Finite Group Databasebeta

← #6434355

Subgroups — #6434355

Order 512.

1920 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35265C211yesmaximal, characteristic, complemented
2#34210C211yesmaximal, characteristic
3#30085C214yesmaximal, complemented
4#30084C214yesmaximal, complemented
5#30077C212yesmaximal, complemented
6#30077C212yesmaximal, complemented
7#30076C212yesmaximal, complemented
8#30075C212yesmaximal, complemented
9#23735C212yesmaximal, complemented
10#23657C211yesmaximal, characteristic, complemented
11#23656C211yesmaximal, characteristic, complemented
12#23632C212yesmaximal, complemented
13#23505C211yesmaximal, characteristic
14C2×#2041C211yesmaximal, characteristic
15#20879C212yesmaximal, complemented
16#20876C212yesmaximal, complemented
17#20774C211yesmaximal, characteristic, complemented
18C22×#127C2211yescharacteristic
19C2×#518C2212yes—
20C2×#428C2212yes—
21C2×#427C2212yes—
22C2×#418C2212yes—
23C2×#418C2212yes—
24C22×#102C2212yes—
25C22×#102C2212yes—
26#2864C2212yescomplemented
27#2859C2212yescomplemented
28#2647C2212yescomplemented
29#2647C2212yescomplemented
30#2509—21no—
31#2508—21no—
32#2507—21no—
33#2506—21no—
34#2505—22no—
35#2473—22no—
36#2366—21no—
37#2366—21no—
38#2366—21no—
39#2366—21no—
40D4×C22⋊C4—22no—
41D4×C22⋊C4—22no—
42#2328C2214yes—
43#2317C2211yescharacteristic
44#2317C2211yescharacteristic
45#2317C2211yescharacteristic
46#2316C2211yescharacteristic
47#2309C2212yes—
48C4×#127C2212yes—
49C4×#127C2212yes—
50#2296C2211yescharacteristic
51C2×#389C2211yescharacteristic
52#2285C2211yescharacteristic
53C4×#117C2211yescharacteristic
54#2094C2214yescomplemented
55#2093C2214yescomplemented
56#2092C2214yescomplemented
57#2092C2214yescomplemented
58#2085C2212yescomplemented
59#2085C2212yescomplemented
60#2085C2214yescomplemented
61#2084C2211yescharacteristic, complemented
62#2084C2211yescharacteristic, complemented
63#2084C2212yescomplemented
64#2083C2211yescharacteristic, complemented
65#2083C2211yescharacteristic, complemented
66#2083C2212yescomplemented
67#2073C2214yescomplemented
68#2073C2214yescomplemented
69#2071C2214yescomplemented
70#2071C2214yescomplemented
71#2041C2212yes—
72#2041C2212yes—
73#2041C2212yes—
74#2041C2212yes—
75#2041C2212yes—
76#2041C2214yes—
77#2041C2212yes—
78#1864C2214yescomplemented
79#1864C2214yescomplemented
80#1864C2214yescomplemented
81#1864C2214yescomplemented
82#1862C2214yescomplemented
83#1862C2214yescomplemented
84#1862C2214yescomplemented
85#1862C2214yescomplemented
86#1840C2214yescomplemented
87#1839C2212yescomplemented
88#1839C2212yescomplemented
89#1736C2212yes—
90#1735C2212yes—
91#1729C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C23×D4—21no—
96C23×D4—21no—
97C23×D4C2312yes—
98C22×M16C2312yes—
99C23×C8C2312yes—
100C2×#127—28no—
101C2×#127—21no—
102C2×#127—21no—
103C2×#127C2312yes—
104C2×#127C2312yes—
105C2×#125—22no—
106#518—44no—
107#518—44no—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—22no—
111C2×C4×D4—22no—
112C2×C4×D4—22no—
113C2×C4×D4—22no—
114C2×C4×D4C2314yes—
115C2×C4×D4C2314yes—
116C2×#117C2312yes—
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C22×C4⋊C4—21no—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—24no—
125C22×C42C2311yescharacteristic
126#428—44no—
127#427—44no—
128C2×#107C2312yes—
129C2×#107C2314yes—
130C2×#107C2312yes—
131C2×#106C2312yes—
132C2×#106C2314yes—
133C2×#106C2312yes—
134#418—28no—
135#418—44no—
136C2×#102—28no—
137C2×#102—44no—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146#400—22no—
