Finite Group Databasebeta

← #6500888

Subgroups — #6500888

Order 512.

1762 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35621C211yesmaximal, characteristic
2#34368C211yesmaximal, characteristic
3#27612C212yesmaximal, complemented
4#27609C212yesmaximal, complemented
5#27583C212yesmaximal, complemented
6#26223C214yesmaximal, complemented
7#26211C214yesmaximal, complemented
8#26148C214yesmaximal, complemented
9#26128C214yesmaximal, complemented
10#24014C212yesmaximal, complemented
11#21771C212yesmaximal, complemented
12#21745C211yesmaximal, characteristic
13#20848C212yesmaximal, complemented
14C2×#536C2211yescharacteristic
15C2×#536C2211yescharacteristic
16C2×#532C2211yescharacteristic
17C2×#520C2211yescharacteristic
18C2×#484C2212yes—
19C2×#483C2212yes—
20C2×#428C2212yes—
21C22×#106C2212yes—
22C2×#408C2212yes—
23#2547—21no—
24#2546—21no—
25#2545—21no—
26#2544—21no—
27#2425—24no—
28#2417C2214yes—
29#2415C2214yes—
30#2379—21no—
31#2379—21no—
32#2369—21no—
33#2366—21no—
34#2354C2214yes—
35D4×C4⋊C4C2214yes—
36#2324—24no—
37#2311C2212yes—
38#2311C2211yescharacteristic
39#2311C2212yes—
40#2296C2211yescharacteristic
41C2×#391C2211yescharacteristic
42#2285C2212yes—
43#2100C2212yes—
44#2100C2212yes—
45#2096C2214yes—
46#2084C2212yes—
47#2083C2212yes—
48#2077C2214yes—
49#2077C2214yes—
50#2072C2214yes—
51#2036C2214yes—
52#2035C2214yes—
53#1927C2214yes—
54#1927C2214yes—
55#1920C2214yes—
56#1918C2214yes—
57#1894C2214yes—
58#1892C2214yes—
59#1873C2214yes—
60#1873C2212yes—
61#1873C2212yes—
62#1872C2214yes—
63#1870C2214yes—
64#1869C2214yes—
65#1854C2214yes—
66#1854C2214yes—
67#1853C2214yes—
68#1853C2214yes—
69#1853C2214yes—
70#1825C2214yes—
71#1825C2214yes—
72#1817C2212yes—
73#1787C2214yes—
74#1780C2212yes—
75#1736C2212yes—
76#1735C2212yes—
77C22×#41C2311yescharacteristic
78C22×#41C2311yescharacteristic
79C23×Q8C2311yescharacteristic
80C22×M16C2312yes—
81C23×C8C2312yes—
82#536—28no—
83#536—42no—
84#536—42no—
85#532—28no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8D411yescharacteristic
89C2×#127D411yescharacteristic
90C2×#124—21no—
91C2×#124—21no—
92C2×#124D411yescharacteristic
93C2×#124D411yescharacteristic
94C2×#123—22no—
95C2×#123—22no—
96C2×C22⋊Q8C2314yes—
97C2×C22⋊Q8C2312yes—
98C2×C22⋊Q8C2312yes—
99C2×#121D414yes—
100#520—42no—
101#520—42no—
102C2×C4×Q8—21no—
103C2×C4×D4—22no—
104C2×C4×D4—21no—
105C2×C4×D4—24no—
106C2×C4×D4C2314yes—
107C2×#117—22no—
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C2×#117C2312yes—
111C2×#117C2312yes—
112C2×#117C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C42C2311yescharacteristic
118#484—44no—
119#483—44no—
120#428—44no—
121C2×#107C2314yes—
122C2×#106—44no—
123C2×#106C2314yes—
124C2×#106C2314yes—
125C2×#106C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130#408—28no—
131C2×C22⋊C8C2314yes—
132C2×C22⋊C8C2314yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2314yes—
136#398—24no—
137#397—24no—
138#396—24no—
139#394—24no—
140#391—22no—
141#391—22no—
142#391—21no—
143#391—21no—
144#391—21no—
145#391C2314yes—
146#390—22no—
