Finite Group Databasebeta

← #6478171

Subgroups — #6478171

Order 512.

1896 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35005C211yesmaximal, characteristic
2C2×#2344C211yesmaximal, characteristic, complemented
3#33146C211yesmaximal, characteristic, complemented
4#33106C211yesmaximal, characteristic, complemented
5#32978C212yesmaximal, complemented
6#29966C211yesmaximal, characteristic, complemented
7#29890C212yesmaximal, complemented
8#29754C211yesmaximal, characteristic, complemented
9#27454C214yesmaximal, complemented
10#27390C214yesmaximal, complemented
11#27326C214yesmaximal, complemented
12#27158C214yesmaximal, complemented
13#22075C212yesmaximal, complemented
14#21902C211yesmaximal, characteristic, complemented
15C4×#480C212yesmaximal, complemented
16C2×#538—21no—
17C22×#117—21no—
18C23×C4⋊C4C2211yescharacteristic
19C2×#480C2212yes—
20C2×#480C2212yes—
21#2857C2211yescharacteristic
22#2856C2211yescharacteristic
23#2847C2212yes—
24#2639—21no—
25#2617—21no—
26#2559—22no—
27#2557—22no—
28#2496C2212yes—
29#2491C2212yes—
30#2428C2212yes—
31#2422C2212yes—
32#2344—22no—
33#2344—22no—
34#2344C2214yes—
35#2344C2214yes—
36#2325—21no—
37C4×#121C2212yes—
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#387—21no—
41C2×#387—21no—
42C2×#381—21no—
43C2×#380C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47#2099C2212yescomplemented
48#2086C2214yescomplemented
49#2083C2212yescomplemented
50#2075C2214yes—
51#2075C2214yes—
52#2067C2214yescomplemented
53#2057C2212yescomplemented
54#2041C2212yescomplemented
55#2033C2214yes—
56#2033C2214yes—
57#1988C2212yescomplemented
58#1982C2212yescomplemented
59#1947C2212yescomplemented
60#1947C2212yescomplemented
61#1942C2214yescomplemented
62#1934C2214yescomplemented
63#1925C2214yescomplemented
64#1925C2214yescomplemented
65#1923C2214yes—
66#1923C2214yes—
67#1918C2214yescomplemented
68#1918C2214yescomplemented
69#1918C2214yescomplemented
70#1874C2214yes—
71#1874C2212yes—
72#1874C2212yes—
73#1874C2214yes—
74#1862C2214yescomplemented
75#1852C2212yes—
76#1852C2214yes—
77#1852C2212yes—
78#1852C2214yes—
79#1823C2212yes—
80#1823C2212yes—
81#1797C2211yescharacteristic
82#1797C2211yescharacteristic
83#1779C2212yes—
84C4×#107C2214yescomplemented
85C4×#102C2214yescomplemented
86C4×C22⋊C8C2212yes—
87C24×C4—21no—
88C24×C4C2311yescharacteristic
89#538—48no—
90C2×C4⋊Q8C2311yescharacteristic
91C2×#127C2311yescharacteristic
92C2×#126—22no—
93C2×#124—22no—
94C2×#123—22no—
95C2×C22⋊Q8—21no—
96C2×#121—21no—
97C2×#121—21no—
98C2×#121—22no—
99C2×#121C2312yes—
100C2×#121C2312yes—
101C2×#121C2312yes—
102C2×#120—22no—
103C2×C4×D4—22no—
104C2×C4×D4C2312yes—
105C2×#117—24no—
106C2×#117—24no—
107C2×#117—48no—
108C22×C4⋊C4—22no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—24no—
111C22×C4⋊C4—24no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4—21no—
120C22×C22⋊C4D411yescharacteristic
121C22×C22⋊C4D411yescharacteristic
122C22×C42—21no—
123C22×C42—21no—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#480—28no—
127#480—28no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2314yes—
141C2×C22⋊C8C2314yes—
142C2×C4×C8C2314yes—
143#400—22no—
144#397—22no—
145#397—22no—
146#394—22no—
147#394—22no—
148#394—22no—
149#393—22no—
150#392—22no—
151#391—21no—
152#390—21no—
153#390—21no—
154#390C2312yes—
155#390C2312yes—
156#389—22no—
157#389—21no—
158#389—21no—
159#389—21no—
160#389C2312yes—
161#389C2312yes—
162#389C2312yes—
163#389C2314yes—
164#389C2312yes—
165#388—22no—
166#388—22no—
167#387—22no—
168#387—24no—
169#387—24no—
170#387—24no—
171#387—24no—
172#386—22no—
173#386—22no—
174#386—22no—
175#386—22no—
176#386—22no—
177#385—22no—
178#384C2311yescharacteristic
179#384C2311yescharacteristic
180#381—24no—
181#381—24no—
182#380—21no—
183#380—22no—
184#380C2312yes—
185#380C2312yes—
186#380C2314yes—
187#380C2312yes—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2314yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C4×C22⋊C4—22no—
196C4×C22⋊C4D412yes—
197C4×C22⋊C4D412yes—
198C4×C22⋊C4C2312yes—
199C4×C22⋊C4C2312yes—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95D411yescharacteristic
206C2×#95D411yescharacteristic
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#336C2312yes—
216#336C2312yes—
217#336C2314yes—
218C25C2×D411yescharacteristic
219C22×Q8—21no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—21no—
224C22×D4—21no—
225C22×D4C2412yes—
226C23×C4—28no—
227C23×C4—21no—
228C23×C4—24no—
229C23×C4—21no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—22no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4—44no—
238C23×C4—44no—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D412yes—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247C22×C8C2414yes—
248C22×C8C2414yes—
249C4⋊Q8—42no—
250#127—42no—
251#126—48no—
252#124—48no—
253#123—48no—
254C22⋊Q8—44no—
255#121—24no—
256#121—24no—
257#121—44no—
258#121—44no—
259#121—48no—
260#121—44no—
261#121—44no—
262#120—48no—
263C4×D4—44no—
264C4×D4—48no—
