Finite Group Databasebeta

← #7541806

Subgroups — #7541806

Order 512.

2005 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3341C212yesmaximal, complemented
2#59822C211yesmaximal, characteristic, complemented
3#36187C211yesmaximal, characteristic, complemented
4#30566C214yesmaximal, complemented
5#28689C211yesmaximal, characteristic, complemented
6#28509C211yesmaximal, characteristic, complemented
7#26235C212yesmaximal, complemented
8#26141C214yesmaximal, complemented
9#25704C212yesmaximal, complemented
10#25384C212yesmaximal, complemented
11#25339C212yesmaximal, complemented
12#25013C212yesmaximal, complemented
13#24117C212yesmaximal, complemented
14#23173C211yesmaximal, characteristic, complemented
15#22657C212yesmaximal, complemented
16C4×#476C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic, complemented
18C2×#555C2212yes—
19C2×#555C2212yes—
20#3341—216no—
21C2×#500C2212yes—
22C2×#482C2214yes—
23C2×#482C2214yes—
24C2×#476C2212yes—
25C2×#476C2212yes—
26C2×#476C2211yescharacteristic
27C2×#466C2214yes—
28C2×#459C2211yescharacteristic
29C2×C4×D8C2212yes—
30C2×C8×D4C2212yes—
31C2×C8×D4C2212yes—
32C2×#427C2214yes—
33#2868C2211yescharacteristic
34#2856—21no—
35#2733—22no—
36#2563C2212yes—
37#2548—22no—
38#2541—21no—
39#2541C2212yes—
40#2537C2212yes—
41#2537C2212yes—
42#2537C2212yes—
43#2500—22no—
44#2491—22no—
45#2491—22no—
46#2414—22no—
47#2414C2214yes—
48#2354—21no—
49#2354C2212yes—
50#2354C2212yes—
51#2354C2212yes—
52C4×C4⋊Q8C2211yescharacteristic
53C4×#127C2211yescharacteristic
54C4×#127C2211yescharacteristic
55#2285—21no—
56C42×D4C2211yescharacteristic
57#2094C2214yes—
58#2083C2214yes—
59#2077C2214yes—
60#2034C2214yes—
61#2034C2214yes—
62#2033C2214yes—
63#2033C2214yes—
64#1999C2214yes—
65#1990C2212yes—
66#1987C2211yescharacteristic
67#1982C2211yescharacteristic
68#1957C2212yes—
69#1954C2212yes—
70#1954C2212yes—
71#1948C2211yescharacteristic
72#1947C2212yes—
73#1947C2212yes—
74#1947C2211yescharacteristic
75#1925C2212yes—
76#1920C2212yes—
77#1893C2214yes—
78#1879C2212yes—
79#1875C2214yes—
80#1869C2214yes—
81#1864C2212yes—
82#1854C2214yes—
83#1854C2214yes—
84#1854C2214yes—
85#1844C2212yes—
86#1807C2214yes—
87#1802C2211yescharacteristic
88#1802C2211yescharacteristic
89C4×#106C2212yes—
90C4×C4⋊C8C2211yescharacteristic
91C4×C4⋊C8C2211yescharacteristic
92C4×#102C2212yes—
93C4×#102C2212yes—
94C22×D8C2312yes—
95C23×C8C2314yes—
96#555—28no—
97#555—28no—
98#555—44no—
99#555—44no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127—22no—
105C2×#127D411yescharacteristic
106C2×#127C2311yescharacteristic
107C2×#124—24no—
108C2×#124—24no—
109C2×#121—22no—
110C2×#121—22no—
111C2×#121—22no—
112C2×#121—22no—
113C2×#121—22no—
114C2×#121C2314yes—
115C2×#121C2314yes—
116C2×#121C2314yes—
117C2×C4×Q8—21no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4C2312yes—
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2312yes—
125C2×C4×D4C2312yes—
126C2×C4×D4C2312yes—
127C2×#117—22no—
128C2×#117—22no—
129C2×#117C2314yes—
130C2×#117C2314yes—
131C22×C42—21no—
132#500—28no—
133#482—216no—
134#482—216no—
135#476—28no—
136#476—44no—
137#476D418yescomplemented
138#466—216no—
139#459—24no—
140C4×D8—28no—
141C8×D4—28no—
142C8×D4—28no—
143#427—216no—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#107C2312yes—
147C2×#106C2314yes—
148C2×#106C2312yes—
149C2×#106C2314yes—
150C2×#106C2312yes—
151C2×C4⋊C8C2311yescharacteristic
152C2×C4⋊C8C2311yescharacteristic
153C2×C4⋊C8C2311yescharacteristic
