Finite Group Databasebeta

← #6635569

Subgroups — #6635569

Order 512.

1638 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37386C211yesmaximal, characteristic
2#35673C211yesmaximal, characteristic
3#32915C212yesmaximal, complemented
4#31955C214yesmaximal, complemented
5#31935C214yesmaximal, complemented
6#31006C214yesmaximal, complemented
7#30990C214yesmaximal, complemented
8#29764C212yesmaximal, complemented
9#28313C211yesmaximal, characteristic, complemented
10#28304C211yesmaximal, characteristic, complemented
11#27980C211yesmaximal, characteristic, complemented
12#27975C211yesmaximal, characteristic, complemented
13#24014C212yesmaximal
14#23928C212yesmaximal
15#21905C211yesmaximal, characteristic
16C2×#536C2212yes—
17C2×#536C2211yescharacteristic
18C2×#519C2211yescharacteristic
19C2×#484C2212yes—
20C2×#483C2212yes—
21C2×#462C2214yes—
22C2×#451C2214yes—
23C2×#451C2212yes—
24C2×#451C2212yes—
25#2857C2211yescharacteristic
26#2857C2211yescharacteristic
27#2734—22no—
28#2724—22no—
29#2717—24no—
30#2621C2212yes—
31#2499C2212yes—
32#2499C2212yes—
33#2498—24no—
34#2448—22no—
35#2446—22no—
36#2434C2212yes—
37#2434C2212yes—
38#2423C2214yes—
39#2421C2214yes—
40#2316C2211yescharacteristic
41#2311C2211yescharacteristic
42#2297C2211yescharacteristic
43C2×#391C2211yescharacteristic
44#2285C2211yescharacteristic
45#2096C2212yes—
46#2095C2212yes—
47#2085C2214yes—
48#2074C2214yes—
49#2072C2214yes—
50#2068C2214yes—
51#2066C2214yes—
52#2043C2214yes—
53#2036C2214yes—
54#2035C2214yes—
55#2016C2212yes—
56#2016C2212yes—
57#2001C2214yes—
58#2001C2214yes—
59#1971C2214yes—
60#1969C2214yes—
61#1960C2212yes—
62#1956C2212yes—
63#1924C2214yes—
64#1923C2214yes—
65#1878C2214yes—
66#1878C2214yes—
67#1873C2212yes—
68#1873C2212yes—
69#1873C2212yes—
70#1873C2212yes—
71#1873C2214yes—
72#1854C2214yes—
73#1836C2212yes—
74#1834C2212yes—
75#1805C2214yes—
76#1798C2212yes—
77#1787C2214yes—
78#1780C2212yes—
79#1749C2211yescharacteristic
80C4×C22⋊C8C2211yescharacteristic
81C22×#41C2311yescharacteristic
82C23×Q8C2311yescharacteristic
83C23×Q8C2312yes—
84C22×Q16C2314yes—
85C22×Q16C2314yes—
86#536—44no—
87#536—44no—
88#536—44no—
89C2×C4⋊Q8—21no—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2312yes—
94C2×#124—21no—
95C2×#124—22no—
96C2×#123—22no—
97C2×#123—22no—
98C2×#123—22no—
99C2×C22⋊Q8C2312yes—
100C2×C22⋊Q8C2312yes—
101C2×C22⋊Q8C2314yes—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2312yes—
104#519—44no—
105C2×C4×Q8—22no—
106C2×C4×Q8C2314yes—
107C2×#117—22no—
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C2×#117C2312yes—
111C2×#117C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C42C2311yescharacteristic
114#484—44no—
115#483—44no—
116#462—48no—
117#451—44no—
118#451—44no—
119#451—48no—
120C2×#107C2314yes—
121C2×#106C2314yes—
122C2×C4⋊C8C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×C22⋊C8C2312yes—
132C2×C22⋊C8C2312yes—
133C2×C22⋊C8C2314yes—
134C2×C22⋊C8C2314yes—
135C2×#97C2312yes—
136C2×C4×C8C2312yes—
137#400—24no—
138#399—24no—
139#398—24no—
140#398—24no—
141#397—24no—
142#397—24no—
143#397—24no—
144#397—24no—
145#395—24no—
146#395—24no—
147#391—21no—
148#391—22no—
149#391D412yes—
150#391C2312yes—
151#391C2312yes—
152#391C2312yes—
153#391C2312yes—
154#391C2312yes—
155#391C2312yes—
156#390—21no—
157#390D412yes—
158#389—22no—
159#389—22no—
160#389—22no—
161#389C2314yes—
162#388—24no—
163#388—24no—
164#387—24no—
165#387—24no—
166#387—24no—
167#387—24no—
168#386—22no—
169#386D414yes—
170#385—24no—
171#385—24no—
172#384C2312yes—
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#382C2312yes—
176#382C2312yes—
177#380—22no—
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—
196C22×Q8—24no—
197C22×Q8—21no—
198C22×Q8—22no—
199C22×Q8—24no—
200C22×Q8—42no—
201C22×Q8—42no—
202C22×Q8—42no—
203C22×Q8—44no—
204C22×Q8—44no—
205C22×Q8C2×D411yescharacteristic
206C22×Q8C2412yes—
207C22×Q8C2412yes—
208C22×Q8C2412yes—
209C22×D4C2×D412yes—
210C22×D4C2×D411yescharacteristic
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—22no—
214C23×C4C2×D412yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C2×Q16—28no—
218C2×Q16—28no—
219C2×Q16—44no—
220C2×Q16—44no—
221C2×Q16—44no—
222C2×Q16—44no—
223C22×C8C2414yes—
224C22×C8C2414yes—
225C4⋊Q8—42no—
226C4⋊Q8—42no—
227C4⋊Q8—44no—
228C4⋊Q8—44no—
229C4⋊Q8—48no—
230#124—48no—
231#124—44no—
232#123—48no—
233#123—48no—
