Finite Group Databasebeta

← #6463922

Subgroups — #6463922

Order 512.

2014 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
1C2×#2859C211yesmaximal, characteristic, complemented
2C4×#532C211yesmaximal, characteristic
3#29132C212yesmaximal, complemented
4#29131C214yesmaximal, complemented
5#29130C212yesmaximal, complemented
6#29123C212yesmaximal, complemented
7#29122C214yesmaximal, complemented
8#29121C212yesmaximal, complemented
9#23747C212yesmaximal, complemented
10#23508C212yesmaximal
11#21093C214yesmaximal, complemented
12#20648C212yesmaximal, complemented
13#20648C212yesmaximal, complemented
14C4×#423C211yesmaximal, characteristic, complemented
15C2×#532C2211yescharacteristic
16C22×#125—21no—
17C2×C4×#41C2212yes—
18C2×#423C2212yes—
19C2×#417C2214yes—
20C2×#417C2214yes—
21C2×C4×M16C2212yes—
22#2859—21no—
23#2859—21no—
24#2859—21no—
25#2859—21no—
26#2859C2212yes—
27#2859C2212yes—
28#2859C2212yes—
29#2859C2212yes—
30C4×C4⋊Q8C2211yescharacteristic
31C4×C4⋊Q8C2211yescharacteristic
32C4×#127C2211yescharacteristic
33C4×#127C2211yescharacteristic
34C4×#124C2212yes—
35C4×#124C2212yes—
36C4×#121—21no—
37C4×#121—21no—
38C4×#121—21no—
39C4×#121—21no—
40#2297C2212yes—
41#2293—21no—
42#2293—21no—
43C2×#389—21no—
44C2×#389—21no—
45C2×#389C2212yes—
46C2×#389C2212yes—
47#2285C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C2×#384C2211yescharacteristic
50C2×#382—21no—
51C42×D4—21no—
52C42×D4—21no—
53C2×C43C2211yescharacteristic
54#2101C2218yes—
55#2100C2214yes—
56#2099C2214yes—
57#2043C2214yes—
58#2042C2218yes—
59#2041C2214yes—
60#1936C2218yes—
61#1935C2218yes—
62#1934C2218yes—
63#1933C2218yes—
64#1803C2214yes—
65#1803C2214yes—
66#1803C2214yes—
67#1803C2214yes—
68#1802C2214yes—
69#1802C2214yes—
70#1802C2214yes—
71#1802C2214yes—
72#1767C2212yes—
73#1767C2212yes—
74C4×C4⋊C8C2212yes—
75C4×C4⋊C8C2212yes—
76#1736C2214yes—
77#1736C2214yes—
78#1728C2212yes—
79C22×#41C2312yes—
80C22×M16C2314yes—
81#532—28no—
82C2×C4⋊Q8C2311yescharacteristic
83C2×#127C2311yescharacteristic
84C2×#125—22no—
85C2×#125—22no—
86C2×#125—22no—
87C2×#125—22no—
88C2×#125—48no—
89C2×#124C2312yes—
90C2×#121—21no—
91C2×#121—21no—
92C4×#41—28no—
93C4×#41—28no—
94C2×C4×Q8C2312yes—
95C2×C4×Q8C2312yes—
96C2×C4×D4—22no—
97C2×C4×D4—22no—
98C2×C4×D4—21no—
99C2×C4×D4—21no—
100C2×C4×D4—21no—
101C2×C4×D4—21no—
102C2×C4×D4—21no—
103C2×C4×D4—21no—
104C2×C4×D4C2312yes—
105C2×C4×D4C2312yes—
106C2×#117—22no—
107C2×#117C2312yes—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—22no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C42—22no—
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#423—28no—
125C2×C4⋊C8C2314yes—
126C2×C4⋊C8C2314yes—
127#417—216no—
128#417—48no—
129C2×#103C2318yes—
130C2×#103C2318yes—
131C2×#102C2318yes—
132C2×#102C2318yes—
133C4×M16—44no—
134C2×#97C2314yes—
135C2×C4×C8C2314yes—
136#391C2312yes—
137#391C2312yes—
138#390—21no—
139#390—21no—
140#390—21no—
141#390—21no—
142#390C2312yes—
143#390C2312yes—
144#389—21no—
145#389—21no—
146#389—21no—
147#389—21no—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389C2314yes—
161#389C2314yes—
162#389C2314yes—
163#389C2314yes—
164#388—22no—
165#388—22no—
166#388—22no—
167#388—22no—
168#386—22no—
169#386—22no—
170#384—21no—
171#384—21no—
172#384—21no—
173#384—21no—
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#384C2312yes—
177#384C2312yes—
178#384C2312yes—
179#384C2312yes—
180#382—22no—
181#382—22no—
182#382—22no—
183#382—22no—
184#382C2312yes—
185#380—21no—
186#380—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—21no—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—22no—
195C4×C22⋊C4—22no—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199C2×#95—22no—
200C2×#95—22no—
201C43D411yescharacteristic
202C43D411yescharacteristic
203C43D411yescharacteristic
204C43D411yescharacteristic
205C43C2311yescharacteristic
206C43C2311yescharacteristic
207C43C2311yescharacteristic
208C43C2311yescharacteristic
209#340C2318yes—
210#340C2318yes—
211#340C2318yes—
212#340C2318yes—
213#336C2314yes—
214#336C2314yes—
215C2×#41—24no—
216C2×#41—24no—
217C2×#41—24no—
218C2×#41—24no—
219C2×#41—24no—
220C2×#41#4118yes—
221C22×Q8C2412yes—
222C22×D4—22no—
223C22×D4—21no—
224C22×D4—21no—
225C22×D4C2412yes—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2411yescharacteristic
238C2×M16—28no—
239C2×M16—28no—
240C2×M16—28no—
241C22×C8C2418yes—
242C4⋊Q8—24no—
243#127—24no—
244#125—48no—
245#125—48no—
