Finite Group Databasebeta

← #6463838

Subgroups — #6463838

Order 512.

1964 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3082C212yesmaximal
2#34272C211yesmaximal, characteristic
3#34209C211yesmaximal, characteristic
4#28216C214yesmaximal, complemented
5#28213C214yesmaximal, complemented
6#28209C212yesmaximal, complemented
7#28209C212yesmaximal, complemented
8#28206C212yesmaximal, complemented
9#28205C212yesmaximal, complemented
10#23657C211yesmaximal, characteristic
11#23656C211yesmaximal, characteristic
12#23507C211yesmaximal, characteristic
13C2×#2043C211yesmaximal, characteristic
14#21066C212yesmaximal
15#21064C212yesmaximal
16#20754C211yesmaximal, characteristic
17#20651C212yesmaximal
18C2×#544C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#519C2212yes—
21#3082—216no—
22C2×#479C2214yes—
23C2×#479C2214yes—
24C2×#477C2214yes—
25C2×#475C2214yes—
26C2×#428C2212yes—
27C2×#427C2212yes—
28C2×#424C2212yes—
29C2×#419C2212yes—
30C2×#419C2212yes—
31C22×#103C2212yes—
32C22×#103C2212yes—
33#2370—21no—
34#2370—21no—
35#2369—21no—
36#2368—22no—
37#2365—21no—
38#2365—21no—
39#2365—21no—
40#2365—21no—
41#2365—21no—
42#2360—22no—
43#2344—22no—
44#2344—22no—
45#2331C2212yes—
46#2330C2212yes—
47#2327C2214yes—
48#2317C2211yescharacteristic
49#2310C2212yes—
50C4×C4⋊Q8C2212yes—
51C4×C4⋊Q8C2212yes—
52C4×C4⋊Q8C2212yes—
53C4×#125C2212yes—
54#2296C2211yescharacteristic
55C2×#389C2211yescharacteristic
56#2285C2211yescharacteristic
57#2274C2211yescharacteristic
58#2274C2211yescharacteristic
59C4×#117C2211yescharacteristic
60#2085C2212yes—
61#2085C2214yes—
62#2085C2212yes—
63#2084C2211yescharacteristic
64#2084C2212yes—
65#2084C2211yescharacteristic
66#2083C2211yescharacteristic
67#2083C2212yes—
68#2083C2211yescharacteristic
69#2043C2212yes—
70#2043C2212yes—
71#2043C2212yes—
72#2043C2212yes—
73#2043C2212yes—
74#2043C2214yes—
75#2043C2212yes—
76#1928C2214yes—
77#1928C2214yes—
78#1928C2214yes—
79#1928C2214yes—
80#1927C2214yes—
81#1927C2214yes—
82#1927C2214yes—
83#1927C2214yes—
84#1832C2214yes—
85#1831C2212yes—
86#1831C2212yes—
87#1805C2214yes—
88#1805C2214yes—
89#1803C2214yes—
90#1803C2214yes—
91#1736C2212yes—
92#1735C2212yes—
93#1729C2212yes—
94#1728C2211yescharacteristic
95C2×#336C2211yescharacteristic
96C23×Q8C2312yes—
97C22×M16C2312yes—
98C23×C8C2312yes—
99#544—24no—
100#544D418yes—
101C2×C4⋊Q8—28no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×#125—21no—
109C2×#125C2312yes—
110C2×C22⋊Q8—22no—
111#519—28no—
112#519—28no—
113C2×C4×Q8—22no—
114C2×C4×Q8—22no—
115C2×C4×Q8—22no—
116C2×C4×Q8C2314yes—
117C2×C4×Q8C2314yes—
118C2×#117—24no—
119C2×#117C2312yes—
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C42C2311yescharacteristic
130#479—216no—
131#479—216no—
132#477—216no—
133#475—216no—
134#428—28no—
135#427—28no—
136C2×#107C2312yes—
137C2×#107C2314yes—
138C2×#107C2312yes—
139C2×#106C2312yes—
140C2×#106C2314yes—
141C2×#106C2312yes—
142#424—28no—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2314yes—
145#419—28no—
146#419—28no—
147C2×#103—28no—
148C2×#103—28no—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157#391—22no—
158#391—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391C2314yes—
163#391C2314yes—
164#389—21no—
165#389—21no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389C2311yescharacteristic
176#389C2312yes—
177#389C2311yescharacteristic
178#389C2311yescharacteristic
179#389C2312yes—
180#389C2311yescharacteristic
181#387—22no—
182#387—22no—
183#387—24no—
184#387—24no—
185#387—24no—
186#387—24no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#385—21no—
192#385—21no—
193#385—22no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#384C2312yes—
197#382C2312yes—
198#382C2312yes—
199#381—24no—
200#380—21no—
201#380—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4D412yes—
228C4×C22⋊C4D412yes—
229#376C2312yes—
230C2×#95—21no—
231C43C2312yes—
232#340C2312yes—
233#340C2312yes—
234#340C2314yes—
235#340C2314yes—
236#340C2312yes—
237#340C2312yes—
238#336C2311yescharacteristic
239#336C2312yes—
240#336C2311yescharacteristic
241#336C2312yes—
242#336C2312yes—
243#336C2314yes—
244#336C2314yes—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—24no—
248C22×Q8—24no—
249C22×Q8—24no—
250C22×Q8—24no—
251C22×Q8—24no—
252C22×Q8#4118yes—
253C22×Q8C2414yes—
254C23×C4—22no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2411yescharacteristic
260C2×M16—24no—
261C2×M16—24no—
262C2×M16—24no—
263C22×C8—24no—
264C22×C8—24no—
265C22×C8—24no—
266C22×C8C2412yes—
267C22×C8C2412yes—
268C22×C8C2414yes—
269C4⋊Q8—28no—
270C4⋊Q8—28no—