234#123—48no—
235C22⋊Q8—28no—
236C22⋊Q8—24no—
237C22⋊Q8—24no—
238C22⋊Q8—24no—
239C22⋊Q8—24no—
240C22⋊Q8—44no—
241C22⋊Q8—42no—
242C22⋊Q8—42no—
243C22⋊Q8—42no—
244C22⋊Q8—42no—
245C4×Q8—28no—
246C4×Q8—44no—
247C4×Q8—48no—
248#117—42no—
249#117—42no—
250#117—42no—
251#117—42no—
252#117—42no—
253#117—48no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—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—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4C2×D414yes—
286C2×C22⋊C4C2×D412yes—
287C2×C22⋊C4C2×D414yes—
288C2×C42—24no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—22no—
292C2×C42—42no—
293C2×C42C2×D412yes—
294C2×C42C2412yes—
295C2×C42C2411yescharacteristic
296C2×C42C2411yescharacteristic
297#107—44no—
298#106—44no—
299C4⋊C8—44no—
300#103—44no—
301#103—44no—
302#103—44no—
303#103—44no—
304#103—44no—
305#103—44no—
306#103—44no—
307#103—44no—
308C22⋊C8—24no—
309C22⋊C8—24no—
310C22⋊C8—44no—
311C22⋊C8—44no—
312#97—42no—
313C4×C8—42no—
314#95—24no—
315#95—24no—
316#95—22no—
317#95—22no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95C2×D414yes—
326#95C2×D414yes—
327C24C22×D412yes—
328C24C22×D411yescharacteristic
329#41—88no—
330C2×Q8—24no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—24no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—24no—
337C2×Q8—24no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—24no—
341C2×Q8—24no—
342C2×Q8—24no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8—416no—
346C2×Q8—42no—
347C2×Q8—44no—
348C2×Q8—42no—
349C2×Q8—82no—
350C2×Q8—82no—
351C2×Q8—88no—
352C2×Q8—88no—
353C2×Q8H(5,𝔽2)14yes—
354C2×D4—24no—
355C2×D4—22no—
356C2×D4—42no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—42no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—44no—
381C22×C4—42no—
382C22×C4—82no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4C22×D412yes—
386C22×C4C22×D414yes—
387C22×C4C22×D412yes—
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D412yes—
390C22×C4H(5,𝔽2)14yes—
391C22×C4C2511yescharacteristic
392Q16—88no—
393Q16—88no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—42no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—42no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—82no—
419C4⋊C4—84no—
420C4⋊C4—82no—
421C4⋊C4—84no—
422C4⋊C4—82no—
423C22⋊C4—28no—
424C22⋊C4—44no—
425C22⋊C4—42no—
426C22⋊C4—48no—
427C22⋊C4—42no—
428C22⋊C4—44no—
429C22⋊C4—42no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—84no—
435C42—24no—
436C42—22no—
437C42—22no—
438C42—42no—
439C42—48no—
440C42—42no—
441C42—42no—
442C42—44no—
443C23—22no—
444C23—21no—
445C23—21no—
446C23—22no—
447C23—24no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—24no—
452C23#53512yes—
453C23C23×D411yescharacteristic
454Q8—28no—
455Q8—48no—
456Q8—44no—
457Q8—44no—
458Q8—44no—
459Q8—44no—
460Q8—44no—
461Q8—82no—
462Q8—88no—
463D4—82no—
464D4—84no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—28no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—48no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—84no—
504C2×C4—88no—
505C2×C4—162no—
506C2×C4#53512yes—
507C2×C4#57814yes—
508C2×C4C2×H(5,𝔽2)12yes—
509C2×C4C2×H(5,𝔽2)12yes—
510C2×C4#58512yes—
511C8—44no—
512C8—44no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—42no—
517C22—42no—
518C22—44no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—44no—
523C22C2×#53511yescharacteristic
524C22#348411yescharacteristic
525C22#348511yescharacteristic
526C22C2×#57812yes—
527C22#358612yes—
528C4—24no—
529C4—22no—
530C4—42no—
531C4—42no—
532C4—44no—
533C4—44no—
534C4—82no—
535C4—82no—
536C4—82no—
537C4—82no—
538C4—82no—
539C4—82no—
540C4—82no—
541C4#353414yes—
542C2—42no—
543C2—82no—
544C2#3672111yescharacteristic
545C2#6082111yescharacteristic
546C2#6082811yescharacteristic
547C2#6168312yes—
548C2#6175712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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
3. #32915C2semidirect
4. #31955C2semidirect
5. #31935C2semidirect
6. #31006C2semidirect
7. #30990C2semidirect
8. #29764C2semidirect
9. #28313C2semidirect
10. #28304C2semidirect
11. #27980C2semidirect
12. #27975C2semidirect

Maximal subgroups

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

subgroupindex
#373862
#356732
#329152
#319552
#319352
#310062
#309902
#297642
#283132
#283042
#279802
#279752
#240142
#239282
#219052