Finite Group Databasebeta

← #6498097

Subgroups — #6498097

Order 512.

1914 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic, complemented
2C2×#2293C211yesmaximal, characteristic
3#33147C212yesmaximal, complemented
4#32953C212yesmaximal
5#29981C212yesmaximal, complemented
6#29864C212yesmaximal, complemented
7#29580C214yesmaximal, complemented
8#29389C214yesmaximal, complemented
9#29301C212yesmaximal, complemented
10#29109C212yesmaximal
11#27614C212yesmaximal, complemented
12#27585C212yesmaximal
13#22116C214yesmaximal, complemented
14#20778C211yesmaximal, characteristic, complemented
15C2×#533—21no—
16C22×C4×D4C2211yescharacteristic
17C22×#117—21no—
18C2×#474C2214yes—
19#2868C2211yescharacteristic, complemented
20#2868C2211yescharacteristic, complemented
21#2862—21no—
22#2859—21no—
23#2859C2212yescomplemented
24#2857—21no—
25#2857—21no—
26#2857C2212yes—
27#2857C2212yes—
28C4×C4⋊Q8—21no—
29C4×C4⋊Q8—21no—
30C4×C4⋊Q8C2212yescomplemented
31C4×C4⋊Q8C2212yescomplemented
32#2293—21no—
33#2293—21no—
34#2293—21no—
35#2293—21no—
36#2293C2212yes—
37#2293C2212yes—
38#2293C2212yes—
39#2293C2212yes—
40C2×#390C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#388—21no—
45C2×#388—21no—
46#2285—21no—
47#2285C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4—21no—
50C2×C43C2211yescharacteristic
51#2100C2214yescomplemented
52#2094C2214yes—
53#2083C2214yescomplemented
54#2083C2214yescomplemented
55#2071C2214yes—
56#2071C2214yes—
57#2058C2214yescomplemented
58#2023C2214yes—
59#1988C2214yescomplemented
60#1948C2214yescomplemented
61#1943C2218yescomplemented
62#1936C2218yes—
63#1926C2218yes—
64#1921C2218yescomplemented
65#1919C2218yescomplemented
66#1862C2218yes—
67#1853C2214yes—
68#1852C2214yes—
69#1839C2212yescomplemented
70#1831C2212yescomplemented
71#1824C2214yes—
72#1817C2212yes—
73#1802C2214yes—
74#1797C2212yes—
75#1786C2214yes—
76#1767C2212yescomplemented
77#1749C2212yes—
78#1745C2212yescomplemented
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81#533—48no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2312yes—
88C2×#125—22no—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—22no—
91C2×C4×Q8—22no—
92C2×C4×Q8—22no—
93C2×C4×D4—28no—
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×C4×D4C2312yes—
97C2×C4×D4C2312yes—
98C2×#117—22no—
99C2×#117—22no—
100C2×#117—22no—
101C2×#117—22no—
102C2×#117—22no—
103C2×#117—48no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C42—21no—
115C22×C42D411yescharacteristic
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#474—216no—
123C2×#107C2318yes—
124C2×#106C2318yes—
125C2×C4⋊C8C2314yes—
126C2×#102C2318yes—
127C2×#102C2318yes—
128C2×C22⋊C8C2314yes—
129C2×#97C2314yes—
130#391—22no—
131#391—22no—
132#391—22no—
133#391—22no—
134#390—21no—
135#390—21no—
136#390—21no—
137#390—21no—
138#390C2312yes—
139#390C2312yes—
140#390C2312yes—
141#390C2312yes—
142#389—22no—
143#389—22no—
144#389—22no—
145#389—22no—
146#389—22no—
147#389—22no—
148#389—21no—
149#389—21no—
150#389—21no—
151#389—21no—
152#389C2314yes—
153#389C2314yes—
154#389C2312yes—
155#389C2312yes—
156#389C2312yes—
157#389C2312yes—
158#388—22no—
159#388—22no—
160#388—22no—
161#388—22no—
162#388—22no—
163#388—22no—
164#388—22no—
165#388—22no—
166#384—21no—
167#384—21no—
168#384—21no—
169#384—21no—
170#384C2312yes—
171#384C2312yes—
172#384C2312yes—
173#384C2312yes—
174#382—22no—
175#382—22no—
176#380—22no—
177#380—22no—
178#380—22no—
179#380C2314yes—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4C2314yes—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C4×C22⋊C4—22no—
189C4×C22⋊C4—22no—
190C4×C22⋊C4—22no—
191C4×C22⋊C4—22no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C43D411yescharacteristic
196C43D411yescharacteristic
197C43D411yescharacteristic
198C43D411yescharacteristic
199C43C2311yescharacteristic
200C43C2311yescharacteristic
201C43C2311yescharacteristic
202C43C2311yescharacteristic
203#340C2318yes—
204#340C2318yes—
205#340C2318yes—
206#340C2318yes—
207#336C2314yes—
208#336C2314yes—
209#336C2314yes—
210#336C2314yes—
211C25—21no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×D4—24no—
215C22×D4—21no—
216C22×D4—24no—
217C22×D4—42no—
218C22×D4C2412yes—
219C23×C4—28no—
220C23×C4—21no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2411yescharacteristic
234C22×C8C2418yes—
235C4⋊Q8—28no—
236C4⋊Q8—44no—
237C4⋊Q8—44no—
238C4⋊Q8—44no—
239C4⋊Q8—44no—
240C4⋊Q8—44no—
241#125—48no—
242C22⋊Q8—48no—
243C22⋊Q8—48no—
244C4×Q8—48no—
245C4×Q8—48no—
246C4×D4—28no—
247C4×D4—28no—
248C4×D4—24no—
249C4×D4—24no—
250C4×D4—24no—
251C4×D4#4118yes—
252#117—48no—
253#117—48no—
