Finite Group Databasebeta

← #7506483

Subgroups — #7506483

Order 512.

1940 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35667C211yesmaximal, characteristic
2#35619C211yesmaximal, characteristic
3#31857C214yesmaximal, complemented
4#31856C214yesmaximal, complemented
5#31841C214yesmaximal, complemented
6#31840C214yesmaximal, complemented
7#23744C211yesmaximal, characteristic, complemented
8#23633C211yesmaximal, characteristic, complemented
9#23542C211yesmaximal, characteristic, complemented
10C2×#2016C211yesmaximal, characteristic, complemented
11#23389C211yesmaximal, characteristic
12#23389C211yesmaximal, characteristic
13#21202C211yesmaximal, characteristic, complemented
14#21094C211yesmaximal, characteristic, complemented
15#21048C211yesmaximal, characteristic
16#20935C211yesmaximal, characteristic, complemented
17#20913C211yesmaximal, characteristic
18#20913C211yesmaximal, characteristic
19#20651C211yesmaximal, characteristic, complemented
20C23×Q16C2211yescharacteristic
21C23×Q16C2211yescharacteristic
22C2×#536C2211yescharacteristic
23C2×#536C2211yescharacteristic
24C2×#536C2211yescharacteristic
25C22×C4⋊Q8C2211yescharacteristic
26C2×#520C2211yescharacteristic
27C2×#519C2211yescharacteristic
28C2×#424C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#419C2211yescharacteristic
32C2×#419C2211yescharacteristic
33C2×#419C2211yescharacteristic
34C2×#419C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C2×#408C2211yescharacteristic
38C2×#408C2211yescharacteristic
39C2×C4×M16C2211yescharacteristic
40#2546—22no—
41#2545—22no—
42#2503C2212yes—
43#2502C2212yes—
44#2499C2212yes—
45#2499C2212yes—
46#2447—22no—
47#2446—22no—
48#2429C2214yes—
49#2429C2214yes—
50#2419—24no—
51#2413—24no—
52#2317C2211yescharacteristic
53#2310C2211yescharacteristic
54#2310C2211yescharacteristic
55#2297C2211yescharacteristic
56#2296C2211yescharacteristic
57C2×#391C2211yescharacteristic
58#2285C2211yescharacteristic
59#2101C2212yes—
60#2101C2212yes—
61#2100C2212yes—
62#2099C2212yes—
63#2074C2214yes—
64#2072C2214yes—
65#2056C2214yes—
66#2056C2214yes—
67#2016C2214yes—
68#2016C2214yes—
69#2001C2214yes—
70#2001C2214yes—
71#2001C2214yes—
72#2001C2214yes—
73#1960C2212yes—
74#1960C2212yes—
75#1956C2212yes—
76#1956C2212yes—
77#1935C2214yes—
78#1933C2214yes—
79#1924C2214yes—
80#1923C2214yes—
81#1878C2214yes—
82#1878C2214yes—
83#1873C2214yes—
84#1873C2214yes—
85#1872C2214yes—
86#1872C2214yes—
87#1805C2214yes—
88#1803C2214yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×Q8C2311yescharacteristic
100C23×Q8C2311yescharacteristic
101C22×Q16—24no—
102C22×Q16—42no—
103C22×Q16—42no—
104C22×Q16—42no—
105C22×Q16—42no—
106C22×Q16—42no—
107C22×Q16C2314yes—
108C22×Q16C2314yes—
109C22×M16C2311yescharacteristic
110C22×M16C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113#536—24no—
114#536—24no—
115#536—42no—
116#536—42no—
117#536—42no—
118#536—42no—
119C2×C4⋊Q8—21no—
120C2×C4⋊Q8—21no—
121C2×C4⋊Q8—21no—
122C2×C4⋊Q8—24no—
123C2×C4⋊Q8—22no—
124C2×C4⋊Q8—42no—
125C2×C4⋊Q8C2312yes—
126C2×C4⋊Q8C2312yes—
127C2×#124—21no—
128C2×#124—21no—
129C2×#124—21no—
130C2×#123—22no—
131C2×#123—22no—
132C2×#123—22no—
133C2×C22⋊Q8C2314yes—
134C2×C22⋊Q8C2314yes—
135C2×C22⋊Q8C2312yes—
136C2×C22⋊Q8C2312yes—
