Finite Group Databasebeta

← #7510524

Subgroups — #7510524

Order 512.

1723 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
1#35662C211yesmaximal, characteristic, complemented
2#35628C211yesmaximal, characteristic
3#32626C214yesmaximal, complemented
4#32625C214yesmaximal, complemented
5#32619C214yesmaximal, complemented
6#32616C214yesmaximal, complemented
7#23738C211yesmaximal, characteristic
8#23633C211yesmaximal, characteristic, complemented
9#23619C211yesmaximal, characteristic, complemented
10#23547C211yesmaximal, characteristic
11#23512C211yesmaximal, characteristic
12#23425C211yesmaximal, characteristic
13#21301C211yesmaximal, characteristic, complemented
14C2×#1971C211yesmaximal, characteristic, complemented
15#21086C211yesmaximal, characteristic
16#21064C211yesmaximal, characteristic, complemented
17#21038C211yesmaximal, characteristic, complemented
18#20935C211yesmaximal, characteristic, complemented
19#20930C211yesmaximal, characteristic
20C22×#137C2211yescharacteristic
21C23×Q16C2211yescharacteristic
22C2×#536C2211yescharacteristic
23C2×#533C2211yescharacteristic
24C2×#519C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C2×#417C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#419C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C2×#416C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#97C2211yescharacteristic
37#2592C2214yes—
38#2582C2214yes—
39#2581C2214yes—
40#2568C2214yes—
41#2492—22no—
42#2489—22no—
43#2488—24no—
44#2453—22no—
45#2451—22no—
46#2438—24no—
47#2317C2211yescharacteristic
48#2316C2211yescharacteristic
49#2311C2211yescharacteristic
50#2310C2211yescharacteristic
51#2310C2211yescharacteristic
52#2297C2211yescharacteristic
53#2296C2211yescharacteristic
54C2×#391C2211yescharacteristic
55C2×#391C2211yescharacteristic
56#2274C2211yescharacteristic
57#2097C2212yes—
58#2096C2212yes—
59#2094C2212yes—
60#2092C2212yes—
61#2074C2214yes—
62#2072C2214yes—
63#2068C2214yes—
64#2066C2214yes—
65#2056C2214yes—
66#2055C2214yes—
67#2043C2214yes—
68#2042C2214yes—
69#2016C2212yes—
70#2016C2212yes—
71#2015C2212yes—
72#2015C2212yes—
73#1985C2214yes—
74#1985C2214yes—
75#1971C2214yes—
76#1971C2214yes—
77#1936C2212yes—
78#1936C2212yes—
79#1935C2212yes—
80#1935C2212yes—
81#1928C2214yes—
82#1927C2214yes—
83#1920C2214yes—
84#1919C2214yes—
85#1880C2214yes—
86#1878C2214yes—
87#1878C2214yes—
88#1878C2214yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96#1729C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×Q8C2311yescharacteristic
100C2×#137—24no—
101C2×#137—42no—
102C2×#137—42no—
103C22×Q16—24no—
104C22×Q16—24no—
105C22×Q16—42no—
106C22×Q16C2312yes—
107C22×Q16C2312yes—
108C22×Q16C2312yes—
109C22×SD16C2312yes—
110C22×M16C2311yescharacteristic
111C22×M16C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C23×C8C2311yescharacteristic
114#536—42no—
115#536—42no—
116#533—42no—
117#533—42no—
118C2×C4⋊Q8D412yes—
119C2×C4⋊Q8C2312yes—
120C2×#125D412yes—
121C2×#124C2312yes—
122C2×#123—22no—
123C2×C22⋊Q8—22no—
124C2×C22⋊Q8D414yes—
125#519—42no—
126#519—42no—
127C2×C4×Q8—22no—
128C2×C4×Q8C2314yes—
129C2×C4×Q8C2314yes—
130C2×#117—22no—
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#428—42no—
140C2×#107C2312yes—
141C2×#106—24no—
142C2×#106C2312yes—
143C2×#106C2312yes—
144C2×#106C2312yes—
145#423—42no—
146C2×C4⋊C8C2314yes—
147#419—24no—
148#419—42no—
149#419—42no—
150#417—42no—
151#416—24no—
152#416—42no—
153C2×#103—42no—
154C2×#103—42no—
155C2×#103C2314yes—
156C2×#103C2311yescharacteristic
157C2×#103C2311yescharacteristic
158C2×#103C2312yes—
159C2×#103C2312yes—
