Finite Group Databasebeta

← #7510516

Subgroups — #7510516

Order 512.

1939 conjugacy classes of subgroups.

The subgroup lattice has 743 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35556C211yesmaximal, characteristic, complemented
2#35515C211yesmaximal, characteristic, complemented
3#32606C214yesmaximal, complemented
4#32605C214yesmaximal, complemented
5#32599C214yesmaximal, complemented
6#32596C214yesmaximal, complemented
7#23737C211yesmaximal, characteristic, complemented
8#23632C211yesmaximal, characteristic, complemented
9#23618C211yesmaximal, characteristic, complemented
10#23546C211yesmaximal, characteristic, complemented
11#23511C211yesmaximal, characteristic, complemented
12#23422C211yesmaximal, characteristic
13#21300C211yesmaximal, characteristic, complemented
14C2×#1970C211yesmaximal, characteristic, complemented
15#21083C211yesmaximal, characteristic, complemented
16#21063C211yesmaximal, characteristic, complemented
17#21037C211yesmaximal, characteristic, complemented
18#20934C211yesmaximal, characteristic, complemented
19#20929C211yesmaximal, characteristic, complemented
20C22×AGL(1,ℤ/8)C2211yescharacteristic
21C23×D8C2211yescharacteristic
22C2×#535C2211yescharacteristic
23C2×#533C2211yescharacteristic
24C2×#518C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C2×#417C2211yescharacteristic
29C2×#418C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C2×#416C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×C4×C8C2211yescharacteristic
37#2592C2214yescomplemented
38#2580C2214yescomplemented
39#2548C2214yescomplemented
40#2538C2214yescomplemented
41#2491—22no—
42#2490—22no—
43#2472—24no—
44#2454—22no—
45#2452—22no—
46#2437—24no—
47#2317C2211yescharacteristic
48#2316C2211yescharacteristic
49#2311C2211yescharacteristic
50#2309C2211yescharacteristic
51#2309C2211yescharacteristic
52#2297C2211yescharacteristic
53#2296C2211yescharacteristic
54C2×#390C2211yescharacteristic
55C2×#390C2211yescharacteristic
56#2274C2211yescharacteristic
57#2097C2212yescomplemented
58#2095C2212yescomplemented
59#2093C2212yescomplemented
60#2092C2212yescomplemented
61#2073C2214yescomplemented
62#2071C2214yescomplemented
63#2067C2214yescomplemented
64#2065C2214yescomplemented
65#2055C2214yescomplemented
66#2054C2214yescomplemented
67#2042C2214yescomplemented
68#2041C2214yescomplemented
69#2012C2212yes—
70#2012C2212yes—
71#2011C2212yes—
72#2011C2212yes—
73#1984C2214yescomplemented
74#1984C2214yescomplemented
75#1970C2214yes—
76#1970C2214yes—
77#1934C2212yescomplemented
78#1934C2212yescomplemented
79#1933C2212yescomplemented
80#1933C2212yescomplemented
81#1926C2214yescomplemented
82#1925C2214yescomplemented
83#1921C2214yescomplemented
84#1918C2214yescomplemented
85#1879C2214yes—
86#1879C2214yes—
87#1879C2214yescomplemented
88#1877C2214yescomplemented
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96#1729C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C2×AGL(1,ℤ/8)—24no—
101C2×AGL(1,ℤ/8)—42no—
102C2×AGL(1,ℤ/8)—42no—
103C22×SD16C2312yes—
104C22×D8—24no—
105C22×D8—24no—
106C22×D8—42no—
107C22×D8C2312yes—
108C22×D8C2312yes—
109C22×D8C2312yes—
110C22×M16C2311yescharacteristic
111C22×M16C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C23×C8C2311yescharacteristic
114#535—42no—
115#535—42no—
116#533—42no—
117#533—42no—
118C2×C4⋊Q8D412yes—
119C2×#127C2312yes—
120C2×#125D412yes—
121C2×#124—24no—
122C2×#124C2312yes—
