Finite Group Databasebeta

← #6531998

Subgroups — #6531998

Order 512.

1672 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2580C211yesmaximal, characteristic
2#34187C211yesmaximal, characteristic, complemented
3#27617C212yesmaximal, complemented
4#27614C211yesmaximal, characteristic, complemented
5#27611C211yesmaximal, characteristic, complemented
6#27588C212yesmaximal
7#27585C211yesmaximal, characteristic
8#27582C211yesmaximal, characteristic
9#25065C214yesmaximal, complemented
10#25064C214yesmaximal, complemented
11#24999C214yesmaximal, complemented
12#24998C214yesmaximal, complemented
13#21983C212yesmaximal, complemented
14#21818C211yesmaximal, characteristic, complemented
15#20733C212yesmaximal
16C22×C4×Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C2×#431C2212yes—
19#2580—22no—
20#2580—22no—
21#2580C2214yes—
22#2580C2214yes—
23C2×#399—21no—
24C2×#393—21no—
25#2356—22no—
26#2356—22no—
27#2355C2212yescomplemented
28#2355C2212yescomplemented
29#2355C2212yescomplemented
30#2355C2212yescomplemented
31#2313—21no—
32#2313—21no—
33#2310—21no—
34C2×#390C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#385—21no—
39C2×#385—21no—
40#2285C2211yescharacteristic
41#2285C2211yescharacteristic
42#2283C2212yes—
43#2283C2212yes—
44C2×#382C2211yescharacteristic
45#2274—21no—
46#2101C2214yescomplemented
47#2100C2212yescomplemented
48#2099C2212yescomplemented
49#2094C2212yes—
50#2093C2212yes—
51#2092C2214yes—
52#2085C2214yescomplemented
53#2084C2212yescomplemented
54#2083C2212yescomplemented
55#2073C2214yes—
56#2071C2212yes—
57#2071C2212yes—
58#1984C2214yescomplemented
59#1984C2214yescomplemented
60#1949C2214yescomplemented
61#1949C2214yescomplemented
62#1926C2214yes—
63#1926C2214yes—
64#1925C2214yes—
65#1925C2214yes—
66#1921C2214yescomplemented
67#1920C2214yescomplemented
68#1919C2214yescomplemented
69#1918C2214yescomplemented
70#1853C2212yes—
71#1853C2212yes—
72#1852C2212yes—
73#1852C2212yes—
74#1846C2214yescomplemented
75#1835C2214yescomplemented
76#1824C2212yes—
77#1824C2212yes—
78#1817C2211yescharacteristic
79#1817C2211yescharacteristic
80#1817C2212yes—
81#1804C2214yes—
82#1804C2214yes—
83#1786C2212yes—
84#1786C2212yes—
85C4×#102C2214yes—
86C4×#102C2214yes—
87C23×Q8D411yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C2×C4×Q8—28no—
91C2×C4×Q8D412yes—
92C2×C4×Q8D412yes—
93C2×C4×Q8C2312yes—
94C2×C4×Q8C2312yes—
95C2×C4×D4—22no—
96C2×C4×D4—28no—
97C2×C4×D4C2314yes—
98C2×#117—21no—
99C2×#117—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4D411yescharacteristic
103C22×C4⋊C4D411yescharacteristic
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116#431—28no—
117C2×#107C2314yes—
118C2×#107C2314yes—
119C2×#106C2314yes—
120C2×#106C2314yes—
121C2×C4⋊C8C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2312yes—
128C2×C4×C8C2314yes—
129#399—24no—
130#399—24no—
131#393—24no—
132#393—24no—
133#391—22no—
134#391—22no—
135#391—22no—
136#391—22no—
137#390—22no—
138#390—21no—
139#390—21no—
140#390C2312yes—
141#390C2312yes—
142#390C2314yes—
143#389—22no—
144#389—21no—
145#389—21no—
146#389C2312yes—
147#389C2312yes—
148#389C2314yes—
149#387—22no—
150#387—22no—
151#387—22no—
152#387—22no—
153#387—22no—
154#387—22no—
155#387—22no—
156#387—22no—
157#385—24no—
158#385—24no—
159#385—24no—
160#385—24no—
161#384C2311yescharacteristic
162#384C2311yescharacteristic
163#384C2311yescharacteristic
164#384C2311yescharacteristic
165#383—22no—
166#383—22no—
167#382—22no—
168#382—21no—
169#382—21no—
170#382C2312yes—
171#382C2312yes—
172#382C2314yes—
173#382C2312yes—
174#382C2312yes—
175#381—22no—
176#381—22no—
177#380—21no—
178#380—21no—
179#380C2312yes—
180#380C2312yes—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C4×C4⋊C4C2312yes—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2312yes—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—22no—
195#376—22no—
196#376—22no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#336C2312yes—
210#336C2312yes—
211#336C2314yes—
212#336C2312yes—
213#336C2312yes—
214#336C2314yes—
215C25—21no—
216C22×Q8—24no—
217C22×Q8—24no—
218C22×Q8—42no—
219C22×Q8C2×D412yes—
220C22×Q8C2×D412yes—
221C22×D4—24no—
222C22×D4—21no—
223C22×D4—24no—
224C22×D4—42no—
225C22×D4C2412yes—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—24no—
230C23×C4—44no—
231C23×C4—82no—
232C23×C4—82no—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2411yescharacteristic
240C22×C8C2414yes—