147#400—22no—
148#400—22no—
149#400—22no—
150#398—22no—
151#398—22no—
152#398—22no—
153#398—22no—
154#390—22no—
155#390—22no—
156#390—22no—
157#390—22no—
158#390—22no—
159#390—22no—
160#390C2314yes—
161#390C2314yes—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—22no—
169#389—22no—
170#389—21no—
171#389—21no—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2311yescharacteristic
177#389C2312yes—
178#389C2311yescharacteristic
179#389C2312yes—
180#389C2311yescharacteristic
181#389C2311yescharacteristic
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#386—24no—
187#386—24no—
188#386—24no—
189#386—24no—
190#385—21no—
191#385—21no—
192#385—22no—
193#385—22no—
194#385—22no—
195#384—21no—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2312yes—
200#382—21no—
201#382—22no—
202#382C2312yes—
203#382C2312yes—
204#382C2312yes—
205#380—21no—
206#380—21no—
207#380—22no—
208#379—24no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D411yescharacteristic
217C4×C22⋊C4D412yes—
218C4×C22⋊C4D412yes—
219#376C2312yes—
220C2×#95—21no—
221C43C2312yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#336C2314yes—
229#336C2314yes—
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2312yes—
234#336C2312yes—
235#336C2314yes—
236#336C2314yes—
237C25—22no—
238C22×D4—28no—
239C22×D4—28no—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—24no—
246C22×D4—24no—
247C22×D4—24no—
248C22×D4—24no—
249C22×D4—24no—
250C22×D4—82no—
251C22×D4—82no—
252C22×D4—82no—
253C22×D4—82no—
254C22×D4#4118yes—
255C22×D4C2414yes—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—24no—
259C23×C4—24no—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2411yescharacteristic
263C2×M16—24no—
264C2×M16—24no—
265C2×M16—24no—
266C22×C8—24no—
267C22×C8—24no—
268C22×C8—24no—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C22×C8C2414yes—
272#127—28no—
273#127—28no—
274#127—28no—
275#127—28no—
276#125—84no—
277C4×D4—44no—
278C4×D4—44no—
279C4×D4—44no—
280C4×D4—44no—
281C4×D4—48no—
282C4×D4—48no—
283C4×D4—84no—
284C4×D4—84no—
285C4×D4—84no—
286C4×D4—84no—
287#117—22no—
288#117—22no—
289#117—24no—
290#117—24no—
291#117—24no—
292#117—24no—
293#117—22no—
294#117—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
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—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—82no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—416no—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2412yes—
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#107—42no—
358#107—42no—
359#107—44no—
360#106—42no—
361#106—42no—
362#106—44no—
363#102—28no—
364#102—28no—
365#102—28no—
366#102—28no—
367#102—44no—
368#102—44no—
369#102—44no—
370#102—44no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24—24no—
392C24—28no—
393C24—44no—
394C24—44no—
395C24—82no—
396C24—82no—
397C24—82no—
398C24—82no—
399C24C22×D411yescharacteristic
400C2×D4—28no—
401C2×D4—28no—
402C2×D4—28no—
403C2×D4—28no—
404C2×D4—28no—
405C2×D4—216no—
406C2×D4—216no—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—82no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4—816no—
425C2×D4—816no—
426C2×D4C2×#4118yes—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—84no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4—84no—
449C22×C4—81no—
450C22×C4—81no—
451C22×C4—81no—
452C22×C4—81no—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C2×#4114yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C42−1+414yes—
466C22×C4C2511yescharacteristic
