Finite Group Databasebeta

← #6488724

Subgroups — #6488724

Order 512.

1680 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35295C211yesmaximal, characteristic
2#34209C211yesmaximal, characteristic, complemented
3#33015C214yesmaximal, complemented
4#30130C212yesmaximal, complemented
5#30129C212yesmaximal, complemented
6#29369C212yesmaximal, complemented
7#29368C212yesmaximal, complemented
8#29205C212yesmaximal, complemented
9#28623C212yesmaximal, complemented
10#28128C214yesmaximal, complemented
11#27963C211yesmaximal, characteristic, complemented
12#27917C211yesmaximal, characteristic
13#25861C211yesmaximal, characteristic, complemented
14#25718C211yesmaximal, characteristic
15#22275C212yesmaximal, complemented
16#22231C212yesmaximal, complemented
17#20784C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×#519C2212yes—
20C2×#492C2212yes—
21C2×#491C2212yes—
22C2×#482C2214yes—
23C2×#448C2214yes—
24#2860C2212yes—
25#2856C2211yescharacteristic
26#2851C2211yescharacteristic
27#2647C2211yescharacteristic
28#2641C2211yescharacteristic
29#2624C2212yes—
30#2597—22no—
31#2559—22no—
32#2479—22no—
33#2476—22no—
34#2365—22no—
35#2365—22no—
36#2344—22no—
37#2344—22no—
38#2330C2211yescharacteristic
39#2328C2211yescharacteristic
40#2327C2212yes—
41#2327C2212yes—
42#2324C2212yes—
43#2310C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48#2293C2212yes—
49#2285C2211yescharacteristic
50C4×#117C2211yescharacteristic
51#2100C2212yes—
52#2099C2212yes—
53#2094C2212yes—
54#2093C2212yes—
55#2068C2214yes—
56#2066C2214yes—
57#2055C2214yes—
58#2055C2214yes—
59#2043C2212yes—
60#2043C2212yes—
61#2013C2212yes—
62#2013C2212yes—
63#1991C2212yes—
64#1961C2212yes—
65#1960C2212yes—
66#1956C2212yes—
67#1928C2214yes—
68#1927C2214yes—
69#1880C2214yes—
70#1880C2214yes—
71#1880C2214yes—
72#1880C2214yes—
73#1877C2214yes—
74#1877C2214yes—
75#1865C2214yes—
76#1865C2214yes—
77#1863C2214yes—
78#1863C2214yes—
79#1847C2212yes—
80#1839C2212yes—
81#1835C2212yes—
82#1831C2212yes—
83#1825C2214yes—
84#1818C2212yes—
85#1798C2212yes—
86#1787C2214yes—
87#1768C2212yes—
88#1749C2211yescharacteristic
89C4×C22⋊C8C2211yescharacteristic
90#1746C2212yes—
91C23×Q8C2312yes—
92C22×SD16C2314yes—
93C22×SD16C2314yes—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2312yes—
99C2×#127C2311yescharacteristic
100C2×#126—22no—
101C2×#125C2311yescharacteristic
102C2×#121—22no—
103C2×#121C2312yes—
104#519—48no—
105C2×C4×Q8—22no—
106C2×C4×Q8—22no—
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2312yes—
111C2×C4×D4C2312yes—
112C2×#117—22no—
113C2×#117—22no—
114C2×#117—22no—
115C2×#117C2312yes—
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C22×C4⋊C4—22no—
119C22×C42C2311yescharacteristic
120#492—28no—
121#491—28no—
122#482—48no—
123#448—48no—
124C2×#107C2314yes—
125C2×#106C2314yes—
126C2×C4⋊C8C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×#97C2312yes—
138C2×C4×C8C2312yes—
139#400—22no—
140#399—22no—
141#398—22no—
142#397—22no—
143#394—22no—
144#394—22no—
145#394—22no—
146#394—22no—
147#391—24no—
148#391C2312yes—
149#391C2312yes—
150#391C2312yes—
151#391C2312yes—
152#390—22no—
153#390—22no—
154#390C2312yes—
155#390C2312yes—
156#390C2312yes—
157#390C2312yes—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#389C2312yes—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#387—22no—
174#387—24no—
175#387—24no—
176#387—24no—
177#387—24no—
178#387—22no—
179#386—22no—
180#386—22no—
181#386—22no—
182#385—22no—
183#385—22no—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#382C2312yes—
188#382C2311yescharacteristic
189#381—24no—
190#380—22no—
191#380C2312yes—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4D412yes—
198C4×C4⋊C4D412yes—
199C4×C4⋊C4C2312yes—
200C4×C22⋊C4—22no—
201C4×C22⋊C4D412yes—
202C4×C22⋊C4D412yes—
203#376C2311yescharacteristic
204#376C2311yescharacteristic
205C43C2311yescharacteristic
206C43C2311yescharacteristic
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#336C2314yes—
212#336C2314yes—
213#336C2314yes—
214#336C2312yes—
215#336C2312yes—
216C22×Q8—24no—
217C22×Q8—24no—
218C22×Q8—22no—
219C22×Q8—28no—
220C22×Q8—44no—
221C22×Q8C2412yes—
222C22×Q8C2412yes—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4C2412yes—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4C2×D412yes—
229C23×C4C2411yescharacteristic
230C2×SD16—28no—
231C2×SD16—28no—
232C2×SD16—44no—
233C2×SD16—44no—
234C2×SD16—44no—
235C2×SD16—44no—
236C22×C8C2414yes—
237C22×C8C2414yes—
238C4⋊Q8—216no—
239C4⋊Q8—24no—
240C4⋊Q8—24no—
241C4⋊Q8—24no—
242C4⋊Q8#4118yes—
243#127—24no—
244#126—84no—
245#125—24no—
246#121—28no—
247#121—84no—
248C4×Q8—44no—
249C4×Q8—44no—