123C2×C22⋊Q8—24no—
124C2×C22⋊Q8D414yes—
125C2×#121—22no—
126C2×#120—22no—
127#518—42no—
128#518—42no—
129C2×C4×Q8C2314yes—
130C2×C4×D4—22no—
131C2×C4×D4C2314yes—
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C22⋊C4—22no—
140C22×C22⋊C4—22no—
141C22×C22⋊C4—22no—
142C22×C42C2311yescharacteristic
143#428—42no—
144C2×#107—24no—
145C2×#107C2312yes—
146C2×#107C2312yes—
147C2×#107C2312yes—
148C2×#106C2312yes—
149#423—42no—
150C2×C4⋊C8C2314yes—
151#418—24no—
152#418—42no—
153#418—42no—
154#417—42no—
155#416—24no—
156#416—42no—
157C2×#103C2311yescharacteristic
158C2×#103C2311yescharacteristic
159C2×#103C2312yes—
160C2×#103C2312yes—
161C2×#102—42no—
162C2×#102—42no—
163C2×#102C2314yes—
164C2×#102C2311yescharacteristic
165C2×#102C2311yescharacteristic
166C2×#102C2312yes—
167C2×#102C2312yes—
168C2×#102C2314yes—
169C2×#102C2312yes—
170C2×#102C2314yes—
171C2×#102C2314yes—
172C2×#102C2312yes—
173C2×C4×C8—42no—
174C2×C4×C8C2312yes—
175C2×C4×C8C2312yes—
176#399—24no—
177#397—24no—
178#396—24no—
179#396—24no—
180#394—24no—
181#394—24no—
182#393—24no—
183#393—24no—
184#391C2312yes—
185#391C2312yes—
186#390—22no—
187#390—22no—
188#390—24no—
189#390—21no—
190#390—21no—
191#390—21no—
192#390—21no—
193#390C2312yes—
194#390C2314yes—
195#390C2312yes—
196#390C2312yes—
197#390C2314yes—
198#390C2312yes—
199#390C2314yes—
200#389—22no—
201#389—22no—
202#389C2314yes—
203#389C2312yes—
204#389C2312yes—
205#388—24no—
206#388—24no—
207#386—22no—
208#386—22no—
209#385—22no—
210#385—24no—
211#385—24no—
212#385—22no—
213#384C2312yes—
214#382C2312yes—
215#380—22no—
216#380—24no—
217#380—22no—
218#379—22no—
219#379—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4C2314yes—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—21no—
226C4×C22⋊C4—21no—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#336C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2312yes—
237#336C2312yes—
238#336C2314yes—
239C25—21no—
240C25—21no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—22no—
247C22×Q8C2×D414yes—
248C22×Q8C2411yescharacteristic
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4—24no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4—22no—
256C22×D4—22no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4#4114yes—
265C22×D4C2412yes—
266C22×D4C2411yescharacteristic
267C22×D4C2412yes—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4—21no—
271C23×C4—24no—
272C23×C4C2×D412yes—
273C23×C4C2411yescharacteristic
274AGL(1,ℤ/8)—416no—
275C2×SD16—24no—
276C2×SD16—42no—
277C2×SD16—42no—
278C2×D8—24no—
279C2×D8—24no—
280C2×D8—24no—
281C2×D8—24no—
282C2×D8—24no—
283C2×D8—42no—
284C2×D8—42no—
285C2×D8—42no—
286C2×D8—42no—
287C2×D8—416no—
288C2×M16—22no—
289C2×M16—22no—
290C2×M16—22no—
291C2×M16—22no—
292C2×M16—22no—
293C2×M16—22no—
294C22×C8—22no—
295C22×C8—22no—
296C22×C8—22no—
297C22×C8—22no—
298C22×C8—22no—
299C22×C8#4114yes—
300C22×C8C2412yes—
301C22×C8C2412yes—
302C22×C8C2412yes—
303C22×C8C2411yescharacteristic
304C22×C8C2411yescharacteristic
305C4⋊Q8—44no—
306#127—42no—
307#127—42no—
308#125—44no—
309#124—42no—
310#124—42no—
311#124—88no—
312C22⋊Q8—48no—
313C22⋊Q8—88no—
314#121—48no—
315#120—48no—
316C4×Q8—44no—
317C4×Q8—44no—
318C4×D4—44no—
319C4×D4—44no—
320C4×D4—48no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117—22no—
327#117—22no—
328#117—22no—
329#117—22no—
330#117—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D414yes—
344C2×C4⋊C4C2×D414yes—