154C2×C4⋊C8C2311yescharacteristic
155C2×#102C2312yes—
156C2×#102C2312yes—
157C2×#102C2314yes—
158C2×#102C2312yes—
159C2×#102C2312yes—
160C2×#102C2314yes—
161C2×#102C2312yes—
162C2×#102C2312yes—
163C2×C22⋊C8C2314yes—
164C2×C22⋊C8C2314yes—
165C2×C22⋊C8C2314yes—
166C2×C4×C8C2312yes—
167C2×C4×C8C2312yes—
168#394—24no—
169#394—24no—
170#394—24no—
171#394—24no—
172#393—24no—
173#392—24no—
174#392—24no—
175#391—21no—
176#390—22no—
177#390—22no—
178#390—22no—
179#390—22no—
180#390—22no—
181#390—24no—
182#390—21no—
183#390—21no—
184#390D412yes—
185#390D412yes—
186#390D412yes—
187#390C2312yes—
188#389—21no—
189#389C2312yes—
190#388—24no—
191#388—24no—
192#388—24no—
193#388—24no—
194#384—21no—
195#384—21no—
196#384D411yescharacteristic
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#384C2312yes—
200#384C2312yes—
201#382—22no—
202#380—22no—
203#380—22no—
204#380—22no—
205#380C2314yes—
206#380C2314yes—
207#380C2314yes—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—21no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219C43C2311yescharacteristic
220#340C2312yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#336C2312yes—
229#336C2312yes—
230C22×Q8—21no—
231C22×Q8—22no—
232C22×D4—24no—
233C22×D4—24no—
234C22×D4—24no—
235C22×D4—21no—
236C22×D4—21no—
237C22×D4C2×D412yes—
238C22×D4C2×D412yes—
239C22×D4C2×D412yes—
240C22×D4C2412yes—
241C22×D4C2411yescharacteristic
242C23×C4—21no—
243C23×C4—22no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4C2414yes—
247C2×D8—24no—
248C2×D8—24no—
249C2×D8—42no—
250C22×C8—28no—
251C22×C8—28no—
252C22×C8—44no—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C4⋊Q8—24no—
258C4⋊Q8—22no—
259C4⋊Q8—24no—
260C4⋊Q8—82no—
261C4⋊Q8—82no—
262C4⋊Q8C2×D414yes—
263#127—24no—
264#127—22no—
265#127—22no—
266#127—84no—
267#124—416no—
268#124—416no—
269#121—28no—
270#121—28no—
271#121—44no—
272#121—44no—
273#121—48no—
274#121—48no—
275#121—44no—
276#121—44no—
277#121—84no—
278#121—84no—
279#121—84no—
280C4×Q8—82no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—22no—
290C4×D4—41no—
291C4×D4—41no—
292C4×D4—42no—
293C4×D4—42no—
294C4×D4—44no—
295C4×D4—82no—
296C4×D4C2×D414yes—
297#117—28no—
298#117—28no—
299#117—44no—
300#117—44no—
301#117—48no—
302#117—48no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4C2×D414yes—
339C2×C22⋊C4C2×D414yes—
340C2×C22⋊C4C2×D414yes—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—44no—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#107—24no—
356#107—28no—
357#107—28no—
358#106—28no—
359#106—42no—
360#106—44no—
361#106C2×D418yes—
362C4⋊C8—22no—
363C4⋊C8—22no—
364C4⋊C8C2×D414yes—
365C4⋊C8#4114yes—
366#102—24no—
367#102—28no—
368#102—24no—
369#102—24no—
370#102—42no—
371#102—42no—
372#102—44no—
373#102C2×D418yes—
374C22⋊C8—28no—
375C22⋊C8—28no—
376C22⋊C8—44no—
377C4×C8—24no—
378C4×C8—24no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—22no—
386#95—22no—
387C24—21no—
388C24—22no—
389C24C22×D414yes—
390C2×Q8—24no—
391C2×Q8—44no—
392C2×Q8—81no—
393C2×Q8—81no—
394C2×Q8—82no—
395C2×Q8—82no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—21no—
409C2×D4—22no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—81no—
423C2×D4—81no—
424C2×D4—84no—
425C2×D4—84no—
426C2×D4—84no—
427C2×D4—84no—
428C2×D4—84no—
429C2×D4—84no—
430C2×D4—84no—
431C2×D4C22×D412yes—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4C22×D414yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C2511yescharacteristic
