Finite Group Databasebeta

← #6500732

Subgroups — #6500732

Order 512.

2502 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#34736C211yesmaximal, characteristic
2#34185C211yesmaximal, characteristic, complemented
3#27590C214yesmaximal, complemented
4#24880C214yesmaximal, complemented
5#24878C214yesmaximal, complemented
6#24864C214yesmaximal, complemented
7#24862C214yesmaximal, complemented
8#24007C214yesmaximal, complemented
9#21943C211yesmaximal, characteristic, complemented
10#21939C211yesmaximal, characteristic, complemented
11#21740C211yesmaximal, characteristic, complemented
12#20600C211yesmaximal, characteristic, complemented
13#20597C211yesmaximal, characteristic, complemented
14C23×#41C2211yescharacteristic
15C2×#525C2212yes—
16C22×C4×D4C2211yescharacteristic
17C2×#484C2214yescomplemented
18C2×#481C2214yescomplemented
19C2×#419C2212yescomplemented
20C2×#418C2212yescomplemented
21C2×#416C2212yescomplemented
22C2×#416C2212yescomplemented
23C2×#407C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25#2409C2212yes—
26#2408C2212yes—
27#2407C2214yes—
28#2406—21no—
29#2405—21no—
30#2404—21no—
31#2403—21no—
32#2388—22no—
33#2353—22no—
34#2353—22no—
35#2352C2214yes—
36#2352C2214yescomplemented
37#2313—21no—
38#2312—21no—
39#2308—21no—
40C4×#120—22no—
41#2297C2212yes—
42#2283C2212yescomplemented
43#2280C2211yescharacteristic
44#2280C2211yescharacteristic
45#2280C2211yescharacteristic
46C2×#380C2212yescomplemented
47#2273—21no—
48#2097C2214yescomplemented
49#2092C2214yescomplemented
50#2074C2214yescomplemented
51#2073C2214yescomplemented
52#2035C2214yescomplemented
53#2035C2214yescomplemented
54#2033C2214yescomplemented
55#2033C2214yescomplemented
56#1928C2214yescomplemented
57#1927C2214yescomplemented
58#1926C2214yescomplemented
59#1925C2214yescomplemented
60#1874C2214yescomplemented
61#1872C2214yescomplemented
62#1870C2214yes—
63#1869C2212yescomplemented
64#1869C2212yescomplemented
65#1854C2214yescomplemented
66#1854C2214yescomplemented
67#1854C2214yescomplemented
68#1854C2214yescomplemented
69#1817C2212yescomplemented
70#1809C2214yescomplemented
71#1808C2212yescomplemented
72#1807C2212yescomplemented
73#1788C2214yes—
74#1788C2214yes—
75#1788C2214yescomplemented
76#1787C2212yescomplemented
77#1786C2212yescomplemented
78#1779C2212yescomplemented
79C22×#41—21no—
80C22×#41—21no—
81C22×#41—24no—
82C22×#41—24no—
83C22×#41—44no—
84C22×#41—44no—
85C22×#41C2312yes—
86C22×#41C2312yes—
87C23×Q8C2311yescharacteristic
88C23×D4—21no—
89C23×D4D411yescharacteristic
90C23×D4C2311yescharacteristic
91C24×C4—21no—
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C22×M16C2312yes—
95C23×C8C2311yescharacteristic
96C23×C8C2311yescharacteristic
97#525—44no—
98#525—44no—
99C2×#123—22no—
100C2×C22⋊Q8C2314yes—
101C2×#121C2314yes—
102C2×#120—22no—
103C2×#120—22no—
104C2×C4×D4—24no—
105C2×C4×D4—24no—
106C2×C4×D4—24no—
107C2×C4×D4D412yes—
108C2×C4×D4D412yes—
109C2×C4×D4C2312yes—
110C2×C4×D4C2312yes—
111C2×#117C2312yes—
112C2×#117C2312yes—
113C2×#117C2312yes—
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2314yes—
116C22×C22⋊C4—22no—
117C22×C22⋊C4D411yescharacteristic
118C22×C22⋊C4D411yescharacteristic
119C22×C42C2311yescharacteristic
120#484—48no—
121#481—48no—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#106C2314yes—
125C2×#106C2314yes—
126#419—44no—
127#418—44no—
128#416—44no—
129#416—44no—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138#407—24no—
139C2×C22⋊C8—24no—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146#395—22no—
147#394—22no—
148#394—24no—
149#393—22no—
150#392—24no—
151#392—22no—
152#391C2314yes—
153#390C2314yes—
154#388—24no—
155#388—24no—
156#387—24no—
157#387—24no—
158#386—24no—
159#386—24no—
160#386—22no—
161#386—22no—
162#386—22no—
163#386—24no—
164#385—24no—
165#385—22no—
166#383—22no—
167#382C2314yes—
168#381—22no—
169#381—21no—
170#381—21no—
171#381—21no—
172#380—22no—
173#380—22no—
174#380C2312yes—