345C2×C4⋊C4C2×D414yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2411yescharacteristic
351C2×C4⋊C4C2411yescharacteristic
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—48no—
369C2×C22⋊C4—84no—
370C2×C22⋊C4—84no—
371C2×C22⋊C4C2×D412yes—
372C2×C22⋊C4C2×D412yes—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—21no—
376C2×C42—21no—
377C2×C42—21no—
378C2×C42—24no—
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2412yes—
382C2×C42C2412yes—
383C2×C42C2412yes—
384#107—24no—
385#107—24no—
386#107—42no—
387#106—42no—
388C4⋊C8—44no—
389#103—22no—
390#103—22no—
391#103—42no—
392#103—42no—
393#102—28no—
394#102—22no—
395#102—22no—
396#102—44no—
397#102—44no—
398#102—42no—
399#102—42no—
400#102—42no—
401#102—42no—
402#102—44no—
403C4×C8—42no—
404C4×C8—42no—
405#95—24no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—24no—
411#95—24no—
412#95—24no—
413C24—22no—
414C24—24no—
415C24—22no—
416C24—21no—
417C24—42no—
418C24—42no—
419C24—42no—
420C24—42no—
421C24—42no—
422C24—82no—
423C24—82no—
424C24—82no—
425C24—82no—
426C24C22×D411yescharacteristic
427#41—44no—
428#41—44no—
429#41—44no—
430#41—44no—
431C2×Q8—28no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×D4—28no—
441C2×D4—24no—
442C2×D4—28no—
443C2×D4—22no—
444C2×D4—21no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—21no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—22no—
456C2×D4—21no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—21no—
460C2×D4—48no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—44no—
467C2×D4—44no—
468C2×D4—44no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—44no—
473C2×D4—44no—
474C2×D4—44no—
475C2×D4—84no—
476C2×D4—84no—
477C2×D4C2×#4114yes—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—42no—
488C22×C4—48no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—84no—
495C22×C4—84no—
496C22×C4—82no—
497C22×C4—82no—
498C22×C4C22×D414yes—
499C22×C4C22×D412yes—
500C22×C4C22×D414yes—
501C22×C4C22×D412yes—
502C22×C4C22×D411yescharacteristic
503C22×C4C2×#4112yes—
504C22×C4C2×#4112yes—
505C22×C4H(5,𝔽2)12yes—
506C22×C4H(5,𝔽2)12yes—
507C22×C4H(5,𝔽2)12yes—
508C22×C42−1+412yes—
509C22×C4C2511yescharacteristic
510SD16—48no—
511D8—48no—
512D8—48no—
513D8—48no—
514M16—44no—
515M16—44no—
516C2×C8—28no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—44no—
531C2×C8C2×#4112yes—
532C2×C8C2×#4112yes—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—22no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—21no—
542C4⋊C4—22no—
543C4⋊C4—21no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—44no—
553C4⋊C4—82no—
554C4⋊C4—84no—
555C4⋊C4—84no—
556C22⋊C4—24no—
557C22⋊C4—24no—
558C22⋊C4—48no—
559C22⋊C4—48no—
560C22⋊C4—42no—
561C22⋊C4—42no—
562C22⋊C4—42no—
563C22⋊C4—48no—
564C22⋊C4—48no—
565C22⋊C4—44no—
566C22⋊C4—82no—
567C22⋊C4—84no—
568C22⋊C4—84no—
569C22⋊C4—84no—
570C22⋊C4—84no—
571C22⋊C4—84no—
572C22⋊C4—84no—
573C22⋊C4—84no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—42no—
583C42—42no—
584C42—42no—
585C42—84no—
586C23—22no—
587C23—22no—
588C23—24no—
589C23—44no—
590C23—42no—
591C23—42no—
592C23—44no—
593C23—48no—
594C23—42no—
595C23—44no—
596C23—42no—
597C23—41no—
598C23—41no—
599C23—42no—
600C23—41no—
601C23—42no—
602C23—42no—
603C23—44no—
604C23—88no—
605C23—88no—
606C23—84no—
607C23—164no—
608C23—164no—
609C23—162no—
610C23—162no—
611C23—162no—
612C23—162no—
613C23#53412yes—
614C23#54812yes—
