Finite Group Databasebeta

← #6502655

Subgroups — #6502655

Order 512.

2088 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3195C211yesmaximal, characteristic
2C22×#467C211yesmaximal, characteristic
3#37453C211yesmaximal, characteristic
4#34380C211yesmaximal, characteristic
5#28588C214yesmaximal, complemented
6#25208C214yesmaximal, complemented
7#25082C214yesmaximal, complemented
8#24764C214yesmaximal, complemented
9#24577C212yesmaximal, complemented
10#24529C212yesmaximal, complemented
11#24480C212yesmaximal, complemented
12#24360C212yesmaximal, complemented
13#21765C211yesmaximal, characteristic
14C2×#1892C211yesmaximal, characteristic
15#20834C211yesmaximal, characteristic
16C23×Q16C2211yescharacteristic
17C24×C8C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21#3195—28no—
22C2×#483C2212yes—
23C2×#483C2212yes—
24C2×#470C2212yes—
25C2×#470C2212yes—
26C2×#467—28no—
27C2×#467C2214yes—
28C2×#467C2214yes—
29C2×#462C2214yescomplemented
30C2×#451C2212yes—
31C2×#451C2212yes—
32C22×#107C2211yescharacteristic
33C2×#424C2212yes—
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2212yes—
37C2×#407C2211yescharacteristic
38#2817—22no—
39#2627—21no—
40#2627—21no—
41#2627—22no—
42#2621C2214yes—
43#2503C2212yes—
44#2503C2212yes—
45#2483—22no—
46#2434C2212yes—
47#2434C2212yes—
48#2378—22no—
49#2360C2214yes—
50#2344C2212yes—
51#2342—22no—
52Q8×C22⋊C4C2212yes—
53#2329—22no—
54#2323C2214yes—
55#2316—21no—
56#2313—21no—
57#2280C2211yescharacteristic
58C2×#380C2211yescharacteristic
59#2085C2214yescomplemented
60#2074C2214yescomplemented
61#2068C2214yescomplemented
62#2066C2214yescomplemented
63#2056C2214yescomplemented
64#2056C2214yescomplemented
65#2043C2214yescomplemented
66#2016C2214yescomplemented
67#1981C2214yescomplemented
68#1949C2214yescomplemented
69#1928C2214yescomplemented
70#1927C2214yescomplemented
71#1894C2212yes—
72#1894C2212yes—
73#1892C2212yes—
74#1892C2212yes—
75#1892C2214yes—
76#1878C2214yescomplemented
77#1878C2214yescomplemented
78#1872C2214yes—
79#1850C2212yes—
80#1850C2214yes—
81#1850C2212yes—
82#1823C2212yes—
83#1805C2214yescomplemented
84#1787C2212yes—
85C2×#340C2212yes—
86#1729C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×Q8C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C22×Q16—24no—
92C22×Q16—44no—
93C22×Q16C2314yes—
94C22×Q16C2312yes—
95C22×Q16C2312yes—
96C22×M16C2311yescharacteristic
97C22×M16C2311yescharacteristic
98C23×C8—24no—
99C23×C8—44no—
100C23×C8C2311yescharacteristic
101C23×C8C2312yes—
102C23×C8C2312yes—
103C23×C8C2311yescharacteristic
104C2×C4⋊Q8C2314yes—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8D418yes—
117C2×C22⋊Q8C2312yes—
118C2×C22⋊Q8C2312yes—
119C2×C22⋊Q8C2312yes—
120C2×C22⋊Q8C2312yes—
121C2×C22⋊Q8C2312yes—
122C2×C22⋊Q8C2312yes—
123C2×C4×Q8C2314yes—
124C2×#117—22no—
125C2×#117C2312yes—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—21no—
136C22×C22⋊C4—21no—
137#483—28no—
138#483—28no—
139#470—28no—
140#470—28no—
141#467—216no—
142#467—216no—
143#462—48no—
144#451—28no—
145#451—28no—
146C2×#107—24no—
147C2×#107C2314yes—
148C2×#106C2312yes—
149C2×#106C2312yes—
150#424—44no—
151C2×C4⋊C8C2314yes—
152C2×C4⋊C8C2314yes—
153C2×#103—24no—
154C2×#103—24no—
155C2×#103—44no—
156C2×#103C2312yes—
157C2×#103C2314yes—
158C2×#103C2312yes—
159C2×#103C2312yes—
160C2×#103C2312yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166#407—24no—
167C2×C22⋊C8C2312yes—
