Finite Group Databasebeta

← #6497273

Subgroups — #6497273

Order 512.

1854 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic
3#33148C212yesmaximal, complemented
4#33146C212yesmaximal, complemented
5#29939C212yesmaximal, complemented
6#29937C212yesmaximal, complemented
7#27966C212yesmaximal, complemented
8#27953C212yesmaximal, complemented
9#27881C214yesmaximal, complemented
10#27612C214yesmaximal, complemented
11#27602C214yesmaximal
12#21988C212yesmaximal, complemented
13#21839C212yesmaximal, complemented
14#20715C211yesmaximal, characteristic
15C2×#533C2211yescharacteristic
16C22×#117—21no—
17C23×C42C2211yescharacteristic
18C2×#432C2212yes—
19#2857C2211yescharacteristic
20#2857C2211yescharacteristic
21#2857C2211yescharacteristic
22#2857C2211yescharacteristic
23#2855—21no—
24#2855—21no—
25#2851C2212yes—
26#2851C2212yes—
27#2848—21no—
28#2848—21no—
29C4×C22⋊Q8—21no—
30C4×C22⋊Q8—21no—
31C4×C22⋊Q8—21no—
32C4×C22⋊Q8—21no—
33#2296C2212yes—
34#2285—21no—
35#2285—21no—
36#2285—21no—
37#2285—21no—
38#2285C2212yes—
39#2285C2212yes—
40#2285C2212yes—
41#2285C2212yes—
42#2285C2211yescharacteristic
43C2×#384C2211yescharacteristic
44C2×#384C2211yescharacteristic
45C2×#382—21no—
46#2280C2212yes—
47C2×#380—21no—
48C2×#380C2212yes—
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4C2212yes—
51#2100C2214yescomplemented
52#2100C2214yescomplemented
53#2084C2214yescomplemented
54#2083C2214yescomplemented
55#2077C2218yes—
56#2076C2218yes—
57#1992C2214yescomplemented
58#1992C2214yescomplemented
59#1988C2214yescomplemented
60#1987C2214yescomplemented
61#1948C2214yescomplemented
62#1947C2214yescomplemented
63#1920C2218yescomplemented
64#1918C2218yescomplemented
65#1854C2218yes—
66#1853C2214yes—
67#1852C2214yes—
68#1847C2214yescomplemented
69#1836C2214yescomplemented
70#1825C2218yes—
71#1823C2214yes—
72#1823C2214yes—
73#1817C2212yes—
74#1797C2212yes—
75#1797C2212yes—
76#1779C2212yes—
77#1749C2212yes—
78#1749C2212yes—
79C24×C4D411yescharacteristic
80C24×C4D411yescharacteristic
81C24×C4C2311yescharacteristic
82#533—28no—
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2311yescharacteristic
85C2×#125C2312yes—
86C2×C22⋊Q8—21no—
87C2×C22⋊Q8—21no—
88C2×C22⋊Q8—21no—
89C2×C22⋊Q8—21no—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8—21no—
92C2×C4×Q8—21no—
93C2×C4×Q8—21no—
94C2×#117—22no—
95C2×#117—22no—
96C2×#117—22no—
97C2×#117—22no—
98C2×#117—48no—
99C2×#117C2312yes—
100C2×#117C2312yes—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4—22no—
108C22×C42—28no—
109C22×C42—22no—
110C22×C42D411yescharacteristic
111C22×C42D411yescharacteristic
112C22×C42D411yescharacteristic
113C22×C42D411yescharacteristic
114C22×C42D411yescharacteristic
115C22×C42D411yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#432—28no—
123C2×#107C2318yes—
124C2×#106C2318yes—
125C2×C4⋊C8C2314yes—
126C2×C22⋊C8C2314yes—
127C2×C22⋊C8C2314yes—
128C2×C22⋊C8C2314yes—
129C2×#97C2314yes—
130#391—21no—
131#391—21no—
132#391—21no—
133#391—21no—
134#389—21no—
135#389—21no—
136#389—21no—
137#389—21no—
138#389C2312yes—
139#389C2312yes—
140#389C2312yes—
141#389C2312yes—
142#387—22no—
143#387—22no—
144#387—22no—
145#387—22no—
146#385—22no—
147#385—22no—
148#385—22no—
149#385—22no—
150#385—22no—
151#385—22no—
152#384—21no—
153#384—21no—
154#384—21no—
155#384—21no—
156#384C2312yes—
157#384C2311yescharacteristic
158#384C2311yescharacteristic
159#384C2312yes—
160#384C2312yes—
161#384C2312yes—
162#382—22no—
163#382—22no—
164#382—22no—
165#382—22no—
166#382C2312yes—
167#381—22no—
168#380—21no—
169#380—21no—
170#380—21no—
171#380—21no—
172#380—21no—
173#380—21no—
174#380—22no—
175#380—22no—
176#380—22no—
177#380—22no—
178#380—22no—
179#380—22no—
180#380C2312yes—
181#380C2314yes—
182#380C2312yes—
183#380C2312yes—
184#380C2312yes—
185#380C2314yes—
186#379—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4C2314yes—
198C4×C4⋊C4C2314yes—
199C4×C22⋊C4—22no—
200C4×C22⋊C4—22no—
201C2×#95—22no—
202C2×#95—22no—
203#340C2318yes—
204#340C2318yes—
205#340C2318yes—
206#340C2318yes—
207#340C2318yes—
208#340C2318yes—
209#336C2314yes—
210#336C2314yes—
