Finite Group Databasebeta

← #6435379

Subgroups — #6435379

Order 512.

1725 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35277C211yesmaximal, characteristic, complemented
2#34280C211yesmaximal, characteristic, complemented
3#30113C212yesmaximal, complemented
4#30105C212yesmaximal, complemented
5#30072C212yesmaximal, complemented
6#30047C212yesmaximal, complemented
7#29980C212yesmaximal, complemented
8#29931C212yesmaximal, complemented
9#29800C212yesmaximal, complemented
10#29675C212yesmaximal, complemented
11#23824C211yesmaximal, characteristic, complemented
12#23820C211yesmaximal, characteristic, complemented
13#23803C211yesmaximal, characteristic, complemented
14#23798C211yesmaximal, characteristic, complemented
15#23579C214yesmaximal, complemented
16#21509C214yesmaximal, complemented
17#20780C211yesmaximal, characteristic, complemented
18C2×#542C2212yes—
19C2×#540C2211yescharacteristic
20C2×#463C2214yes—
21C2×#440C2214yes—
22C2×C4×D8C2214yes—
23#2868C2211yescharacteristic
24#2858C2211yescharacteristic
25#2857C2211yescharacteristic
26#2856C2211yescharacteristic
27#2640C2212yes—
28#2620C2212yes—
29#2512—22no—
30#2491—22no—
31#2452—22no—
32#2422—22no—
33#2374—22no—
34#2364—22no—
35#2363—22no—
36#2359—22no—
37#2330C2211yescharacteristic
38#2328C2211yescharacteristic
39#2326C2212yes—
40#2324C2212yes—
41#2314C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×#127C2211yescharacteristic
44C4×#127C2211yescharacteristic
45C4×#125C2211yescharacteristic
46C4×#121C2212yes—
47#2293C2212yes—
48#2275C2212yes—
49C4×#117C2211yescharacteristic
50#2099C2212yes—
51#2099C2212yes—
52#2095C2212yes—
53#2093C2212yes—
54#2093C2212yes—
55#2093C2212yes—
56#2074C2212yes—
57#2073C2212yes—
58#2057C2214yes—
59#2043C2212yes—
60#2042C2214yes—
61#2041C2212yes—
62#2036C2214yes—
63#2033C2214yes—
64#2026C2212yes—
65#2025C2212yes—
66#2023C2214yes—
67#2014C2212yes—
68#2013C2212yes—
69#2012C2214yes—
70#2011C2212yes—
71#2011C2212yes—
72#2000C2214yes—
73#1999C2214yes—
74#1983C2214yes—
75#1947C2214yes—
76#1928C2214yes—
77#1925C2214yes—
78#1879C2214yes—
79#1877C2214yes—
80#1840C2212yes—
81#1831C2212yes—
82#1817C2212yes—
83#1802C2214yes—
84#1798C2212yes—
85#1768C2212yes—
86#1762C2214yes—
87C4×#103C2214yes—
88#1750C2212yes—
89#1745C2211yescharacteristic
90#1744C2211yescharacteristic
91C22×SD16C2314yes—
92C22×D8C2314yes—
93#542—48no—
94#540—28no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2312yes—
98C2×#127C2311yescharacteristic
99C2×#127C2311yescharacteristic
100C2×#126—22no—
101C2×#125C2311yescharacteristic
102C2×#124—22no—
103C2×#124C2312yes—
104C2×#124C2312yes—
105C2×#123—22no—
106C2×#123—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8C2312yes—
109C2×#121—22no—
110C2×#121—22no—
111C2×#121C2312yes—
112C2×#121C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4C2312yes—
118C2×C4×D4C2312yes—
119C2×C4×D4C2312yes—
120C2×C4×D4C2312yes—
121C2×#117—22no—
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#463—216no—
126#440—48no—
127C4×D8—48no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×C4⋊C8C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×C22⋊C8C2314yes—
140C2×#97C2312yes—
141C2×C4×C8C2312yes—
142#397—24no—
143#396—24no—
144#394—24no—
145#393—24no—
146#391C2312yes—
147#391C2312yes—
148#390—22no—
149#390—22no—
150#390—22no—
151#390—22no—
152#390—22no—
153#390—22no—
154#390C2312yes—
155#390C2312yes—
156#390C2312yes—
157#390C2312yes—
158#389—22no—
159#389—22no—
160#389—22no—
161#389C2312yes—
162#389C2312yes—
163#388—22no—
164#388—22no—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—24no—
169#387—22no—
170#387—22no—
171#387—22no—
172#386—22no—
173#386—22no—
174#386—22no—
175#386—22no—
176#385—22no—
177#385—24no—
178#384—22no—
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#384C2312yes—
183#382C2311yescharacteristic
184#380—22no—
185#380C2312yes—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4D412yes—
189C4×C4⋊C4D412yes—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2312yes—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—22no—
195C4×C22⋊C4—22no—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4D412yes—
199C4×C22⋊C4D412yes—
200#376C2311yescharacteristic
201#376C2311yescharacteristic
202C2×#95—22no—
203C43C2311yescharacteristic
204C43C2311yescharacteristic
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#336C2312yes—
210#336C2312yes—
211#336C2314yes—
212#336C2312yes—
213#336C2312yes—
214#336C2314yes—
215C22×Q8—22no—
216C22×Q8C2412yes—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4C2412yes—
222C22×D4C2412yes—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4C2×D412yes—
228C23×C4C2411yescharacteristic
229C2×SD16—28no—
230C2×SD16—44no—
231C2×SD16—44no—
232C2×D8—28no—
233C2×D8—44no—
234C2×D8—44no—