246#125—48no—
247#125—48no—
248#124—24no—
249#124—24no—
250#121—28no—
251#121—28no—
252C4×Q8—24no—
253C4×Q8—24no—
254C4×Q8—24no—
255C4×Q8—24no—
256C4×D4—28no—
257C4×D4—28no—
258C4×D4—24no—
259C4×D4—24no—
260C4×D4—24no—
261C4×D4—24no—
262C4×D4—44no—
263C4×D4—44no—
264C4×D4—48no—
265C4×D4—48no—
266C4×D4—44no—
267C4×D4—44no—
268#117—24no—
269#117—24no—
270#117—48no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—48no—
301C2×C4⋊C4—48no—
302C2×C4⋊C4C2414yes—
303C2×C4⋊C4C2414yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
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—22no—
314C2×C22⋊C4—22no—
315C2×C42—24no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—42no—
324C2×C42—42no—
325C2×C42—48no—
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×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42#4118yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C4⋊C8—28no—
356C4⋊C8—28no—
357#103—216no—
358#103—48no—
359#102—216no—
360#102—48no—
361#97—44no—
362C4×C8—44no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373C24—22no—
374C24C22×D411yescharacteristic
375#41—216no—
376#41—216no—
377#41—48no—
378#41—48no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8C2×#4114yes—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—28no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4C2×#4114yes—
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—28no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4#12414yes—
437C22×C4#12414yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
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×C4C2×#4114yes—
452C22×C4C2×#4114yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C42−1+412yes—
455C22×C4C2511yescharacteristic
456M16—416no—
457C2×C8—28no—
458C2×C8—28no—
459C2×C8—28no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—28no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C22⋊C4—28no—
490C22⋊C4—28no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—41no—
522C42—44no—
523C42—44no—
524C42—44no—
525C42#12414yes—
526C42#12414yes—
527C42AGL(1,ℤ/8)14yes—
528C42#13714yes—
529C42C2×#4114yes—
530C42C2×#4114yes—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—24no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23#52512yes—
543C23C2×#12412yes—
544C23#53712yes—
545C23C23×D411yescharacteristic
546Q8—24no—
547Q8—24no—
548Q8—42no—
549Q8—42no—
550D4—24no—
551D4—24no—
552D4—48no—
553D4—48no—
554D4—416no—
555D4—42no—
556D4—42no—
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—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—24no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—416no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—88no—
604C2×C4—88no—
605C2×C4—88no—
606C2×C4—88no—
607C2×C4#52512yes—
608C2×C4C2×#12412yes—
609C2×C4C2×#12412yes—
610C2×C4C2×#12412yes—
611C2×C4C2×#12412yes—
612C2×C4C2×#12412yes—
613C2×C4#53712yes—
614C2×C4#55218yes—
615C2×C4#55318yes—
616C2×C4C2×AGL(1,ℤ/8)12yes—
617C2×C4C2×AGL(1,ℤ/8)12yes—
618C2×C4C2×AGL(1,ℤ/8)12yes—
619C2×C4C2×#13712yes—
620C2×C4C2×#13712yes—
621C2×C4C2×#13712yes—
622C2×C4#57512yes—
623C2×C4#57512yes—
624C2×C4C22×#4112yes—
625C2×C4C22×#4112yes—
626C2×C4C2×H(5,𝔽2)11yescharacteristic
627C2×C4C2×2−1+411yescharacteristic
628C8—48no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22#256214yes—
643C22C2×#52511yescharacteristic
644C22C22×#12411yescharacteristic
645C22C2×#53711yescharacteristic
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—44no—
657C4—41no—
658C4—41no—
659C4—41no—
660C4—41no—
661C4—84no—
662C4—82no—
663C4—82no—
664C4—84no—
665C4#256214yes—
666C4#314114yes—
667C4#314214yes—
668C4C2×#55212yes—
669C4C2×#55312yes—
670C2—24no—
671C2—84no—
672C2C2×#256211yescharacteristic
673C2#6056111yescharacteristic
674C2#6056211yescharacteristic
675C2C2×#314111yescharacteristic
676C2C2×#314211yescharacteristic
677C2#6091611yescharacteristic
678C2#6091811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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. C2×#2859C2semidirect
3. #29132C2semidirect
4. #29131C2semidirect
5. #29130C2semidirect
6. #29123C2semidirect
7. #29122C2semidirect
8. #29121C2semidirect
9. #23747C2semidirect
11. #21093C2semidirect
12. #20648C2semidirect
13. #20648C2semidirect
14. C4×#423C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28592
C4×#5322
#291322
#291312
#291302
#291232
#291222
#291212
#237472
#235082
#210932
#206482
C4×#4232