175#380C2312yes—
176#380C2312yes—
177#380C2312yes—
178#380C2314yes—
179#380C2312yes—
180#380C2312yes—
181#379—21no—
182#379—21no—
183#379—21no—
184#379—22no—
185C4×C4⋊C4C2314yes—
186C4×C22⋊C4—22no—
187C4×C22⋊C4—21no—
188C4×C22⋊C4—21no—
189C4×C22⋊C4—21no—
190C4×C22⋊C4—21no—
191C4×C22⋊C4—21no—
192C4×C22⋊C4—21no—
193C4×C22⋊C4—24no—
194C4×C22⋊C4—24no—
195C4×C22⋊C4—24no—
196#376—22no—
197C2×#95—22no—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#336C2314yes—
203#336C2314yes—
204C25—21no—
205C25C2×D411yescharacteristic
206C25C2×D411yescharacteristic
207C2×#41—24no—
208C2×#41—24no—
209C2×#41—24no—
210C2×#41—24no—
211C2×#41—24no—
212C2×#41—24no—
213C2×#41—24no—
214C2×#41—24no—
215C2×#41—42no—
216C2×#41—42no—
217C2×#41—42no—
218C2×#41—42no—
219C2×#41—48no—
220C2×#41—416no—
221C2×#41—48no—
222C2×#41—416no—
223C2×#41—432no—
224C2×#41—44no—
225C2×#41—44no—
226C22×Q8—21no—
227C22×Q8—24no—
228C22×Q8—22no—
229C22×Q8—22no—
230C22×Q8—42no—
231C22×Q8C2412yes—
232C22×D4—24no—
233C22×D4—24no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4—24no—
237C22×D4—21no—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—42no—
241C22×D4—44no—
242C22×D4—48no—
243C22×D4—44no—
244C22×D4C2×D412yes—
245C22×D4C2×D412yes—
246C22×D4C2412yes—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—21no—
251C23×C4—24no—
252C23×C4—22no—
253C23×C4—42no—
254C23×C4—42no—
255C23×C4—42no—
256C23×C4—42no—
257C23×C4—48no—
258C23×C4—44no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D412yes—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4#4114yes—
267C23×C4#4114yes—
268C23×C4C2412yes—
269C23×C4C2411yescharacteristic
270C23×C4C2411yescharacteristic
271C23×C4C2411yescharacteristic
272C2×M16—24no—
273C2×M16—42no—
274C2×M16—42no—
275C22×C8—22no—
276C22×C8—22no—
277C22×C8—42no—
278C22×C8—42no—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2414yes—
282#123—48no—
283C22⋊Q8—28no—
284C22⋊Q8—44no—
285#121—28no—
286#121—44no—
287#120—48no—
288#120—48no—
289C4×D4—28no—
290C4×D4—28no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—416no—
296#117—42no—
297#117—42no—
298#117—42no—
299#117—42no—
300#117—42no—
301#117—42no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2414yes—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—48no—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—24no—
339C2×C42C2×D412yes—
340C2×C42C2412yes—
341#107—44no—
342#107—44no—
343#106—44no—
344#106—44no—
345#103—28no—
346#103—28no—
347#103—44no—
348#103—44no—
349#102—28no—
350#102—28no—
351#102—44no—
352#102—44no—
353C22⋊C8—24no—
354C22⋊C8—24no—
355C22⋊C8—24no—
356C22⋊C8—24no—
357C22⋊C8—44no—
358C22⋊C8—44no—
359#95—22no—
360#95—22no—
361#95—24no—
362#95—24no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95—22no—
367#95—22no—
368#95—24no—
369#95—24no—
370#95—24no—
371C24—22no—
372C24—24no—
373C24—22no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—44no—
380C24—44no—
381C24—42no—
382C24—82no—
383C24#12314yes—
384C24C22×D411yescharacteristic
385C24C22×D411yescharacteristic
386C24C22×D411yescharacteristic
387C24C22×D412yes—
388#41—216no—
389#41—48no—
390#41—48no—
391#41—48no—
392#41—48no—
393#41—48no—
394#41—48no—
395#41—48no—
396#41—432no—
397#41—432no—
398C2×Q8—22no—
399C2×Q8—28no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—44no—
405C2×Q8—48no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8H(5,𝔽2)14yes—
409C2×D4—28no—
410C2×D4—28no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—48no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—416no—
428C2×D4—44no—
429C2×D4—432no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—48no—
433C2×D4—48no—
434C2×D4—42no—
435C2×D4—84no—
436C2×D4H(5,𝔽2)14yes—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—28no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—48no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—432no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—48no—
