Finite Group Databasebeta

← #6510727

Subgroups — #6510727

Order 512.

1630 conjugacy classes of subgroups.

The subgroup lattice has 609 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35079C211yesmaximal, characteristic
2C2×#2503C211yesmaximal, characteristic
3#29112C212yesmaximal, complemented
4#29105C212yesmaximal, complemented
5#27969C212yesmaximal, complemented
6#27964C212yesmaximal, complemented
7#27723C214yesmaximal, complemented
8#27713C214yesmaximal, complemented
9#27705C214yesmaximal, complemented
10#27699C214yesmaximal, complemented
11#21765C211yesmaximal, characteristic
12#21762C211yesmaximal, characteristic
13#20700C211yesmaximal, characteristic
14C2×#1824C211yesmaximal, characteristic
15#20525C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C2×#407C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20#2857C2212yes—
21#2851C2212yes—
22#2667—21no—
23#2665—21no—
24#2569—21no—
25#2549—21no—
26#2503—22no—
27#2503—21no—
28#2503—21no—
29#2503C2212yes—
30#2503C2214yes—
31#2503C2212yes—
32#2442C2212yes—
33#2441C2212yes—
34#2440C2212yes—
35#2439C2212yes—
36#2437—22no—
37C2×#400—21no—
38C2×#400—21no—
39#2329—22no—
40C2×#391C2212yes—
41C2×#382C2212yes—
42#2280C2211yescharacteristic
43C2×#381—22no—
44C2×#380C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46#2101C2214yescomplemented
47#2101C2214yescomplemented
48#2101C2214yescomplemented
49#2101C2214yescomplemented
50#2074C2214yescomplemented
51#2072C2214yescomplemented
52#2026C2214yescomplemented
53#2024C2214yescomplemented
54#1993C2214yescomplemented
55#1991C2214yescomplemented
56#1935C2214yescomplemented
57#1935C2214yescomplemented
58#1933C2214yescomplemented
59#1933C2214yescomplemented
60#1894C2212yes—
61#1894C2212yes—
62#1892C2214yes—
63#1873C2214yes—
64#1872C2214yes—
65#1836C2214yescomplemented
66#1835C2214yescomplemented
67#1824C2212yes—
68#1824C2212yes—
69#1824C2212yes—
70#1824C2212yes—
71#1824C2212yes—
72#1824C2212yes—
73#1823C2211yescharacteristic
74#1823C2211yescharacteristic
75#1823C2212yes—
76#1823C2214yes—
77#1787C2211yescharacteristic
78#1787C2211yescharacteristic
79#1787C2212yes—
80#1753C2214yes—
81#1752C2212yes—
82C8×C22⋊C4C2212yes—
83#1749C2212yes—
84C4×C22⋊C8C2212yes—
85#1736C2214yes—
86C2×#340C2214yes—
87C23×Q8C2311yescharacteristic
88C24×C4—21no—
89C24×C4C2311yescharacteristic
90C22×M16C2312yes—
91C23×C8C2312yes—
92C2×C4⋊Q8C2312yes—
93C2×#125C2312yes—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8C2312yes—
101C2×C22⋊Q8C2312yes—
102C2×#117C2312yes—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4D412yes—
117C22×C42C2312yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122#407—24no—
123C2×C22⋊C8—24no—
124C2×C22⋊C8C2312yes—
125C2×C22⋊C8C2311yescharacteristic
126C2×C22⋊C8C2311yescharacteristic
127C2×C22⋊C8C2312yes—
128C2×C22⋊C8C2312yes—
129C2×#97C2314yes—
130C2×C4×C8C2314yes—
131#400—22no—
132#400—24no—
133#400—24no—
134#400—22no—
135#400—22no—
136#400—22no—
137#400—22no—
138#399—22no—
139#398—22no—
140#398—22no—
141#396—22no—
142#396—22no—
143#395—22no—
144#393—22no—
145#391—22no—
146#391—22no—
147#391—22no—
148#391—22no—
149#391C2314yes—
150#391C2314yes—
151#389—22no—
152#389—22no—
153#389C2312yes—
154#389C2312yes—
155#389C2312yes—
156#389C2312yes—
157#387—22no—
158#387—22no—
159#387—22no—
160#387—22no—
161#385—22no—
162#385—22no—
163#385—22no—
164#385—22no—
165#384C2312yes—
166#382—22no—
167#382—22no—
168#382C2312yes—
169#382C2314yes—
170#382C2314yes—
171#381—21no—
172#381—24no—
173#381—24no—
174#381—24no—
175#381—24no—
176#380—22no—
177#380—21no—
178#380—21no—
179#380—22no—
180#380C2312yes—
181#380C2311yescharacteristic
182#380C2312yes—
183#380C2311yescharacteristic
184#380C2311yescharacteristic
185#380C2311yescharacteristic
186#379—21no—
187C4×C22⋊C4—21no—
188C4×C22⋊C4—21no—
189C4×C22⋊C4D412yes—
190C4×C22⋊C4D411yescharacteristic
191C4×C22⋊C4D411yescharacteristic
192C4×C22⋊C4C2312yes—
193C4×C22⋊C4C2311yescharacteristic
194C4×C22⋊C4C2311yescharacteristic
195C2×#95—21no—
196C2×#95—22no—
197C2×#95—22no—
198C2×#95—22no—
199C2×#95—22no—
200C2×#95D411yescharacteristic
201C2×#95D411yescharacteristic
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2314yes—
215#336C2314yes—
216C25C2×D411yescharacteristic
217C22×Q8—21no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—44no—
222C22×Q8C2412yes—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—44no—
230C23×C4—48no—
231C23×C4—44no—
232C23×C4C2×D412yes—
233C23×C4C2×D412yes—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4#4114yes—
238C23×C4#4114yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C2×M16—24no—