265#117—416no—
266#117—416no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—216no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4#4114yes—
293C2×C4⋊C4#4114yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4D818yes—
316C2×C22⋊C4C2×D412yes—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C42—24no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—42no—
325C2×C42—42no—
326C2×C42—44no—
327C2×C42—44no—
328C2×C42C2×D412yes—
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2412yes—
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334#107—28no—
335#107—28no—
336#107—44no—
337#107—44no—
338C4⋊C8—42no—
339C4⋊C8—42no—
340#102—28no—
341#102—28no—
342#102—44no—
343#102—44no—
344C22⋊C8—24no—
345C22⋊C8—24no—
346C22⋊C8—44no—
347C22⋊C8—44no—
348C4×C8—44no—
349#95—24no—
350#95—22no—
351#95—22no—
352#95—24no—
353#95—24no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95C2×D412yes—
363#95C2×D412yes—
364#95C2×D412yes—
365#95C2×D412yes—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—41no—
370C24—41no—
371C24—41no—
372C24—41no—
373C24C2×D814yes—
374C24C22×D412yes—
375C24C22×D411yescharacteristic
376C2×Q8—42no—
377C2×Q8—81no—
378C2×Q8—81no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—24no—
382C2×D4—24no—
383C2×D4—42no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—42no—
390C2×D4—81no—
391C2×D4—81no—
392C2×D4—81no—
393C2×D4—81no—
394C2×D4—82no—
395C2×D4—82no—
396C2×D4—82no—
397C2×D4—82no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—216no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—432no—
433C22×C4—48no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—48no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4C2×D814yes—
449C22×C4C2×D814yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2511yescharacteristic
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C4⋊C4—216no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—81no—
490C4⋊C4—81no—
491C4⋊C4C2×#4114yes—
492C4⋊C4C2×#4114yes—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—44no—
496C22⋊C4—48no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C22⋊C4—82no—
506C22⋊C4—82no—
507C22⋊C4C2×D818yes—
508C22⋊C4C2×D818yes—
509C42—24no—
510C42—22no—
511C42—22no—
512C42—42no—
513C42—41no—
514C42—41no—
515C42—48no—
516C42—44no—
517C42—44no—
518C42—44no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—22no—
524C23—22no—
525C23—28no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23#52512yes—
544C23C22×D814yes—
545C23C22×D812yes—
546C23#57512yes—
547C23C23×D411yescharacteristic
548Q8—162no—
549D4—44no—
550D4—44no—
551D4—88no—
552D4—162no—
553D4—162no—
554D4—164no—
555D4—164no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—44no—
584C2×C4—432no—
585C2×C4—48no—
586C2×C4—48no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4—164no—
613C2×C4#52512yes—
614C2×C4C22×D814yes—
615C2×C4C22×D812yes—
616C2×C4C22×D812yes—
617C2×C4C22×#4111yescharacteristic
618C2×C4C22×#4111yescharacteristic
619C2×C4#58512yes—
620C2×C4#58512yes—
621C8—44no—
622C8—44no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—24no—
630C22—24no—
631C22—24no—
632C22—416no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22#310014yes—
645C22C2×#52511yescharacteristic
646C22#348511yescharacteristic
647C22#348511yescharacteristic
648C22C23×D811yescharacteristic
649C22C2×#57511yescharacteristic
650C22#358611yescharacteristic
651C22#358611yescharacteristic
652C4—22no—
653C4—24no—
654C4—22no—
655C4—22no—
656C4—42no—
657C4—41no—
658C4—41no—
659C4—42no—
660C4—44no—
661C4—44no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4#345512yes—
668C4#349312yes—
669C2—24no—
670C2—44no—
671C2—162no—
672C2#3509311yescharacteristic
673C2C2×#310011yescharacteristic
674C2#6056611yescharacteristic
675C2#6169411yescharacteristic
676C2#6169411yescharacteristic
677C2#6171111yescharacteristic
678C2#6171611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35005C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2344C2semidirect
3. #33146C2semidirect
4. #33106C2semidirect
5. #32978C2semidirect
6. #29966C2semidirect
7. #29890C2semidirect
8. #29754C2semidirect
9. #27454C2semidirect
10. #27390C2semidirect
11. #27326C2semidirect
12. #27158C2semidirect
13. #22075C2semidirect
14. #21902C2semidirect
15. C4×#480C2semidirect
47. #2099C22semidirect
48. #2086C22semidirect
49. #2083C22semidirect
52. #2067C22semidirect
53. #2057C22semidirect
54. #2041C22semidirect
57. #1988C22semidirect
58. #1982C22semidirect
59. #1947C22semidirect
60. #1947C22semidirect
61. #1942C22semidirect
62. #1934C22semidirect
63. #1925C22semidirect
64. #1925C22semidirect
67. #1918C22semidirect
68. #1918C22semidirect
69. #1918C22semidirect
74. #1862C22semidirect
84. C4×#107C22semidirect
85. C4×#102C22semidirect

Maximal subgroups

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

subgroupindex
#350052
C2×#23442
#331462
#331062
#329782
#299662
#298902
#297542
#274542
#273902
#273262
#271582
#220752
#219022
C4×#4802