Finite Group Databasebeta

← #6461910

Subgroups — #6461910

Order 512.

1854 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35338C211yesmaximal, characteristic
2#34254C211yesmaximal, characteristic, complemented
3#29571C214yesmaximal, complemented
4#29567C214yesmaximal, complemented
5#29563C214yesmaximal, complemented
6#29557C214yesmaximal, complemented
7#23551C211yesmaximal, characteristic, complemented
8#23506C211yesmaximal, characteristic
9#23455C211yesmaximal, characteristic, complemented
10C2×#2015C211yesmaximal, characteristic
11#21232C211yesmaximal, characteristic, complemented
12#21231C211yesmaximal, characteristic
13#21091C211yesmaximal, characteristic
14#21090C211yesmaximal, characteristic, complemented
15#21066C211yesmaximal, characteristic, complemented
16#21065C211yesmaximal, characteristic
17#20936C211yesmaximal, characteristic
18#20935C211yesmaximal, characteristic, complemented
19#20368C211yesmaximal, characteristic, complemented
20C23×Q16C2211yescharacteristic
21C23×SD16C2211yescharacteristic
22C2×#533C2211yescharacteristic
23C22×#124C2211yescharacteristic
24C2×#519C2211yescharacteristic
25C2×#427C2211yescharacteristic
26C2×#427C2211yescharacteristic
27C2×#419C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C2×#405C2211yescharacteristic
36C2×#405C2211yescharacteristic
37#2861C2214yes—
38#2640C2214yes—
39#2592—22no—
40#2592—22no—
41#2561—24no—
42#2356—22no—
43#2355—22no—
44#2343—24no—
45#2320C2214yescomplemented
46#2317C2211yescharacteristic
47#2310C2211yescharacteristic
48#2310C2211yescharacteristic
49#2310C2211yescharacteristic
50#2309C2211yescharacteristic
51#2309C2211yescharacteristic
52#2296C2211yescharacteristic
53#2295C2214yescomplemented
54C2×#391C2211yescharacteristic
55#2274C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2059C2212yescomplemented
58#2059C2212yescomplemented
59#2058C2212yescomplemented
60#2057C2212yescomplemented
61#2042C2214yes—
62#2042C2214yes—
63#2026C2214yescomplemented
64#2024C2214yescomplemented
65#2015C2212yes—
66#2015C2212yes—
67#2015C2212yes—
68#2015C2212yes—
69#1973C2214yes—
70#1971C2214yescomplemented
71#1969C2214yescomplemented
72#1968C2214yes—
73#1936C2214yes—
74#1935C2214yescomplemented
75#1934C2214yes—
76#1933C2214yescomplemented
77#1928C2214yescomplemented
78#1927C2214yescomplemented
79#1926C2214yes—
80#1925C2214yes—
81#1880C2214yes—
82#1880C2214yes—
83#1878C2214yescomplemented
84#1878C2214yescomplemented
85#1746C2212yescomplemented
86#1746C2212yescomplemented
87#1745C2212yescomplemented
88#1744C2212yescomplemented
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C23×Q8C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C22×Q16—24no—
101C22×Q16—24no—
102C22×Q16—42no—
103C22×Q16C2312yes—
104C22×Q16C2312yes—
105C22×SD16—24no—
106C22×SD16—24no—
107C22×SD16—42no—
108C22×SD16C2312yes—
109C22×SD16C2312yes—
110C22×M16C2311yescharacteristic
111C22×M16C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C23×C8C2311yescharacteristic
114#533—42no—
115#533—42no—
116C2×C4⋊Q8C2312yes—
117C2×#125C2312yes—
118C2×#124—22no—
119C2×#124—42no—
120C2×#124—42no—
121C2×#124C2312yes—
122C2×#124C2312yes—
123C2×C22⋊Q8—22no—
124#519—24no—
125#519—24no—
126C2×C4×Q8—22no—
127C2×C4×Q8—22no—
128C2×C4×Q8C2314yes—
129C2×#117—22no—
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C22×C4⋊C4C2311yescharacteristic
137C22×C22⋊C4—22no—
138C22×C42C2311yescharacteristic
139#427—42no—
140#427—42no—
141C2×#107C2312yes—
142C2×#107C2312yes—
143C2×#106C2312yes—
144C2×#106C2312yes—
145#419—24no—
146#419—42no—
147#419—42no—
148#419—42no—
149#418—24no—
150#418—42no—
151C2×#103—24no—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#102—24no—
159C2×#102C2312yes—
