Finite Group Databasebeta

← #6502275

Subgroups — #6502275

Order 512.

2078 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2445C211yesmaximal, characteristic
2#34213C211yesmaximal, characteristic, complemented
3#29240C212yesmaximal, complemented
4#27585C211yesmaximal, characteristic, complemented
5#27585C211yesmaximal, characteristic, complemented
6#27194C214yesmaximal, complemented
7#26692C214yesmaximal, complemented
8#26588C214yesmaximal, complemented
9#26163C214yesmaximal, complemented
10#24211C212yesmaximal
11#23933C211yesmaximal, characteristic, complemented
12#23933C211yesmaximal, characteristic, complemented
13#22043C212yesmaximal, complemented
14#21912C211yesmaximal, characteristic, complemented
15C4×#450C212yesmaximal, complemented
16C2×#533C2211yescharacteristic
17C22×#127C2211yescharacteristic
18C22×#123—21no—
19C22×C4×D4C2211yescharacteristic
20C2×#460C2214yes—
21C2×#450C2212yes—
22C2×#450C2212yes—
23C2×C4×SD16C2214yescomplemented
24#2445—22no—
25#2445—22no—
26#2445C2214yes—
27#2445C2214yes—
28C2×#394—21no—
29C2×#394—21no—
30#2370—22no—
31#2369C2212yescomplemented
32#2365C2212yescomplemented
33#2355—22no—
34#2355C2212yescomplemented
35Q8×C4⋊C4C2212yescomplemented
36#2329—21no—
37#2329—21no—
38#2317—21no—
39C4×C22⋊Q8C2212yes—
40C4×C22⋊Q8C2212yes—
41C2×#390C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#386—21no—
46#2285C2211yescharacteristic
47#2274—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49#2094C2212yescomplemented
50#2094C2212yescomplemented
51#2088C2214yescomplemented
52#2071C2212yescomplemented
53#2071C2212yescomplemented
54#2066C2214yescomplemented
55#2041C2214yes—
56#2023C2212yescomplemented
57#2023C2212yescomplemented
58#2014C2214yes—
59#2013C2212yescomplemented
60#2013C2212yescomplemented
61#1982C2214yes—
62#1968C2214yescomplemented
63#1934C2214yescomplemented
64#1926C2214yescomplemented
65#1926C2214yescomplemented
66#1926C2214yescomplemented
67#1919C2214yescomplemented
68#1916C2214yescomplemented
69#1916C2214yescomplemented
70#1880C2214yescomplemented
71#1880C2214yescomplemented
72#1880C2214yescomplemented
73#1877C2214yescomplemented
74#1872C2212yes—
75#1872C2212yes—
76#1865C2214yescomplemented
77#1864C2214yescomplemented
78#1853C2212yes—
79#1853C2212yes—
80#1817C2211yescharacteristic
81#1817C2211yescharacteristic
82#1797C2212yes—
83#1786C2212yes—
84#1786C2212yes—
85C4×#102C2214yescomplemented
86C4×C22⋊C8C2212yes—
87C23×D4—21no—
88C23×D4—21no—
89C23×D4C2311yescharacteristic
90C23×D4C2311yescharacteristic
91C24×C4—21no—
92C22×SD16C2314yes—
93C22×SD16C2314yes—
94#533—28no—
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×#127—22no—
98C2×#127—44no—
99C2×#127C2314yes—
100C2×#125—21no—
101C2×#123—24no—
102C2×#123—24no—
103C2×#123—48no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2312yes—
106C2×C4×Q8—21no—
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2312yes—
110C2×C4×D4—22no—
111C2×C4×D4—44no—
112C2×C4×D4C2314yes—
113C2×#117—21no—
114C2×#117—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130#460—48no—
131#450—28no—
132#450—28no—
133C4×SD16—48no—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C4×C8C2314yes—
148#394—24no—
149#394—24no—
150#394—24no—
151#394—24no—
152#391—22no—
153#391—22no—
154#391C2312yes—
155#391C2312yes—
156#390—21no—
157#390—21no—
158#390—21no—
159#390—21no—
160#390—22no—
161#390C2312yes—
162#390C2312yes—
163#390C2314yes—
164#390C2312yes—
165#390C2312yes—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389C2314yes—
172#389C2312yes—
173#389C2312yes—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#386—24no—
181#386—24no—
182#385—22no—
183#385—22no—
184#384—22no—
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#382—22no—
188#382—21no—
189#380—21no—
190#380C2312yes—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4D412yes—
199C4×C4⋊C4D412yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C22⋊C4—22no—
206C4×C22⋊C4—22no—
207C2×#95—21no—
208C2×#95D411yescharacteristic
209C2×#95D411yescharacteristic
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2314yes—
215#336C2312yes—
216#336C2312yes—
217#336C2312yes—
218#336C2312yes—
219#336C2314yes—
220C25—21no—
221C25—21no—
222C22×Q8—21no—
223C22×Q8C2412yes—
224C22×D4—24no—
225C22×D4—21no—
226C22×D4—21no—
227C22×D4—24no—
228C22×D4—24no—
229C22×D4—24no—
230C22×D4—24no—
231C22×D4—42no—
232C22×D4—42no—
233C22×D4—42no—
234C22×D4—44no—
235C22×D4—44no—
236C22×D4—44no—
237C22×D4—44no—
238C22×D4—82no—
239C22×D4—82no—
240C22×D4C2412yes—
241C22×D4C2412yes—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—24no—
245C23×C4—44no—
246C23×C4—44no—
247C23×C4—82no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255C2×SD16—28no—
256C2×SD16—28no—
257C2×SD16—28no—
258C2×SD16—44no—
259C2×SD16—44no—
260C2×SD16—44no—
261C22×C8C2414yes—
262C22×C8C2414yes—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265#127—28no—
266#127—44no—
267#127—48no—
268#125—28no—
269#123—416no—
270#123—416no—
271C22⋊Q8—28no—
272C22⋊Q8—24no—
273C22⋊Q8#4118yes—
274C4×Q8—28no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×D4—28no—
280C4×D4—44no—
281C4×D4—48no—
282#117—28no—
