Finite Group Databasebeta

← #6992710

Subgroups — #6992710

Order 512.

1635 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36445C211yesmaximal, characteristic
2#35280C211yesmaximal, characteristic
3#32762C212yesmaximal, complemented
4#31388C214yesmaximal, complemented
5#30707C214yesmaximal, complemented
6#30187C212yesmaximal, complemented
7#30169C212yesmaximal, complemented
8#29664C211yesmaximal, characteristic, complemented
9#28575C212yesmaximal, complemented
10#28142C212yesmaximal, complemented
11#27972C211yesmaximal, characteristic, complemented
12#27648C211yesmaximal, characteristic, complemented
13#25386C212yesmaximal, complemented
14#24996C212yesmaximal, complemented
15#24191C212yesmaximal, complemented
16#23571C211yesmaximal, characteristic, complemented
17#22469C211yesmaximal, characteristic
18C2×#557C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#484C2212yes—
22C2×#467C2214yes—
23C2×#462C2212yes—
24C2×C4×Q16C2212yes—
25C2×C8×D4C2212yes—
26#2857C2212yes—
27#2857C2211yescharacteristic
28#2857C2212yes—
29#2857C2212yes—
30#2629—22no—
31#2582—21no—
32#2582C2212yes—
33#2542C2211yescharacteristic
34#2542C2212yes—
35#2538—21no—
36#2538C2212yes—
37#2538C2212yes—
38#2538C2211yescharacteristic
39#2454—22no—
40#2454—21no—
41#2441—21no—
42#2441—22no—
43#2417—22no—
44#2417C2214yes—
45Q8×C4⋊C4—21no—
46Q8×C4⋊C4C2211yescharacteristic
47#2341—21no—
48#2316—21no—
49#2311—21no—
50C4×C22⋊Q8C2211yescharacteristic
51C4×C22⋊Q8C2212yes—
52C2×#384C2211yescharacteristic
53#2100C2214yes—
54#2083C2212yes—
55#2076C2214yes—
56#2072C2212yes—
57#2072C2212yes—
58#2072C2212yes—
59#2072C2214yes—
60#2057C2211yescharacteristic
61#2043C2211yescharacteristic
62#2043C2214yes—
63#2036C2212yes—
64#2036C2214yes—
65#2024C2211yescharacteristic
66#2024C2212yes—
67#2016C2211yescharacteristic
68#2016C2214yes—
69#2016C2212yes—
70#2001C2212yes—
71#2001C2214yes—
72#2001C2214yes—
73#2001C2214yes—
74#1984C2212yes—
75#1956C2212yes—
76#1947C2212yes—
77#1928C2214yes—
78#1921C2214yes—
79#1892C2214yes—
80#1892C2214yes—
81#1892C2214yes—
82#1878C2214yes—
83#1872C2214yes—
84#1869C2214yes—
85#1863C2214yes—
86#1834C2212yes—
87#1824C2212yes—
88#1817C2212yes—
89#1803C2212yes—
90#1803C2212yes—
91#1797C2211yescharacteristic
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C2×#336C2212yes—
95C22×#41D411yescharacteristic
96C22×Q16C2314yes—
97C22×Q16C2312yes—
98C23×C8C2314yes—
99#557—44no—
100#533—44no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×#125—21no—
106C2×#125—22no—
107C2×#125—21no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2311yescharacteristic
111C2×C22⋊Q8C2312yes—
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2312yes—
114C2×C22⋊Q8C2312yes—
115C2×C22⋊Q8C2314yes—
116C2×C22⋊Q8C2312yes—
117C4×#41—44no—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8D411yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2311yescharacteristic
126C2×C4×D4D412yes—
127C2×C4×D4D411yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2312yes—
130C2×#117—21no—
131C2×#117D411yescharacteristic
132C2×#117D412yes—
133C22×C4⋊C4—21no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138#484—44no—
139#467—48no—
140#462—44no—
141C4×Q16—44no—
142C8×D4—44no—
143C2×#107C2314yes—
144C2×#107C2312yes—
145C2×C4⋊C8C2312yes—
146C2×#103C2312yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2312yes—
158C2×C4×C8C2312yes—
159#399—22no—
160#399—22no—
161#399—22no—
