Finite Group Databasebeta

← #6436167

Subgroups — #6436167

Order 512.

1862 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3081C212yesmaximal
2#35275C211yesmaximal, characteristic, complemented
3#34255C211yesmaximal, characteristic
4#29874C214yesmaximal, complemented
5#29872C214yesmaximal, complemented
6#29870C212yesmaximal, complemented
7#29870C212yesmaximal, complemented
8#29868C212yesmaximal, complemented
9#29864C212yesmaximal, complemented
10#23662C212yesmaximal, complemented
11#23658C211yesmaximal, characteristic, complemented
12#23656C211yesmaximal, characteristic, complemented
13#23632C212yesmaximal
14#23628C211yesmaximal, characteristic
15C2×#2071C211yesmaximal, characteristic
16#21315C212yesmaximal, complemented
17#20754C211yesmaximal, characteristic, complemented
18C2×#544C2212yes—
19C2×#518C2211yescharacteristic
20#3081—216no—
21C2×#478C2214yes—
22C2×#478C2214yes—
23C2×#476C2214yes—
24C2×#474C2214yes—
25C2×#428C2212yes—
26C2×#427C2212yes—
27C2×#424C2212yes—
28C2×#418C2212yes—
29C22×#102C2212yes—
30#2868C2212yescomplemented
31#2859C2212yescomplemented
32#2660C2214yescomplemented
33#2552—21no—
34#2552—21no—
35#2551—21no—
36#2551—21no—
37#2440—22no—
38#2440—22no—
39#2357—21no—
40#2357—21no—
41#2354—21no—
42#2354—21no—
43#2342—22no—
44#2342—22no—
45#2319C2212yes—
46#2319C2212yes—
47#2317C2211yescharacteristic
48#2317C2212yes—
49#2309C2211yescharacteristic
50#2296C2211yescharacteristic
51#2293C2212yes—
52#2293C2212yes—
53C2×#390C2211yescharacteristic
54#2285C2211yescharacteristic
55C2×#384C2211yescharacteristic
56#2274C2211yescharacteristic
57C4×#117C2211yescharacteristic
58#2085C2212yescomplemented
59#2085C2212yescomplemented
60#2085C2214yescomplemented
61#2085C2214yescomplemented
62#2084C2211yescharacteristic, complemented
63#2084C2211yescharacteristic, complemented
64#2084C2212yescomplemented
65#2084C2212yescomplemented
66#2084C2214yescomplemented
67#2083C2211yescharacteristic, complemented
68#2083C2211yescharacteristic, complemented
69#2083C2212yescomplemented
70#2083C2212yescomplemented
71#2083C2214yescomplemented
72#2073C2214yes—
73#2073C2214yes—
74#2073C2214yes—
75#2071C2212yes—
76#2071C2212yes—
77#2071C2214yes—
78#2071C2212yes—
79#2071C2212yes—
80#2071C2214yes—
81#2071C2214yes—
82#1989C2214yescomplemented
83#1989C2214yescomplemented
84#1988C2214yescomplemented
85#1987C2214yescomplemented
86#1832C2214yescomplemented
87#1831C2212yescomplemented
88#1831C2212yescomplemented
89#1736C2212yes—
90#1736C2212yes—
91#1735C2212yes—
92C2×#340C2212yes—
93#1729C2212yes—
94#1728C2211yescharacteristic
95C2×#336C2211yescharacteristic
96C23×D4C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#544—28no—
100#544—28no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8C2314yes—
105C2×#125—22no—
106C2×#125C2314yes—
107C2×C22⋊Q8—24no—
108#518—24no—
109#518D418yes—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4C2312yes—
113C2×C4×D4C2312yes—
114C2×#117—22no—
115C2×#117—22no—
116C2×#117C2311yescharacteristic
117C2×#117C2312yes—
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C22⋊C4—22no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C42C2311yescharacteristic
129#478—216no—
130#478—216no—
131#476—216no—
132#474—216no—
133#428—28no—
134#427—28no—
135C2×#107C2312yes—
136C2×#107C2314yes—
137C2×#107C2312yes—
138C2×#106C2312yes—
139C2×#106C2314yes—
140C2×#106C2312yes—
141#424—28no—
