Finite Group Databasebeta

← #7505405

Subgroups — #7505405

Order 512.

2024 conjugacy classes of subgroups.

The subgroup lattice has 729 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35668C211yesmaximal, characteristic, complemented
2#35554C211yesmaximal, characteristic
3#31662C214yesmaximal, complemented
4#31661C214yesmaximal, complemented
5#31638C214yesmaximal, complemented
6#31637C214yesmaximal, complemented
7#23744C211yesmaximal, characteristic, complemented
8#23632C211yesmaximal, characteristic, complemented
9#23541C211yesmaximal, characteristic, complemented
10#23456C211yesmaximal, characteristic, complemented
11#23456C211yesmaximal, characteristic, complemented
12C2×#2014C211yesmaximal, characteristic
13#21203C211yesmaximal, characteristic
14#21094C211yesmaximal, characteristic, complemented
15#21022C211yesmaximal, characteristic, complemented
16#20936C211yesmaximal, characteristic, complemented
17#20936C211yesmaximal, characteristic, complemented
18#20933C211yesmaximal, characteristic, complemented
19#20650C211yesmaximal, characteristic, complemented
20C23×SD16C2211yescharacteristic
21C23×SD16C2211yescharacteristic
22C2×#544C2211yescharacteristic
23C22×#127C2211yescharacteristic
24C2×#519C2211yescharacteristic
25C2×#518C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#419C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#418C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C2×C4×M16C2211yescharacteristic
36#2568C2212yescomplemented
37#2567C2212yescomplemented
38#2565C2212yescomplemented
39#2565C2212yescomplemented
40#2453—22no—
41#2451—22no—
42#2445—22no—
43#2445—22no—
44#2445C2214yes—
45#2445C2214yes—
46#2438—24no—
47#2405—24no—
48#2317C2211yescharacteristic
49#2310C2211yescharacteristic
50#2309C2211yescharacteristic
51#2309C2211yescharacteristic
52#2309C2211yescharacteristic
53#2296C2211yescharacteristic
54#2296C2211yescharacteristic
55C2×#391C2211yescharacteristic
56#2285C2211yescharacteristic
57#2101C2212yescomplemented
58#2101C2212yescomplemented
59#2100C2212yescomplemented
60#2099C2212yescomplemented
61#2073C2214yescomplemented
62#2071C2214yescomplemented
63#2055C2214yescomplemented
64#2055C2214yescomplemented
65#2025C2214yescomplemented
66#2025C2214yescomplemented
67#2023C2214yescomplemented
68#2023C2214yescomplemented
69#2014C2214yes—
70#2014C2214yes—
71#1961C2212yes—
72#1961C2212yes—
73#1957C2212yes—
74#1957C2212yes—
75#1935C2214yescomplemented
76#1933C2214yescomplemented
77#1917C2214yescomplemented
78#1916C2214yescomplemented
79#1880C2214yescomplemented
80#1880C2214yescomplemented
81#1880C2214yescomplemented
82#1880C2214yescomplemented
83#1877C2214yescomplemented
84#1877C2214yescomplemented
85#1804C2214yescomplemented
86#1802C2214yescomplemented
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1735C2211yescharacteristic
90#1735C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C23×Q8C2311yescharacteristic
98C23×D4—22no—
99C23×D4C2311yescharacteristic
100C23×D4C2311yescharacteristic
101C22×SD16—24no—
102C22×SD16—24no—
103C22×SD16—24no—
104C22×SD16—42no—
105C22×SD16—42no—
106C22×SD16—42no—
107C22×SD16C2314yes—
108C22×SD16C2314yes—
109C22×M16C2311yescharacteristic
