Finite Group Databasebeta

← #6479561

Subgroups — #6479561

Order 512.

1564 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2625C211yesmaximal, characteristic
2#34180C211yesmaximal, characteristic
3#33104C214yesmaximal, complemented
4#29918C214yesmaximal, complemented
5#27872C214yesmaximal, complemented
6#27870C214yesmaximal, complemented
7#27616C212yesmaximal, complemented
8#27610C212yesmaximal, complemented
9#27589C212yesmaximal, complemented
10#27586C212yesmaximal, complemented
11#21944C211yesmaximal, characteristic
12#21940C211yesmaximal, characteristic
13#21765C211yesmaximal, characteristic
14#21762C211yesmaximal, characteristic
15#20685C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C2×#407C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20#2625—22no—
21#2625—21no—
22#2625—21no—
23#2625C2212yes—
24#2625C2214yes—
25#2625C2212yes—
26C2×#400—21no—
27C2×#398—21no—
28C2×#397—22no—
29#2347—21no—
30#2346—21no—
31#2345—21no—
32#2344—21no—
33#2343—22no—
34#2342—22no—
35#2317C2212yes—
36#2314C2212yes—
37#2314C2212yes—
38C2×#391C2212yes—
39C2×#389C2212yes—
40#2285C2212yes—
41#2283C2212yes—
42#2280C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45#2274C2212yes—
46#2101C2214yescomplemented
47#2099C2214yescomplemented
48#2097C2214yescomplemented
49#2095C2214yescomplemented
50#2085C2214yescomplemented
51#2083C2214yescomplemented
52#2074C2214yescomplemented
53#2072C2214yescomplemented
54#1928C2214yescomplemented
55#1928C2214yescomplemented
56#1928C2214yescomplemented
57#1927C2214yescomplemented
58#1921C2214yescomplemented
59#1920C2214yescomplemented
60#1918C2214yescomplemented
61#1918C2214yescomplemented
62#1894C2212yes—
63#1894C2212yes—
64#1892C2214yes—
65#1873C2214yes—
66#1872C2214yes—
67#1854C2214yes—
68#1853C2212yes—
69#1852C2214yes—
70#1852C2212yes—
71#1852C2212yes—
72#1852C2212yes—
73#1823C2211yescharacteristic
74#1823C2211yescharacteristic
75#1823C2212yes—
76#1818C2212yes—
77#1818C2212yes—
78#1817C2212yes—
79#1817C2212yes—
80#1809C2214yes—
81#1808C2212yes—
82#1808C2212yes—
83#1787C2211yescharacteristic
84#1787C2211yescharacteristic
85#1787C2212yes—
86#1729C2214yes—
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C22×M16C2312yes—
90C23×C8C2312yes—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8C2312yes—
96C2×C22⋊Q8C2312yes—
97C2×#117—24no—
98C2×#117—24no—
99C2×#117—22no—
100C2×#117D412yes—
101C2×#117D411yescharacteristic
102C2×#117D411yescharacteristic
103C2×#117C2312yes—
104C2×#117C2312yes—
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C22×C4⋊C4—21no—
108C22×C4⋊C4—22no—
109C22×C4⋊C4D411yescharacteristic
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4D412yes—
118C22×C42C2312yes—
119C2×#107C2314yes—
120C2×#107C2314yes—
121C2×#107C2314yes—
122C2×#106C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127#407—24no—
128C2×C22⋊C8—24no—
129C2×C22⋊C8C2311yescharacteristic
130C2×C22⋊C8C2311yescharacteristic
131C2×C22⋊C8C2312yes—
132C2×C22⋊C8C2312yes—
133C2×C22⋊C8C2312yes—
134#400—22no—
135#400—24no—
136#400—22no—
137#398—22no—
138#398—24no—
139#398—22no—
140#397—24no—
141#397—24no—
142#397—24no—
143#397—24no—
144#391—22no—
145#391—22no—
146#391C2314yes—
147#391C2314yes—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389C2314yes—
153#389C2314yes—
154#388—22no—
155#388—22no—
156#388—22no—
157#388—22no—
158#387—22no—
159#387—22no—
160#387—22no—
161#387—22no—
162#385—22no—
163#385—22no—
164#384C2312yes—
165#384C2312yes—
166#383—22no—
167#382—22no—
168#382C2312yes—
169#382C2312yes—
170#381—22no—
171#381—21no—
172#380—22no—
173#380—21no—
174#380—21no—
175#380—21no—
176#380—21no—
177#380C2311yescharacteristic
178#380C2311yescharacteristic
179#380C2312yes—
180#380C2312yes—
181#380C2312yes—
182#380C2311yescharacteristic
183#380C2311yescharacteristic
184#380C2312yes—
185#379—21no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4C2312yes—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2312yes—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—21no—
195C4×C22⋊C4—21no—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198#376—22no—
199C2×#95—21no—
200C2×#95—22no—
201C2×#95—22no—
202C2×#95—22no—
203C2×#95—21no—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#336C2314yes—
213#336C2314yes—
214#336C2314yes—
215#336C2314yes—
216C25C2×D411yescharacteristic
217C22×Q8—21no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—44no—
221C22×Q8C2412yes—
222C23×C4—21no—
223C23×C4—22no—
224C23×C4—21no—
225C23×C4—22no—
226C23×C4—44no—
227C23×C4C2×D412yes—
228C23×C4C2×D412yes—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4#4114yes—
233C23×C4#4114yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C2×M16—24no—