211C25C2×D411yescharacteristic
212C22×Q8—21no—
213C22×Q8—21no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C22×C8C2418yes—
244C4⋊Q8—24no—
245C4⋊Q8—24no—
246#125—28no—
247C22⋊Q8—44no—
248C22⋊Q8—44no—
249C22⋊Q8—44no—
250C22⋊Q8—44no—
251C22⋊Q8—44no—
252C22⋊Q8—44no—
253C4×Q8—44no—
254C4×Q8—44no—
255#117—28no—
256#117—44no—
257#117—48no—
258#117—48no—
259#117—48no—
260#117—48no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4C2414yes—
290C2×C4⋊C4C2414yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—48no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—48no—
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42#4118yes—
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337#107—48no—
338#106—48no—
339C4⋊C8—28no—
340C22⋊C8—28no—
341C22⋊C8—28no—
342C22⋊C8—44no—
343#97—28no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—41no—
364C24—41no—
365C24C22×D411yescharacteristic
366C24C22×D411yescharacteristic
367C24C22×D411yescharacteristic
368C2×Q8—42no—
369C2×Q8—42no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—42no—
373C2×Q8—42no—
374C22×C4—232no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—416no—
430C22×C4—416no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4#12314yes—
438C22×C4#12314yes—
439C22×C4AGL(1,ℤ/8)14yes—
440C22×C4#13714yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C2×#4114yes—
453C22×C42−1+414yes—
454C22×C4C2511yescharacteristic
455C2×C8—28no—
456C2×C8—28no—
457C2×C8—28no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—28no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—44no—
508C42—44no—
509C42—48no—
510C42—48no—
511C42#12314yes—
512C42#12314yes—
513C42C2×#4114yes—
514C42C2×#4114yes—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—28no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23C2×#12312yes—
539C23#53612yes—
540C23C2×AGL(1,ℤ/8)12yes—
541C23C2×#13712yes—
542C23#57512yes—
543C23C23×D411yescharacteristic
544Q8—84no—
545Q8—84no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
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—44no—
603C2×C4—44no—
604C2×C4—88no—
605C2×C4—88no—
606C2×C4C2×#12312yes—
607C2×C4C2×#12312yes—
608C2×C4C2×#12312yes—
609C2×C4C2×#12312yes—
610C2×C4C2×#12312yes—
611C2×C4#53612yes—
612C2×C4C2×AGL(1,ℤ/8)12yes—
613C2×C4C2×AGL(1,ℤ/8)12yes—
614C2×C4C2×#13712yes—
615C2×C4C2×#13712yes—
616C2×C4#57512yes—
617C2×C4C22×#4112yes—
618C2×C4C2×2−1+412yes—
619C2×C4#58512yes—
620C8—48no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—416no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22#323514yes—
643C22C22×#12311yescharacteristic
644C22C2×#53611yescharacteristic
645C22#344911yescharacteristic
646C22C22×AGL(1,ℤ/8)11yescharacteristic
647C22C22×#13711yescharacteristic
648C22C2×#57511yescharacteristic
649C22C2×#57511yescharacteristic
650C4—22no—
651C4—22no—
652C4—22no—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—41no—
659C4—41no—
660C4—82no—
661C4—84no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—84no—
666C4#309214yes—
667C4#349514yes—
668C2—24no—
669C2—42no—
670C2—42no—
671C2#3531111yescharacteristic
672C2C2×#309211yescharacteristic
673C2#6051111yescharacteristic
674C2C2×#323511yescharacteristic
675C2#6086511yescharacteristic
676C2#6095111yescharacteristic
677C2#6096311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33148C2semidirect
4. #33146C2semidirect
5. #29939C2semidirect
6. #29937C2semidirect
7. #27966C2semidirect
8. #27953C2semidirect
9. #27881C2semidirect
10. #27612C2semidirect
12. #21988C2semidirect
13. #21839C2semidirect
51. #2100C22semidirect
52. #2100C22semidirect
53. #2084C22semidirect
54. #2083C22semidirect
57. #1992C22semidirect
58. #1992C22semidirect
59. #1988C22semidirect
60. #1987C22semidirect
61. #1948C22semidirect
62. #1947C22semidirect
63. #1920C22semidirect
64. #1918C22semidirect
68. #1847C22semidirect
69. #1836C22semidirect

Maximal subgroups

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

subgroupindex
#366252
C2×#22852
#331482
#331462
#299392
#299372
#279662
#279532
#278812
#276122
#276022
#219882
#218392
#207152