Finite Group Databasebeta

← #7510495

Subgroups — #7510495

Order 512.

1820 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35515C211yesmaximal, characteristic
2#35320C211yesmaximal, characteristic, complemented
3#32621C214yesmaximal, complemented
4#32615C214yesmaximal, complemented
5#32604C214yesmaximal, complemented
6#32598C214yesmaximal, complemented
7#23736C211yesmaximal, characteristic, complemented
8#23633C211yesmaximal, characteristic, complemented
9#23615C211yesmaximal, characteristic, complemented
10#23541C211yesmaximal, characteristic
11#23507C211yesmaximal, characteristic, complemented
12C2×#2013C211yesmaximal, characteristic
13#21304C211yesmaximal, characteristic, complemented
14#21231C211yesmaximal, characteristic
15#21088C211yesmaximal, characteristic, complemented
16#21060C211yesmaximal, characteristic
17#21032C211yesmaximal, characteristic, complemented
18#20933C211yesmaximal, characteristic
19#20933C211yesmaximal, characteristic
20C23×SD16C2211yescharacteristic
21C23×SD16C2211yescharacteristic
22C2×#533C2211yescharacteristic
23C22×C4⋊Q8C2211yescharacteristic
24C2×#518C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#427C2211yescharacteristic
27C2×#424C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#419C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C2×#418C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C2×#405C2211yescharacteristic
37#2862—22no—
38#2642—22no—
39#2580C2214yes—
40#2568C2214yescomplemented
41#2568C2214yescomplemented
42#2538C2214yes—
43#2503—24no—
44#2454—22no—
45#2452—22no—
46#2437—24no—
47#2317C2211yescharacteristic
48#2316C2211yescharacteristic
49#2310C2211yescharacteristic
50#2310C2211yescharacteristic
51#2310C2211yescharacteristic
52#2310C2211yescharacteristic
53#2309C2211yescharacteristic
54#2296C2211yescharacteristic
55C2×#390C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2097C2212yescomplemented
58#2097C2212yescomplemented
59#2096C2212yescomplemented
60#2095C2212yescomplemented
61#2074C2214yescomplemented
62#2072C2214yescomplemented
63#2068C2214yescomplemented
64#2066C2214yescomplemented
65#2055C2214yes—
66#2055C2214yes—
67#2043C2214yescomplemented
68#2043C2214yescomplemented
69#2013C2214yes—
70#2013C2214yes—
71#1985C2214yescomplemented
72#1984C2214yescomplemented
73#1973C2212yes—
74#1973C2212yes—
75#1968C2212yes—
76#1968C2212yes—
77#1935C2214yescomplemented
78#1933C2214yescomplemented
79#1926C2214yes—
80#1925C2214yes—
81#1921C2212yescomplemented
82#1920C2212yescomplemented
83#1919C2212yescomplemented
84#1918C2212yescomplemented
85#1877C2214yes—
86#1877C2214yes—
87#1877C2214yes—
88#1877C2214yes—
89#1736C2211yescharacteristic
90#1735C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C23×Q8C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C22×SD16—24no—
101C22×SD16—24no—
102C22×SD16—24no—
103C22×SD16—42no—
104C22×SD16—42no—
105C22×SD16—42no—
106C22×SD16C2312yes—
107C22×SD16C2312yes—
108C22×SD16C2314yes—
109C22×M16C2311yescharacteristic
110C22×M16C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113#533—42no—
114#533—42no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—42no—
118C2×C4⋊Q8—42no—
119C2×C4⋊Q8D412yes—
120C2×C4⋊Q8C2314yes—
121C2×#125D412yes—
122C2×C22⋊Q8—24no—
123C2×C22⋊Q8D414yes—
124#518—42no—
125#518—42no—
126C2×C4×Q8C2314yes—
127C2×C4×D4—22no—
128C2×C4×D4C2314yes—
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C22×C4⋊C4—22no—
136C22×C4⋊C4C2311yescharacteristic
137C22×C22⋊C4—22no—
138C22×C22⋊C4—22no—
139C22×C42C2311yescharacteristic
140#428—42no—
141#427—24no—
142C2×#107C2311yescharacteristic
143C2×#107C2311yescharacteristic
144C2×#107C2312yes—
145C2×#106C2311yescharacteristic
146C2×#106C2311yescharacteristic