465D8—48no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—416no—
477C2×C8C22×D414yes—
478C2×C8C2×#4114yes—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—81no—
501C4⋊C4—81no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4C22×D412yes—
506C4⋊C4C22×D414yes—
507C22⋊C4—24no—
508C22⋊C4—28no—
509C22⋊C4—28no—
510C22⋊C4—44no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—42no—
516C22⋊C4—44no—
517C22⋊C4—42no—
518C22⋊C4—44no—
519C22⋊C4—82no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C42—22no—
525C42—22no—
526C42—21no—
527C42—21no—
528C42—41no—
529C42—48no—
530C42—48no—
531C42—42no—
532C42—42no—
533C42—42no—
534C42—44no—
535C42#13514yes—
536C42C22×D412yes—
537C42C2×#4112yes—
538C42H(5,𝔽2)12yes—
539C422−1+412yes—
540C23—24no—
541C23—24no—
542C23—22no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23C23×D411yescharacteristic
562Q8—84no—
563Q8—164no—
564D4—22no—
565D4—22no—
566D4—48no—
567D4—42no—
568D4—42no—
569D4—41no—
570D4—42no—
571D4—48no—
572D4—41no—
573D4—42no—
574D4—84no—
575D4—84no—
576D4—84no—
577D4—168no—
578D4—168no—
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—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—21no—
591C2×C4—24no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—48no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4—88no—
621C2×C4D4212yes—
622C2×C4D4212yes—
623C2×C4#54712yes—
624C2×C4#54712yes—
625C2×C4C2×#13512yes—
626C2×C4C2×#13512yes—
627C2×C4C2×#13512yes—
628C2×C4#57612yes—
629C2×C4#57712yes—
630C2×C4C23×D411yescharacteristic
631C2×C4C22×#4111yescharacteristic
632C2×C4C2×H(5,𝔽2)11yescharacteristic
633C2×C4C2×2−1+411yescharacteristic
634C2×C4#58511yescharacteristic
635C2×C4#58511yescharacteristic
636C8—24no—
637C8—24no—
638C8—42no—
639C8—42no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—42no—
645C22—44no—
646C22—44no—
647C22—42no—
648C22—44no—
649C22—42no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22C2×D4211yescharacteristic
662C22C2×#54711yescharacteristic
663C22#348511yescharacteristic
664C22C22×#13511yescharacteristic
665C22C2×#57611yescharacteristic
666C22C2×#57711yescharacteristic
667C22#358511yescharacteristic
668C4—22no—
669C4—22no—
670C4—22no—
671C4—41no—
672C4—42no—
673C4—44no—
674C4—44no—
675C4—81no—
676C4—81no—
677C4—82no—
678C4—82no—
679C4—84no—
680C4#328414yes—
681C4#334114yescomplemented
682C4D4×#4111yescharacteristic
683C4#349711yescharacteristic
684C4#351711yescharacteristic
685C4#355411yescharacteristic
686C2—84no—
687C2—82no—
688C2—162no—
689C2#5906611yescharacteristic
690C2C2×#328411yescharacteristic
691C2#6109311yescharacteristic
692C2C2×#334111yescharacteristic
693C2#6127911yescharacteristic
694C2#6170011yescharacteristic
695C2#6178811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
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
1. C2×#3341C2semidirect
2. #59822C2semidirect
3. #36187C2semidirect
4. #30566C2semidirect
5. #28689C2semidirect
6. #28509C2semidirect
7. #26235C2semidirect
8. #26141C2semidirect
9. #25704C2semidirect
10. #25384C2semidirect
11. #25339C2semidirect
12. #25013C2semidirect
13. #24117C2semidirect
14. #23173C2semidirect
15. #22657C2semidirect
16. C4×#476C2semidirect
17. C4×#459C2semidirect
137. #476D4semidirect
681. C4#3341semidirect

Maximal subgroups

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

subgroupindex
C2×#33412
#598222
#361872
#305662
#286892
#285092
#262352
#261412
#257042
#253842
#253392
#250132
#241172
#231732
#226572
C4×#4762
C4×#4592