Finite Group Databasebeta

← #6502639

Subgroups — #6502639

Order 512.

2102 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3195C211yesmaximal, characteristic
2C22×#470C211yesmaximal, characteristic
3#35011C211yesmaximal, characteristic
4#34245C211yesmaximal, characteristic
5#28581C214yesmaximal, complemented
6#25219C214yesmaximal, complemented
7#25095C214yesmaximal, complemented
8#24760C214yesmaximal, complemented
9#24580C212yesmaximal, complemented
10#24532C212yesmaximal, complemented
11#24480C212yesmaximal, complemented
12#24360C212yesmaximal, complemented
13#21768C211yesmaximal, characteristic
14#20972C211yesmaximal, characteristic
15#20837C211yesmaximal, characteristic
16C23×Q16C2211yescharacteristic
17C23×M16C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21#3195—28no—
22C2×#483C2212yes—
23C2×#483C2212yes—
24C2×#470—28no—
25C2×#470C2214yes—
26C2×#470C2214yes—
27C2×#470C2212yes—
28C2×#470C2212yes—
29C2×#462C2214yescomplemented
30C2×#451C2212yes—
31C2×#451C2212yes—
32C22×#106C2211yescharacteristic
33C2×#423C2212yes—
34C2×#419C2212yes—
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C2×#407C2211yescharacteristic
38#2857C2214yes—
39#2659—22no—
40#2583—21no—
41#2583—22no—
42#2583—21no—
43#2502C2212yes—
44#2502C2212yes—
45#2434C2212yes—
46#2434C2212yes—
47#2368—22no—
48#2362C2214yes—
49#2344C2212yes—
50#2342—22no—
51Q8×C22⋊C4C2212yes—
52#2332—22no—
53#2315—22no—
54#2313—21no—
55C4×C22⋊Q8C2214yes—
56#2280C2211yescharacteristic
57#2274—21no—
58C2×C4×C22⋊C4C2211yescharacteristic
59#2084C2214yescomplemented
60#2072C2214yescomplemented
61#2066C2214yescomplemented
62#2066C2214yescomplemented
63#2056C2214yescomplemented
64#2056C2214yescomplemented
65#2043C2214yescomplemented
66#2016C2214yescomplemented
67#1981C2214yescomplemented
68#1948C2214yescomplemented
69#1927C2214yescomplemented
70#1927C2214yescomplemented
71#1894C2212yes—
72#1894C2212yes—
73#1894C2214yes—
74#1892C2212yes—
75#1892C2212yes—
76#1878C2214yescomplemented
77#1878C2214yescomplemented
78#1873C2214yes—
79#1851C2214yes—
80#1850C2212yes—
81#1850C2212yes—
82#1824C2212yes—
83#1803C2214yescomplemented
84#1787C2212yes—
85#1736C2212yes—
86#1728C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×Q8C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C22×Q16—24no—
92C22×Q16—44no—
93C22×Q16C2314yes—
94C22×Q16C2312yes—
95C22×Q16C2312yes—
96C22×M16—24no—
97C22×M16—44no—
98C22×M16C2312yes—
99C22×M16C2312yes—
100C22×M16C2311yescharacteristic
101C22×M16C2311yescharacteristic
102C23×C8C2311yescharacteristic
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×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4D412yes—
136C22×C42C2312yes—
137#483—28no—
138#483—28no—
139#470—28no—
140#470—28no—
141#470—216no—
142#470—216no—
143#462—48no—
144#451—28no—
145#451—28no—
146C2×#106—24no—
147C2×#106C2312yes—
148C2×#106C2312yes—
149C2×#106C2314yes—
150#423—44no—
151C2×C4⋊C8C2314yes—
152C2×C4⋊C8C2314yes—
153#419—44no—
154C2×#103—24no—
155C2×#103—24no—
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#398—22no—
170#398—22no—
171#398—24no—
172#395—22no—
173#395—22no—
174#395—24no—
175#391—24no—
176#391C2314yes—
177#391C2314yes—
178#391C2314yes—
179#389—24no—
180#389—24no—
181#389C2314yes—
182#388—24no—
183#388—24no—
184#387—24no—
185#387—24no—
186#387—24no—
187#385—24no—
188#385—22no—
189#385—24no—
190#385—22no—
191#385—22no—
192#385—24no—
193#385—22no—
194#384C2314yes—
195#381—21no—
196#381—22no—
197#380—24no—
198#380—24no—
199#380—22no—
200#380C2312yes—
201#380C2312yes—
202#379—21no—
203C4×C4⋊C4—24no—
204C4×C4⋊C4—24no—
205C4×C4⋊C4C2314yes—
206C4×C22⋊C4—21no—
207C4×C22⋊C4—24no—
208C4×C22⋊C4—21no—
209C4×C22⋊C4—22no—
210C4×C22⋊C4D411yescharacteristic
211C4×C22⋊C4D412yes—
212C4×C22⋊C4D411yescharacteristic
213C4×C22⋊C4C2312yes—
214C4×C22⋊C4C2312yes—
215C2×#95D412yes—
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—42no—
248C23×C4—42no—
249C23×C4C2×D414yes—
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—24no—
267C2×M16—24no—
268C2×M16—22no—
269C2×M16—22no—
270C2×M16—24no—
271C2×M16—24no—
272C2×M16—42no—
273C2×M16—42no—
274C2×M16—42no—
275C2×M16—42no—
276C2×M16—416no—
277C22×C8—22no—
278C22×C8—24no—
279C22×C8—22no—
280C22×C8—42no—
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—48no—
308#117—42no—
309#117—42no—
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—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4—44no—