147#390—22no—
148#390—21no—
149#390—24no—
150#389—24no—
151#389—24no—
152#389C2312yes—
153#389C2312yes—
154#388—24no—
155#388—24no—
156#388—24no—
157#386—22no—
158#386—22no—
159#386—22no—
160#386—22no—
161#386—22no—
162#386—24no—
163#385—22no—
164#385—24no—
165#385—24no—
166#384C2312yes—
167#384C2312yes—
168#382—22no—
169#380—22no—
170#380—24no—
171#380C2314yes—
172#380C2314yes—
173#380C2314yes—
174#380C2314yes—
175C4×C4⋊C4—24no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—21no—
178C4×C4⋊C4—21no—
179C4×C4⋊C4C2314yes—
180C2×#95—22no—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#336C2314yes—
190#336C2312yes—
191#336C2312yes—
192C2×#41—24no—
193C2×#41—41no—
194C2×#41—41no—
195C2×#41—42no—
196C2×#41—41no—
197C2×#41—41no—
198C2×#41—42no—
199C2×#41#4118yes—
200C22×Q8—21no—
201C22×Q8—24no—
202C22×Q8—41no—
203C22×Q8—41no—
204C22×Q8—42no—
205C22×Q8C2×D411yescharacteristic
206C22×Q8C2×D411yescharacteristic
207C22×Q8C2412yes—
208C22×D4C2×D414yes—
209C22×D4C2×D412yes—
210C22×D4C2×D411yescharacteristic
211C22×D4C2×D412yes—
212C22×D4C2×D411yescharacteristic
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4C2×D412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C2×M16—24no—
223C2×M16—42no—
224C2×M16—42no—
225C22×C8—24no—
226C22×C8—42no—
227C22×C8—42no—
228C22×C8C2414yes—
229C22×C8C2414yes—
230C4⋊Q8—24no—
231C4⋊Q8—44no—
232C4⋊Q8—44no—
233#127—24no—
234#124—24no—
235#124—24no—
236#124—44no—
237#124—44no—
238#123—48no—
239#123—48no—
240C22⋊Q8—28no—
241C22⋊Q8—28no—
242C22⋊Q8—24no—
243C22⋊Q8—24no—
244C22⋊Q8—42no—
245C22⋊Q8—42no—
246#121—216no—
247C4×Q8—44no—
248C4×D4—28no—
249C4×D4—28no—
250C4×D4—44no—
251C4×D4—48no—
252C4×D4—88no—
253#117—22no—
254#117—22no—
255#117—24no—
256#117—24no—
257#117—42no—
258#117—42no—
259#117—42no—
260#117—44no—
261#117—84no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
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—42no—
277C2×C4⋊C4—41no—
278C2×C4⋊C4—41no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2414yes—
286C2×C4⋊C4C2414yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2411yescharacteristic
289C2×C4⋊C4C2411yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4C2×D414yes—
303C2×C22⋊C4C2×D414yes—
304C2×C42—21no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—21no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2412yes—
314#107—28no—
315#106—28no—
316#106—44no—
317#106—44no—
318#103—44no—
319#103—44no—
320#103—44no—
321#103—44no—
322C22⋊C8—28no—
323C22⋊C8—28no—
324C22⋊C8—44no—
325C22⋊C8—42no—
326C22⋊C8—42no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—22no—
333#95—22no—
334#95—24no—
335#95—24no—
336C24C22×D412yes—
337C24C22×D412yes—
338C24C22×D411yescharacteristic
339#41—48no—
340#41—48no—
341#41—84no—
342#41—84no—
343C2×Q8—22no—
344C2×Q8—24no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—24no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8—84no—
356C2×Q8—84no—
357C2×Q8#12314yes—
358C2×D4—28no—
359C2×D4—24no—
360C2×D4—22no—
361C2×D4—28no—
362C2×D4—41no—
363C2×D4—42no—
364C2×D4—44no—
