Finite Group Databasebeta

← #7511142

Subgroups — #7511142

Order 512.

1728 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35662C211yesmaximal, characteristic, complemented
2#35280C211yesmaximal, characteristic
3#32667C214yesmaximal, complemented
4#32661C214yesmaximal, complemented
5#32656C214yesmaximal, complemented
6#32649C214yesmaximal, complemented
7#23738C211yesmaximal, characteristic
8C2×#2096C211yesmaximal, characteristic, complemented
9#23661C211yesmaximal, characteristic, complemented
10#23658C211yesmaximal, characteristic, complemented
11#23615C211yesmaximal, characteristic, complemented
12#23610C211yesmaximal, characteristic
13#21304C211yesmaximal, characteristic, complemented
14#21234C211yesmaximal, characteristic
15#21157C211yesmaximal, characteristic, complemented
16#20930C211yesmaximal, characteristic
17#20927C211yesmaximal, characteristic
18#20880C211yesmaximal, characteristic, complemented
19#20872C211yesmaximal, characteristic
20C22×#137C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C2×#519C2211yescharacteristic
23C2×C4×#41C2211yescharacteristic
24C2×#428C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#427C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C2×#424C2211yescharacteristic
30C2×#417C2211yescharacteristic
31C2×#419C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C2×#416C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C2×#405C2211yescharacteristic
37#2857—22no—
38#2629—22no—
39#2582C2214yes—
40#2582C2214yes—
41#2581C2214yes—
42#2538C2214yes—
43#2453—22no—
44#2451—22no—
45#2438—24no—
46#2417—24no—
47#2317C2211yescharacteristic
48#2316C2211yescharacteristic
49#2316C2211yescharacteristic
50#2316C2211yescharacteristic
51#2311C2211yescharacteristic
52#2311C2211yescharacteristic
53#2310C2211yescharacteristic
54#2296C2211yescharacteristic
55C2×#391C2211yescharacteristic
56C2×#384C2211yescharacteristic
57#2097C2212yes—
58#2096C2214yes—
59#2096C2214yes—
60#2096C2212yes—
61#2094C2212yes—
62#2092C2212yes—
63#2085C2214yes—
64#2084C2214yes—
65#2084C2214yes—
66#2083C2214yes—
67#2068C2214yes—
68#2066C2214yes—
69#2066C2214yes—
70#2065C2214yes—
71#1985C2214yes—
72#1984C2214yes—
73#1973C2212yes—
74#1971C2212yes—
75#1969C2212yes—
76#1968C2212yes—
77#1944C2214yes—
78#1941C2214yes—
79#1880C2214yes—
80#1878C2214yes—
81#1878C2214yes—
82#1877C2214yes—
83#1865C2212yes—
84#1865C2214yes—
85#1864C2212yes—
86#1863C2212yes—
87#1863C2214yes—
88#1862C2212yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C2×#137—42no—
100C2×#137—42no—
101C2×#137Q818yescomplemented
102C22×Q16C2312yes—
103C22×SD16C2312yes—
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#533—42no—
109#533—42no—
110C2×C4⋊Q8D412yes—
111C2×#125D412yes—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8D414yes—
114#519—42no—
115#519—42no—
116C4×#41—42no—
117C4×#41—42no—
118C2×C4×Q8—22no—
119C2×C4×Q8C2314yes—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2314yes—
122C2×C4×D4—22no—
123C2×C4×D4C2312yes—
124C2×#117—22no—
125C2×#117—21no—
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C22×C4⋊C4—22no—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137#428—42no—
138#428—42no—
139#427—42no—
140C2×#107C2312yes—
141C2×#107C2312yes—
142C2×#107C2312yes—
143C2×#106—24no—
144C2×#106—42no—
145C2×#106C2312yes—
146C2×#106C2312yes—
147C2×#106C2314yes—
148C2×#106C2312yes—
149C2×#106C2312yes—
150C2×#106C2312yes—
151#424—42no—
152C2×C4⋊C8C2314yes—
153#419—42no—
154#417—42no—
155#416—24no—
156#416—42no—
157C2×#103—24no—
158C2×#103—42no—
159C2×#103C2312yes—
160C2×#103C2311yescharacteristic
161C2×#103C2312yes—
162C2×#103C2311yescharacteristic