475C22×C4—82no—
476C22×C4—84no—
477C22×C4—82no—
478C22×C4—84no—
479C22×C4—84no—
480C22×C4#12314yes—
481C22×C4#12314yes—
482C22×C4#12314yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C4C2511yescharacteristic
496M16—84no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—24no—
502C2×C8—24no—
503C2×C8—24no—
504C2×C8—82no—
505C2×C8—82no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—48no—
515C4⋊C4—84no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—48no—
528C22⋊C4—84no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C42—24no—
535C42—22no—
536C42—22no—
537C42—44no—
538C42—84no—
539C42—82no—
540C42—82no—
541C23—28no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—24no—
546C23—22no—
547C23—22no—
548C23—24no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—24no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—48no—
560C23—44no—
561C23—42no—
562C23—42no—
563C23—44no—
564C23—82no—
565C23—82no—
566C23—88no—
567C23—84no—
568C23—84no—
569C23—88no—
570C23—816no—
571C23C2×#12312yes—
572C23C2×#12312yes—
573C23C2×#12312yes—
574C23#57514yes—
575C23C23×D411yescharacteristic
576Q8—24no—
577Q8—24no—
578Q8—48no—
579Q8—42no—
580Q8—42no—
581D4—24no—
582D4—24no—
583D4—48no—
584D4—48no—
585D4—48no—
586D4—416no—
587D4—416no—
588D4—42no—
589D4—42no—
590D4—816no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—416no—
615C2×C4—416no—
616C2×C4—44no—
617C2×C4—48no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—84no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—168no—
636C2×C4C2×#12312yes—
637C2×C4C2×#12312yes—
638C2×C4C2×#12312yes—
639C2×C4C22×#4111yescharacteristic
640C2×C4C22×#4111yescharacteristic
641C2×C4C2×H(5,𝔽2)12yes—
642C2×C4C2×H(5,𝔽2)12yes—
643C8—42no—
644C8—42no—
645C8—44no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—24no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—28no—
657C22—42no—
658C22—42no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—48no—
665C22—88no—
666C22—88no—
667C22—816no—
668C22—82no—
669C22—82no—
670C22—84no—
671C22C22×#12311yescharacteristic
672C22#348412yes—
673C22C2×#57512yes—
674C22#358412yes—
675C4—22no—
676C4—22no—
677C4—22no—
678C4—42no—
679C4—44no—
680C4—42no—
681C4—42no—
682C4—84no—
683C4—84no—
684C4—84no—
685C4#349011yescharacteristic
686C4#349012yes—
687C4#349011yescharacteristic
688C2—42no—
689C2—44no—
690C2—42no—
691C2—42no—
692C2—84no—
693C2#3535011yescharacteristic
694C2#6051111yescharacteristic
695C2#6051611yescharacteristic
696C2#6160614yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#34736C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34185C2semidirect
3. #27590C2semidirect
4. #24880C2semidirect
5. #24878C2semidirect
6. #24864C2semidirect
7. #24862C2semidirect
8. #24007C2semidirect
9. #21943C2semidirect
10. #21939C2semidirect
11. #21740C2semidirect
12. #20600C2semidirect
13. #20597C2semidirect
17. C2×#484C22semidirect
18. C2×#481C22semidirect
19. C2×#419C22semidirect
20. C2×#418C22semidirect
21. C2×#416C22semidirect
22. C2×#416C22semidirect
36. #2352C22semidirect
42. #2283C22semidirect
46. C2×#380C22semidirect
48. #2097C22semidirect
49. #2092C22semidirect
50. #2074C22semidirect
51. #2073C22semidirect
52. #2035C22semidirect
53. #2035C22semidirect
54. #2033C22semidirect
55. #2033C22semidirect
56. #1928C22semidirect
57. #1927C22semidirect
58. #1926C22semidirect
59. #1925C22semidirect
60. #1874C22semidirect
61. #1872C22semidirect
63. #1869C22semidirect
64. #1869C22semidirect
65. #1854C22semidirect
66. #1854C22semidirect
67. #1854C22semidirect
68. #1854C22semidirect
69. #1817C22semidirect
70. #1809C22semidirect
71. #1808C22semidirect
72. #1807C22semidirect
75. #1788C22semidirect
76. #1787C22semidirect
77. #1786C22semidirect
78. #1779C22semidirect

Maximal subgroups

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

subgroupindex
#347362
#341852
#275902
#248802
#248782
#248642
#248622
#240072
#219432
#219392
#217402
#206002
#205972