Finite Group Databasebeta

← #6435382

Subgroups — #6435382

Order 512.

1621 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35277C211yesmaximal, characteristic
2#34281C211yesmaximal, characteristic, complemented
3#30116C212yesmaximal, complemented
4#30108C212yesmaximal
5#30074C212yesmaximal, complemented
6#30050C212yesmaximal, complemented
7#29983C212yesmaximal, complemented
8#29932C212yesmaximal, complemented
9#29801C212yesmaximal, complemented
10#29676C212yesmaximal, complemented
11#23824C211yesmaximal, characteristic, complemented
12#23820C211yesmaximal, characteristic, complemented
13#23805C211yesmaximal, characteristic, complemented
14#23799C211yesmaximal, characteristic, complemented
15#23581C214yesmaximal, complemented
16#21511C214yesmaximal, complemented
17#20780C211yesmaximal, characteristic, complemented
18C2×#542C2212yes—
19C2×#541C2211yescharacteristic
20C2×#464C2214yes—
21C2×#441C2214yes—
22C2×C4×SD16C2214yes—
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#2373—22no—
34#2373—22no—
35#2371—22no—
36#2370—22no—
37#2330C2211yescharacteristic
38#2327C2211yescharacteristic
39#2326C2212yes—
40#2323C2212yes—
41#2314C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×#127C2211yescharacteristic
45C4×#125C2211yescharacteristic
46C4×C22⋊Q8C2212yes—
47#2295C2212yes—
48#2276C2212yes—
49C4×#117C2211yescharacteristic
50#2099C2212yes—
51#2099C2212yes—
52#2096C2212yes—
53#2096C2212yes—
54#2096C2212yes—
55#2094C2212yes—
56#2074C2212yes—
57#2073C2212yes—
58#2058C2214yes—
59#2043C2212yes—
60#2042C2214yes—
61#2041C2212yes—
62#2035C2214yes—
63#2034C2214yes—
64#2026C2212yes—
65#2025C2212yes—
66#2024C2214yes—
67#2016C2212yes—
68#2016C2212yes—
69#2015C2214yes—
70#2014C2212yes—
71#2013C2212yes—
72#2001C2214yes—
73#2000C2214yes—
74#1983C2214yes—
75#1948C2214yes—
76#1927C2214yes—
77#1926C2214yes—
78#1880C2214yes—
79#1878C2214yes—
80#1840C2212yes—
81#1831C2212yes—
82#1817C2212yes—
83#1803C2214yes—
84#1798C2212yes—
85#1768C2212yes—
86#1762C2214yes—
87C4×#103C2214yes—
88#1750C2212yes—
89#1745C2211yescharacteristic
90#1744C2211yescharacteristic
91C22×Q16C2314yes—
92C22×SD16C2314yes—
93#542—48no—
94#541—28no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2312yes—
99C2×#127C2311yescharacteristic
100C2×#126—22no—
101C2×#126—22no—
102C2×#125—22no—
103C2×#125C2311yescharacteristic
104C2×#124—22no—
105C2×#124C2312yes—
106C2×#124C2312yes—
107C2×#123—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8C2312yes—
110C2×C22⋊Q8C2312yes—
111C2×#121—22no—
112C2×#121C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2312yes—
117C2×C4×D4—22no—
118C2×C4×D4C2312yes—
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#464—216no—
123#441—48no—
124C4×SD16—48no—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×C4⋊C8C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×C22⋊C8C2314yes—
137C2×#97C2312yes—
138C2×C4×C8C2312yes—
139#397—24no—
140#396—24no—
141#394—24no—
142#393—24no—
143#391C2312yes—
144#391C2312yes—
145#391C2312yes—
146#391C2312yes—
147#390—22no—
148#390—22no—
149#390—22no—
150#390—22no—
151#390C2312yes—
152#390C2312yes—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389C2312yes—
159#389C2312yes—
160#388—22no—
161#388—22no—
162#388—22no—
163#388—24no—
164#387—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#385—22no—
174#385—24no—
175#384—22no—
176#384C2311yescharacteristic
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#384C2312yes—
180#382C2311yescharacteristic
181#380—22no—
182#380C2312yes—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4D412yes—
192C4×C4⋊C4D412yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C22⋊C4—22no—
197C4×C22⋊C4D412yes—
198C4×C22⋊C4D412yes—
199#376—22no—
200#376—22no—
201#376C2311yescharacteristic
202#376C2311yescharacteristic
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×Q8C2412yes—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4C2412yes—
221C23×C4—22no—
222C23×C4C2×D412yes—
223C23×C4C2411yescharacteristic
224C2×Q16—28no—
225C2×Q16—44no—
226C2×Q16—44no—
227C2×SD16—28no—
228C2×SD16—44no—
229C2×SD16—44no—
230C22×C8C2414yes—
231C22×C8C2414yes—
232C4⋊Q8—24no—
233C4⋊Q8—44no—
234C4⋊Q8—42no—
235C4⋊Q8—84no—
236#127—42no—
237#126—216no—
238#126—48no—
239#125—216no—
240#125—24no—
241#124—28no—
