Finite Group Databasebeta

← #6491249

Subgroups — #6491249

Order 512.

3367 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#540C211yesmaximal, characteristic
2C2×#3260C212yesmaximal, complemented
3#35310C211yesmaximal, characteristic, complemented
4#32959C212yesmaximal, complemented
5#32905C212yesmaximal, complemented
6#29700C212yesmaximal, complemented
7#29673C212yesmaximal, complemented
8#25507C214yesmaximal, complemented
9#24385C212yesmaximal, complemented
10#24379C212yesmaximal, complemented
11#24307C212yesmaximal, complemented
12#24211C212yesmaximal, complemented
13#24164C214yesmaximal, complemented
14#21897C212yesmaximal, complemented
15#20628C211yesmaximal, characteristic, complemented
16C2×#541C2211yescharacteristic, complemented
17C2×#540—216no—
18C2×#540—22no—
19C2×#540—22no—
20C2×#540C2212yes—
21C2×#540C2212yes—
22C2×#540C2212yes—
23C2×#540C2212yes—
24C22×#127C2211yescharacteristic
25C22×#125C2211yescharacteristic
26C22×#123—22no—
27C22×#121—22no—
28C22×#121C2212yes—
29C22×C4×D4C2212yes—
30C22×#117C2211yescharacteristic
31#3260—216no—
32C2×#479C2214yescomplemented
33C2×#478C2214yescomplemented
34C2×#468C2214yescomplemented
35C2×#465C2214yescomplemented
36C2×#460C2214yescomplemented
37C2×#461C2214yescomplemented
38C2×#450C2212yes—
39C2×#450C2212yes—
40C2×#424C2212yescomplemented
41C2×#407C2211yescharacteristic
42C22×C22⋊C8C2211yescharacteristic
43#2857C2211yescharacteristic
44#2851C2211yescharacteristic
45#2848C2212yes—
46#2629C2211yescharacteristic, complemented
47#2625C2212yescomplemented
48#2621C2211yescharacteristic, complemented
49#2480—22no—
50#2478—22no—
51#2475—22no—
52#2473—22no—
53#2323C2212yescomplemented
54#2314C2211yescharacteristic, complemented
55C4×C22⋊Q8C2212yes—
56#2059C2214yescomplemented
57#2041C2214yescomplemented
58#2034C2214yescomplemented
59#2033C2214yescomplemented
60#2026C2214yescomplemented
61#2014C2214yescomplemented
62#2000C2214yescomplemented
63#2000C2214yescomplemented
64#1989C2214yescomplemented
65#1982C2214yescomplemented
66#1895C2214yescomplemented
67#1893C2214yescomplemented
68#1877C2214yescomplemented
69#1877C2214yescomplemented
70#1872C2212yes—
71#1872C2212yes—
72#1864C2214yescomplemented
73#1862C2214yescomplemented
74#1853C2212yes—
75#1852C2212yes—
76#1804C2214yescomplemented
77#1802C2214yescomplemented
78#1798C2212yescomplemented
79#1798C2212yescomplemented
80#1797C2212yes—
81#1797C2212yes—
82#1786C2212yes—
83#1786C2212yes—
84#1729C2212yescomplemented
85C23×D4—22no—
86C23×D4—22no—
87C23×D4C2312yes—
88C24×C4—22no—
89C24×C4C2311yescharacteristic
90C22×SD16C2314yes—
91C22×SD16C2314yes—
92C22×M16C2312yes—
93C23×C8C2312yes—
94#541—28no—
95#540—232no—
96#540—232no—
97#540—28no—
98#540—28no—
99#540—216no—
100#540—216no—
101#540—28no—
102#540D4116yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127—28no—
105C2×#127—22no—
106C2×#127C2312yes—
107C2×#127C2312yes—
108C2×#126—22no—
109C2×#125—28no—
110C2×#125—22no—
111C2×#125—22no—
112C2×#125C2311yescharacteristic
113C2×#125C2312yes—
114C2×#125C2312yes—
115C2×#123—232no—
116C2×#123—24no—
117C2×#123—24no—
118C2×#123—24no—
119C2×#123—24no—