467M16—48no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—48no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—81no—
509C4⋊C4—81no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—42no—
520C22⋊C4—44no—
521C22⋊C4—42no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C42—22no—
527C42—22no—
528C42—24no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—24no—
535C42—42no—
536C42—42no—
537C42—42no—
538C42—42no—
539C42—44no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—28no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—416no—
552C23—88no—
553C23—88no—
554C23—164no—
555C23—164no—
556C23—164no—
557C23—164no—
558C23#52512yes—
559C23#53512yes—
560C23#53612yes—
561C23C23×D411yescharacteristic
562D4—28no—
563D4—28no—
564D4—416no—
565D4—416no—
566D4—416no—
567D4—44no—
568D4—44no—
569D4—88no—
570D4—88no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—84no—
620C2×C4—168no—
621C2×C4—168no—
622C2×C4—162no—
623C2×C4#52512yes—
624C2×C4#53512yes—
625C2×C4#53612yes—
626C2×C4#53918yes—
627C2×C4#53918yes—
628C2×C4#57612yes—
629C2×C4#57712yes—
630C2×C4C22×#4112yes—
631C2×C4C2×H(5,𝔽2)11yescharacteristic
632C2×C4C2×H(5,𝔽2)11yescharacteristic
633C2×C4C2×2−1+412yes—
634C8—42no—
635C8—42no—
636C8—44no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—48no—
644C22—48no—
645C22—48no—
646C22—84no—
647C22—84no—
648C22—84no—
649C22—84no—
650C22—1616no—
651C22#263314yes—
652C22C2×#52511yescharacteristic
653C22C2×#53511yescharacteristic
654C22C2×#53611yescharacteristic
655C22C2×#57611yescharacteristic
656C22C2×#57711yescharacteristic
657C22#358511yescharacteristic
658C22#358511yescharacteristic
659C4—22no—
660C4—22no—
661C4—42no—
662C4—42no—
663C4—41no—
664C4—41no—
665C4—42no—
666C4—42no—
667C4—81no—
668C4—82no—
669C4—82no—
670C4—81no—
671C4—82no—
672C4—81no—
673C4—82no—
674C4—81no—
675C4#263314yes—
676C4C2×#53912yes—
677C4C2×#53912yes—
678C2—24no—
679C2—88no—
680C2C2×#263311yescharacteristic
681C2#6057811yescharacteristic
682C2#6057911yescharacteristic
683C2#6083211yescharacteristic
684C2#6083311yescharacteristic
685C2#6087711yescharacteristic
686C2#6088311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
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
1. #35265C2semidirect
3. #30085C2semidirect
4. #30084C2semidirect
5. #30077C2semidirect
6. #30077C2semidirect
7. #30076C2semidirect
8. #30075C2semidirect
9. #23735C2semidirect
10. #23657C2semidirect
11. #23656C2semidirect
12. #23632C2semidirect
15. #20879C2semidirect
16. #20876C2semidirect
17. #20774C2semidirect
26. #2864C22semidirect
27. #2859C22semidirect
28. #2647C22semidirect
29. #2647C22semidirect
54. #2094C22semidirect
55. #2093C22semidirect
56. #2092C22semidirect
57. #2092C22semidirect
58. #2085C22semidirect
59. #2085C22semidirect
60. #2085C22semidirect
61. #2084C22semidirect
62. #2084C22semidirect
63. #2084C22semidirect
64. #2083C22semidirect
65. #2083C22semidirect
66. #2083C22semidirect
67. #2073C22semidirect
68. #2073C22semidirect
69. #2071C22semidirect
70. #2071C22semidirect
78. #1864C22semidirect
79. #1864C22semidirect
80. #1864C22semidirect
81. #1864C22semidirect
82. #1862C22semidirect
83. #1862C22semidirect
84. #1862C22semidirect
85. #1862C22semidirect
86. #1840C22semidirect
87. #1839C22semidirect
88. #1839C22semidirect

Maximal subgroups

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

subgroupindex
#352652
#342102
#300852
#300842
#300772
#300762
#300752
#237352
#236572
#236562
#236322
#235052
C2×#20412
#208792
#208762
#207742