168C2×C22⋊C8C2312yes—
169#400—22no—
170#400—24no—
171#400—22no—
172#398—24no—
173#398—24no—
174#397—24no—
175#397—24no—
176#395—22no—
177#395—24no—
178#395—22no—
179#391—24no—
180#391C2314yes—
181#391C2314yes—
182#391C2314yes—
183#389—24no—
184#389—24no—
185#389C2314yes—
186#388—24no—
187#388—24no—
188#387—24no—
189#387—24no—
190#387—24no—
191#385—24no—
192#385—22no—
193#385—22no—
194#385—24no—
195#382C2314yes—
196#381—22no—
197#381—22no—
198#381—22no—
199#381—24no—
200#381D412yes—
201#380—21no—
202#380—21no—
203#380—22no—
204#380D411yescharacteristic
205#380D411yescharacteristic
206#380D412yes—
207#380C2312yes—
208#380C2312yes—
209#379D412yes—
210C4×C4⋊C4C2314yes—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—24no—
213C4×C22⋊C4C2312yes—
214C4×C22⋊C4C2312yes—
215C2×#95—21no—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#336C2314yes—
221#336C2314yes—
222C25C2×D411yescharacteristic
223C22×Q8—21no—
224C22×Q8—21no—
225C22×Q8—21no—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8—42no—
232C22×Q8—42no—
233C22×Q8—42no—
234C22×Q8—42no—
235C22×Q8—42no—
236C22×Q8—42no—
237C22×Q8C2×D414yes—
238C22×Q8C2412yes—
239C22×Q8C2412yes—
240C22×Q8C2412yes—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4—42no—
249C23×C4—42no—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D414yes—
254C23×C4#4114yes—
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257C23×C4C2411yescharacteristic
258C2×Q16—28no—
259C2×Q16—24no—
260C2×Q16—24no—
261C2×Q16—28no—
262C2×Q16—44no—
263C2×Q16—416no—
264C2×Q16—44no—
265C2×Q16—44no—
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—42no—
269C2×M16—42no—
270C22×C8—22no—
271C22×C8—24no—
272C22×C8—22no—
273C22×C8—24no—
274C22×C8—24no—
275C22×C8—24no—
276C22×C8—24no—
277C22×C8—42no—
278C22×C8—42no—
279C22×C8—42no—
280C22×C8—416no—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C4⋊Q8—44no—
286C4⋊Q8—44no—
287C22⋊Q8—28no—
288C22⋊Q8—28no—
289C22⋊Q8—24no—
290C22⋊Q8—24no—
291C22⋊Q8—24no—
292C22⋊Q8—24no—
293C22⋊Q8—24no—
294C22⋊Q8—24no—
295C22⋊Q8—24no—
296C22⋊Q8—24no—
297C22⋊Q8—24no—
298C22⋊Q8—24no—
299C22⋊Q8—44no—
300C22⋊Q8—44no—
301C22⋊Q8—44no—
302C22⋊Q8—44no—
303C22⋊Q8C2×D418yes—
304C22⋊Q8C2×D418yes—
305C4×Q8—44no—
306C4×Q8—44no—
307#117—42no—
308#117—42no—
309#117—84no—
310C2×C4⋊C4—28no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—42no—
331C2×C4⋊C4—42no—
332C2×C4⋊C4—44no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4C2×D414yes—
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2414yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C22⋊C4—28no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—44no—
351C2×C22⋊C4—44no—
352C2×C22⋊C4—82no—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D414yes—
356C2×C22⋊C4C2×D414yes—
357C2×C42—24no—
358C2×C42—24no—
359C2×C42C2414yes—
360#107—28no—
361#106—24no—
362#106—24no—
363C4⋊C8—28no—
364C4⋊C8—44no—
365#103—28no—
366#103—24no—
367#103—24no—
368#103—28no—
369#103—28no—
370#103—24no—
371#103—24no—
372#103—44no—
373#103—44no—
374#103—44no—
375C22⋊C8—24no—
376C22⋊C8—24no—
377#95—22no—
378#95—22no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95C2×D412yes—
390#95C2×D414yes—
391#95C2×D412yes—
392C24—22no—
393C24—24no—
394C24—22no—
395C24—42no—
396C24#12114yes—
397C24C22×D411yescharacteristic
398C24C22×D411yescharacteristic
399C24C22×D411yescharacteristic
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—24no—