160C2×#102C2312yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163#405—24no—
164#405—24no—
165C2×#97C2312yes—
166C2×#97C2312yes—
167C2×C4×C8C2312yes—
168C2×C4×C8C2312yes—
169#397—24no—
170#397—24no—
171#393—24no—
172#393—24no—
173#391—22no—
174#391—21no—
175#391—21no—
176#391—21no—
177#391—21no—
178#391—22no—
179#391C2314yes—
180#391C2314yes—
181#391C2314yes—
182#391C2312yes—
183#391C2312yes—
184#390—22no—
185#390—21no—
186#390—21no—
187#390C2314yes—
188#390C2314yes—
189#389C2312yes—
190#389C2312yes—
191#389C2314yes—
192#388—24no—
193#388—24no—
194#388—24no—
195#388—24no—
196#387—24no—
197#387—24no—
198#387—24no—
199#387—24no—
200#385—22no—
201#385—22no—
202#384—21no—
203#383—24no—
204#382—21no—
205#382—24no—
206#381—22no—
207#381—22no—
208#381—22no—
209#379—22no—
210#379—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—24no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4D412yes—
215C4×C4⋊C4D412yes—
216C4×C4⋊C4C2314yes—
217C4×C22⋊C4—22no—
218C4×C22⋊C4D414yes—
219#376—24no—
220#376C2312yes—
221C2×#95—22no—
222C43C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2312yes—
234#336C2312yes—
235#336C2312yes—
236#336C2312yes—
237#336C2312yes—
238#336C2312yes—
239C25—21no—
240C22×Q8—21no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8#4114yes—
255C22×Q8C2412yes—
256C22×Q8C2412yes—
257C22×D4—21no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4C2412yes—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4C2×D412yes—
269C23×C4C2411yescharacteristic
270C2×Q16—24no—
271C2×Q16—24no—
272C2×Q16—24no—
273C2×Q16—24no—
274C2×Q16—42no—
275C2×Q16—42no—
276C2×Q16—416no—
277C2×SD16—24no—
278C2×SD16—24no—
279C2×SD16—24no—
280C2×SD16—24no—
281C2×SD16—42no—
282C2×SD16—42no—
283C2×SD16—416no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—22no—
289C2×M16—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8#4114yes—
295C22×C8#4114yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2411yescharacteristic
299C22×C8C2411yescharacteristic
300C22×C8C2411yescharacteristic
301C22×C8C2411yescharacteristic
302C4⋊Q8—42no—
303C4⋊Q8—42no—
304#125—42no—
305#125—42no—
306#124—42no—
307#124—42no—
308#124—42no—
309#124—42no—
310#124—48no—
311C22⋊Q8—48no—
312C4×Q8—216no—
313C4×Q8—28no—
314C4×Q8—28no—
315C4×Q8—48no—
316#117—216no—
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—22no—
328#117#4114yes—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—48no—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C42—24no—
361C2×C42—21no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—21no—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—24no—
368C2×C42C2×D412yes—
369C2×C42C2×D414yes—
370C2×C42C2×D414yes—
371C2×C42C2412yes—
372C2×C42C2412yes—
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375#107—42no—
376#107—42no—
377#106—42no—
378#106—42no—
379#103—24no—
380#103—24no—
381#103—28no—
382#103—44no—
383#103—44no—
384#103—44no—
385#102—24no—
386#102—24no—
387#102—28no—
388#102—44no—
389#97—24no—
390#97—24no—
391C4×C8—24no—
392C4×C8—24no—
393#95—24no—
394#95—24no—
395#95—22no—
396#95—24no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95—22no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95C2×D414yes—
406#95C2×D414yes—
407C24—22no—
408C24—42no—
409C24—42no—
410C24—42no—
411C24—82no—
412C24—82no—
413C24C22×D411yescharacteristic
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—28no—
417C2×Q8—28no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—44no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8—82no—
441C2×Q8—82no—
442C2×Q8C2×#4114yes—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—44no—
453C2×D4—44no—
454C2×D4—44no—
