Finite Group Databasebeta

← #6635574

Subgroups — #6635574

Order 512.

1720 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37296C211yesmaximal, characteristic
2#35673C211yesmaximal, characteristic, complemented
3#32918C212yesmaximal, complemented
4#31956C214yesmaximal, complemented
5#31932C214yesmaximal, complemented
6#31005C214yesmaximal, complemented
7#30991C214yesmaximal, complemented
8#29768C212yesmaximal, complemented
9#28313C211yesmaximal, characteristic, complemented
10#28305C211yesmaximal, characteristic, complemented
11#27981C211yesmaximal, characteristic, complemented
12#27975C211yesmaximal, characteristic, complemented
13#24012C212yesmaximal, complemented
14#23927C212yesmaximal, complemented
15#21905C211yesmaximal, characteristic, complemented
16C2×#537C2212yes—
17C2×#536C2211yescharacteristic
18C2×#519C2211yescharacteristic
19C2×#481C2212yes—
20C2×#483C2212yes—
21C2×#464C2214yes—
22C2×#452C2214yes—
23C2×#451C2212yes—
24C2×#448C2212yes—
25#2857C2211yescharacteristic
26#2856C2211yescharacteristic
27#2732—22no—
28#2682—22no—
29#2671—24no—
30#2620C2212yes—
31#2502C2212yes—
32#2501C2212yes—
33#2498—24no—
34#2448—22no—
35#2446—22no—
36#2432C2212yes—
37#2431C2212yes—
38#2423C2214yes—
39#2421C2214yes—
40#2316C2211yescharacteristic
41#2311C2211yescharacteristic
42#2297C2211yescharacteristic
43C2×#391C2211yescharacteristic
44#2285C2211yescharacteristic
45#2096C2212yes—
46#2096C2212yes—
47#2084C2214yes—
48#2074C2214yes—
49#2071C2214yes—
50#2066C2214yes—
51#2065C2214yes—
52#2042C2214yes—
53#2035C2214yes—
54#2034C2214yes—
55#2016C2212yes—
56#2014C2212yes—
57#2001C2214yes—
58#2000C2214yes—
59#1971C2214yes—
60#1968C2214yes—
61#1961C2212yes—
62#1956C2212yes—
63#1924C2214yes—
64#1923C2214yes—
65#1880C2214yes—
66#1878C2214yes—
67#1875C2212yes—
68#1875C2212yes—
69#1873C2212yes—
70#1873C2212yes—
71#1872C2214yes—
72#1853C2214yes—
73#1836C2212yes—
74#1834C2212yes—
75#1803C2214yes—
76#1798C2212yes—
77#1788C2214yes—
78#1780C2212yes—
79#1749C2211yescharacteristic
80C4×C22⋊C8C2211yescharacteristic
81C22×#41C2312yes—
82C22×#41C2311yescharacteristic
83C23×Q8C2311yescharacteristic
84C22×Q16C2314yes—
85C22×SD16C2314yes—
86#537—44no—
87#537—44no—
88#536—44no—
89C2×C4⋊Q8—21no—
90C2×C4⋊Q8C2311yescharacteristic
91C2×#127C2311yescharacteristic
92C2×#124—21no—
93C2×#124C2312yes—
94C2×#123—22no—
95C2×#123—24no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8C2312yes—
98C2×C22⋊Q8C2314yes—
99C2×C22⋊Q8C2312yes—
100C2×#121—22no—
101C2×#121—22no—
102C2×#121—22no—
103C2×#121—22no—
104C2×#121C2312yes—
105C2×#121C2312yes—
106#519—44no—
107C2×C4×Q8—22no—
108C2×C4×Q8C2314yes—
109C2×#117—22no—
110C2×#117C2311yescharacteristic
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C42C2311yescharacteristic
116#483—44no—
117#481—44no—
118#464—48no—
119#452—48no—
120#451—44no—
121#448—44no—
122C2×#106C2314yes—
123C2×#106C2314yes—
124C2×C4⋊C8C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2314yes—
136C2×C22⋊C8C2314yes—
137C2×#97C2312yes—
138C2×C4×C8C2312yes—
139#400—24no—
140#399—24no—
141#399—24no—
142#398—24no—
143#398—24no—
144#397—24no—
145#396—24no—