120C2×C22⋊Q8C2312yes—
121C2×C22⋊Q8C2312yes—
122C2×C22⋊Q8C2312yes—
123C2×#121—232no—
124C2×#121—22no—
125C2×#121—22no—
126C2×#121—216no—
127C2×#121—24no—
128C2×#121—24no—
129C2×#121—24no—
130C2×#121—24no—
131C2×#121C2314yes—
132C2×#121C2314yes—
133C2×C4×Q8C2312yes—
134C2×C4×D4—216no—
135C2×C4×D4—22no—
136C2×C4×D4—22no—
137C2×C4×D4C2314yes—
138C2×C4×D4C2314yes—
139C2×#117—28no—
140C2×#117D412yes—
141C2×#117D412yes—
142C2×#117C2311yescharacteristic
143C2×#117C2311yescharacteristic
144C2×#117C2311yescharacteristic
145C2×#117C2311yescharacteristic
146C22×C4⋊C4—22no—
147C22×C4⋊C4D412yes—
148C22×C4⋊C4C2312yes—
149C22×C22⋊C4—22no—
150C22×C22⋊C4—22no—
151C22×C22⋊C4D412yes—
152C22×C42C2311yescharacteristic
153C22×C42C2311yescharacteristic
154#479—216no—
155#478—216no—
156#468—216no—
157#465—216no—
158#461—216no—
159#460—216no—
160#450—44no—
161#450—44no—
162C2×#107C2314yes—
163C2×#106C2314yes—
164#424—28no—
165C2×C4⋊C8C2314yes—
166C2×C4⋊C8C2314yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#102C2314yes—
170C2×#102C2314yes—
171C2×#102C2314yes—
172C2×#102C2314yes—
173C2×#102C2314yes—
174C2×#102C2314yes—
175#407—42no—
176C2×C22⋊C8—42no—
177C2×C22⋊C8C2312yes—
178C2×C22⋊C8C2312yes—
179C2×C22⋊C8C2312yes—
180C2×C22⋊C8C2312yes—
181#400—22no—
182#398—22no—
183#398—22no—
184#398—22no—
185#397—22no—
186#397—22no—
187#397—22no—
188#396—22no—
189#391C2312yes—
190#391C2312yes—
191#389—22no—
192#389—22no—
193#389C2312yes—
194#389C2312yes—
195#388—22no—
196#388—22no—
197#388—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#385—22no—
203#385—22no—
204#385—22no—
205#385—22no—
206#385—22no—
207#384C2311yescharacteristic
208#382C2311yescharacteristic
209#380—22no—
210#380C2312yes—
211#380C2312yes—
212#380C2312yes—
213#379—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—22no—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#336C2314yes—
222#336C2314yes—
223C25—22no—
224C25C2×D411yescharacteristic
225C22×Q8C2412yes—
226C22×D4—216no—
227C22×D4—216no—
228C22×D4—24no—
229C22×D4—24no—
230C22×D4—28no—
231C22×D4—22no—
232C22×D4—24no—
233C22×D4—24no—
234C22×D4—24no—
235C22×D4—24no—
236C22×D4—44no—
237C22×D4—84no—
238C22×D4—84no—
239C22×D4—84no—
240C22×D4—84no—
241C22×D4C2414yes—
242C23×C4—22no—
243C23×C4—216no—
244C23×C4—24no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—24no—
248C23×C4—24no—
249C23×C4—42no—
250C23×C4—84no—
251C23×C4—84no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D412yes—
254C23×C4C2×D412yes—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2411yescharacteristic
258C23×C4C2411yescharacteristic
259C23×C4C2411yescharacteristic
260C2×SD16—28no—
261C2×SD16—28no—
262C2×SD16—44no—
263C2×SD16—44no—
264C2×SD16—44no—
265C2×SD16—44no—
266C2×M16—24no—
267C2×M16—42no—
268C2×M16—42no—
269C22×C8—24no—
270C22×C8—42no—
271C22×C8—42no—
272C22×C8C2414yes—
273C22×C8C2414yes—
274C4⋊Q8—24no—
275#127—216no—
276#127—28no—
277#127—24no—
278#127C2×D418yes—
279#126—48no—
280#125—216no—
281#125—28no—
282#125—24no—
283#125—24no—
284#125—48no—
285#125C2×D418yes—
286#123—232no—
287#123—232no—
288#123—232no—
289#123—232no—
290C22⋊Q8—28no—
291C22⋊Q8—24no—