137#520—42no—
138#520—42no—
139#519—24no—
140#519—42no—
141C2×C4×Q8—21no—
142C2×C4×Q8—22no—
143C2×C4×Q8C2314yes—
144C2×C4×D4—22no—
145C2×C4×D4—21no—
146C2×#117—22no—
147C2×#117C2311yescharacteristic
148C2×#117C2311yescharacteristic
149C2×#117C2311yescharacteristic
150C2×#117C2311yescharacteristic
151C2×#117C2312yes—
152C2×#117C2311yescharacteristic
153C22×C4⋊C4—22no—
154C22×C4⋊C4C2311yescharacteristic
155C22×C4⋊C4C2312yes—
156C22×C42C2311yescharacteristic
157#424—42no—
158C2×C4⋊C8C2314yes—
159#419—24no—
160#419—42no—
161#419—42no—
162#419—42no—
163#419—42no—
164#419—42no—
165C2×#103—42no—
166C2×#103—42no—
167C2×#103C2314yes—
168C2×#103C2312yes—
169C2×#103C2312yes—
170C2×#103C2314yes—
171C2×#103C2312yes—
172C2×#103C2314yes—
173C2×#103C2312yes—
174C2×#103C2312yes—
175C2×#103C2314yes—
176C2×#103C2312yes—
177C2×#103C2312yes—
178C2×#103C2312yes—
179#408—42no—
180#408—42no—
181C2×C22⋊C8C2314yes—
182C2×C22⋊C8C2314yes—
183C4×M16—42no—
184C2×#97C2312yes—
185C2×C4×C8C2312yes—
186#400—24no—
187#398—24no—
188#398—24no—
189#398—24no—
190#397—24no—
191#397—24no—
192#396—24no—
193#396—24no—
194#391—22no—
195#391—22no—
196#391—22no—
197#391D412yes—
198#391C2314yes—
199#391C2312yes—
200#391C2312yes—
201#391C2312yes—
202#391C2312yes—
203#390D412yes—
204#389—22no—
205#389—24no—
206#389—24no—
207#389C2312yes—
208#389C2312yes—
209#388—24no—
210#388—24no—
211#387—24no—
212#387—24no—
213#386D414yes—
214#385—22no—
215#385—22no—
216#385—24no—
217#384C2312yes—
218#384C2311yescharacteristic
219#384C2311yescharacteristic
220#382C2312yes—
221#382C2311yescharacteristic
222#382C2311yescharacteristic
223#381—22no—
224#381—22no—
225#381—24no—
226#380—22no—
227C4×C4⋊C4—22no—
228C4×C4⋊C4—22no—
229C4×C22⋊C4C2312yes—
230C4×C22⋊C4C2312yes—
231C2×#95—22no—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#340C2314yes—
236#340C2312yes—
237#340C2312yes—
238#340C2314yes—
239#340C2312yes—
240#340C2312yes—
241#340C2312yes—
242#336C2314yes—
243#336C2312yes—
244#336C2312yes—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41—22no—
251C22×Q8—21no—
252C22×Q8—21no—
253C22×Q8—21no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8—22no—
259C22×Q8—22no—
260C22×Q8—22no—
261C22×Q8—22no—
262C22×Q8—22no—
263C22×Q8—22no—
264C22×Q8—22no—
265C22×Q8—22no—
266C22×Q8—22no—
267C22×Q8—22no—
268C22×Q8—22no—
269C22×Q8—22no—
270C22×Q8—22no—
271C22×Q8C2×D411yescharacteristic
272C22×Q8#4114yes—
273C22×Q8C2412yes—
274C22×Q8C2412yes—
275C22×Q8C2412yes—
276C22×D4C2×D412yes—
277C22×D4C2×D411yescharacteristic
278C23×C4—21no—
279C23×C4—21no—
280C23×C4—21no—
281C23×C4—21no—
282C23×C4—22no—
283C23×C4C2×D412yes—
284C23×C4C2412yes—
285C23×C4C2411yescharacteristic
286C2×Q16—28no—
287C2×Q16—44no—
288C2×Q16—44no—
289C2×Q16—44no—
290C2×Q16—44no—
291C2×Q16—44no—
292C2×Q16—416no—
293C2×Q16—416no—
294C2×M16—22no—
295C2×M16—22no—
296C2×M16—22no—
297C2×M16—22no—
298C2×M16—22no—
299C2×M16—22no—
300C22×C8—22no—
301C22×C8—22no—
302C22×C8—22no—
303C22×C8—22no—
304C22×C8—22no—
305C22×C8—22no—
306C22×C8C2412yes—
307C22×C8C2412yes—
308C22×C8C2412yes—
309C22×C8C2412yes—
310C4⋊Q8—24no—
311C4⋊Q8—24no—
312C4⋊Q8—42no—
313C4⋊Q8—42no—
314C4⋊Q8—48no—
315C4⋊Q8—44no—
316C4⋊Q8—44no—
317C4⋊Q8—44no—
318#124—44no—
319#124—44no—