146#394—24no—
147#394—24no—
148#392—24no—
149#391—21no—
150#391—22no—
151#391D412yes—
152#391C2312yes—
153#391C2312yes—
154#391C2312yes—
155#391C2312yes—
156#390—21no—
157#390—22no—
158#390D412yes—
159#390C2312yes—
160#390C2312yes—
161#389—22no—
162#389—22no—
163#389C2314yes—
164#388—24no—
165#388—24no—
166#387—24no—
167#387—24no—
168#386—22no—
169#386—24no—
170#386D414yes—
171#385—24no—
172#385—24no—
173#384C2312yes—
174#384C2312yes—
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#382C2312yes—
178#380—22no—
179#380—22no—
180#380—22no—
181#380C2314yes—
182C4×C4⋊C4—22no—
183C4×C22⋊C4C2312yes—
184C4×C22⋊C4C2312yes—
185C2×#95—22no—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#336C2314yes—
191#336C2312yes—
192#336C2312yes—
193C2×#41—24no—
194C2×#41—42no—
195C2×#41—42no—
196C2×#41—42no—
197C2×#41—42no—
198C2×#41—44no—
199C2×#41—44no—
200C22×Q8—24no—
201C22×Q8—21no—
202C22×Q8—24no—
203C22×Q8—42no—
204C22×Q8C2×D411yescharacteristic
205C22×Q8C2412yes—
206C22×Q8C2412yes—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4C2×D412yes—
212C22×D4C2×D411yescharacteristic
213C22×D4C2412yes—
214C23×C4—21no—
215C23×C4—22no—
216C23×C4—21no—
217C23×C4—22no—
218C23×C4C2×D412yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C2×Q16—28no—
222C2×Q16—44no—
223C2×Q16—44no—
224C2×SD16—28no—
225C2×SD16—44no—
226C2×SD16—44no—
227C22×C8C2414yes—
228C22×C8C2414yes—
229C4⋊Q8—42no—
230C4⋊Q8—44no—
231#127—42no—
232#124—44no—
233#124—44no—
234#123—48no—
235#123—88no—
236C22⋊Q8—28no—
237C22⋊Q8—24no—
238C22⋊Q8—24no—
239C22⋊Q8—44no—
240C22⋊Q8—42no—
241C22⋊Q8—42no—
242C22⋊Q8—48no—
243#121—24no—
244#121—24no—
245#121—42no—
246#121—42no—
247#121—48no—
248#121—48no—
249#121—48no—
250#121—84no—
251C4×Q8—28no—
252C4×Q8—44no—
253C4×Q8—48no—
254#117—42no—
255#117—42no—
256#117—42no—
257#117—48no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4C2×D414yes—
292C2×C22⋊C4C2×D412yes—
293C2×C22⋊C4C2×D414yes—
294C2×C42—24no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—42no—
298C2×C42C2×D412yes—
299C2×C42C2412yes—
300C2×C42C2411yescharacteristic
301C2×C42C2411yescharacteristic
302#106—44no—
303#106—44no—
304C4⋊C8—44no—
305#103—44no—
306#103—44no—
307#103—44no—
308#103—44no—
309#103—44no—
310#102—44no—
311#102—44no—
312#102—44no—
313C22⋊C8—24no—
314C22⋊C8—24no—
315C22⋊C8—44no—
316C22⋊C8—44no—
317#97—42no—
318C4×C8—42no—
319#95—24no—
320#95—24no—
321#95—22no—
322#95—22no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95C2×D414yes—
328#95C2×D414yes—
329C24—22no—
330C24C22×D412yes—
331C24C22×D411yescharacteristic
332#41—88no—
333#41—88no—
334#41—88no—
335C2×Q8—24no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—24no—
339C2×Q8—22no—
340C2×Q8—24no—
341C2×Q8—22no—
342C2×Q8—24no—
343C2×Q8—24no—
344C2×Q8—41no—
345C2×Q8—41no—
346C2×Q8—416no—
347C2×Q8—42no—
348C2×Q8—42no—
349C2×Q8H(5,𝔽2)14yes—
350C2×D4—24no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—24no—
354C2×D4—22no—
355C2×D4—24no—
356C2×D4—42no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—42no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—82no—
365C2×D4—82no—
366C2×D4—82no—