411C2×Q8—24no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—48no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—82no—
420C2×Q8—82no—
421C2×Q8—82no—
422C2×Q8—82no—
423C2×Q8—82no—
424C2×Q8—84no—
425C2×Q8—84no—
426C2×Q8C22×D414yes—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—48no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4#12114yes—
464C22×C4#12114yes—
465C22×C4#12114yes—
466C22×C4C22×D414yes—
467C22×C4C22×D414yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2511yescharacteristic
478Q16—416no—
479Q16—84no—
480Q16—84no—
481M16—82no—
482M16—82no—
483C2×C8—28no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—24no—
491C2×C8—22no—
492C2×C8—24no—
493C2×C8—48no—
494C2×C8—48no—
495C2×C8—44no—
496C2×C8H(5,𝔽2)14yes—
497C4⋊C4—28no—
498C4⋊C4—28no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—82no—
516C4⋊C4—84no—
517C4⋊C4—84no—
518C4⋊C4—84no—
519C4⋊C4—84no—
520C4⋊C4C22×D414yes—
521C22⋊C4—28no—
522C22⋊C4—28no—
523C22⋊C4—24no—
524C22⋊C4—28no—
525C22⋊C4—24no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—82no—
533C42—24no—
534C42—24no—
535C42—84no—
536C42—84no—
537C23—22no—
538C23—22no—
539C23—21no—
540C23—22no—
541C23—21no—
542C23—22no—
543C23—28no—
544C23—28no—
545C23—21no—
546C23—22no—
547C23—21no—
548C23—44no—
549C23—48no—
550C23—48no—
551C23C2×#12112yes—
552C23C2×#12112yes—
553C23C2×#12112yes—
554C23#57712yes—
555C23#57814yes—
556C23#57812yes—
557C23C23×D411yescharacteristic
558Q8—24no—
559Q8—24no—
560Q8—44no—
561Q8—42no—
562Q8—42no—
563Q8—42no—
564Q8—42no—
565Q8—44no—
566Q8—44no—
567Q8—84no—
568Q8—84no—
569Q8—84no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—48no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—164no—
615C2×C4C2×#12112yes—
616C2×C4C2×#12112yes—
617C2×C4C2×#12112yes—
618C2×C4C23×D411yescharacteristic
619C2×C4C22×#4111yescharacteristic
620C2×C4C2×H(5,𝔽2)12yes—
621C2×C4#58512yes—
622C8—24no—
623C8—42no—
624C8—42no—
625C8—42no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—24no—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—416no—
638C22—42no—
639C22—42no—
640C22#304414yes—
641C22#304614yes—
642C22C22×#12111yescharacteristic
643C22#349612yes—
644C22C2×#57711yescharacteristic
645C22C2×#57812yes—
646C22C2×#57811yescharacteristic
647C4—22no—
648C4—22no—
649C4—22no—
650C4—44no—
651C4—44no—
652C4—44no—
653C4—44no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4—84no—
659C4#348411yescharacteristic
660C4#348511yescharacteristic
661C4#350512yes—
662C2—24no—
663C2—24no—
664C2—42no—
665C2#3535811yescharacteristic
666C2C2×#304411yescharacteristic
667C2C2×#304611yescharacteristic
668C2#6227112yes—
669C2#6228112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #28588C2semidirect
6. #25208C2semidirect
7. #25082C2semidirect
8. #24764C2semidirect
9. #24577C2semidirect
10. #24529C2semidirect
11. #24480C2semidirect
12. #24360C2semidirect
29. C2×#462C22semidirect
59. #2085C22semidirect
60. #2074C22semidirect
61. #2068C22semidirect
62. #2066C22semidirect
63. #2056C22semidirect
64. #2056C22semidirect
65. #2043C22semidirect
66. #2016C22semidirect
67. #1981C22semidirect
68. #1949C22semidirect
69. #1928C22semidirect
70. #1927C22semidirect
76. #1878C22semidirect
77. #1878C22semidirect
83. #1805C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31952
C22×#4672
#374532
#343802
#285882
#252082
#250822
#247642
#245772
#245292
#244802
#243602
#217652
C2×#18922
#208342