Finite Group Databasebeta

← #7534969

Subgroups — #7534969

Order 512.

1993 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#59815C211yesmaximal, characteristic
2#36594C211yesmaximal, characteristic, complemented
3#31269C214yesmaximal, complemented
4#29141C211yesmaximal, characteristic, complemented
5#29139C211yesmaximal, characteristic, complemented
6#28142C212yesmaximal, complemented
7#28141C212yesmaximal, complemented
8#24972C212yesmaximal, complemented
9#24968C212yesmaximal, complemented
10#24242C212yesmaximal, complemented
11#24241C212yesmaximal, complemented
12#23968C214yesmaximal, complemented
13#23173C211yesmaximal, characteristic, complemented
14#22418C212yesmaximal, complemented
15#22417C212yesmaximal, complemented
16C4×#462C211yesmaximal, characteristic, complemented
17C4×#460C211yesmaximal, characteristic, complemented
18C2×#556C2212yes—
19C2×#550C2212yes—
20C2×#547C2212yes—
21C2×#462C2211yescharacteristic
22C2×#460C2211yescharacteristic
23C2×#452C2214yes—
24C2×C4×Q16C2212yes—
25C2×C4×SD16C2212yes—
26C2×C8×D4C2212yes—
27C2×C8×D4C2212yes—
28C2×#416C2214yes—
29#2868C2211yescharacteristic
30#2858—21no—
31#2682—22no—
32#2547C2212yes—
33#2545—22no—
34#2541—21no—
35#2538C2212yes—
36#2538C2212yes—
37#2538C2212yes—
38#2537C2212yes—
39#2512—22no—
40#2512—22no—
41#2445—22no—
42#2416—22no—
43#2407C2214yes—
44#2354C2212yes—
45#2354C2212yes—
46D4×C4⋊C4—21no—
47C4×C4⋊Q8—21no—
48C4×C4⋊Q8C2211yescharacteristic
49C4×#127C2211yescharacteristic
50C4×#127C2211yescharacteristic
51C42×Q8C2211yescharacteristic
52#2077C2214yes—
53#2076C2214yes—
54#2035C2214yes—
55#2033C2214yes—
56#2024C2214yes—
57#2023C2214yes—
58#2001C2214yes—
59#2000C2214yes—
60#1982C2211yescharacteristic
61#1981C2211yescharacteristic
62#1957C2212yes—
63#1956C2212yes—
64#1954C2212yes—
65#1954C2212yes—
66#1948C2211yescharacteristic
67#1947C2211yescharacteristic
68#1928C2212yes—
69#1926C2212yes—
70#1893C2214yes—
71#1892C2214yes—
72#1892C2214yes—
73#1880C2212yes—
74#1878C2212yes—
75#1875C2214yes—
76#1873C2214yes—
77#1869C2214yes—
78#1834C2212yes—
79#1834C2212yes—
80#1825C2214yes—
81#1803C2212yes—
82#1803C2212yes—
83#1803C2211yescharacteristic
84#1803C2211yescharacteristic
85#1802C2212yes—
86#1802C2212yes—
87#1788C2214yes—
88#1788C2214yes—
89C4×C4⋊C8C2211yescharacteristic
90C4×C4⋊C8C2211yescharacteristic
91C4×#103C2212yes—
92C4×#102C2212yes—
93C22×#41C2314yes—
94C22×Q16C2312yes—
95C22×SD16C2312yes—
96C23×C8C2314yes—
97#556—44no—
98#556—44no—
99#550—28no—
100#550—44no—
101#547—28no—
102#547—44no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×#127—22no—
108C2×#127D411yescharacteristic
109C2×#127C2311yescharacteristic
110C2×#126—24no—
111C2×#124—22no—
112C2×#124—22no—
113C2×#123—24no—
114C2×C22⋊Q8C2314yes—
115C2×C22⋊Q8C2314yes—
116C2×#121—22no—
117C2×#121—22no—
118C2×#121—24no—
119C2×#121C2314yes—
120C2×C4×Q8—22no—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2311yescharacteristic
125C2×C4×D4—22no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4C2312yes—
129C2×C4×D4C2312yes—
130C2×C4×D4C2312yes—
131C2×C4×D4C2312yes—
132C2×#117C2314yes—
133C22×C4⋊C4—22no—
134#462—24no—
135#460—24no—
136#452—48no—
137C4×Q16—44no—
138C4×SD16—44no—
139C8×D4—44no—
140C8×D4—44no—
141C2×#107C2312yes—
142C2×#106C2312yes—
143C2×C4⋊C8C2311yescharacteristic
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2311yescharacteristic