367C2×D4—82no—
368C2×D4—84no—
369C2×D4—84no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—42no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4—84no—
395C22×C4—82no—
396C22×C4C22×D412yes—
397C22×C4C22×D412yes—
398C22×C4C22×D414yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D412yes—
401C22×C4H(5,𝔽2)14yes—
402C22×C4C2511yescharacteristic
403Q16—88no—
404SD16—88no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—42no—
419C4⋊C4—48no—
420C4⋊C4—48no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—42no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—82no—
428C4⋊C4—84no—
429C4⋊C4—82no—
430C4⋊C4—84no—
431C4⋊C4—82no—
432C22⋊C4—28no—
433C22⋊C4—44no—
434C22⋊C4—42no—
435C22⋊C4—48no—
436C22⋊C4—44no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—48no—
443C22⋊C4—84no—
444C22⋊C4—84no—
445C22⋊C4—84no—
446C42—24no—
447C42—22no—
448C42—22no—
449C42—42no—
450C42—48no—
451C42—42no—
452C42—42no—
453C23—22no—
454C23—21no—
455C23—21no—
456C23—22no—
457C23—24no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—24no—
462C23—44no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23#53512yes—
468C23C23×D411yescharacteristic
469Q8—28no—
470Q8—48no—
471Q8—44no—
472Q8—44no—
473Q8—44no—
474Q8—82no—
475D4—44no—
476D4—44no—
477D4—82no—
478D4—84no—
479D4—88no—
480D4—164no—
481D4—164no—
482D4—164no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—28no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—48no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
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—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—84no—
519C2×C4—88no—
520C2×C4—162no—
521C2×C4—164no—
522C2×C4#53512yes—
523C2×C4#57714yes—
524C2×C4C2×H(5,𝔽2)12yes—
525C2×C4C2×H(5,𝔽2)12yes—
526C2×C4#58512yes—
527C8—44no—
528C8—44no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—42no—
533C22—42no—
534C22—44no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—44no—
539C22—84no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22—84no—
544C22C2×#53511yescharacteristic
545C22#348411yescharacteristic
546C22#348511yescharacteristic
547C22C2×#57712yes—
548C22#358612yes—
549C4—24no—
550C4—22no—
551C4—42no—
552C4—42no—
553C4—44no—
554C4—44no—
555C4—82no—
556C4—82no—
557C4—82no—
558C4—82no—
559C4—82no—
560C4—82no—
561C4#353414yes—
562C2—42no—
563C2—82no—
564C2—162no—
565C2#3672111yescharacteristic
566C2#6081811yescharacteristic
567C2#6082811yescharacteristic
568C2#6168612yes—
569C2#6175512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#37296C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35673C2semidirect
3. #32918C2semidirect
4. #31956C2semidirect
5. #31932C2semidirect
6. #31005C2semidirect
7. #30991C2semidirect
8. #29768C2semidirect
9. #28313C2semidirect
10. #28305C2semidirect
11. #27981C2semidirect
12. #27975C2semidirect
13. #24012C2semidirect
14. #23927C2semidirect
15. #21905C2semidirect

Maximal subgroups

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

subgroupindex
#372962
#356732
#329182
#319562
#319322
#310052
#309912
#297682
#283132
#283052
#279812
#279752
#240122
#239272
#219052