147C2×#106C2312yes—
148#424—42no—
149C2×C4⋊C8C2314yes—
150#419—42no—
151#419—42no—
152#419—42no—
153#419—42no—
154#418—24no—
155#418—42no—
156C2×#103—24no—
157C2×#103—42no—
158C2×#103C2312yes—
159C2×#103C2312yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2312yes—
165C2×#103C2312yes—
166C2×#103C2314yes—
167C2×#102C2314yes—
168C2×#102C2312yes—
169C2×#102C2312yes—
170#405—42no—
171C2×#97C2312yes—
172C2×C4×C8C2312yes—
173#400—24no—
174#400—24no—
175#400—24no—
176#400—24no—
177#399—24no—
178#396—24no—
179#396—24no—
180#393—24no—
181#391—24no—
182#391—24no—
183#391—22no—
184#391—22no—
185#391—22no—
186#391—22no—
187#391C2314yes—
188#391C2312yes—
189#391C2312yes—
190#391C2314yes—
191#391C2312yes—
192#391C2312yes—
193#390—22no—
194#390—21no—
195#390—21no—
196#390C2312yes—
197#390C2312yes—
198#390C2314yes—
199#389—22no—
200#389—22no—
201#389C2314yes—
202#389C2312yes—
203#389C2312yes—
204#385—22no—
205#385—24no—
206#385—24no—
207#385—22no—
208#384C2312yes—
209#382—24no—
210#382—24no—
211#382C2312yes—
212#381—22no—
213#381—22no—
214#381—22no—
215#381—22no—
216#380—24no—
217#380—22no—
218#379—22no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2312yes—
223C4×C22⋊C4—22no—
224#376—21no—
225C43—21no—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2312yes—
237#336C2312yes—
238#336C2314yes—
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×Q8C2×D414yes—
255C22×Q8C2412yes—
256C22×Q8C2412yes—
257C22×D4—21no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4#4114yes—
264C22×D4C2412yes—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—24no—
269C23×C4C2×D412yes—
270C23×C4C2411yescharacteristic
271C2×SD16—24no—
272C2×SD16—24no—
273C2×SD16—28no—
274C2×SD16—24no—
275C2×SD16—24no—
276C2×SD16—42no—
277C2×SD16—42no—
278C2×SD16—44no—
279C2×SD16—44no—
280C2×SD16—416no—
281C2×SD16—416no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C22×C8—22no—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8#4114yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2411yescharacteristic
298C22×C8C2411yescharacteristic
299C4⋊Q8—44no—
300C4⋊Q8—44no—
301C4⋊Q8—44no—
302C4⋊Q8—44no—
303C4⋊Q8—44no—
304#125—44no—
305C22⋊Q8—48no—
306C22⋊Q8—88no—
307C4×Q8—44no—
308C4×Q8—44no—
309C4×D4—44no—
310C4×D4—44no—
311C4×D4—48no—
312#117—22no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—48no—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2×D414yes—
337C2×C4⋊C4C2×D414yes—
338C2×C4⋊C4C2×D414yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2411yescharacteristic
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—48no—
356C2×C22⋊C4—84no—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—21no—
364C2×C42—21no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2412yes—
370C2×C42C2412yes—
371C2×C42C2412yes—
372#107—22no—
373#107—22no—
374#107—42no—
375#106—22no—
376#106—22no—
377#106—42no—
378C4⋊C8—44no—
379#103—24no—
380#103—24no—
381#103—44no—
382#103—42no—
383#103—42no—
384#103—42no—
385#103—42no—
386#103—44no—
387#103—44no—
388#102—28no—
389#102—42no—
390#102—42no—
391#97—42no—
392C4×C8—42no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—24no—
407C24—22no—
408C24—24no—
409C24—42no—
410C24—42no—
411C24—82no—
412C24—82no—
413C24C22×D411yescharacteristic
414C2×Q8—28no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
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—44no—
428C2×Q8—44no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—44no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×D4—28no—
444C2×D4—24no—
445C2×D4—28no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—42no—
452C2×D4—44no—
453C2×D4—42no—
454C2×D4—44no—
455C2×D4C2×#4114yes—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—42no—