146C2×C4⋊C8C2311yescharacteristic
147#416—48no—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#102C2312yes—
156C2×#102C2312yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2312yes—
160C2×C22⋊C8C2314yes—
161C2×C22⋊C8C2314yes—
162C2×C22⋊C8C2314yes—
163C2×C4×C8C2312yes—
164C2×C4×C8C2312yes—
165#397—24no—
166#396—24no—
167#396—24no—
168#396—24no—
169#394—24no—
170#394—24no—
171#392—24no—
172#391—22no—
173#391C2312yes—
174#390—22no—
175#390—22no—
176#390—22no—
177#390—22no—
178#390—22no—
179#390—21no—
180#390—21no—
181#390D412yes—
182#390D412yes—
183#390D412yes—
184#390C2312yes—
185#389—22no—
186#389—22no—
187#389—22no—
188#389—21no—
189#389—21no—
190#388—24no—
191#388—24no—
192#386—24no—
193#385—24no—
194#384—21no—
195#384—21no—
196#384D411yescharacteristic
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#384C2312yes—
200#384C2312yes—
201#380—22no—
202#380C2314yes—
203#380C2314yes—
204#380C2314yes—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C43—21no—
216C43C2311yescharacteristic
217#340C2314yes—
218#340C2314yes—
219#340C2312yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#336C2312yes—
226#336C2312yes—
227C2×#41—28no—
228C2×#41—28no—
229C2×#41—44no—
230C2×#41—44no—
231C2×#41—44no—
232C2×#41—44no—
233C22×Q8—22no—
234C22×Q8C2412yes—
235C22×Q8C2411yescharacteristic
236C22×D4—24no—
237C22×D4—24no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C2×D412yes—
241C22×D4C2×D412yes—
242C22×D4C2×D412yes—
243C22×D4C2412yes—
244C23×C4—22no—
245C23×C4—24no—
246C23×C4C2414yes—
247C2×Q16—24no—
248C2×Q16—24no—
249C2×Q16—42no—
250C2×SD16—24no—
251C2×SD16—24no—
252C2×SD16—42no—
253C22×C8—44no—
254C22×C8—44no—
255C22×C8—44no—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C4⋊Q8—22no—
261C4⋊Q8—42no—
262C4⋊Q8—42no—
263C4⋊Q8—44no—
264C4⋊Q8—82no—
265C4⋊Q8#4114yes—
266#127—24no—
267#127—22no—
268#127—48no—
269#127#4114yes—
270#126—416no—
271#124—48no—
272#124—84no—
273#123—416no—
274C22⋊Q8—28no—
275C22⋊Q8—28no—
276C22⋊Q8—44no—
277C22⋊Q8—44no—
278#121—28no—
279#121—416no—
280#121—44no—
281#121—48no—
282#121—84no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8—24no—
286C4×Q8—22no—
287C4×Q8—22no—
288C4×Q8—24no—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×Q8—48no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—42no—
299C4×D4—42no—
300C4×D4—48no—
301C4×D4—44no—
302C4×D4—82no—
303#117—28no—
304#117—44no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—48no—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4C2×D414yes—
343C2×C22⋊C4C2×D414yes—
344C2×C22⋊C4C2×D414yes—
345C2×C42—21no—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360#107—42no—
361#106—42no—
362C4⋊C8—22no—
363C4⋊C8—22no—
364C4⋊C8—22no—
365C4⋊C8—22no—
366#103—24no—
367#103—24no—
368#103—44no—
369#103—42no—
370#103—42no—
371#103—42no—
372#103—44no—
373#102—24no—
374#102—24no—
375#102—44no—
376#102—42no—
377#102—44no—
378C22⋊C8—44no—
379C22⋊C8—44no—
380C22⋊C8—44no—
381C4×C8—24no—
382C4×C8—24no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95—24no—
387#95—24no—
388#95—22no—
389#95—22no—
390#95—24no—
391C24—22no—
392C24C22×D414yes—
393#41—416no—
394#41—88no—
395#41—88no—
396#41—88no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8C2×#4114yes—
411C2×Q8H(5,𝔽2)12yes—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—48no—
429C2×D4—48no—