242#124—44no—
243#124—48no—
244#123—48no—
245C22⋊Q8—24no—
246C22⋊Q8—24no—
247C22⋊Q8—44no—
248C22⋊Q8—48no—
249#121—44no—
250#121—84no—
251C4×Q8—24no—
252C4×Q8—28no—
253C4×Q8—28no—
254C4×Q8—44no—
255C4×Q8#4118yes—
256C4×D4—28no—
257C4×D4—84no—
258#117—24no—
259#117—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
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—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4C2×D411yescharacteristic
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D411yescharacteristic
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4C2×D411yescharacteristic
295C2×C22⋊C4C2×D411yescharacteristic
296C2×C22⋊C4C2×D412yes—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—42no—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2×D412yes—
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312C2×C42C2411yescharacteristic
313C2×C42C2411yescharacteristic
314C2×C42C2411yescharacteristic
315C2×C42C2411yescharacteristic
316#106—44no—
317#106—44no—
318C4⋊C8—28no—
319#103—28no—
320#103—44no—
321#103—44no—
322#103—44no—
323#103—44no—
324#102—28no—
325#102—44no—
326#102—44no—
327C22⋊C8—44no—
328#97—24no—
329C4×C8—24no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—24no—
342#95—24no—
343#95C2×D412yes—
344#95C2×D412yes—
345#95C2×D412yes—
346#95C2×D412yes—
347C24—22no—
348C24C22×D411yescharacteristic
349C2×Q8—22no—
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—44no—
357C2×Q8—84no—
358C2×Q8C2×#4114yes—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—84no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—44no—
382C22×C4—84no—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D412yes—
385C22×C4C22×D412yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D412yes—
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D412yes—
392C22×C4C2511yescharacteristic
393Q16—416no—
394SD16—416no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C4⋊C4—28no—
402C4⋊C4—28no—
403C4⋊C4—28no—
404C4⋊C4—28no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—82no—
423C4⋊C4—82no—
424C4⋊C4—84no—
425C4⋊C4C2×#4114yes—
426C22⋊C4—28no—
427C22⋊C4—24no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—48no—
435C22⋊C4—44no—
436C22⋊C4—84no—
437C22⋊C4—84no—
438C22⋊C4—82no—
439C42—28no—
440C42—24no—
441C42—24no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—42no—
447C42—42no—
448C42—42no—
449C42—44no—
450C42—44no—
451C42—44no—
452C42—82no—
453C42—82no—
454C42—82no—
455C42C2×#4114yes—
456C42H(5,𝔽2)14yes—
457C23—22no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—22no—
462C23—44no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23C23×D411yescharacteristic
468Q8—28no—
469Q8—44no—
470Q8—44no—
471Q8—44no—
472Q8—164no—
473D4—44no—
474D4—44no—
475D4—88no—
476D4—164no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—84no—
517C2×C4—164no—
518C2×C4#52514yes—
519C2×C4#53512yes—
520C2×C4#53512yes—
521C2×C4#57714yes—
522C2×C4C22×#4112yes—
523C2×C4C2×H(5,𝔽2)12yes—
524C2×C4#58512yes—
525C8—44no—
526C8—44no—
527C22—41no—
528C22—42no—
529C22—41no—
530C22—41no—
531C22—42no—
532C22—82no—
533C22—84no—
534C22—82no—
535C22—82no—
536C22—84no—
537C22C2×#52511yescharacteristic
538C22C2×#53511yescharacteristic
539C22#344911yescharacteristic
540C22C2×#57712yes—
541C22#358612yes—
542C4—24no—
543C4—22no—
544C4—22no—
545C4—24no—
546C4—41no—
547C4—44no—
548C4—42no—
549C4—41no—
550C4—41no—
551C4—42no—
552C4—44no—
553C4—82no—
554C4—82no—
555C4—82no—
556C4—82no—
557C4—82no—
558C4#346014yes—
559C2—81no—
560C2—162no—
561C2#3460911yescharacteristic
562C2#6059912yes—
563C2#6081811yescharacteristic
564C2#6083111yescharacteristic
565C2#6097712yes—
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. #34281C2semidirect
3. #30116C2semidirect
5. #30074C2semidirect
6. #30050C2semidirect
7. #29983C2semidirect
8. #29932C2semidirect
9. #29801C2semidirect
10. #29676C2semidirect
11. #23824C2semidirect
12. #23820C2semidirect
13. #23805C2semidirect
14. #23799C2semidirect
15. #23581C2semidirect
16. #21511C2semidirect
17. #20780C2semidirect

Maximal subgroups

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

subgroupindex
#352772
#342812
#301162
#301082
#300742
#300502
#299832
#299322
#298012
#296762
#238242
#238202
#238052
#237992
#235812
#215112
#207802