160C2×#103C2314yes—
161C2×#103C2312yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2312yes—
165C2×#102C2311yescharacteristic
166C2×#102C2311yescharacteristic
167C2×#102C2312yes—
168C2×#102C2312yes—
169C2×#97—42no—
170C2×#97C2312yes—
171C2×#97C2312yes—
172#400—24no—
173#400—24no—
174#400—24no—
175#398—24no—
176#398—24no—
177#397—24no—
178#395—24no—
179#393—24no—
180#391—24no—
181#391—24no—
182#391—22no—
183#391—22no—
184#391—21no—
185#391—21no—
186#391—21no—
187#391—21no—
188#391C2314yes—
189#391C2312yes—
190#391C2312yes—
191#391C2314yes—
192#391C2312yes—
193#391C2312yes—
194#391C2314yes—
195#390C2312yes—
196#390C2312yes—
197#389—22no—
198#389—22no—
199#389—24no—
200#389C2314yes—
201#389C2312yes—
202#389C2312yes—
203#388—24no—
204#388—24no—
205#387—24no—
206#387—24no—
207#386—22no—
208#386—22no—
209#385—22no—
210#385—22no—
211#384C2312yes—
212#382—24no—
213#382C2312yes—
214#381—22no—
215#381—22no—
216#380—22no—
217#380—22no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4C2314yes—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—21no—
225C2×#95—22no—
226C2×#95—22no—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#336C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2312yes—
237#336C2312yes—
238#336C2314yes—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C22×Q8—21no—
246C22×Q8—21no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8C2×D414yes—
259C22×Q8#4114yes—
260C22×Q8C2412yes—
261C22×Q8C2411yescharacteristic
262C22×Q8C2412yes—
263C22×D4—21no—
264C22×D4C2411yescharacteristic
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4C2×D412yes—
271C23×C4C2411yescharacteristic
272#137—416no—
273C2×Q16—24no—
274C2×Q16—24no—
275C2×Q16—24no—
276C2×Q16—24no—
277C2×Q16—24no—
278C2×Q16—42no—
279C2×Q16—42no—
280C2×Q16—42no—
281C2×Q16—42no—
282C2×Q16—416no—
283C2×SD16—24no—
284C2×SD16—42no—
285C2×SD16—42no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—22no—
289C2×M16—22no—
290C2×M16—22no—
291C2×M16—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8—22no—
295C22×C8—22no—
296C22×C8—22no—
297C22×C8#4114yes—
298C22×C8C2412yes—
299C22×C8C2412yes—
300C22×C8C2412yes—
301C22×C8C2411yescharacteristic
302C22×C8C2411yescharacteristic
303C4⋊Q8—44no—
304C4⋊Q8—42no—
305C4⋊Q8—42no—
306#125—44no—
307#124—42no—
308#124—42no—
309#123—48no—
310C22⋊Q8—48no—
311C22⋊Q8—48no—
312C4×Q8—44no—
313C4×Q8—44no—
314C4×Q8—44no—
315C4×Q8—44no—
316C4×Q8—48no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117—22no—
327#117—48no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D414yes—
343C2×C4⋊C4C2×D414yes—
344C2×C4⋊C4C2×D414yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2411yescharacteristic
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—24no—
363C2×C42—21no—
364C2×C42—21no—
365C2×C42—21no—
366C2×C42—24no—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2412yes—
370C2×C42C2412yes—
371C2×C42C2412yes—
372#107—42no—
373#106—24no—
374#106—24no—
375#106—42no—
376C4⋊C8—44no—
377#103—28no—
378#103—22no—
379#103—22no—
380#103—44no—
381#103—44no—
382#103—42no—
383#103—42no—
384#103—42no—
385#103—42no—
386#103—44no—
387#102—22no—
388#102—22no—
389#102—42no—
390#102—42no—
391#97—42no—
392#97—42no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95—24no—
409C24—21no—
410C24C22×D411yescharacteristic
411#41—44no—
412#41—44no—
413#41—44no—
414#41—44no—
415C2×Q8—24no—
416C2×Q8—22no—
417C2×Q8—28no—
418C2×Q8—28no—
419C2×Q8—22no—
420C2×Q8—21no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—21no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—22no—