162#398—24no—
163#398—24no—
164#396—24no—
165#396—24no—
166#396—22no—
167#396—22no—
168#396—22no—
169#391—22no—
170#391—22no—
171#391C2314yes—
172#391C2311yescharacteristic
173#391C2312yes—
174#390—22no—
175#390—21no—
176#390—22no—
177#390—21no—
178#390—22no—
179#389—22no—
180#389—22no—
181#389—21no—
182#389—22no—
183#389—21no—
184#389—22no—
185#389—22no—
186#389—22no—
187#389—22no—
188#389—21no—
189#389—22no—
190#389—21no—
191#389—22no—
192#389C2311yescharacteristic
193#389C2314yes—
194#387—21no—
195#387—22no—
196#386—24no—
197#385—21no—
198#385—22no—
199#384—21no—
200#384—22no—
201#384—21no—
202#384—21no—
203#384C2311yescharacteristic
204#384C2312yes—
205#384C2312yes—
206#384C2312yes—
207#380—22no—
208#380—22no—
209#380—21no—
210#380—22no—
211#380—22no—
212#380—21no—
213#380—22no—
214#380—24no—
215#380C2312yes—
216#380C2314yes—
217#380C2311yescharacteristic
218C4×C4⋊C4—21no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C22⋊C4—21no—
223C4×C22⋊C4—22no—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234#336C2314yes—
235#336C2314yes—
236C2×#41—42no—
237C2×#41—42no—
238C2×#41Q1618yescomplemented
239C22×Q8—21no—
240C22×Q8C2×D411yescharacteristic
241C22×Q8C2411yescharacteristic
242C22×Q8C2412yes—
243C22×Q8C2412yes—
244C22×D4C2×D411yescharacteristic
245C22×D4C2×D412yes—
246C23×C4—22no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2411yescharacteristic
252C23×C4C2412yes—
253C2×Q16—42no—
254C2×Q16—42no—
255C2×Q16—42no—
256C2×Q16—44no—
257C2×Q16—44no—
258C2×Q16—44no—
259C22×C8—44no—
260C22×C8—44no—
261C22×C8—44no—
262C22×C8C2412yes—
263C22×C8C2414yes—
264C22×C8C2412yes—
265C4⋊Q8—24no—
266C4⋊Q8—42no—
267C4⋊Q8—42no—
268C4⋊Q8—44no—
269#125—44no—
270#125—48no—
271#125—44no—
272C22⋊Q8—24no—
273C22⋊Q8—22no—
274C22⋊Q8—41no—
275C22⋊Q8—42no—
276C22⋊Q8—42no—
277C22⋊Q8—42no—
278C22⋊Q8—44no—
279C22⋊Q8—44no—
280C22⋊Q8—44no—
281C22⋊Q8—44no—
282C22⋊Q8—44no—
283C22⋊Q8—44no—
284C22⋊Q8—84no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—42no—
290C4×Q8—44no—
291C4×Q8—44no—
292C4×Q8—44no—
293C4×Q8—44no—
294C4×D4—24no—
295C4×D4—42no—
296C4×D4—44no—
297C4×D4—44no—
298#117—44no—
299#117—42no—
300#117—44no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—42no—
330C2×C4⋊C4—42no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4—82no—
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2414yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2411yescharacteristic
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C42—24no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—21no—
360C2×C42—22no—
361C2×C42—21no—
362C2×C42—44no—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D412yes—
366C2×C42C2×D412yes—
367C2×C42C2412yes—
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370#107—28no—
371#107—42no—
372C4⋊C8—24no—
373#103—24no—
374#103—44no—
375#103—42no—
376#103—44no—
377#103—42no—
378#103—42no—
379#103—44no—
380#103—44no—
381#103—44no—
382C22⋊C8—42no—
383C22⋊C8—44no—
384C22⋊C8—42no—
385C4×C8—24no—
386#95—21no—
387#95—21no—
388#95—22no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—21no—
393#95—22no—
394#95—21no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95—24no—
399C24C22×D412yes—
400C24C22×D411yescharacteristic
401#41—416no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—24no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—24no—
416C2×Q8—21no—