455C2×D4—44no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—82no—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D414yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C4C2511yescharacteristic
487Q16—48no—
488Q16—48no—
489SD16—48no—
490SD16—48no—
491M16—44no—
492M16—44no—
493C2×C8—28no—
494C2×C8—28no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8C2×#4112yes—
510C2×C8C2×#4112yes—
511C2×C8C2×#4112yes—
512C2×C8C2×#4112yes—
513C4⋊C4—216no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—84no—
532C4⋊C4—82no—
533C4⋊C4C2×#4114yes—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—48no—
538C22⋊C4—48no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—84no—
543C22⋊C4—84no—
544C22⋊C4—84no—
545C22⋊C4—84no—
546C42—216no—
547C42—24no—
548C42—24no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—44no—
558C42—44no—
559C42—42no—
560C42—42no—
561C42—42no—
562C42—84no—
563C23—22no—
564C23—22no—
565C23—44no—
566C23—44no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—88no—
571C23—84no—
572C23—162no—
573C23—162no—
574C23—164no—
575C23#52512yes—
576C23#53212yes—
577C23#53512yes—
578C23C23×D411yescharacteristic
579Q8—24no—
580Q8—24no—
581Q8—48no—
582Q8—42no—
583Q8—42no—
584Q8—42no—
585Q8—42no—
586Q8—42no—
587Q8—42no—
588Q8—88no—
589Q8—84no—
590D4—42no—
591D4—42no—
592D4—42no—
593D4—42no—
594D4—84no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—48no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
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—84no—
630C2×C4—84no—
631C2×C4—162no—
632C2×C4#52512yes—
633C2×C4#53212yes—
634C2×C4#53512yes—
635C2×C4#55114yes—
636C2×C4#55114yes—
637C2×C4#55514yes—
638C2×C4#55514yes—
639C2×C4C22×#4111yescharacteristic
640C2×C4C22×#4111yescharacteristic
641C2×C4C22×#4111yescharacteristic
642C2×C4C22×#4111yescharacteristic
643C2×C4C2×H(5,𝔽2)11yescharacteristic
644C2×C4C2×H(5,𝔽2)11yescharacteristic
645C8—22no—
646C8—22no—
647C8—22no—
648C8—22no—
649C8—42no—
650C8—42no—
651C22—22no—
652C22—22no—
653C22—24no—
654C22—24no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—84no—
659C22—84no—
660C22—168no—
661C22#261014yes—
662C22C2×#52511yescharacteristic
663C22C2×#53211yescharacteristic
664C22C2×#53511yescharacteristic
665C22#358412yes—
666C22#358612yes—
667C4—42no—
668C4—42no—
669C4—44no—
670C4—42no—
671C4—42no—
672C4—44no—
673C4—44no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4#261014yes—
680C4C2×#55111yescharacteristic
681C4C2×#55111yescharacteristic
682C4C2×#55511yescharacteristic
683C4C2×#55511yescharacteristic
684C2—24no—
685C2—162no—
686C2C2×#261011yescharacteristic
687C2#6059512yes—
688C2#6071311yescharacteristic
689C2#6072011yescharacteristic
690C2#6081612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34254C2semidirect
3. #29571C2semidirect
4. #29567C2semidirect
5. #29563C2semidirect
6. #29557C2semidirect
7. #23551C2semidirect
9. #23455C2semidirect
11. #21232C2semidirect
14. #21090C2semidirect
15. #21066C2semidirect
18. #20935C2semidirect
19. #20368C2semidirect
45. #2320C22semidirect
53. #2295C22semidirect
57. #2059C22semidirect
58. #2059C22semidirect
59. #2058C22semidirect
60. #2057C22semidirect
63. #2026C22semidirect
64. #2024C22semidirect
70. #1971C22semidirect
71. #1969C22semidirect
74. #1935C22semidirect
76. #1933C22semidirect
77. #1928C22semidirect
78. #1927C22semidirect
83. #1878C22semidirect
84. #1878C22semidirect
85. #1746C22semidirect
86. #1746C22semidirect
87. #1745C22semidirect
88. #1744C22semidirect

Maximal subgroups

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

subgroupindex
#353382
#342542
#295712
#295672
#295632
#295572
#235512
#235062
#234552
C2×#20152
#212322
#212312
#210912
#210902
#210662
#210652
#209362
#209352
#203682