365C2×D4—42no—
366C2×D4—41no—
367C2×D4—42no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—42no—
371C2×D4#12318yes—
372C2×D4#12314yes—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—42no—
394C22×C4—44no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4—84no—
402C22×C4—84no—
403C22×C4—84no—
404C22×C4—82no—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C2×#4114yes—
411C22×C4C2×#4114yes—
412C22×C4C2×#4114yes—
413C22×C4C2511yescharacteristic
414M16—48no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—48no—
422C4⋊C4—28no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—42no—
437C4⋊C4—48no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C4⋊C4—82no—
443C4⋊C4—82no—
444C4⋊C4—82no—
445C4⋊C4H(5,𝔽2)12yes—
446C4⋊C4H(5,𝔽2)12yes—
447C22⋊C4—24no—
448C22⋊C4—28no—
449C22⋊C4—28no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—42no—
454C22⋊C4—48no—
455C22⋊C4—84no—
456C22⋊C4—82no—
457C22⋊C4—82no—
458C22⋊C4—84no—
459C22⋊C4—84no—
460C22⋊C4—84no—
461C42—24no—
462C42—22no—
463C42—22no—
464C42—24no—
465C42—44no—
466C42—42no—
467C42—48no—
468C42—82no—
469C42—82no—
470C23—22no—
471C23—24no—
472C23—21no—
473C23—22no—
474C23—22no—
475C23—21no—
476C23—22no—
477C23—24no—
478C23—24no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23C2×#12314yes—
484C23C2×#12312yes—
485C23C23×D411yescharacteristic
486Q8—44no—
487Q8—44no—
488Q8—44no—
489Q8—82no—
490Q8—84no—
491D4—44no—
492D4—48no—
493D4—416no—
494D4—82no—
495D4—84no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—48no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—84no—
536C2×C4—164no—
537C2×C4—162no—
538C2×C4—164no—
539C2×C4—164no—
540C2×C4C2×#12312yes—
541C2×C4C2×#12314yes—
542C2×C4#57714yes—
543C2×C4C22×#4112yes—
544C2×C4C2×H(5,𝔽2)11yescharacteristic
545C2×C4C2×H(5,𝔽2)11yescharacteristic
546C2×C4#58512yes—
547C8—44no—
548C8—44no—
549C22—22no—
550C22—24no—
551C22—24no—
552C22—28no—
553C22—42no—
554C22—44no—
555C22—42no—
556C22—41no—
557C22—42no—
558C22—41no—
559C22—42no—
560C22—44no—
561C22—44no—
562C22—42no—
563C22—42no—
564C22C22×#12311yescharacteristic
565C22#348411yescharacteristic
566C22#348511yescharacteristic
567C22C2×#57712yes—
568C22#358612yes—
569C4—22no—
570C4—24no—
571C4—24no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—82no—
577C4—82no—
578C4—84no—
579C4—84no—
580C4—82no—
581C4#350512yes—
582C4#350512yes—
583C2—42no—
584C2—44no—
585C2—82no—
586C2#3535111yescharacteristic
587C2#6052212yes—
588C2#6166411yescharacteristic
589C2#6167411yescharacteristic
590C2#6180912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#532 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27612C2semidirect
4. #27609C2semidirect
5. #27583C2semidirect
6. #26223C2semidirect
7. #26211C2semidirect
8. #26148C2semidirect
9. #26128C2semidirect
10. #24014C2semidirect
11. #21771C2semidirect
13. #20848C2semidirect

Maximal subgroups

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

subgroupindex
#356212
#343682
#276122
#276092
#275832
#262232
#262112
#261482
#261282
#240142
#217712
#217452
#208482