110C22×M16C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113#544—42no—
114#544—42no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8C2312yes—
117C2×#127—24no—
118C2×#127—22no—
119C2×#127—42no—
120C2×#127C2312yes—
121C2×#127C2312yes—
122C2×#125—21no—
123C2×#125C2312yes—
124C2×#123—24no—
125C2×#123—24no—
126C2×C22⋊Q8—22no—
127#519—42no—
128#519—42no—
129#518—24no—
130#518—42no—
131C2×C4×Q8—21no—
132C2×C4×Q8—21no—
133C2×C4×Q8C2312yes—
134C2×C4×Q8C2312yes—
135C2×C4×D4—22no—
136C2×C4×D4C2314yes—
137C2×#117—22no—
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C2×#117C2311yescharacteristic
143C2×#117C2311yescharacteristic
144C22×C4⋊C4C2311yescharacteristic
145C22×C22⋊C4—22no—
146C22×C42C2311yescharacteristic
147#424—42no—
148C2×C4⋊C8C2314yes—
149#419—24no—
150#419—42no—
151#418—42no—
152#418—42no—
153#418—42no—
154#418—42no—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#102—42no—
158C2×#102—42no—
159C2×#102C2312yes—
160C2×#102C2312yes—
161C2×#102C2312yes—
162C2×#102C2314yes—
163C2×#102C2312yes—
164C2×#102C2312yes—
165C2×#102C2314yes—
166C2×#102C2312yes—
167C2×#102C2312yes—
168C2×#102C2312yes—
169C4×M16—42no—
170C2×#97C2312yes—
171C2×C4×C8C2312yes—
172#400—24no—
173#398—24no—
174#398—24no—
175#395—24no—
176#394—24no—
177#394—24no—
178#394—24no—
179#394—24no—
180#391—22no—
181#391—24no—
182#391—22no—
183#391C2312yes—
184#391C2312yes—
185#391C2312yes—
186#391C2312yes—
187#390—21no—
188#390—21no—
189#390—22no—
190#390—21no—
191#390—21no—
192#390C2314yes—
193#390C2314yes—
194#390C2314yes—
195#389—24no—
196#389—22no—
197#389D412yes—
198#389D412yes—
199#389C2312yes—
200#389C2312yes—
201#387—24no—
202#387—24no—
203#386—24no—
204#386—24no—
205#385—22no—
206#385—22no—
207#385D414yes—
208#384C2311yescharacteristic
209#384C2312yes—
210#384C2311yescharacteristic
211#382C2311yescharacteristic
212#382C2312yes—
213#382C2311yescharacteristic
214#381—22no—
215#380—22no—
216#379—22no—
217#379—22no—
218#379—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2312yes—
223C2×#95—22no—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#336C2314yes—
235#336C2312yes—
236#336C2312yes—
237#336C2314yes—
238#336C2314yes—
239C25—21no—
240C25—21no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8#4114yes—
249C22×Q8C2412yes—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4—24no—
253C22×D4—24no—
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—22no—
265C22×D4—48no—
266C22×D4—84no—
267C22×D4—84no—
268C22×D4#4114yes—
269C22×D4C2412yes—
270C22×D4C2412yes—
271C23×C4—21no—
272C23×C4—21no—
273C23×C4—24no—
274C23×C4C2×D412yes—
275C23×C4C2411yescharacteristic
276C2×SD16—28no—
277C2×SD16—28no—
278C2×SD16—28no—
279C2×SD16—44no—
280C2×SD16—44no—
281C2×SD16—44no—
282C2×SD16—416no—
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—22no—
295C22×C8—22no—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C22×C8C2412yes—
300C4⋊Q8—42no—
301C4⋊Q8—42no—
302C4⋊Q8—44no—
303#127—24no—
304#127—24no—
305#127—42no—
306#127—42no—
307#127—48no—
308#125—42no—
309#125—42no—
310#125—44no—
311#123—88no—
312#123—88no—
313C22⋊Q8—48no—
314C4×Q8—42no—
315C4×Q8—42no—
316C4×Q8—42no—
317C4×Q8—42no—
318C4×Q8—44no—
319C4×Q8—44no—
320C4×D4—28no—
321C4×D4—44no—
322C4×D4—48no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117—22no—
