Finite Group Databasebeta

← #6463963

Subgroups — #6463963

Order 512.

1870 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#533C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic
3#29277C212yesmaximal, complemented
4#29276C212yesmaximal, complemented
5#29275C214yesmaximal, complemented
6#29274C214yesmaximal, complemented
7#29273C212yesmaximal, complemented
8#29272C212yesmaximal, complemented
9#23747C212yesmaximal
10#23554C212yesmaximal
11#21313C212yesmaximal
12#21171C212yesmaximal
13#21033C214yesmaximal
14#20752C211yesmaximal, characteristic
15C2×#533C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×Q8—21no—
18C2×#428C2214yes—
19C2×#423C2212yes—
20C4×C4⋊Q8—21no—
21C4×C4⋊Q8—21no—
22C4×C4⋊Q8—21no—
23C4×C4⋊Q8—21no—
24C4×C4⋊Q8C2211yescharacteristic
25C4×C4⋊Q8C2212yes—
26C4×C4⋊Q8C2211yescharacteristic
27C4×C4⋊Q8C2212yes—
28C4×C4⋊Q8C2212yes—
29C4×C4⋊Q8C2212yes—
30C4×C4⋊Q8C2211yescharacteristic
31C4×C4⋊Q8C2211yescharacteristic
32C4×#125C2212yes—
33C4×#125C2212yes—
34C4×C22⋊Q8—21no—
35C4×C22⋊Q8—21no—
36C4×C22⋊Q8—21no—
37C4×C22⋊Q8—21no—
38#2296C2212yes—
39#2295—21no—
40#2295—21no—
41C2×#391—21no—
42C2×#389—21no—
43C2×#389C2212yes—
44#2285C2211yescharacteristic
45C2×#384C2211yescharacteristic
46C42×Q8—21no—
47C42×Q8—21no—
48C4×#117C2212yes—
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4C2212yes—
51C2×C43C2211yescharacteristic
52#2101C2218yes—
53#2100C2214yes—
54#2099C2214yes—
55#2059C2218yes—
56#2058C2214yes—
57#2057C2214yes—
58#1988C2214yes—
59#1988C2214yes—
60#1987C2214yes—
61#1987C2214yes—
62#1948C2214yes—
63#1948C2214yes—
64#1947C2214yes—
65#1947C2214yes—
66#1921C2218yes—
67#1920C2218yes—
68#1919C2218yes—
69#1918C2218yes—
70#1831C2212yes—
71#1831C2212yes—
72#1831C2212yes—
73#1831C2212yes—
74#1736C2214yes—
75#1736C2214yes—
76#1736C2214yes—
77#1728C2212yes—
78#1728C2212yes—
79C23×Q8—21no—
80C22×M16C2314yes—
81#533—28no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8Q818yescomplemented
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×#125C2312yes—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—21no—
93C2×C4×Q8—21no—
94C2×C4×Q8—21no—
95C2×C4×Q8—22no—
96C2×C4×Q8—22no—
97C2×C4×Q8—22no—
98C2×C4×Q8—22no—
99C2×C4×Q8—21no—
100C2×C4×Q8—21no—
101C2×C4×Q8—21no—
102C2×C4×Q8—21no—
103C2×C4×Q8—48no—
104C2×#117C2312yes—
105C2×#117C2312yes—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—22no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C42—22no—
113C22×C42D411yescharacteristic
114C22×C42D411yescharacteristic
115C22×C42D411yescharacteristic
116C22×C42D411yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120#428—216no—
121C2×#107C2318yes—
122C2×#106C2318yes—
123#423—28no—
124C2×C4⋊C8C2314yes—
125C2×C4⋊C8C2314yes—
126#391—22no—
127#391—22no—
128#391—22no—
129#391—22no—
130#391—21no—
131#391—21no—
132#391—21no—
133#391—21no—
134#389—22no—
135#389—22no—
136#389—22no—
137#389—22no—
138#389—22no—
139#389—22no—
140#389—21no—
141#389—21no—
142#389—21no—
143#389—21no—
144#389C2312yes—
145#389C2312yes—
146#389C2312yes—
147#389C2312yes—
148#389C2314yes—
149#389C2314yes—
150#387—22no—
151#387—22no—
152#387—22no—
153#387—22no—
154#385—22no—
155#385—22no—
156#384—21no—
157#384—21no—
158#384C2311yescharacteristic
159#384C2311yescharacteristic
160#384C2312yes—
161#384C2312yes—
162#382C2312yes—
163#380—21no—
164#380—21no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—21no—
176C4×C4⋊C4—21no—
177C4×C4⋊C4—21no—
178C4×C4⋊C4—21no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2314yes—
188C4×C4⋊C4C2314yes—
189C4×C22⋊C4—22no—
190C4×C22⋊C4—22no—
191#376—22no—
192C2×#95—22no—
193C2×#95—22no—
194C43—22no—
195C43D411yescharacteristic
196C43D411yescharacteristic
197C43D411yescharacteristic
198C43D411yescharacteristic
199C43C2311yescharacteristic
200C43C2311yescharacteristic
201C43C2311yescharacteristic
202C43C2311yescharacteristic
203#340C2318yes—
204#340C2318yes—
205#340C2318yes—
206#340C2318yes—
207#340C2318yes—
208#340C2318yes—
209#336C2314yes—
210#336C2314yes—
211#336C2314yes—
212#336C2314yes—
213C22×Q8—28no—
214C22×Q8—21no—
215C22×Q8—21no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—44no—
219C22×Q8—44no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2411yescharacteristic
229C2×M16—28no—
230C2×M16—28no—
231C2×M16—28no—
232C22×C8C2418yes—
233C4⋊Q8—24no—
234C4⋊Q8—24no—
235C4⋊Q8—28no—
236C4⋊Q8—28no—
237C4⋊Q8—24no—
238C4⋊Q8—24no—
239C4⋊Q8C2×Q818yes—