254#117—48no—
255#117—48no—
256#117—48no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2414yes—
284C2×C4⋊C4C2414yes—
285C2×C4⋊C4C2412yes—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—42no—
311C2×C42—42no—
312C2×C42—44no—
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
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×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341#107—216no—
342#106—48no—
343C4⋊C8—28no—
344#102—216no—
345#102—48no—
346C22⋊C8—28no—
347#97—44no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358C24—24no—
359C24—22no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24C22×D411yescharacteristic
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—44no—
372C2×Q8—44no—
373C2×D4—24no—
374C2×D4—22no—
375C2×D4—24no—
376C2×D4—24no—
377C2×D4—416no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4C2×#4114yes—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—28no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—28no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—44no—
408C22×C4—432no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4AGL(1,ℤ/8)14yes—
421C22×C4#13714yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C42−1+414yes—
436C22×C4C2511yescharacteristic
437C2×C8—28no—
438C2×C8—28no—
439C2×C8—28no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—28no—
443C4⋊C4—24no—
444C4⋊C4—28no—
445C4⋊C4—24no—
446C4⋊C4—42no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—48no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4C2×#4114yes—
465C22⋊C4—28no—
466C22⋊C4—28no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—48no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—41no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42—44no—
503C42—48no—
504C42—44no—
505C42AGL(1,ℤ/8)14yes—
506C42C2×#4114yes—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—24no—
512C23—48no—
513C23—48no—
514C23—48no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—84no—
522C23—84no—
523C23—84no—
524C23—84no—
525C23#53612yes—
526C23C2×AGL(1,ℤ/8)12yes—
527C23C2×#13712yes—
528C23C23×D411yescharacteristic
529Q8—88no—
530Q8—88no—
531D4—28no—
532D4—48no—
533D4—44no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—28no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—21no—
554C2×C4—24no—
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—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—416no—
571C2×C4—48no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—416no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4#52512yes—
590C2×C4#52512yes—
591C2×C4#53612yes—
592C2×C4C2×AGL(1,ℤ/8)12yes—
593C2×C4C2×AGL(1,ℤ/8)12yes—
594C2×C4C2×AGL(1,ℤ/8)12yes—
595C2×C4C2×AGL(1,ℤ/8)12yes—
596C2×C4C2×AGL(1,ℤ/8)12yes—
597C2×C4C2×#13712yes—
598C2×C4C2×#13712yes—
599C2×C4#57512yes—
600C2×C4C22×#4112yes—
601C2×C4C2×2−1+412yes—
602C2×C4#58512yes—
603C8—48no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—816no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22#323914yes—
619C22C2×#52511yescharacteristic
620C22C2×#53611yescharacteristic
621C22#344911yescharacteristic
622C22C22×AGL(1,ℤ/8)11yescharacteristic
623C22C22×AGL(1,ℤ/8)11yescharacteristic
624C22C22×#13711yescharacteristic
625C22C2×#57511yescharacteristic
626C4—22no—
627C4—22no—
628C4—22no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—41no—
635C4—41no—
636C4—44no—
637C4—41no—
638C4—41no—
639C4—84no—
640C4—82no—
641C4—84no—
642C4—82no—
643C4#310914yes—
644C4#349514yes—
645C2—24no—
646C2—84no—
647C2#3532511yescharacteristic
648C2C2×#310911yescharacteristic
649C2#6056211yescharacteristic
650C2C2×#323911yescharacteristic
651C2#6086411yescharacteristic
652C2#6094911yescharacteristic
653C2#6096911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36645C2semidirect
3. #33147C2semidirect
5. #29981C2semidirect
6. #29864C2semidirect
7. #29580C2semidirect
8. #29389C2semidirect
9. #29301C2semidirect
11. #27614C2semidirect
13. #22116C2semidirect
14. #20778C2semidirect
19. #2868C22semidirect
20. #2868C22semidirect
23. #2859C22semidirect
30. C4×C4⋊Q8C22semidirect
31. C4×C4⋊Q8C22semidirect
51. #2100C22semidirect
53. #2083C22semidirect
54. #2083C22semidirect
57. #2058C22semidirect
59. #1988C22semidirect
60. #1948C22semidirect
61. #1943C22semidirect
64. #1921C22semidirect
65. #1919C22semidirect
69. #1839C22semidirect
70. #1831C22semidirect
76. #1767C22semidirect
78. #1745C22semidirect

Maximal subgroups

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

subgroupindex
#366452
C2×#22932
#331472
#329532
#299812
#298642
#295802
#293892
#293012
#291092
#276142
#275852
#221162
#207782