292C22⋊Q8—24no—
293C22⋊Q8—44no—
294#121—232no—
295#121—232no—
296#121—216no—
297#121—232no—
298#121—216no—
299#121—28no—
300#121—28no—
301#121—28no—
302#121—232no—
303#121C2×D4116yes—
304C4×Q8—28no—
305C4×D4—216no—
306C4×D4—216no—
307C4×D4—28no—
308C4×D4—28no—
309C4×D4—28no—
310C4×D4C2×D4116yes—
311#117—24no—
312#117—28no—
313#117—28no—
314#117—24no—
315#117—24no—
316#117C2×D418yes—
317C2×C4⋊C4—216no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—28no—
322C2×C4⋊C4—28no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2414yes—
340C2×C22⋊C4—216no—
341C2×C22⋊C4—216no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—28no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C22⋊C4C2×D412yes—
357C2×C42—24no—
358C2×C42—24no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42C2×D412yes—
362C2×C42C2×D412yes—
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367#107—28no—
368#106—28no—
369C4⋊C8—28no—
370C4⋊C8—28no—
371#103—28no—
372#103—28no—
373#102—28no—
374#102—28no—
375#102—44no—
376#102—44no—
377#102—44no—
378#102—44no—
379C22⋊C8—24no—
380C22⋊C8—24no—
381C22⋊C8—42no—
382C22⋊C8—42no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394C24—28no—
395C24—22no—
396C24—24no—
397C24—22no—
398C24—24no—
399C24—41no—
400C24—42no—
401C24—44no—
402C24—44no—
403C24—44no—
404C24—44no—
405C24—44no—
406C24—44no—
407C24C22×D411yescharacteristic
408C24C22×D411yescharacteristic
409C24C22×D411yescharacteristic
410C2×Q8—24no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—24no—
414C2×D4—28no—
415C2×D4—216no—
416C2×D4—216no—
417C2×D4—216no—
418C2×D4—216no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—416no—
425C2×D4—416no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—84no—
429C2×D4—84no—
430C2×D4—84no—
431C2×D4—84no—
432C2×D4—84no—
433C2×D4—84no—
434C2×D4—84no—
435C2×D4—84no—
436C2×D4—832no—
437C2×D4—832no—
438C2×D4C22×D418yes—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—28no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—216no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—216no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—84no—
476C22×C4—84no—
477C22×C4—84no—
478C22×C4—84no—
479C22×C4—832no—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D414yes—
487C22×C4C22×D411yescharacteristic
488C22×C4H(5,𝔽2)12yes—
489C22×C42−1+412yes—
490C22×C4C2511yescharacteristic
491SD16—88no—
492SD16—88no—
493M16—84no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—84no—
501C4⋊C4—28no—
502C4⋊C4—216no—
503C4⋊C4—216no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—24no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4#12018yes—
522C4⋊C4C22×D418yes—
523C22⋊C4—216no—
524C22⋊C4—216no—
525C22⋊C4—24no—
526C22⋊C4—216no—
527C22⋊C4—216no—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C22⋊C4#12018yes—
537C42—24no—
538C42—24no—
539C42—22no—
540C42—22no—
541C42—44no—
542C42—82no—
543C42C22×D414yes—
544C42C22×D414yes—
545C23—22no—
546C23—21no—
547C23—21no—
548C23—22no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—22no—
553C23—21no—
554C23—28no—
555C23—21no—
556C23—21no—
557C23—22no—
558C23—44no—
559C23—48no—