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8—21no—
435C2×Q8—44no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—44no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—44no—
445C2×Q8—44no—
446C2×Q8—44no—
447C2×Q8—44no—
448C2×Q8—44no—
449C2×Q8C2×#4114yes—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—42no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—82no—
483C22×C4—82no—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D414yes—
487C22×C4C22×D414yes—
488C22×C4C22×D411yescharacteristic
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C4H(5,𝔽2)12yes—
493C22×C4H(5,𝔽2)12yes—
494C22×C42−1+412yes—
495C22×C4C2511yescharacteristic
496Q16—48no—
497Q16—48no—
498Q16—48no—
499SD16—48no—
500M16—44no—
501M16—44no—
502C2×C8—28no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—44no—
517C2×C8C2×#4112yes—
518C2×C8C2×#4112yes—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—22no—
530C4⋊C4—21no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—48no—
538C4⋊C4—48no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—82no—
542C4⋊C4—84no—
543C4⋊C4—84no—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—44no—
547C22⋊C4—48no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—42no—
551C22⋊C4—82no—
552C22⋊C4—84no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—48no—
562C42—42no—
563C42—42no—
564C42—42no—
565C42—84no—
566C23—22no—
567C23—22no—
568C23—42no—
569C23—42no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23#53412yes—
574C23#54812yes—
575C23#55012yes—
576C23C23×D411yescharacteristic
577Q8—24no—
578Q8—24no—
579Q8—42no—
580Q8—48no—
581Q8—42no—
582Q8—41no—
583Q8—42no—
584Q8—42no—
585Q8—41no—
586Q8—42no—
587Q8—41no—
588Q8—42no—
589Q8—41no—
590Q8—88no—
591Q8—84no—
592D4—41no—
593D4—41no—
594D4—41no—
595D4—41no—
596D4—84no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—48no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4—162no—
637C2×C4—162no—
638C2×C4#53412yes—
639C2×C4#53814yes—
640C2×C4#54214yes—
641C2×C4#54812yes—
642C2×C4#55012yes—
643C2×C4#55114yes—
644C2×C4#56214yes—
645C2×C4C22×#4111yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4C2×H(5,𝔽2)11yescharacteristic
648C2×C4C2×H(5,𝔽2)11yescharacteristic
649C2×C4C2×H(5,𝔽2)11yescharacteristic
650C2×C4C2×2−1+411yescharacteristic
651C8—22no—
652C8—22no—
653C8—42no—
654C8—42no—
655C8—42no—
656C22—24no—
657C22—24no—
658C22—22no—
659C22—22no—
660C22—82no—
661C22—82no—
662C22—81no—
663C22—81no—
664C22—81no—
665C22#273014yes—
666C22C2×#53411yescharacteristic
667C22C2×#54811yescharacteristic
668C22C2×#55011yescharacteristic
669C22#358612yes—
670C22#358612yes—
671C4—42no—
672C4—42no—
673C4—44no—
674C4—44no—
675C4—44no—
676C4—82no—
677C4—82no—
678C4—81no—
679C4—82no—
680C4—81no—
681C4—81no—
682C4—82no—
683C4—82no—
684C4#273014yes—
685C4C2×#53811yescharacteristic
686C4C2×#54211yescharacteristic
687C4C2×#55111yescharacteristic
688C4C2×#56211yescharacteristic
689C2—24no—
690C2—161no—
691C2C2×#273011yescharacteristic
692C2#6077312yes—
693C2#6133512yes—
694C2#6141811yescharacteristic
695C2#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. #32626C2semidirect
4. #32625C2semidirect
5. #32619C2semidirect
6. #32616C2semidirect
8. #23633C2semidirect
9. #23619C2semidirect
13. #21301C2semidirect
14. C2×#1971C2semidirect
16. #21064C2semidirect
17. #21038C2semidirect
18. #20935C2semidirect

Maximal subgroups

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

subgroupindex
#356622
#356282
#326262
#326252
#326192
#326162
#237382
#236332
#236192
#235472
#235122
#234252
#213012
C2×#19712
#210862
#210642
#210382
#209352
#209302