327#117—22no—
328#117—22no—
329#117—22no—
330#117—22no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—48no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4C2×D412yes—
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D414yes—
353C2×C4⋊C4C2×D414yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2411yescharacteristic
358C2×C4⋊C4C2411yescharacteristic
359C2×C4⋊C4C2411yescharacteristic
360C2×C4⋊C4C2412yes—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—48no—
373C2×C22⋊C4C2×D412yes—
374C2×C22⋊C4C2×D412yes—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42C2×D412yes—
384C2×C42C2412yes—
385C2×C42C2412yes—
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388C4⋊C8—44no—
389#103—28no—
390#103—44no—
391#102—44no—
392#102—44no—
393#102—42no—
394#102—42no—
395#102—42no—
396#102—42no—
397#102—42no—
398#102—42no—
399#102—42no—
400#102—42no—
401#97—42no—
402C4×C8—42no—
403#95—24no—
404#95—24no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—24no—
410#95—24no—
411#95—24no—
412#95C2×D414yes—
413#95C2×D414yes—
414C24—22no—
415C24—24no—
416C24—22no—
417C24—42no—
418C24—42no—
419C24—42no—
420C24—42no—
421C24—42no—
422C24—82no—
423C24—82no—
424C24—82no—
425C24—82no—
426C24C22×D411yescharacteristic
427C2×Q8—24no—
428C2×Q8—22no—
429C2×Q8—28no—
430C2×Q8—28no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—44no—
436C2×Q8—42no—
437C2×Q8—44no—
438C2×Q8—42no—
439C2×Q8—44no—
440C2×Q8—84no—
441C2×Q8C2×#4114yes—
442C2×D4—28no—
443C2×D4—24no—
444C2×D4—28no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—48no—
457C2×D4—48no—
458C2×D4—48no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—44no—
462C2×D4—44no—
463C2×D4—44no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—44no—
468C2×D4—44no—
469C2×D4—44no—
470C2×D4—48no—
471C2×D4—84no—
472C2×D4—84no—
473C2×D4—832no—
474C2×D4C2×#4114yes—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—42no—
487C22×C4—48no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—84no—
492C22×C4—84no—
493C22×C4—82no—
494C22×C4C22×D412yes—
495C22×C4C22×D414yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D411yescharacteristic
500C22×C4C2×#4112yes—
501C22×C4C2×#4112yes—
502C22×C4H(5,𝔽2)12yes—
503C22×C4H(5,𝔽2)12yes—
504C22×C4H(5,𝔽2)12yes—
505C22×C42−1+412yes—
506C22×C4C2511yescharacteristic
507SD16—416no—
508SD16—416no—
509M16—44no—
510M16—44no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—44no—
524C2×C8—44no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—24no—
528C4⋊C4—28no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—42no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—82no—
554C4⋊C4—84no—
555C22⋊C4—24no—
556C22⋊C4—24no—
557C22⋊C4—48no—
558C22⋊C4—42no—
559C22⋊C4—42no—
560C22⋊C4—42no—
561C22⋊C4—42no—
562C22⋊C4—84no—
563C22⋊C4—84no—
564C22⋊C4—84no—
565C22⋊C4—84no—
566C22⋊C4—84no—
567C22⋊C4—84no—
568C42—24no—
569C42—21no—
570C42—21no—
571C42—22no—
572C42—21no—
573C42—21no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—44no—
578C42—44no—
579C42—42no—
580C42—42no—
581C42—42no—
582C42—42no—
583C23—22no—
584C23—22no—
585C23—24no—
586C23—44no—
587C23—42no—
588C23—42no—
589C23—44no—
590C23—48no—
591C23—44no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—44no—