417C2×Q8—22no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8C2×Q1614yes—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4C2×Q1614yes—
427C2×D4C2×Q1618yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—42no—
451C22×C4—41no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4C2×Q1614yes—
459C22×C4C2×Q1618yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C2511yescharacteristic
469Q16—48no—
470Q16—88no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—88no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—81no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—84no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—84no—
521C4⋊C4H(5,𝔽2)12yes—
522C22⋊C4—24no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—44no—
526C22⋊C4—42no—
527C22⋊C4—44no—
528C22⋊C4—81no—
529C22⋊C4—82no—
530C22⋊C4—84no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4—84no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—24no—
539C42—42no—
540C42—42no—
541C42—44no—
542C42—44no—
543C42—44no—
544C42—42no—
545C42—42no—
546C42H(5,𝔽2)14yes—
547C23—21no—
548C23—22no—
549C23—22no—
550C23—21no—
551C23—22no—
552C23—21no—
553C23—21no—
554C23—22no—
555C23—21no—
556C23—22no—
557C23—22no—
558C23—21no—
559C23C22×Q1612yes—
560C23C22×Q1614yes—
561C23C23×D411yescharacteristic
562Q8—28no—
563Q8—42no—
564Q8—42no—
565Q8—44no—
566Q8—42no—
567Q8—42no—
568Q8—44no—
569Q8—42no—
570Q8—42no—
571Q8—84no—
572D4—28no—
573D4—48no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—28no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—48no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—41no—
625C2×C4—84no—
626C2×C4—164no—
627C2×C4#53412yes—
628C2×C4#53412yes—
629C2×C4C22×Q1612yes—
630C2×C4C22×Q1614yes—
631C2×C4C22×Q1612yes—
632C2×C4#57512yes—
633C2×C4#57712yes—
634C2×C4C2×H(5,𝔽2)12yes—
635C2×C4C2×H(5,𝔽2)11yescharacteristic
636C2×C4#58511yescharacteristic
637C2×C4#58512yes—
638C8—42no—
639C8—42no—
640C8—44no—
641C22—24no—
642C22—22no—
643C22—22no—
644C22—24no—
645C22—24no—
646C22—22no—
647C22—41no—
648C22—42no—
649C22—41no—
650C22—42no—
651C22—41no—
652C22—42no—
653C22—42no—
654C22—41no—
655C22C2×#53411yescharacteristic
656C22#348811yescharacteristic
657C22#350011yescharacteristic
658C22C23×Q1611yescharacteristic
659C22C2×#57511yescharacteristic
660C22C2×#57711yescharacteristic
661C22#358611yescharacteristic
662C4—22no—
663C4—24no—
664C4—22no—
665C4—22no—
666C4—42no—
667C4—42no—
668C4—41no—
669C4—42no—
670C4—84no—
671C4—81no—
672C4—82no—
673C4—82no—
674C4—81no—
675C4—82no—
676C4—81no—
677C4—82no—
678C4#318914yes—
679C4#353912yes—
680C4#354612yes—
681C2—42no—
682C2—44no—
683C2#4298511yescharacteristic
684C2C2×#318911yescharacteristic
685C2#6076311yescharacteristic
686C2#6195211yescharacteristic
687C2#6198611yescharacteristic
688C2#6253011yescharacteristic
689C2#6258711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32762C2semidirect
4. #31388C2semidirect
5. #30707C2semidirect
6. #30187C2semidirect
7. #30169C2semidirect
8. #29664C2semidirect
9. #28575C2semidirect
10. #28142C2semidirect
11. #27972C2semidirect
12. #27648C2semidirect
13. #25386C2semidirect
14. #24996C2semidirect
15. #24191C2semidirect
16. #23571C2semidirect
238. C2×#41Q16semidirect

Maximal subgroups

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

subgroupindex
#364452
#352802
#327622
#313882
#307072
#301872
#301692
#296642
#285752
#281422
#279722
#276482
#253862
#249962
#241912
#235712
#224692