacteristic
643C8—42no—
644C8—44no—
645C8—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—44no—
651C22—42no—
652C22—44no—
653C22—42no—
654C22—42no—
655C22—44no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22C2×#53711yescharacteristic
661C22C2×#54611yescharacteristic
662C22#348911yescharacteristic
663C22C22×AGL(1,ℤ/8)11yescharacteristic
664C22C22×#13711yescharacteristic
665C22C2×#57711yescharacteristic
666C22#358511yescharacteristic
667C4—22no—
668C4—22no—
669C4—22no—
670C4—24no—
671C4—41no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—81no—
676C4—82no—
677C4—81no—
678C4—82no—
679C4—81no—
680C4—84no—
681C4—82no—
682C4#325414yes—
683C4#333114yes—
684C4#351511yescharacteristic
685C4#352011yescharacteristic
686C4#354611yescharacteristic
687C4#355211yescharacteristic
688C2—82no—
689C2—84no—
690C2#3890811yescharacteristic
691C2C2×#325411yescharacteristic
692C2#6092911yescharacteristic
693C2C2×#333111yescharacteristic
694C2#6115511yescharacteristic
695C2#6211411yescharacteristic
696C2#6215011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36127C2
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
2. #34941C2semidirect
3. #33198C2semidirect
4. #31626C2semidirect
5. #31566C2semidirect
6. #29938C2semidirect
7. #29218C2semidirect
8. #28280C2semidirect
9. #28150C2semidirect
10. #26106C2semidirect
11. #25346C2semidirect
12. #24846C2semidirect
13. #24815C2semidirect
14. #24156C2semidirect
15. #22650C2semidirect
16. #21893C2semidirect
17. C4×#460C2semidirect

Maximal subgroups

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

subgroupindex
#361272
#349412
#331982
#316262
#315662
#299382
#292182
#282802
#281502
#261062
#253462
#248462
#248152
#241562
#226502
#218932
C4×#4602