142C2×C4⋊C8C2314yes—
143C2×C4⋊C8C2314yes—
144#418—28no—
145C2×#102—28no—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150#398—24no—
151#398—24no—
152#395—24no—
153#395—24no—
154#390—21no—
155#390—21no—
156#390—21no—
157#390—21no—
158#390C2312yes—
159#390C2312yes—
160#390C2312yes—
161#390C2312yes—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389C2312yes—
169#389C2312yes—
170#389C2311yescharacteristic
171#389C2311yescharacteristic
172#389C2311yescharacteristic
173#389C2311yescharacteristic
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#385—21no—
183#385—21no—
184#385—22no—
185#385—24no—
186#384—21no—
187#384—22no—
188#384—22no—
189#384—21no—
190#384—21no—
191#384C2314yes—
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2311yescharacteristic
195#384C2312yes—
196#382—21no—
197#382C2314yes—
198#382C2312yes—
199#381—22no—
200#381—22no—
201#380—21no—
202#380—21no—
203#380—22no—
204#380—22no—
205#379—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4D412yes—
223#376C2312yes—
224C43C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#336C2311yescharacteristic
236#336C2312yes—
237#336C2311yescharacteristic
238#336C2312yes—
239#336C2314yes—
240#336C2314yes—
241#336C2312yes—
242C25—21no—
243C22×Q8—24no—
244C22×D4—21no—
245C22×D4—22no—
246C22×D4—24no—
247C22×D4—24no—
248C22×D4C2×D414yes—
249C22×D4C2412yes—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2411yescharacteristic
257C2×M16—24no—
258C2×M16—24no—
259C2×M16—24no—
260C22×C8—24no—
261C22×C8—24no—
262C22×C8—24no—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2414yes—
266C4⋊Q8—28no—
267C4⋊Q8—28no—
268C4⋊Q8—48no—
269C4⋊Q8—84no—
270C4⋊Q8—84no—
271#125—28no—
272#125—28no—
273#125—48no—
274C22⋊Q8—416no—
275C4×D4—28no—
276C4×D4—28no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4C2×D418yes—
280C4×D4C2×D418yes—
281#117—22no—
282#117—24no—
283#117—24no—
284#117—24no—
285#117—24no—
286#117—84no—
287#117—84no—
288#117C2×D414yes—
289#117C2×D414yes—
290#117#4114yes—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—82no—
316C2×C4⋊C4—82no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—216no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—82no—
344C2×C22⋊C4—82no—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C42—21no—
348C2×C42—24no—
349C2×C42—22no—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D412yes—
358C2×C42C2×D412yes—
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362C2×C42C2412yes—
363#107—24no—
364#107—24no—
365#107—28no—
366#106—24no—
367#106—24no—
368#106—28no—
369C4⋊C8—28no—
370C4⋊C8—28no—
371#102—28no—
372#102—28no—
373#102—28no—
374#102—28no—
375#95—21no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—21no—
384#95—21no—
385#95—24no—
386#95—24no—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24—22no—
392C24—24no—
393C24—42no—
394C24—42no—
395C24—81no—
396C24—81no—
397C24—81no—
398C24—81no—
399C24C22×D411yescharacteristic
400C2×Q8—48no—
401C2×Q8—84no—
402C2×Q8—84no—
403C2×D4—24no—
404C2×D4—216no—
405C2×D4—22no—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—48no—
411C2×D4C22×D414yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4—81no—
435C22×C4—81no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C2×#4112yes—
449C22×C42−1+414yes—
450C22×C42−1+414yes—
451C22×C4C2511yescharacteristic