430C2×D4—48no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—81no—
434C2×D4—81no—
435C2×D4—84no—
436C2×D4—84no—
437C2×D4—84no—
438C2×D4C2×#4114yes—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—84no—
459C22×C4C22×D414yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C2511yescharacteristic
467Q16—48no—
468SD16—48no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—88no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—42no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—81no—
510C4⋊C4—81no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4H(5,𝔽2)12yes—
514C22⋊C4—28no—
515C22⋊C4—28no—
516C22⋊C4—44no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—82no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C42—24no—
528C42—22no—
529C42—22no—
530C42—21no—
531C42—21no—
532C42—41no—
533C42—48no—
534C42—42no—
535C42—44no—
536C42—42no—
537C42—42no—
538C42—42no—
539C42—42no—
540C42#13514yes—
541C42C2×#4112yes—
542C42C2×#4112yes—
543C42H(5,𝔽2)12yes—
544C42H(5,𝔽2)12yes—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23C23×D411yescharacteristic
560Q8—24no—
561Q8—22no—
562Q8—22no—
563Q8—24no—
564Q8—41no—
565Q8—42no—
566Q8—41no—
567Q8—42no—
568Q8—88no—
569D4—24no—
570D4—24no—
571D4—42no—
572D4—42no—
573D4—48no—
574D4—48no—
575D4—48no—
576D4—84no—
577D4—84no—
578D4—816no—
579D4—168no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—816no—
621C2×C4—88no—
622C2×C4#54612yes—
623C2×C4#54612yes—
624C2×C4#54612yes—
625C2×C4#54612yes—
626C2×C4C2×#13512yes—
627C2×C4C2×#13512yes—
628C2×C4C2×#13512yes—
629C2×C4#57712yes—
630C2×C4#57812yes—
631C2×C4C22×#4111yescharacteristic
632C2×C4C22×#4111yescharacteristic
633C2×C4C2×H(5,𝔽2)11yescharacteristic
634C2×C4C2×H(5,𝔽2)11yescharacteristic
635C2×C4C2×H(5,𝔽2)11yescharacteristic
636C2×C4C2×H(5,𝔽2)11yescharacteristic
637C8—42no—
638C8—42no—
639C8—42no—
640C8—42no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22C2×#54611yescharacteristic
656C22C2×#54611yescharacteristic
657C22#348411yescharacteristic
658C22C22×#13511yescharacteristic
659C22C2×#57711yescharacteristic
660C22C2×#57811yescharacteristic
661C22#358511yescharacteristic
662C4—22no—
663C4—22no—
664C4—22no—
665C4—44no—
666C4—44no—
667C4—42no—
668C4—41no—
669C4—42no—
670C4—81no—
671C4—82no—
672C4—81no—
673C4—84no—
674C4—82no—
675C4#330214yes—
676C4#330414yes—
677C4#348111yescharacteristic
678C4#351911yescharacteristic
679C4#351911yescharacteristic
680C4#354911yescharacteristic
681C2—84no—
682C2—84no—
683C2#5904111yescharacteristic
684C2C2×#330211yescharacteristic
685C2C2×#330411yescharacteristic
686C2#6120011yescharacteristic
687C2#6120311yescharacteristic
688C2#6156611yescharacteristic
689C2#6166711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#59815C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36594C2semidirect
3. #31269C2semidirect
4. #29141C2semidirect
5. #29139C2semidirect
6. #28142C2semidirect
7. #28141C2semidirect
8. #24972C2semidirect
9. #24968C2semidirect
10. #24242C2semidirect
11. #24241C2semidirect
12. #23968C2semidirect
13. #23173C2semidirect
14. #22418C2semidirect
15. #22417C2semidirect
16. C4×#462C2semidirect
17. C4×#460C2semidirect

Maximal subgroups

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

subgroupindex
#598152
#365942
#312692
#291412
#291392
#281422
#281412
#249722
#249682
#242422
#242412
#239682
#231732
#224182
#224172
C4×#4622
C4×#4602