560C23—416no—
561C23—416no—
562C23—416no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—84no—
571C23—82no—
572C23—88no—
573C23—88no—
574C23—88no—
575C23—88no—
576C23C2×#12012yes—
577C23#53712yes—
578C23#57714yes—
579C23C23×D411yescharacteristic
580Q8—44no—
581Q8—44no—
582D4—28no—
583D4—28no—
584D4—416no—
585D4—44no—
586D4—44no—
587D4—816no—
588D4—816no—
589D4—816no—
590D4—816no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—28no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—48no—
617C2×C4—48no—
618C2×C4—416no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—816no—
636C2×C4—816no—
637C2×C4C2×#12014yes—
638C2×C4C2×#12014yes—
639C2×C4C2×#12012yes—
640C2×C4#53712yes—
641C2×C4C23×D412yes—
642C2×C4C2×H(5,𝔽2)11yescharacteristic
643C2×C4C2×2−1+411yescharacteristic
644C2×C4#58512yes—
645C8—44no—
646C8—44no—
647C22—24no—
648C22—22no—
649C22—24no—
650C22—22no—
651C22—22no—
652C22—24no—
653C22—42no—
654C22—41no—
655C22—41no—
656C22—42no—
657C22—41no—
658C22—48no—
659C22—48no—
660C22—42no—
661C22—48no—
662C22—41no—
663C22—41no—
664C22—88no—
665C22—832no—
666C22—84no—
667C22—84no—
668C22—84no—
669C22—84no—
670C22#326014yescomplemented
671C22C22×#12011yescharacteristic
672C22C2×#53711yescharacteristic
673C22#344911yescharacteristic
674C22C2×#57712yes—
675C22#358512yes—
676C4—22no—
677C4—22no—
678C4—24no—
679C4—24no—
680C4—41no—
681C4—42no—
682C4—42no—
683C4—48no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4#348812yes—
689C4#348912yes—
690C2—24no—
691C2—42no—
692C2—81no—
693C2—88no—
694C2#3525111yescharacteristic
695C2#6037312yes—
696C2C2×#326011yescharacteristic
697C2#6092011yescharacteristic
698C2#6097912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#540C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3260C2semidirect
3. #35310C2semidirect
4. #32959C2semidirect
5. #32905C2semidirect
6. #29700C2semidirect
7. #29673C2semidirect
8. #25507C2semidirect
9. #24385C2semidirect
10. #24379C2semidirect
11. #24307C2semidirect
12. #24211C2semidirect
13. #24164C2semidirect
14. #21897C2semidirect
15. #20628C2semidirect
16. C2×#541C22semidirect
32. C2×#479C22semidirect
33. C2×#478C22semidirect
34. C2×#468C22semidirect
35. C2×#465C22semidirect
36. C2×#460C22semidirect
37. C2×#461C22semidirect
40. C2×#424C22semidirect
46. #2629C22semidirect
47. #2625C22semidirect
48. #2621C22semidirect
53. #2323C22semidirect
54. #2314C22semidirect
56. #2059C22semidirect
57. #2041C22semidirect
58. #2034C22semidirect
59. #2033C22semidirect
60. #2026C22semidirect
61. #2014C22semidirect
62. #2000C22semidirect
63. #2000C22semidirect
64. #1989C22semidirect
65. #1982C22semidirect
66. #1895C22semidirect
67. #1893C22semidirect
68. #1877C22semidirect
69. #1877C22semidirect
72. #1864C22semidirect
73. #1862C22semidirect
76. #1804C22semidirect
77. #1802C22semidirect
78. #1798C22semidirect
79. #1798C22semidirect
84. #1729C22semidirect
670. C22#3260semidirect

Maximal subgroups

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

subgroupindex
C22×#5402
C2×#32602
#353102
#329592
#329052
#297002
#296732
#255072
#243852
#243792
#243072
#242112
#241642
#218972
#206282