241C22×C8C2414yes—
242C4×Q8—28no—
243C4×Q8—28no—
244C4×Q8—24no—
245C4×Q8—24no—
246C4×Q8—24no—
247C4×Q8—24no—
248C4×D4—28no—
249C4×D4—28no—
250C4×D4—48no—
251#117—44no—
252#117—44no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—82no—
274C2×C4⋊C4—82no—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—82no—
301C2×C22⋊C4—82no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42C2×D412yes—
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D412yes—
314C2×C42#4118yes—
315C2×C42C2412yes—
316C2×C42C2411yescharacteristic
317C2×C42C2411yescharacteristic
318C2×C42C2412yes—
319#107—44no—
320#107—44no—
321#106—44no—
322#106—44no—
323C4⋊C8—28no—
324#102—28no—
325#102—28no—
326#102—44no—
327#102—44no—
328C22⋊C8—24no—
329C22⋊C8—24no—
330C4×C8—28no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352C24—24no—
353C24—22no—
354C24—42no—
355C24—42no—
356C24—42no—
357C24—42no—
358C24—42no—
359C24—42no—
360C24C22×D411yescharacteristic
361C2×Q8—24no—
362C2×Q8—24no—
363C2×Q8—24no—
364C2×Q8—24no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—416no—
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—22no—
371C2×D4—24no—
372C2×D4—416no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D42−1+414yes—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—48no—
410C22×C4—84no—
411C22×C4—84no—
412C22×C4—816no—
413C22×C4#12314yes—
414C22×C4#12314yes—
415C22×C4#13514yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C2×#4114yes—
426C22×C4C2511yescharacteristic
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C4⋊C4—81no—
457C4⋊C4—81no—
458C4⋊C4—84no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C22⋊C4—48no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—48no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—84no—
468C22⋊C4—82no—
469C22⋊C4—82no—
470C42—22no—
471C42—22no—
472C42—24no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—44no—
478C42—44no—
479C42—82no—
480C42—82no—
481C42—82no—
482C42—82no—
483C42#12318yes—
484C42C2×#4114yes—
485C42C2×#4114yes—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—24no—
491C23—48no—
492C23—48no—
493C23—48no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—84no—
501C23—84no—
502C23—84no—
503C23—84no—
504C23C2×#12312yes—
505C23C2×#13512yes—
506C23#57512yes—
507C23C23×D411yescharacteristic
508Q8—48no—
509Q8—48no—
510D4—28no—
511D4—48no—
512D4—44no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—28no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—816no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4C2×#12312yes—
568C2×C4C2×#12314yes—
569C2×C4C2×#12314yes—
570C2×C4C2×#13512yes—
571C2×C4C2×#13512yes—
572C2×C4C22×#4112yes—
573C2×C4C2×2−1+412yes—
574C2×C4#58512yes—
575C8—44no—
576C8—44no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—816no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22#309114yes—
592C22C22×#12311yescharacteristic
593C22#348611yescharacteristic
594C22#348711yescharacteristic
595C22C22×#13511yescharacteristic
596C22C2×#57511yescharacteristic
597C22#358511yescharacteristic
598C22#358511yescharacteristic
599C4—22no—
600C4—24no—
601C4—22no—
602C4—22no—
603C4—24no—
604C4—42no—
605C4—42no—
606C4—41no—
607C4—41no—
608C4—82no—
609C4—82no—
610C4—82no—
611C4—82no—
612C4—82no—
613C4—82no—
614C4—82no—
615C4#349514yes—
616C2—24no—
617C2—84no—
618C2#3567211yescharacteristic
619C2C2×#309111yescharacteristic
620C2#6051811yescharacteristic
621C2#6181911yescharacteristic
622C2#6183111yescharacteristic
623C2#6188511yescharacteristic
624C2#6189511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34187C2semidirect
3. #27617C2semidirect
4. #27614C2semidirect
5. #27611C2semidirect
9. #25065C2semidirect
10. #25064C2semidirect
11. #24999C2semidirect
12. #24998C2semidirect
13. #21983C2semidirect
14. #21818C2semidirect
27. #2355C22semidirect
28. #2355C22semidirect
29. #2355C22semidirect
30. #2355C22semidirect
46. #2101C22semidirect
47. #2100C22semidirect
48. #2099C22semidirect
52. #2085C22semidirect
53. #2084C22semidirect
54. #2083C22semidirect
58. #1984C22semidirect
59. #1984C22semidirect
60. #1949C22semidirect
61. #1949C22semidirect
66. #1921C22semidirect
67. #1920C22semidirect
68. #1919C22semidirect
69. #1918C22semidirect
74. #1846C22semidirect
75. #1835C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25802
#341872
#276172
#276142
#276112
#275882
#275852
#275822
#250652
#250642
#249992
#249982
#219832
#218182
#207332