615C23#55012yes—
616C23C23×D411yescharacteristic
617Q8—41no—
618Q8—41no—
619Q8—41no—
620Q8—41no—
621Q8—88no—
622D4—24no—
623D4—24no—
624D4—41no—
625D4—42no—
626D4—42no—
627D4—42no—
628D4—41no—
629D4—42no—
630D4—41no—
631D4—48no—
632D4—42no—
633D4—42no—
634D4—41no—
635D4—84no—
636D4—84no—
637D4—168no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—22no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—22no—
648C2×C4—24no—
649C2×C4—48no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—41no—
655C2×C4—42no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—41no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—84no—
668C2×C4—168no—
669C2×C4—162no—
670C2×C4—162no—
671C2×C4#53412yes—
672C2×C4#53814yes—
673C2×C4#54214yes—
674C2×C4#54812yes—
675C2×C4#55012yes—
676C2×C4#55114yes—
677C2×C4#56214yes—
678C2×C4#57614yes—
679C2×C4C22×#4111yescharacteristic
680C2×C4C22×#4111yescharacteristic
681C2×C4C2×H(5,𝔽2)11yescharacteristic
682C2×C4C2×H(5,𝔽2)11yescharacteristic
683C2×C4C2×H(5,𝔽2)11yescharacteristic
684C2×C4C2×2−1+411yescharacteristic
685C8—22no—
686C8—22no—
687C8—42no—
688C8—42no—
689C8—42no—
690C22—24no—
691C22—24no—
692C22—22no—
693C22—22no—
694C22—48no—
695C22—44no—
696C22—84no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—84no—
701C22—82no—
702C22—84no—
703C22—81no—
704C22—82no—
705C22—81no—
706C22—82no—
707C22—81no—
708C22—82no—
709C22—168no—
710C22—168no—
711C22#273014yes—
712C22C2×#53411yescharacteristic
713C22C2×#54811yescharacteristic
714C22C2×#55011yescharacteristic
715C22C2×#57612yes—
716C22#358412yes—
717C4—24no—
718C4—42no—
719C4—44no—
720C4—44no—
721C4—82no—
722C4—82no—
723C4—81no—
724C4—81no—
725C4—82no—
726C4—82no—
727C4—81no—
728C4#273014yes—
729C4C2×#53811yescharacteristic
730C4C2×#54211yescharacteristic
731C4C2×#55111yescharacteristic
732C4C2×#56211yescharacteristic
733C2—24no—
734C2—84no—
735C2—161no—
736C2—162no—
737C2C2×#273011yescharacteristic
738C2#6076812yes—
739C2#6132512yes—
740C2#6140911yescharacteristic
741C2#6141311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#23422C2
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. #35556C2semidirect
2. #35515C2semidirect
3. #32606C2semidirect
4. #32605C2semidirect
5. #32599C2semidirect
6. #32596C2semidirect
7. #23737C2semidirect
8. #23632C2semidirect
9. #23618C2semidirect
10. #23546C2semidirect
11. #23511C2semidirect
13. #21300C2semidirect
14. C2×#1970C2semidirect
15. #21083C2semidirect
16. #21063C2semidirect
17. #21037C2semidirect
18. #20934C2semidirect
19. #20929C2semidirect
37. #2592C22semidirect
38. #2580C22semidirect
39. #2548C22semidirect
40. #2538C22semidirect
57. #2097C22semidirect
58. #2095C22semidirect
59. #2093C22semidirect
60. #2092C22semidirect
61. #2073C22semidirect
62. #2071C22semidirect
63. #2067C22semidirect
64. #2065C22semidirect
65. #2055C22semidirect
66. #2054C22semidirect
67. #2042C22semidirect
68. #2041C22semidirect
73. #1984C22semidirect
74. #1984C22semidirect
77. #1934C22semidirect
78. #1934C22semidirect
79. #1933C22semidirect
80. #1933C22semidirect
81. #1926C22semidirect
82. #1925C22semidirect
83. #1921C22semidirect
84. #1918C22semidirect
87. #1879C22semidirect
88. #1877C22semidirect

Maximal subgroups

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

subgroupindex
#355562
#355152
#326062
#326052
#325992
#325962
#237372
#236322
#236182
#235462
#235112
#234222
#213002
C2×#19702
#210832
#210632
#210372
#209342
#209292