235C22×C8C2414yes—
236C22×C8C2414yes—
237C4⋊Q8—44no—
238C4⋊Q8—42no—
239C4⋊Q8—84no—
240#127—24no—
241#127—42no—
242#126—48no—
243#125—24no—
244#124—28no—
245#124—44no—
246#124—48no—
247#123—216no—
248#123—48no—
249C22⋊Q8—44no—
250C22⋊Q8—48no—
251#121—216no—
252#121—24no—
253#121—24no—
254#121—44no—
255#121—84no—
256C4×Q8—28no—
257C4×D4—24no—
258C4×D4—28no—
259C4×D4—28no—
260C4×D4—44no—
261C4×D4—84no—
262C4×D4—84no—
263C4×D4—84no—
264C4×D4#4118yes—
265#117—24no—
266#117—24no—
267#117—84no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4C2×D411yescharacteristic
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D411yescharacteristic
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4C2×D412yes—
305C2×C22⋊C4C2×D411yescharacteristic
306C2×C22⋊C4C2×D411yescharacteristic
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—42no—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2411yescharacteristic
318C2×C42C2411yescharacteristic
319C2×C42C2411yescharacteristic
320C2×C42C2411yescharacteristic
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323#107—44no—
324#107—44no—
325C4⋊C8—28no—
326#103—28no—
327#103—44no—
328#103—44no—
329#102—28no—
330#102—44no—
331#102—44no—
332#102—44no—
333#102—44no—
334C22⋊C8—44no—
335#97—24no—
336C4×C8—24no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—24no—
345#95—24no—
346#95C2×D412yes—
347#95C2×D412yes—
348#95C2×D412yes—
349#95C2×D412yes—
350C24—22no—
351C24—22no—
352C24C22×D411yescharacteristic
353C2×Q8—24no—
354C2×Q8—24no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—44no—
358C2×Q8—84no—
359C2×D4—28no—
360C2×D4—28no—
361C2×D4—22no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—82no—
373C2×D4—82no—
374C2×D4—82no—
375C2×D4—82no—
376C2×D4—84no—
377C2×D4C2×#4114yes—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—28no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—82no—
395C22×C4—82no—
396C22×C4—82no—
397C22×C4—82no—
398C22×C4—84no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D412yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D412yes—
410C22×C4C2511yescharacteristic
411SD16—416no—
412D8—416no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C4⋊C4—28no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—82no—
436C4⋊C4—82no—
437C4⋊C4—84no—
438C4⋊C4C2×#4114yes—
439C22⋊C4—28no—
440C22⋊C4—28no—
441C22⋊C4—24no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—48no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—82no—
455C22⋊C4—82no—
456C22⋊C4—84no—
457C22⋊C4—84no—
458C22⋊C4—82no—
459C42—24no—
460C42—24no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—42no—
466C42—42no—
467C42—42no—
468C42—44no—
469C42—82no—
470C42—82no—
471C42—82no—
472C42C2×#4114yes—
473C42H(5,𝔽2)14yes—
474C23—22no—
475C23—21no—
476C23—24no—
477C23—21no—
478C23—21no—
479C23—22no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23C23×D411yescharacteristic
492Q8—44no—
493Q8—44no—
494Q8—164no—
495D4—28no—
496D4—44no—
497D4—44no—
498D4—44no—
499D4—88no—
500D4—88no—
501D4—88no—
502D4—164no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—24no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—84no—
540C2×C4—88no—
541C2×C4—164no—
542C2×C4—164no—
543C2×C4—164no—
544C2×C4#52514yes—
545C2×C4#53512yes—
546C2×C4#53512yes—
547C2×C4#57614yes—
548C2×C4C22×#4112yes—
549C2×C4C2×H(5,𝔽2)12yes—
550C2×C4#58512yes—
551C8—44no—
552C8—44no—
553C22—42no—
554C22—41no—
555C22—44no—
556C22—41no—
557C22—41no—
558C22—44no—
559C22—42no—
560C22—44no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—84no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—84no—
570C22C2×#52511yescharacteristic
571C22C2×#53511yescharacteristic
572C22#344911yescharacteristic
573C22C2×#57612yes—
574C22#358612yes—
575C4—24no—
576C4—22no—
577C4—22no—
578C4—24no—
579C4—42no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—42no—
584C4—44no—
585C4—82no—
586C4—82no—
587C4—82no—
588C4—82no—
589C4—82no—
590C4#346014yes—
591C2—81no—
592C2—84no—
593C2—162no—
594C2#3460911yescharacteristic
595C2#6059612yes—
596C2#6081711yescharacteristic
597C2#6083111yescharacteristic
598C2#6097612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35277C2semidirect
2. #34280C2semidirect
3. #30113C2semidirect
4. #30105C2semidirect
5. #30072C2semidirect
6. #30047C2semidirect
7. #29980C2semidirect
8. #29931C2semidirect
9. #29800C2semidirect
10. #29675C2semidirect
11. #23824C2semidirect
12. #23820C2semidirect
13. #23803C2semidirect
14. #23798C2semidirect
15. #23579C2semidirect
16. #21509C2semidirect
17. #20780C2semidirect

Maximal subgroups

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

subgroupindex
#352772
#342802
#301132
#301052
#300722
#300472
#299802
#299312
#298002
#296752
#238242
#238202
#238032
#237982
#235792
#215092
#207802