163C2×#103C2312yes—
164C2×#103C2312yes—
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×#102C2311yescharacteristic
168C2×#102C2311yescharacteristic
169C2×#102C2312yes—
170C2×#102C2312yes—
171#405—42no—
172C2×#97C2312yes—
173C2×C4×C8C2312yes—
174#400—24no—
175#398—24no—
176#398—24no—
177#398—24no—
178#398—24no—
179#396—24no—
180#396—24no—
181#395—24no—
182#391—24no—
183#391—24no—
184#391—22no—
185#391—21no—
186#391—21no—
187#391C2312yes—
188#391C2312yes—
189#391C2312yes—
190#391C2312yes—
191#391C2314yes—
192#390C2312yes—
193#390C2312yes—
194#389—22no—
195#389—22no—
196#389—22no—
197#389—22no—
198#389—24no—
199#389—24no—
200#389C2314yes—
201#389C2314yes—
202#389C2314yes—
203#389C2312yes—
204#389C2312yes—
205#387—24no—
206#387—24no—
207#386—22no—
208#386—22no—
209#385—22no—
210#385—22no—
211#384—21no—
212#384—21no—
213#384C2312yes—
214#384C2314yes—
215#382C2312yes—
216#381—22no—
217#380—22no—
218#380—22no—
219#380—24no—
220#380—22no—
221#380—22no—
222#380—22no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—21no—
225C2×#95—22no—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234#336C2312yes—
235#336C2312yes—
236#336C2312yes—
237C2×#41—22no—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41C2×Q814yes—
243C22×Q8—21no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8C2×D414yes—
250C22×Q8C2×Q814yes—
251C22×Q8C2412yes—
252C22×Q8C2411yescharacteristic
253C22×D4—21no—
254C22×D4C2411yescharacteristic
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—22no—
261C23×C4C2×D412yes—
262C23×C4C2411yescharacteristic
263#137—416no—
264C2×Q16—42no—
265C2×Q16—42no—
266C2×Q16C2×Q818yes—
267C2×SD16—42no—
268C2×SD16—42no—
269C2×SD16C2×Q818yes—
270C2×M16—22no—
271C2×M16—22no—
272C2×M16—22no—
273C2×M16—22no—
274C2×M16—22no—
275C2×M16C2×Q814yes—
276C22×C8—22no—
277C22×C8—22no—
278C22×C8—22no—
279C22×C8—22no—
280C22×C8—22no—
281C22×C8#4114yes—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C22×C8C2411yescharacteristic
286C22×C8C2411yescharacteristic
287C4⋊Q8—44no—
288#125—44no—
289C22⋊Q8—48no—
290C22⋊Q8—88no—
291C4×Q8—44no—
292C4×Q8—44no—
293C4×Q8—42no—
294C4×Q8—42no—
295C4×Q8—44no—
296C4×Q8—44no—
297C4×Q8—48no—
298C4×D4—42no—
299C4×D4—42no—
300C4×D4—48no—
301#117—22no—
302#117—22no—
303#117—22no—
304#117—22no—
305#117—22no—
306#117—22no—
307#117—22no—
308#117—22no—
309#117—22no—
310#117—22no—
311#117—44no—
312#117—48no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—84no—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D414yes—
336C2×C4⋊C4C2×D414yes—
337C2×C4⋊C4C2×D414yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—21no—
356C2×C42—24no—
357C2×C42—21no—
358C2×C42—21no—
359C2×C42—22no—
360C2×C42—44no—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2412yes—
364C2×C42C2412yes—
365C2×C42C2412yes—
366#107—42no—
367#107—42no—
368#107—42no—
369#106—24no—
370#106—24no—
371#106—44no—
372#106—42no—
373#106—42no—
374#106—42no—
375C4⋊C8—44no—
376#103—24no—
377#103—24no—
378#103—22no—
379#103—22no—
380#103—44no—
381#103—42no—
382#103—42no—
383#103—44no—
384#102—22no—
385#102—22no—
386#102—42no—
387#102—42no—
388#97—42no—
389C4×C8—42no—
390#95—24no—
391#95—24no—
392#95—22no—
393#95—22no—
394#95—24no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—24no—
402C24—21no—
403C24C22×D411yescharacteristic
404#41—28no—
405#41—28no—
406#41—44no—
407#41—44no—
408C2×Q8—24no—
409C2×Q8—28no—
410C2×Q8—22no—
411C2×Q8—21no—
412C2×Q8—28no—
413C2×Q8—28no—
414C2×Q8—22no—
415C2×Q8—21no—
416C2×Q8—22no—
417C2×Q8—21no—
418C2×Q8—22no—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8—21no—
422C2×Q8—44no—
423C2×Q8—44no—
424C2×Q8—42no—
425C2×Q8—44no—