238C2×M16—44no—
239C22×C8—24no—
240C22×C8—44no—
241C22×C8C2412yes—
242C22×C8C2412yes—
243C22×C8C2414yes—
244C22⋊Q8—24no—
245C22⋊Q8—24no—
246C22⋊Q8—24no—
247C22⋊Q8—24no—
248C22⋊Q8—44no—
249C22⋊Q8—44no—
250#117—28no—
251#117—24no—
252#117—24no—
253#117—22no—
254#117—22no—
255#117—22no—
256#117—22no—
257#117—24no—
258#117—24no—
259#117—44no—
260#117—84no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—84no—
280C2×C4⋊C4C2×D411yescharacteristic
281C2×C4⋊C4C2×D411yescharacteristic
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D411yescharacteristic
285C2×C4⋊C4C2×D411yescharacteristic
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2414yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—82no—
306C2×C22⋊C4—82no—
307C2×C22⋊C4C2×D412yes—
308C2×C22⋊C4C2×D412yes—
309C2×C22⋊C4C2×D412yes—
310C2×C22⋊C4C2×D412yes—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2412yes—
321C2×C42C2412yes—
322#107—44no—
323#107—44no—
324#107—44no—
325#106—44no—
326#103—44no—
327#103—44no—
328#103—44no—
329#103—44no—
330C22⋊C8—22no—
331C22⋊C8—22no—
332C22⋊C8—24no—
333C22⋊C8—24no—
334C22⋊C8—24no—
335#95—22no—
336#95—21no—
337#95—21no—
338#95—22no—
339#95—22no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—21no—
347#95—22no—
348#95—22no—
349#95—21no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357C24—22no—
358C24—22no—
359C24—42no—
360C24—42no—
361C24#12314yes—
362C24C22×D411yescharacteristic
363C24C22×D411yescharacteristic
364C24C22×D411yescharacteristic
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—24no—
368C2×Q8—24no—
369C2×Q8—44no—
370C2×Q8—42no—
371C2×Q8—84no—
372C2×Q8—84no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—44no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—44no—
403C22×C4—82no—
404C22×C4—82no—
405C22×C4#12314yes—
406C22×C4#12314yes—
407C22×C4#12314yes—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C2×#4112yes—
416C22×C4C2×#4112yes—
417C22×C4C2×#4112yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C4C2511yescharacteristic
422M16—84no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—84no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—42no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C4⋊C4—84no—
453C4⋊C4—84no—
454C4⋊C4—82no—
455C4⋊C4—82no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—82no—
467C22⋊C4—81no—
468C22⋊C4—81no—
469C22⋊C4—82no—
470C22⋊C4—82no—
471C42—24no—
472C42—24no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—44no—
478C42—82no—
479C42—82no—
480C42—82no—
481C42—82no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—28no—
493C23—44no—
494C23—44no—
495C23—84no—
496C23—84no—
497C23C2×#12312yes—
498C23C2×#12312yes—
499C23C2×#12312yes—
500C23#57612yes—
501C23#57812yes—
502C23C23×D411yescharacteristic
503Q8—44no—
504Q8—42no—
505Q8—42no—
506Q8—84no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
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—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—48no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—84no—
553C2×C4—88no—
554C2×C4—84no—
555C2×C4—84no—
556C2×C4—164no—
557C2×C4C2×#12312yes—
558C2×C4C2×#12312yes—
559C2×C4C2×#12312yes—
560C2×C4C22×#4111yescharacteristic
561C2×C4C22×#4111yescharacteristic
562C2×C4#58514yes—
563C8—42no—
564C8—42no—
565C8—44no—
566C22—22no—
567C22—24no—
568C22—24no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—88no—
581C22#309814yes—
582C22C22×#12311yescharacteristic
583C22#344912yes—
584C22C2×#57611yescharacteristic
585C22C2×#57811yescharacteristic
586C22#358612yes—
587C4—22no—
588C4—22no—
589C4—22no—
590C4—42no—
591C4—42no—
592C4—42no—
593C4—42no—
594C4—82no—
595C4—82no—
596C4—84no—
597C4—82no—
598C4—82no—
599C4—82no—
600C4—82no—
601C4#349112yes—
602C4#349112yes—
603C2—24no—
604C2—42no—
605C2—42no—
606C2#3509711yescharacteristic
607C2C2×#309811yescharacteristic
608C2#6051911yescharacteristic
609C2#6097612yes—
610C2#6098312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33104C2semidirect
4. #29918C2semidirect
5. #27872C2semidirect
6. #27870C2semidirect
7. #27616C2semidirect
8. #27610C2semidirect
9. #27589C2semidirect
10. #27586C2semidirect
46. #2101C22semidirect
47. #2099C22semidirect
48. #2097C22semidirect
49. #2095C22semidirect
50. #2085C22semidirect
51. #2083C22semidirect
52. #2074C22semidirect
53. #2072C22semidirect
54. #1928C22semidirect
55. #1928C22semidirect
56. #1928C22semidirect
57. #1927C22semidirect
58. #1921C22semidirect
59. #1920C22semidirect
60. #1918C22semidirect
61. #1918C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26252
#341802
#331042
#299182
#278722
#278702
#276162
#276102
#275892
#275862
#219442
#219402
#217652
#217622
#206852