271C4⋊Q8—28no—
272C4⋊Q8—28no—
273C4⋊Q8—28no—
274C4⋊Q8—24no—
275C4⋊Q8C2×D418yes—
276#125—28no—
277#125—24no—
278#125C2×D418yes—
279C22⋊Q8—216no—
280C4×Q8—28no—
281C4×Q8—28no—
282C4×Q8—28no—
283C4×Q8—28no—
284C4×Q8—48no—
285C4×Q8—48no—
286C4×Q8—84no—
287#117—22no—
288#117—24no—
289#117—24no—
290#117—24no—
291#117—24no—
292#117—416no—
293#117C2×D414yes—
294#117C2×D414yes—
295#117#4114yes—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—82no—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D414yes—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2411yescharacteristic
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C42—21no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42C2×D412yes—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361C2×C42C2412yes—
362#107—24no—
363#107—24no—
364#107—28no—
365#106—24no—
366#106—24no—
367#106—28no—
368C4⋊C8—28no—
369C4⋊C8—28no—
370#103—28no—
371#103—28no—
372#103—28no—
373#103—28no—
374#103—28no—
375#103—28no—
376#103—28no—
377#103—28no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—21no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95—22no—
393#95—22no—
394#95—21no—
395#95—21no—
396#95C2×D412yes—
397#95C2×D412yes—
398#95C2×D412yes—
399#95C2×D412yes—
400C24C22×D411yescharacteristic
401C2×Q8—28no—
402C2×Q8—28no—
403C2×Q8—24no—
404C2×Q8—216no—
405C2×Q8—216no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—48no—
413C2×Q8—48no—
414C2×Q8—82no—
415C2×Q8—82no—
416C2×Q8C2×#4118yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—28no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4—81no—
440C22×C4—81no—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D412yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4114yes—
453C22×C42−1+414yes—
454C22×C4C2511yescharacteristic
455M16—48no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—24no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—48no—
466C4⋊C4—28no—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—28no—
470C4⋊C4—24no—
471C4⋊C4—28no—
472C4⋊C4—28no—
473C4⋊C4—28no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—82no—
497C4⋊C4—81no—
498C4⋊C4—81no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4#12114yes—
502C4⋊C4#12114yes—
503C4⋊C4C22×D414yes—
504C4⋊C4C22×D412yes—
505C4⋊C4C22×D412yes—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—44no—
509C22⋊C4—42no—
510C22⋊C4#12118yes—
511C42—24no—
512C42—22no—
513C42—22no—
514C42—24no—
515C42—24no—
516C42—22no—
517C42—22no—
518C42—24no—
519C42—42no—
520C42—42no—
521C42—44no—
522C42—48no—
523C42—48no—
524C42—42no—
525C42—82no—
526C42—82no—
527C42—82no—
528C42—82no—
529C42C22×D414yes—
530C42C2×#4114yes—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23C2×#12112yes—
536C23#52512yes—
537C23#53612yes—
538C23C23×D411yescharacteristic
539Q8—28no—
540Q8—28no—
541Q8—44no—
542Q8—44no—
543Q8—416no—
544Q8—88no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—28no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—416no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—84no—
596C2×C4—162no—
597C2×C4C2×#12114yes—
598C2×C4C2×#12114yes—
599C2×C4C2×#12112yes—
600C2×C4#52512yes—
601C2×C4#53612yes—
602C2×C4#54818yes—
603C2×C4#55218yes—
604C2×C4#57712yes—
605C2×C4#57812yes—
606C2×C4C23×D411yescharacteristic
607C2×C4C22×#4112yes—
608C2×C4C22×#4111yescharacteristic
609C2×C4C2×2−1+412yes—
610C8—42no—
611C8—42no—
612C8—44no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22#253614yes—
620C22C22×#12111yescharacteristic
621C22C2×#52511yescharacteristic
622C22C2×#53611yescharacteristic
623C22C2×#57711yescharacteristic
624C22C2×#57811yescharacteristic
625C22#358611yescharacteristic
626C22#358611yescharacteristic
627C4—24no—
628C4—24no—
629C4—22no—
630C4—22no—
631C4—44no—
632C4—42no—
633C4—42no—
634C4—48no—
635C4—41no—
636C4—41no—
637C4—42no—
638C4—42no—
639C4—82no—
640C4—82no—
641C4—81no—
642C4—82no—
643C4—81no—
644C4#253614yes—
645C4C2×#54812yes—
646C4C2×#55212yes—
647C2—24no—
648C2C2×#253611yescharacteristic
649C2#6042911yescharacteristic
650C2#6043211yescharacteristic
651C2#6058211yescharacteristic
652C2#6058311yescharacteristic
653C2#6087811yescharacteristic
654C2#6088411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #28216C2semidirect
5. #28213C2semidirect
6. #28209C2semidirect
7. #28209C2semidirect
8. #28206C2semidirect
9. #28205C2semidirect

Maximal subgroups

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

subgroupindex
C2×#30822
#342722
#342092
#282162
#282132
#282092
#282062
#282052
#236572
#236562
#235072
C2×#20432
#210662
#210642
#207542
#206512