452M16—48no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—48no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—81no—
490C4⋊C4—81no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—84no—
496C4⋊C4—84no—
497C4⋊C4#12114yes—
498C4⋊C4#12114yes—
499C4⋊C4C22×D412yes—
500C4⋊C4C22×D412yes—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—28no—
504C22⋊C4—28no—
505C22⋊C4—44no—
506C22⋊C4—42no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—82no—
510C22⋊C4—82no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C22⋊C4#12118yes—
516C42—24no—
517C42—22no—
518C42—22no—
519C42—24no—
520C42—24no—
521C42—22no—
522C42—22no—
523C42—24no—
524C42—42no—
525C42—42no—
526C42—44no—
527C42—42no—
528C42—82no—
529C42—82no—
530C42C22×D414yes—
531C42C2×#4114yes—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—24no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—48no—
544C23—84no—
545C23—84no—
546C23—162no—
547C23—162no—
548C23—162no—
549C23—162no—
550C23C2×#12112yes—
551C23#53612yes—
552C23#53612yes—
553C23C23×D411yescharacteristic
554Q8—168no—
555D4—28no—
556D4—48no—
557D4—44no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—88no—
605C2×C4—88no—
606C2×C4—164no—
607C2×C4—162no—
608C2×C4C2×#12114yes—
609C2×C4C2×#12114yes—
610C2×C4C2×#12112yes—
611C2×C4#53612yes—
612C2×C4#53612yes—
613C2×C4#53618yes—
614C2×C4#54118yes—
615C2×C4#57612yes—
616C2×C4#57712yes—
617C2×C4C23×D411yescharacteristic
618C2×C4C22×#4111yescharacteristic
619C2×C4C2×2−1+412yes—
620C2×C4C2×2−1+412yes—
621C8—42no—
622C8—42no—
623C8—44no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—168no—
638C22#262114yes—
639C22C22×#12111yescharacteristic
640C22C2×#53611yescharacteristic
641C22C2×#53611yescharacteristic
642C22C2×#57611yescharacteristic
643C22C2×#57711yescharacteristic
644C22#358511yescharacteristic
645C22#358611yescharacteristic
646C4—24no—
647C4—24no—
648C4—22no—
649C4—22no—
650C4—42no—
651C4—42no—
652C4—41no—
653C4—41no—
654C4—42no—
655C4—42no—
656C4—84no—
657C4—82no—
658C4—82no—
659C4—81no—
660C4—81no—
661C4—82no—
662C4#262114yes—
663C4C2×#53612yes—
664C4C2×#54112yes—
665C2—24no—
666C2—84no—
667C2C2×#262111yescharacteristic
668C2#6042811yescharacteristic
669C2#6042911yescharacteristic
670C2#6087111yescharacteristic
671C2#6087311yescharacteristic
672C2#6087711yescharacteristic
673C2#6088511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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
2. #35275C2semidirect
4. #29874C2semidirect
5. #29872C2semidirect
6. #29870C2semidirect
7. #29870C2semidirect
8. #29868C2semidirect
9. #29864C2semidirect
10. #23662C2semidirect
11. #23658C2semidirect
12. #23656C2semidirect
16. #21315C2semidirect
17. #20754C2semidirect
30. #2868C22semidirect
31. #2859C22semidirect
32. #2660C22semidirect
58. #2085C22semidirect
59. #2085C22semidirect
60. #2085C22semidirect
61. #2085C22semidirect
62. #2084C22semidirect
63. #2084C22semidirect
64. #2084C22semidirect
65. #2084C22semidirect
66. #2084C22semidirect
67. #2083C22semidirect
68. #2083C22semidirect
69. #2083C22semidirect
70. #2083C22semidirect
71. #2083C22semidirect
82. #1989C22semidirect
83. #1989C22semidirect
84. #1988C22semidirect
85. #1987C22semidirect
86. #1832C22semidirect
87. #1831C22semidirect
88. #1831C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30812
#352752
#342552
#298742
#298722
#298702
#298682
#298642
#236622
#236582
#236562
#236322
#236282
C2×#20712
#213152
#207542