320#124—44no—
321#123—48no—
322#123—48no—
323#123—48no—
324C22⋊Q8—28no—
325C22⋊Q8—28no—
326C22⋊Q8—44no—
327C22⋊Q8—44no—
328C22⋊Q8—42no—
329C22⋊Q8—42no—
330C22⋊Q8—42no—
331C22⋊Q8—42no—
332C4×Q8—28no—
333C4×Q8—44no—
334C4×Q8—48no—
335C4×Q8—44no—
336C4×D4—48no—
337C4×D4—44no—
338#117—22no—
339#117—22no—
340#117—22no—
341#117—22no—
342#117—22no—
343#117—22no—
344#117—22no—
345#117—22no—
346#117—22no—
347#117—48no—
348#117—42no—
349#117—42no—
350#117#4114yes—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—24no—
361C2×C4⋊C4—24no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—24no—
367C2×C4⋊C4—42no—
368C2×C4⋊C4—42no—
369C2×C4⋊C4—48no—
370C2×C4⋊C4C2×D412yes—
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2411yescharacteristic
373C2×C4⋊C4C2411yescharacteristic
374C2×C4⋊C4C2412yes—
375C2×C4⋊C4C2411yescharacteristic
376C2×C4⋊C4C2411yescharacteristic
377C2×C4⋊C4C2412yes—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—24no—
386C2×C22⋊C4—24no—
387C2×C22⋊C4C2×D412yes—
388C2×C22⋊C4C2×D414yes—
389C2×C22⋊C4C2×D414yes—
390C2×C42—21no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42—24no—
394C2×C42—21no—
395C2×C42—22no—
396C2×C42—21no—
397C2×C42—21no—
398C2×C42C2×D412yes—
399C2×C42C2412yes—
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402C4⋊C8—44no—
403#103—28no—
404#103—44no—
405#103—44no—
406#103—42no—
407#103—42no—
408#103—42no—
409#103—42no—
410#103—44no—
411#103—42no—
412#103—42no—
413#103—42no—
414#103—42no—
415C22⋊C8—44no—
416C22⋊C8—44no—
417#97—42no—
418C4×C8—42no—
419#95—24no—
420#95—24no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—24no—
426#95—24no—
427#95—24no—
428#95—24no—
429#95—24no—
430#95C2×D414yes—
431#95C2×D414yes—
432C24C22×D412yes—
433C24C22×D411yescharacteristic
434#41—44no—
435#41—44no—
436#41—44no—
437#41—44no—
438C2×Q8—22no—
439C2×Q8—24no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—28no—
444C2×Q8—28no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—42no—
462C2×Q8—42no—
463C2×Q8—42no—
464C2×Q8—44no—
465C2×Q8—42no—
466C2×Q8—42no—
467C2×Q8—42no—
468C2×Q8—42no—
469C2×Q8—42no—
470C2×Q8—44no—
471C2×Q8—44no—
472C2×Q8—44no—
473C2×Q8—44no—
474C2×Q8—44no—
475C2×Q8—44no—
476C2×Q8—44no—
477C2×Q8—44no—
478C2×Q8—44no—
479C2×Q8C2×#4114yes—
480C2×D4—24no—
481C2×D4—24no—
482C2×D4—24no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—22no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—24no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—21no—
499C22×C4—22no—
500C22×C4—21no—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—22no—
504C22×C4—21no—
505C22×C4—42no—
506C22×C4—42no—
507C22×C4—42no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4—42no—
514C22×C4—44no—
515C22×C4—42no—
516C22×C4—82no—
517C22×C4—84no—
518C22×C4C22×D414yes—
519C22×C4C22×D412yes—
520C22×C4C22×D412yes—
521C22×C4C22×D412yes—
522C22×C4C22×D411yescharacteristic
523C22×C4C2×#4112yes—
524C22×C4C2×#4112yes—
525C22×C4H(5,𝔽2)12yes—
526C22×C4H(5,𝔽2)12yes—
527C22×C4H(5,𝔽2)12yes—
528C22×C4H(5,𝔽2)12yes—
529C22×C4C2511yescharacteristic
530Q16—416no—
531Q16—416no—
532M16—44no—
533M16—44no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—22no—
543C2×C8—22no—
544C2×C8—22no—
545C2×C8—22no—
546C2×C8—44no—
547C2×C8—44no—