426C2×Q8—42no—
427C2×Q8—44no—
428C2×Q8C22×Q814yes—
429C2×Q8C22×Q812yes—
430C2×D4—24no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4C22×Q812yes—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—42no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4C22×D414yes—
467C22×C4C22×D414yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×Q812yes—
472C22×C4C2×#4112yes—
473C22×C4H(5,𝔽2)12yes—
474C22×C4H(5,𝔽2)12yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C42−1+412yes—
477C22×C4C2511yescharacteristic
478Q16—48no—
479SD16—48no—
480M16—28no—
481M16—44no—
482C2×C8—28no—
483C2×C8—21no—
484C2×C8—22no—
485C2×C8—21no—
486C2×C8—22no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—44no—
496C2×C8C22×Q814yes—
497C2×C8C2×#4112yes—
498C2×C8C2×#4112yes—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—21no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—48no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—48no—
525C4⋊C4—82no—
526C4⋊C4—84no—
527C4⋊C4—84no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C4⋊C4—84no—
531C22⋊C4—24no—
532C22⋊C4—24no—
533C22⋊C4—44no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—84no—
538C22⋊C4—84no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—44no—
548C42—48no—
549C42—42no—
550C42—42no—
551C42—42no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—42no—
556C23—41no—
557C23—41no—
558C23—42no—
559C23#53412yes—
560C23D4×Q812yes—
561C23#55012yes—
562C23C23×D411yescharacteristic
563Q8—24no—
564Q8—22no—
565Q8—24no—
566Q8—22no—
567Q8—42no—
568Q8—41no—
569Q8—48no—
570Q8—42no—
571Q8—41no—
572Q8—88no—
573D4—22no—
574D4—22no—
575D4—48no—
576D4—41no—
577D4—41no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—48no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—48no—
597C2×C4—48no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—88no—
615C2×C4—162no—
616C2×C4—162no—
617C2×C4—162no—
618C2×C4#53412yes—
619C2×C4#53914yes—
620C2×C4#54314yes—
621C2×C4D4×Q812yes—
622C2×C4#55012yes—
623C2×C4#55414yes—
624C2×C4#56214yes—
625C2×C4#57714yes—
626C2×C4C23×Q811yescharacteristic
627C2×C4C22×#4111yescharacteristic
628C2×C4C2×H(5,𝔽2)11yescharacteristic
629C2×C4C2×H(5,𝔽2)11yescharacteristic
630C2×C4C2×H(5,𝔽2)11yescharacteristic
631C2×C4C2×2−1+411yescharacteristic
632C8—22no—
633C8—22no—
634C8—24no—
635C8—42no—
636C8—42no—
637C22—24no—
638C22—24no—
639C22—22no—
640C22—22no—
641C22—42no—
642C22—44no—
643C22—82no—
644C22—81no—
645C22—81no—
646C22#273214yes—
647C22C2×#53411yescharacteristic
648C22C2×D4×Q811yescharacteristic
649C22C2×#55011yescharacteristic
650C22C2×#57712yes—
651C22#358612yes—
652C4—24no—
653C4—42no—
654C4—44no—
655C4—44no—
656C4—42no—
657C4—44no—
658C4—82no—
659C4—82no—
660C4—81no—
661C4—82no—
662C4—82no—
663C4—81no—
664C4—82no—
665C4#273214yes—
666C4C2×#53911yescharacteristic
667C4C2×#54311yescharacteristic
668C4C2×#55411yescharacteristic
669C4C2×#56211yescharacteristic
670C2—24no—
671C2—82no—
672C2C2×#273211yescharacteristic
673C2#6076312yes—
674C2#6138312yes—
675C2#6141011yescharacteristic
676C2#6141811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35662C2semidirect
3. #32667C2semidirect
4. #32661C2semidirect
5. #32656C2semidirect
6. #32649C2semidirect
8. C2×#2096C2semidirect
9. #23661C2semidirect
10. #23658C2semidirect
11. #23615C2semidirect
13. #21304C2semidirect
15. #21157C2semidirect
18. #20880C2semidirect
101. C2×#137Q8semidirect

Maximal subgroups

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

subgroupindex
#356622
#352802
#326672
#326612
#326562
#326492
#237382
C2×#20962
#236612
#236582
#236152
#236102
#213042
#212342
#211572
#209302
#209272
#208802
#208722