240C4⋊Q8C2×Q818yes—
241#125—24no—
242#125—24no—
243C22⋊Q8—28no—
244C22⋊Q8—28no—
245C4×Q8—28no—
246C4×Q8—28no—
247C4×Q8—44no—
248C4×Q8—44no—
249C4×Q8—48no—
250C4×Q8—48no—
251C4×Q8—48no—
252C4×Q8—44no—
253C4×Q8—48no—
254C4×Q8—44no—
255#117—24no—
256#117—28no—
257#117#4118yes—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—22no—
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—24no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—48no—
286C2×C4⋊C4C2×Q818yes—
287C2×C4⋊C4C2414yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2414yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42—48no—
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
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×Q814yes—
333C2×C42#4114yes—
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340#107—216no—
341#106—216no—
342C4⋊C8—28no—
343C4⋊C8—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—
358C24C22×D411yescharacteristic
359C2×Q8—24no—
360C2×Q8—24no—
361C2×Q8—28no—
362C2×Q8—28no—
363C2×Q8—44no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—44no—
367C2×Q8—432no—
368C2×Q8—44no—
369C2×Q8—42no—
370C2×Q8—42no—
371C2×Q8—44no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—28no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4C22⋊Q814yes—
401C22×C4C22⋊Q814yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×Q812yes—
416C22×C4C2×#4112yes—
417C22×C4C2×#4114yes—
418C22×C42−1+414yes—
419C22×C4C2511yescharacteristic
420M16—416no—
421C2×C8—28no—
422C2×C8—28no—
423C2×C8—28no—
424C4⋊C4—28no—
425C4⋊C4—28no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—28no—
429C4⋊C4—28no—
430C4⋊C4—28no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—28no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—48no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—48no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—42no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4C22×Q818yes—
453C4⋊C4C2×#4114yes—
454C4⋊C4C2×#4114yes—
455C22⋊C4—28no—
456C22⋊C4—44no—
457C42—22no—
458C42—22no—
459C42—28no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42—41no—
480C42—48no—
481C42—44no—
482C42—48no—
483C42—44no—
484C42—44no—
485C42—44no—
486C42—44no—
487C42C22⋊Q814yes—
488C42C22⋊Q814yes—
489C42AGL(1,ℤ/8)14yes—
490C42#13714yes—
491C42C22×Q812yes—
492C42C22×Q812yes—
493C42C2×#4112yes—
494C42C2×#4112yes—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23C2×C22⋊Q812yes—
500C23#52512yes—
501C23#53612yes—
502C23C23×D411yescharacteristic
503Q8—48no—
504Q8—48no—
505Q8—416no—
506Q8—416no—
507C2×C4—28no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—416no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—88no—
557C2×C4C2×C22⋊Q812yes—
558C2×C4C2×C22⋊Q812yes—
559C2×C4C2×C22⋊Q812yes—
560C2×C4C2×C22⋊Q812yes—
561C2×C4C2×C22⋊Q812yes—
562C2×C4#52512yes—
563C2×C4#53612yes—
564C2×C4#55618yes—
565C2×C4#55718yes—
566C2×C4C2×AGL(1,ℤ/8)12yes—
567C2×C4C2×AGL(1,ℤ/8)12yes—
568C2×C4C2×AGL(1,ℤ/8)12yes—
569C2×C4C2×#13712yes—
570C2×C4C2×#13712yes—
571C2×C4C2×#13712yes—
572C2×C4#57512yes—
573C2×C4#57512yes—
574C2×C4C23×Q811yescharacteristic
575C2×C4C22×#4111yescharacteristic
576C2×C4C22×#4112yes—
577C2×C4C2×2−1+412yes—
578C8—48no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22#258114yes—
586C22C22×C22⋊Q811yescharacteristic
587C22C2×#52511yescharacteristic
588C22C2×#53611yescharacteristic
589C22C22×AGL(1,ℤ/8)11yescharacteristic
590C22C22×#13711yescharacteristic
591C22C2×#57511yescharacteristic
592C22C2×#57511yescharacteristic
593C4—22no—
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—22no—
599C4—48no—
600C4—41no—
601C4—44no—
602C4—41no—
603C4—44no—
604C4—41no—
605C4—41no—
606C4—84no—
607C4—82no—
608C4—82no—
609C4#258114yes—
610C4#307714yes—
611C4#307814yes—
612C4C2×#55612yes—
613C4C2×#55712yes—
614C2—24no—
615C2C2×#258111yescharacteristic
616C2C2×#307711yescharacteristic
617C2C2×#307811yescharacteristic
618C2#6056111yescharacteristic
619C2#6056211yescharacteristic
620C2#6086011yescharacteristic
621C2#6086311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
3. #29277C2semidirect
4. #29276C2semidirect
5. #29275C2semidirect
6. #29274C2semidirect
7. #29273C2semidirect
8. #29272C2semidirect
85. C2×C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C4×#5332
C2×C4×C4⋊Q82
#292772
#292762
#292752
#292742
#292732
#292722
#237472
#235542
#213132
#211712
#210332
#207522