283#117—44no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4SD1618yescomplemented
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—82no—
332C2×C22⋊C4—82no—
333C2×C22⋊C4—82no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—42no—
342C2×C42—42no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351#106—44no—
352#106—44no—
353C4⋊C8—44no—
354#103—44no—
355#103—44no—
356#102—28no—
357#102—28no—
358#102—28no—
359#102—44no—
360#102—44no—
361#102—44no—
362C22⋊C8—24no—
363C22⋊C8—24no—
364C4×C8—44no—
365#95—24no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380C24—24no—
381C24—22no—
382C24—22no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24C22×D411yescharacteristic
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—24no—
401C2×Q8—81no—
402C2×Q8—81no—
403C2×Q8C2×#4114yes—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—416no—
413C2×D4—42no—
414C2×D4—48no—
415C2×D4—48no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—48no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—48no—
422C2×D4—88no—
423C2×D4—84no—
424C2×D4—84no—
425C2×D4—816no—
426C2×D4—816no—
427C2×D4H(5,𝔽2)14yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—48no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—48no—
459C22×C4—84no—
460C22×C4—816no—
461C22×C4C2×SD1614yes—
462C22×C4C2×SD1614yes—
463C22×C4C2×SD1614yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C2×#4114yes—
474C22×C4C2511yescharacteristic
475SD16—416no—
476SD16—88no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—28no—
486C4⋊C4—24no—
487C4⋊C4—28no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—48no—
500C4⋊C4—82no—
501C4⋊C4—81no—
502C4⋊C4—81no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4C2×SD1618yes—
507C4⋊C4C2×SD1618yes—
508C4⋊C4C2×#4114yes—
509C22⋊C4—28no—
510C22⋊C4—48no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—48no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—82no—
518C22⋊C4—84no—
519C22⋊C4—82no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C42—28no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—24no—
528C42—24no—
529C42—42no—
530C42—42no—
531C42—44no—
532C42—82no—
533C42—82no—
534C42—82no—
535C42—82no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—24no—
541C23—48no—
542C23—48no—
543C23—48no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—84no—
558C23—84no—
559C23—84no—
560C23—84no—
561C23—84no—
562C23—84no—
563C23—84no—
564C23—84no—
565C23—84no—
566C23—84no—
567C23—84no—
568C23#52512yes—
569C23C22×SD1612yes—
570C23#57512yes—
571C23C23×D411yescharacteristic
572Q8—28no—
573Q8—44no—
574Q8—84no—
575D4—28no—
576D4—48no—
577D4—44no—
578D4—44no—
579D4—44no—
580D4—816no—
581D4—84no—
582D4—816no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—28no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—28no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
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—816no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4#52512yes—
634C2×C4C22×SD1612yes—
635C2×C4C22×SD1612yes—
636C2×C4C22×SD1614yes—
637C2×C4C22×SD1614yes—
638C2×C4C22×#4112yes—
639C2×C4C2×H(5,𝔽2)12yes—
640C2×C4#58512yes—
641C8—44no—
642C8—44no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—816no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—168no—
665C22#310314yes—
666C22C2×#52511yescharacteristic
667C22#348411yescharacteristic
668C22#348711yescharacteristic
669C22C23×SD1611yescharacteristic
670C22C2×#57511yescharacteristic
671C22#358511yescharacteristic
672C22#358511yescharacteristic
673C4—22no—
674C4—22no—
675C4—22no—
676C4—24no—
677C4—24no—
678C4—44no—
679C4—42no—
680C4—42no—
681C4—41no—
682C4—41no—
683C4—44no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4#351314yes—
689C2—24no—
690C2—84no—
691C2—162no—
692C2#3535611yescharacteristic
693C2C2×#310311yescharacteristic
694C2#6056811yescharacteristic
695C2#6156011yescharacteristic
696C2#6157611yescharacteristic
697C2#6188111yescharacteristic
698C2#6189511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34213C2semidirect
3. #29240C2semidirect
4. #27585C2semidirect
5. #27585C2semidirect
6. #27194C2semidirect
7. #26692C2semidirect
8. #26588C2semidirect
9. #26163C2semidirect
11. #23933C2semidirect
12. #23933C2semidirect
13. #22043C2semidirect
14. #21912C2semidirect
15. C4×#450C2semidirect
23. C2×C4×SD16C22semidirect
31. #2369C22semidirect
32. #2365C22semidirect
34. #2355C22semidirect
35. Q8×C4⋊C4C22semidirect
49. #2094C22semidirect
50. #2094C22semidirect
51. #2088C22semidirect
52. #2071C22semidirect
53. #2071C22semidirect
54. #2066C22semidirect
56. #2023C22semidirect
57. #2023C22semidirect
59. #2013C22semidirect
60. #2013C22semidirect
62. #1968C22semidirect
63. #1934C22semidirect
64. #1926C22semidirect
65. #1926C22semidirect
66. #1926C22semidirect
67. #1919C22semidirect
68. #1916C22semidirect
69. #1916C22semidirect
70. #1880C22semidirect
71. #1880C22semidirect
72. #1880C22semidirect
73. #1877C22semidirect
76. #1865C22semidirect
77. #1864C22semidirect
85. C4×#102C22semidirect
304. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#24452
#342132
#292402
#275852
#271942
#266922
#265882
#261632
#242112
#239332
#220432
#219122
C4×#4502