468C22×C4—48no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—84no—
477C22×C4—84no—
478C22×C4—82no—
479C22×C4C22×D414yes—
480C22×C4C22×D414yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C42−1+412yes—
490C22×C4C2511yescharacteristic
491SD16—48no—
492SD16—48no—
493SD16—416no—
494M16—44no—
495M16—44no—
496C2×C8—28no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—44no—
511C2×C8C2×#4112yes—
512C2×C8C2×#4112yes—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—48no—
533C4⋊C4—48no—
534C4⋊C4—42no—
535C4⋊C4—44no—
536C4⋊C4—82no—
537C4⋊C4—84no—
538C4⋊C4—84no—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—48no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—48no—
547C22⋊C4—84no—
548C22⋊C4—84no—
549C22⋊C4—84no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—44no—
559C42—42no—
560C42—42no—
561C42—42no—
562C42—42no—
563C42—44no—
564C23—22no—
565C23—22no—
566C23—24no—
567C23—44no—
568C23—42no—
569C23—42no—
570C23—44no—
571C23—48no—
572C23—88no—
573C23—164no—
574C23—162no—
575C23—162no—
576C23#53412yes—
577C23#54812yes—
578C23#55012yes—
579C23C23×D411yescharacteristic
580Q8—42no—
581Q8—42no—
582Q8—42no—
583Q8—42no—
584Q8—42no—
585Q8—42no—
586Q8—42no—
587Q8—42no—
588Q8—88no—
589Q8—84no—
590Q8—84no—
591D4—24no—
592D4—24no—
593D4—42no—
594D4—48no—
595D4—42no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—48no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4—168no—
633C2×C4—162no—
634C2×C4#53412yes—
635C2×C4#53814yes—
636C2×C4#54214yes—
637C2×C4#54812yes—
638C2×C4#55012yes—
639C2×C4#55114yes—
640C2×C4#56214yes—
641C2×C4#57714yes—
642C2×C4C22×#4111yescharacteristic
643C2×C4C22×#4111yescharacteristic
644C2×C4C2×H(5,𝔽2)11yescharacteristic
645C2×C4C2×H(5,𝔽2)11yescharacteristic
646C2×C4C2×H(5,𝔽2)11yescharacteristic
647C2×C4C2×2−1+411yescharacteristic
648C8—22no—
649C8—22no—
650C8—42no—
651C8—42no—
652C8—42no—
653C22—24no—
654C22—24no—
655C22—22no—
656C22—22no—
657C22—48no—
658C22—44no—
659C22—84no—
660C22—82no—
661C22—82no—
662C22—168no—
663C22#273014yes—
664C22C2×#53411yescharacteristic
665C22C2×#54811yescharacteristic
666C22C2×#55011yescharacteristic
667C22C2×#57712yes—
668C22#358612yes—
669C4—24no—
670C4—42no—
671C4—44no—
672C4—44no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—81no—
678C4—81no—
679C4—82no—
680C4—81no—
681C4—81no—
682C4#273014yes—
683C4C2×#53811yescharacteristic
684C4C2×#54211yescharacteristic
685C4C2×#55111yescharacteristic
686C4C2×#56211yescharacteristic
687C2—24no—
688C2—84no—
689C2C2×#273011yescharacteristic
690C2#6075812yes—
691C2#6134012yes—
692C2#6141011yescharacteristic
693C2#6141811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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
2. #35320C2semidirect
3. #32621C2semidirect
4. #32615C2semidirect
5. #32604C2semidirect
6. #32598C2semidirect
7. #23736C2semidirect
8. #23633C2semidirect
9. #23615C2semidirect
11. #23507C2semidirect
13. #21304C2semidirect
15. #21088C2semidirect
17. #21032C2semidirect
40. #2568C22semidirect
41. #2568C22semidirect
57. #2097C22semidirect
58. #2097C22semidirect
59. #2096C22semidirect
60. #2095C22semidirect
61. #2074C22semidirect
62. #2072C22semidirect
63. #2068C22semidirect
64. #2066C22semidirect
67. #2043C22semidirect
68. #2043C22semidirect
71. #1985C22semidirect
72. #1984C22semidirect
77. #1935C22semidirect
78. #1933C22semidirect
81. #1921C22semidirect
82. #1920C22semidirect
83. #1919C22semidirect
84. #1918C22semidirect

Maximal subgroups

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

subgroupindex
#355152
#353202
#326212
#326152
#326042
#325982
#237362
#236332
#236152
#235412
#235072
C2×#20132
#213042
#212312
#210882
#210602
#210322
#209332