250C4×Q8—44no—
251C4×Q8—44no—
252C4×Q8—48no—
253C4×Q8—48no—
254C4×D4—42no—
255C4×D4—42no—
256#117—24no—
257#117—28no—
258#117—42no—
259#117—48no—
260#117—48no—
261#117—84no—
262C2×C4⋊C4—216no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
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—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4C2×D411yescharacteristic
279C2×C4⋊C4C2×D411yescharacteristic
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C22⋊C4—22no—
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⋊C4C2×D411yescharacteristic
299C2×C22⋊C4C2×D411yescharacteristic
300C2×C22⋊C4C2×D412yes—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42C2×D412yes—
309C2×C42C2×D412yes—
310C2×C42C2×D412yes—
311C2×C42C2×D412yes—
312C2×C42C2×D412yes—
313C2×C42C2411yescharacteristic
314C2×C42C2411yescharacteristic
315C2×C42C2411yescharacteristic
316C2×C42C2411yescharacteristic
317C2×C42C2411yescharacteristic
318C2×C42C2411yescharacteristic
319#107—44no—
320#106—44no—
321C4⋊C8—44no—
322#103—28no—
323#103—28no—
324#103—44no—
325#103—44no—
326#103—44no—
327#103—44no—
328#102—44no—
329#102—44no—
330C22⋊C8—24no—
331C22⋊C8—24no—
332#97—24no—
333C4×C8—24no—
334#95—22no—
335#95—22no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95C2×D412yes—
347#95C2×D412yes—
348#95C2×D412yes—
349#95C2×D412yes—
350C24—22no—
351C24C22×D411yescharacteristic
352C2×Q8—28no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—24no—
356C2×Q8—24no—
357C2×Q8—24no—
358C2×Q8—24no—
359C2×Q8—416no—
360C2×Q8—416no—
361C2×Q8—44no—
362C2×Q8—44no—
363C2×Q8C2×#4114yes—
364C2×Q8C2×#4114yes—
365C2×D4—24no—
366C2×D4—24no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—82no—
372C2×D4—82no—
373C2×D4—82no—
374C2×D4—82no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—28no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4C22×D412yes—
394C22×C4C22×D412yes—
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D412yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D412yes—
402C22×C4H(5,𝔽2)14yes—
403C22×C4C2511yescharacteristic
404SD16—88no—
405SD16—88no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C4⋊C4—28no—
413C4⋊C4—28no—
414C4⋊C4—216no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—42no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—82no—
433C4⋊C4—82no—
434C4⋊C4—82no—
435C4⋊C4—82no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—42no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—82no—
445C22⋊C4—82no—
446C22⋊C4—82no—
447C22⋊C4—82no—
448C22⋊C4—82no—
449C22⋊C4—82no—
450C22⋊C4—82no—
451C42—24no—
452C42—24no—
453C42—24no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—42no—
460C42—42no—
461C42—44no—
462C42—44no—
463C42—44no—
464C42—44no—
465C42—44no—
466C42—82no—
467C42C2×#4114yes—
468C23—22no—
469C23—21no—
470C23—21no—
471C23—22no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23#53512yes—
480C23C23×D411yescharacteristic
481Q8—28no—
482Q8—28no—
483Q8—416no—
484Q8—44no—
485Q8—44no—
486D4—44no—
487D4—44no—
488D4—164no—
489D4—164no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—28no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—48no—
514C2×C4—416no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—88no—
533C2×C4#52514yes—
534C2×C4#53512yes—
535C2×C4#57714yes—
536C2×C4C22×#4112yes—
537C2×C4C2×H(5,𝔽2)12yes—
538C2×C4#58512yes—
539C8—44no—
540C8—44no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—42no—
545C22—42no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22C2×#52511yescharacteristic
554C22C2×#53511yescharacteristic
555C22#344911yescharacteristic
556C22C2×#57712yes—
557C22#358612yes—
558C4—22no—
559C4—24no—
560C4—24no—
561C4—42no—
562C4—42no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—44no—
568C4—44no—
569C4—82no—
570C4—82no—
571C4—82no—
572C4—82no—
573C4—82no—
574C4#351314yes—
575C2—42no—
576C2—162no—
577C2#3523311yescharacteristic
578C2#6058212yes—
579C2#6082011yescharacteristic
580C2#6083111yescharacteristic
581C2#6099112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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
2. #34209C2semidirect
3. #33015C2semidirect
4. #30130C2semidirect
5. #30129C2semidirect
6. #29369C2semidirect
7. #29368C2semidirect
8. #29205C2semidirect
9. #28623C2semidirect
10. #28128C2semidirect
11. #27963C2semidirect
13. #25861C2semidirect
15. #22275C2semidirect
16. #22231C2semidirect
17. #20784C2semidirect

Maximal subgroups

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

subgroupindex
#352952
#342092
#330152
#301302
#301292
#293692
#293682
#292052
#286232
#281282
#279632
#279172
#258612
#257182
#222752
#222312
#207842