548C4⋊C4—24no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—21no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—21no—
561C4⋊C4—22no—
562C4⋊C4—48no—
563C4⋊C4—48no—
564C4⋊C4—48no—
565C4⋊C4—48no—
566C4⋊C4—48no—
567C4⋊C4—44no—
568C4⋊C4—44no—
569C4⋊C4—44no—
570C4⋊C4—44no—
571C4⋊C4—42no—
572C4⋊C4—44no—
573C4⋊C4—44no—
574C4⋊C4—82no—
575C4⋊C4—84no—
576C4⋊C4C2×#4114yes—
577C22⋊C4—24no—
578C22⋊C4—24no—
579C22⋊C4—28no—
580C22⋊C4—44no—
581C22⋊C4—48no—
582C22⋊C4—48no—
583C22⋊C4—44no—
584C22⋊C4—44no—
585C22⋊C4—44no—
586C22⋊C4—42no—
587C22⋊C4—42no—
588C22⋊C4—42no—
589C42—24no—
590C42—24no—
591C42—21no—
592C42—21no—
593C42—22no—
594C42—21no—
595C42—21no—
596C42—22no—
597C42—48no—
598C42—44no—
599C42—42no—
600C42—42no—
601C42—42no—
602C23—22no—
603C23—22no—
604C23—24no—
605C23—22no—
606C23—22no—
607C23—22no—
608C23—24no—
609C23#53512yes—
610C23#54612yes—
611C23#54612yes—
612C23C23×D411yescharacteristic
613Q8—24no—
614Q8—24no—
615Q8—42no—
616Q8—48no—
617Q8—44no—
618Q8—42no—
619Q8—42no—
620Q8—42no—
621Q8—42no—
622Q8—42no—
623Q8—42no—
624Q8—42no—
625Q8—42no—
626Q8—42no—
627Q8—84no—
628Q8—84no—
629D4—48no—
630D4—44no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—24no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—24no—
643C2×C4—22no—
644C2×C4—48no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—48no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—42no—
662C2×C4—41no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—84no—
674C2×C4—84no—
675C2×C4—162no—
676C2×C4—164no—
677C2×C4#53512yes—
678C2×C4#53814yes—
679C2×C4#54014yes—
680C2×C4#54612yes—
681C2×C4#54612yes—
682C2×C4#55314yes—
683C2×C4#56014yes—
684C2×C4#57814yes—
685C2×C4C22×#4111yescharacteristic
686C2×C4C22×#4111yescharacteristic
687C2×C4C2×H(5,𝔽2)11yescharacteristic
688C2×C4C2×H(5,𝔽2)11yescharacteristic
689C2×C4C2×H(5,𝔽2)11yescharacteristic
690C2×C4C2×H(5,𝔽2)11yescharacteristic
691C8—42no—
692C8—42no—
693C8—42no—
694C8—42no—
695C22—24no—
696C22—24no—
697C22—22no—
698C22—22no—
699C22—44no—
700C22—44no—
701C22—42no—
702C22—42no—
703C22—42no—
704C22#270314yes—
705C22C2×#53511yescharacteristic
706C22C2×#54611yescharacteristic
707C22C2×#54611yescharacteristic
708C22C2×#57812yes—
709C22#358612yes—
710C4—24no—
711C4—42no—
712C4—44no—
713C4—44no—
714C4—42no—
715C4—44no—
716C4—82no—
717C4—81no—
718C4—81no—
719C4—81no—
720C4—82no—
721C4—81no—
722C4—82no—
723C4—82no—
724C4#270314yes—
725C4C2×#53811yescharacteristic
726C4C2×#54011yescharacteristic
727C4C2×#55311yescharacteristic
728C4C2×#56011yescharacteristic
729C2—24no—
730C2—82no—
731C2C2×#270311yescharacteristic
732C2#6082111yescharacteristic
733C2#6082611yescharacteristic
734C2#6118612yes—
735C2#6121012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #31857C2semidirect
4. #31856C2semidirect
5. #31841C2semidirect
6. #31840C2semidirect
7. #23744C2semidirect
8. #23633C2semidirect
9. #23542C2semidirect
10. C2×#2016C2semidirect
13. #21202C2semidirect
14. #21094C2semidirect
16. #20935C2semidirect
19. #20651C2semidirect

Maximal subgroups

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

subgroupindex
#356672
#356192
#318572
#318562
#318412
#318402
#237442
#236332
#235422
C2×#20162
#233892
#212022
#210942
#210482
#209352
#209132
#206512