243C2×M16—44no—
244C22×C8—24no—
245C22×C8—44no—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2414yes—
249C4⋊Q8—44no—
250#125—44no—
251C22⋊Q8—24no—
252C22⋊Q8—24no—
253C22⋊Q8—24no—
254C22⋊Q8—24no—
255C22⋊Q8—48no—
256C22⋊Q8—44no—
257C22⋊Q8—44no—
258C22⋊Q8—84no—
259#117—44no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4C2414yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—82no—
307C2×C22⋊C4C2×D412yes—
308C2×C22⋊C4C2×D412yes—
309C2×C22⋊C4C2×D412yes—
310C2×C22⋊C4C2×D412yes—
311C2×C42—22no—
312C2×C42—44no—
313C2×C42C2×D412yes—
314C2×C42C2×D412yes—
315C2×C42C2412yes—
316C2×C42C2412yes—
317#103—44no—
318#103—44no—
319#103—44no—
320#103—44no—
321C22⋊C8—22no—
322C22⋊C8—22no—
323C22⋊C8—24no—
324C22⋊C8—24no—
325C22⋊C8—24no—
326#97—44no—
327C4×C8—44no—
328#95—21no—
329#95—22no—
330#95—21no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—21no—
340#95—21no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95C2×D412yes—
349#95C2×D412yes—
350#95C2×D411yescharacteristic
351#95C2×D411yescharacteristic
352#95C2×D411yescharacteristic
353#95C2×D411yescharacteristic
354C24—24no—
355C24—22no—
356C24—22no—
357C24—42no—
358C24#12414yes—
359C24C22×D411yescharacteristic
360C24C22×D411yescharacteristic
361C24C22×D411yescharacteristic
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—44no—
367C2×Q8—44no—
368C2×Q8—42no—
369C2×Q8—82no—
370C2×Q8—82no—
371C2×Q8—84no—
372C2×Q8—84no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—48no—
386C22×C4—48no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—44no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—48no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—82no—
407C22×C4—82no—
408C22×C4—816no—
409C22×C4#12414yes—
410C22×C4#12414yes—
411C22×C4#12414yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C4C2×#4112yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4C2511yescharacteristic
426M16—84no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—84no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—82no—
453C4⋊C4—82no—
454C4⋊C4—82no—
455C4⋊C4—82no—
456C4⋊C4—82no—
457C4⋊C4—82no—
458C4⋊C4—82no—
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—81no—
470C22⋊C4—81no—
471C22⋊C4—82no—
472C42—24no—
473C42—24no—
474C42—42no—
475C42—42no—
476C42—82no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—21no—
481C23—21no—
482C23—28no—
483C23—28no—
484C23—22no—
485C23—22no—
486C23—21no—
487C23—21no—
488C23—44no—
489C23—48no—
490C23—48no—
491C23C2×#12412yes—
492C23C2×#12412yes—
493C23C2×#12412yes—
494C23#57514yes—
495C23#57712yes—
496C23#57812yes—
497C23C23×D411yescharacteristic
498Q8—44no—
499Q8—42no—
500Q8—42no—
501Q8—84no—
502Q8—164no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—48no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—84no—
547C2×C4—84no—
548C2×C4—88no—
549C2×C4—88no—
550C2×C4—84no—
551C2×C4—84no—
552C2×C4—88no—
553C2×C4—164no—
554C2×C4C2×#12412yes—
555C2×C4C2×#12412yes—
556C2×C4C2×#12412yes—
557C2×C4C22×#4111yescharacteristic
558C2×C4C22×#4111yescharacteristic
559C2×C4#58514yes—
560C8—42no—
561C8—42no—
562C8—44no—
563C22—24no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—416no—
575C22—42no—
576C22—42no—
577C22#313714yes—
578C22#314014yes—
579C22C22×#12411yescharacteristic
580C22#348512yes—
581C22C2×#57512yes—
582C22C2×#57711yescharacteristic
583C22C2×#57811yescharacteristic
584C4—22no—
585C4—22no—
586C4—22no—
587C4—42no—
588C4—42no—
589C4—42no—
590C4—42no—
591C4—82no—
592C4—82no—
593C4—82no—
594C4—82no—
595C4—82no—
596C4—82no—
597C4—84no—
598C4#349112yes—
599C4#349112yes—
600C2—24no—
601C2—24no—
602C2—42no—
603C2#3545511yescharacteristic
604C2C2×#313711yescharacteristic
605C2C2×#314011yescharacteristic
606C2#6172912yes—
607C2#6173812yes—
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
3. #29112C2semidirect
4. #29105C2semidirect
5. #27969C2semidirect
6. #27964C2semidirect
7. #27723C2semidirect
8. #27713C2semidirect
9. #27705C2semidirect
10. #27699C2semidirect
46. #2101C22semidirect
47. #2101C22semidirect
48. #2101C22semidirect
49. #2101C22semidirect
50. #2074C22semidirect
51. #2072C22semidirect
52. #2026C22semidirect
53. #2024C22semidirect
54. #1993C22semidirect
55. #1991C22semidirect
56. #1935C22semidirect
57. #1935C22semidirect
58. #1933C22semidirect
59. #1933C22semidirect
65. #1836C22semidirect
66. #1835C22semidirect

Maximal subgroups

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

subgroupindex
#350792
C2×#25032
#291122
#291052
#279692
#279642
#277232
#277132
#277052
#276992
#217652
#217622
#207002
C2×#18242
#205252