333C2×C4⋊C4C2×D414yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2414yes—
337C2×C4⋊C4C2412yes—
338C2×C22⋊C4—28no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
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—44no—
350C2×C22⋊C4—44no—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D414yes—
353C2×C22⋊C4C2×D412yes—
354C2×C42—24no—
355C2×C42—22no—
356C2×C42—24no—
357C2×C42—42no—
358C2×C42—42no—
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2414yes—
362#106—28no—
363#106—24no—
364#106—24no—
365C4⋊C8—28no—
366C4⋊C8—44no—
367#103—28no—
368#103—24no—
369#103—24no—
370#103—28no—
371#103—28no—
372#103—24no—
373#103—24no—
374#103—44no—
375#103—44no—
376#103—44no—
377C22⋊C8—24no—
378C22⋊C8—24no—
379#95—22no—
380#95—24no—
381#95—24no—
382#95—22no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D414yes—
392C24—24no—
393C24—42no—
394C24—42no—
395C24#12114yes—
396C24C22×D411yescharacteristic
397C24C22×D411yescharacteristic
398C24C22×D411yescharacteristic
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—24no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—48no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8—82no—
419C2×Q8—82no—
420C2×Q8—82no—
421C2×Q8—82no—
422C2×Q8—82no—
423C2×Q8—84no—
424C2×Q8—84no—
425C2×Q8C22×D414yes—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—28no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
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—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4#12114yes—
461C22×C4#12114yes—
462C22×C4#12114yes—
463C22×C4C22×D414yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D414yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4C2511yescharacteristic
475Q16—416no—
476Q16—84no—
477Q16—84no—
478M16—48no—
479M16—48no—
480M16—82no—
481M16—82no—
482C2×C8—28no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8—44no—
493C2×C8H(5,𝔽2)14yes—
494C4⋊C4—28no—
495C4⋊C4—28no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—48no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C4⋊C4C22×D414yes—
517C22⋊C4—28no—
518C22⋊C4—28no—
519C22⋊C4—24no—
520C22⋊C4—28no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—82no—
528C22⋊C4#13718yes—
529C42—24no—
530C42—24no—
531C42—42no—
532C42—42no—
533C42—48no—
534C42—82no—
535C42—84no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—28no—
545C23—44no—
546C23—44no—
547C23—88no—
548C23C2×#12112yes—
549C23C2×#12112yes—
550C23C2×#12112yes—
551C23C2×#13714yes—
552C23C23×D411yescharacteristic
553Q8—24no—
554Q8—24no—
555Q8—44no—
556Q8—42no—
557Q8—42no—
558Q8—42no—
559Q8—42no—
560Q8—44no—
561Q8—44no—
562Q8—84no—
563Q8—84no—
564Q8—84no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—48no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—88no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4C2×#12112yes—
610C2×C4C2×#12112yes—
611C2×C4C2×#12112yes—
612C2×C4C2×#13718yes—
613C2×C4C23×D411yescharacteristic
614C2×C4C22×#4111yescharacteristic
615C2×C4C2×H(5,𝔽2)12yes—
616C2×C4#58512yes—
617C8—24no—
618C8—42no—
619C8—42no—
620C8—42no—
621C22—22no—
622C22—22no—
623C22—28no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—44no—
631C22—44no—
632C22—88no—
633C22#303914yes—
634C22C22×#12111yescharacteristic
635C22#349612yes—
636C22C22×#13712yes—
637C22#358612yes—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—28no—
642C4—44no—
643C4—44no—
644C4—44no—
645C4—82no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4—84no—
650C4#348411yescharacteristic
651C4#348511yescharacteristic
652C4#350512yes—
653C2—24no—
654C2—42no—
655C2—42no—
656C2#3535811yescharacteristic
657C2C2×#303911yescharacteristic
658C2#6042411yescharacteristic
659C2#6230014yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #28581C2semidirect
6. #25219C2semidirect
7. #25095C2semidirect
8. #24760C2semidirect
9. #24580C2semidirect
10. #24532C2semidirect
11. #24480C2semidirect
12. #24360C2semidirect
29. C2×#462C22semidirect
59. #2084C22semidirect
60. #2072C22semidirect
61. #2066C22semidirect
62. #2066C22semidirect
63. #2056C22semidirect
64. #2056C22semidirect
65. #2043C22semidirect
66. #2016C22semidirect
67. #1981C22semidirect
68. #1948C22semidirect
69. #1927C22semidirect
70. #1927C22semidirect
76. #1878C22semidirect
77. #1878C22semidirect
83. #1803C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31952
C22×#4702
#350112
#342452
#285812
#252192
#250952
#247602
#245802
#245322
#244802
#243602
#217682
#209722
#208372