596C23—88no—
597C23—88no—
598C23—84no—
599C23—162no—
600C23—162no—
601C23—162no—
602C23—162no—
603C23—164no—
604C23—164no—
605C23#53512yes—
606C23#54612yes—
607C23#54812yes—
608C23C23×D411yescharacteristic
609Q8—24no—
610Q8—24no—
611Q8—42no—
612Q8—48no—
613Q8—42no—
614Q8—164no—
615D4—24no—
616D4—24no—
617D4—42no—
618D4—42no—
619D4—42no—
620D4—48no—
621D4—42no—
622D4—42no—
623D4—42no—
624D4—816no—
625D4—816no—
626D4—84no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—22no—
640C2×C4—48no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—48no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—168no—
661C2×C4—162no—
662C2×C4#53512yes—
663C2×C4#53814yes—
664C2×C4#53914yes—
665C2×C4#54612yes—
666C2×C4#54812yes—
667C2×C4#55114yes—
668C2×C4#56214yes—
669C2×C4#57714yes—
670C2×C4C22×#4111yescharacteristic
671C2×C4C22×#4111yescharacteristic
672C2×C4C2×H(5,𝔽2)11yescharacteristic
673C2×C4C2×H(5,𝔽2)11yescharacteristic
674C2×C4C2×H(5,𝔽2)11yescharacteristic
675C2×C4C2×2−1+411yescharacteristic
676C8—42no—
677C8—42no—
678C8—42no—
679C8—42no—
680C22—24no—
681C22—22no—
682C22—22no—
683C22—24no—
684C22—48no—
685C22—44no—
686C22—82no—
687C22—82no—
688C22—84no—
689C22—84no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—84no—
694C22—168no—
695C22—168no—
696C22#269614yes—
697C22C2×#53511yescharacteristic
698C22C2×#54611yescharacteristic
699C22C2×#54811yescharacteristic
700C22C2×#57712yes—
701C22#358512yes—
702C4—24no—
703C4—42no—
704C4—44no—
705C4—44no—
706C4—42no—
707C4—42no—
708C4—82no—
709C4—81no—
710C4—81no—
711C4—81no—
712C4—82no—
713C4—81no—
714C4—82no—
715C4#269614yes—
716C4C2×#53811yescharacteristic
717C4C2×#53911yescharacteristic
718C4C2×#55111yescharacteristic
719C4C2×#56211yescharacteristic
720C2—24no—
721C2—84no—
722C2—162no—
723C2C2×#269611yescharacteristic
724C2#6082011yescharacteristic
725C2#6082511yescharacteristic
726C2#6118412yes—
727C2#6134112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35668C2semidirect
3. #31662C2semidirect
4. #31661C2semidirect
5. #31638C2semidirect
6. #31637C2semidirect
7. #23744C2semidirect
8. #23632C2semidirect
9. #23541C2semidirect
10. #23456C2semidirect
11. #23456C2semidirect
14. #21094C2semidirect
15. #21022C2semidirect
16. #20936C2semidirect
17. #20936C2semidirect
18. #20933C2semidirect
19. #20650C2semidirect
36. #2568C22semidirect
37. #2567C22semidirect
38. #2565C22semidirect
39. #2565C22semidirect
57. #2101C22semidirect
58. #2101C22semidirect
59. #2100C22semidirect
60. #2099C22semidirect
61. #2073C22semidirect
62. #2071C22semidirect
63. #2055C22semidirect
64. #2055C22semidirect
65. #2025C22semidirect
66. #2025C22semidirect
67. #2023C22semidirect
68. #2023C22semidirect
75. #1935C22semidirect
76. #1933C22semidirect
77. #1917C22semidirect
78. #1916C22semidirect
79. #1880C22semidirect
80. #1880C22semidirect
81. #1880C22semidirect
82. #1880C22semidirect
83. #1877C22semidirect
84. #1877C22semidirect
85. #1804C22semidirect
86. #1802C22semidirect

Maximal subgroups

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

subgroupindex
#356682
#355542
#316622
#316612
#316382
#316372
#237442
#236322
#235412
#234562
C2×#20142
#212032
#210942
#210222
#209362
#209332
#206502