Finite Group Databasebeta

← #6798589

Subgroups — #6798589

Order 512.

1656 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36188C211yesmaximal, characteristic, complemented
2#36136C211yesmaximal, characteristic
3#33230C211yesmaximal, characteristic, complemented
4#31841C214yesmaximal, complemented
5#31839C214yesmaximal, complemented
6#29815C212yesmaximal, complemented
7#29226C211yesmaximal, characteristic, complemented
8#28447C211yesmaximal, characteristic, complemented
9#27979C212yesmaximal, complemented
10#25604C212yesmaximal, complemented
11#25580C214yesmaximal, complemented
12#25337C214yesmaximal, complemented
13#21868C212yesmaximal, complemented
14#21657C211yesmaximal, characteristic, complemented
15#21271C211yesmaximal, characteristic, complemented
16C2×#556C2212yes—
17C2×#529C2212yes—
18C2×#497C2212yes—
19C2×#491C2211yescharacteristic
20C2×#489C2214yes—
21C2×#464C2214yes—
22C2×#432C2212yes—
23#2868C2211yescharacteristic
24#2858C2211yescharacteristic
25#2857C2212yes—
26#2850C2212yes—
27#2568C2212yes—
28#2554—22no—
29#2547—22no—
30#2540C2212yes—
31#2499C2214yes—
32#2496—22no—
33#2496—22no—
34#2479—22no—
35#2456—22no—
36#2429C2212yes—
37#2428C2212yes—
38#2367C2212yes—
39#2365—22no—
40#2363—22no—
41C4×C4⋊Q8C2211yescharacteristic
42C4×#125C2211yescharacteristic
43#2285C2212yes—
44C42×D4C2211yescharacteristic
45#2099C2214yes—
46#2074C2214yes—
47#2073C2214yes—
48#2056C2214yes—
49#2055C2214yes—
50#2016C2212yes—
51#2014C2212yes—
52#2001C2214yes—
53#2001C2214yes—
54#2000C2214yes—
55#2000C2214yes—
56#1992C2212yes—
57#1992C2211yescharacteristic
58#1987C2214yes—
59#1960C2211yescharacteristic
60#1959C2212yes—
61#1959C2212yes—
62#1959C2212yes—
63#1957C2212yes—
64#1956C2212yes—
65#1955C2212yes—
66#1940C2212yes—
67#1933C2214yes—
68#1923C2214yes—
69#1878C2214yes—
70#1877C2214yes—
71#1875C2214yes—
72#1875C2214yes—
73#1873C2214yes—
74#1873C2214yes—
75#1836C2212yes—
76#1836C2211yescharacteristic
77#1805C2214yes—
78#1804C2214yes—
79#1767C2211yescharacteristic
80#1763C2214yes—
81#1749C2212yes—
82C4×#97C2211yescharacteristic
83C22×Q16C2314yes—
84C22×SD16C2314yes—
85#556—48no—
86C2×C4⋊Q8—22no—
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2312yes—
90C2×#127C2311yescharacteristic
91#529—28no—
92#529—44no—
93C2×#126—24no—
94C2×#126—22no—
95C2×#125C2311yescharacteristic
96C2×#124—22no—
97C2×#124C2312yes—
98C2×#124C2312yes—
99C2×#123—22no—
100C2×C22⋊Q8C2312yes—
101C2×C22⋊Q8C2312yes—
102C2×C22⋊Q8C2314yes—
103C2×#121—22no—
104C2×#121C2312yes—
105C2×#121C2312yes—
106C2×C4×Q8—22no—
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2312yes—
109C2×C4×D4—22no—
110C2×C4×D4D412yes—
111C2×C4×D4C2312yes—
112C2×C4×D4C2311yescharacteristic
113C2×#117—22no—
114C2×#117C2314yes—
115C22×C42C2312yes—
116#497—44no—
117#491—24no—
118#489—48no—
119#464—48no—
120#432—44no—
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×C22⋊C8C2314yes—
132C2×C22⋊C8C2314yes—
133C2×#97C2311yescharacteristic
134C2×#97C2312yes—
135C2×#97C2312yes—
136C2×#97C2311yescharacteristic
137C2×C4×C8C2312yes—
138#400—22no—
139#400—22no—
140#399—22no—
141#398—22no—
142#397—24no—
143#397—24no—
144#396—22no—
145#394—22no—
146#392—22no—
147#392—22no—
148#391—24no—
149#391C2312yes—
150#391C2314yes—
151#391C2312yes—
152#390—22no—
153#390—22no—
154#390—22no—
155#390—22no—
156#390C2312yes—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389C2312yes—
162#388—22no—
163#388—22no—
164#388—22no—
165#388—22no—
166#388—22no—
167#387—22no—
168#387—24no—
169#387—24no—
170#386—22no—
171#386—22no—
172#386—22no—
173#385—22no—
174#384—22no—
175#384C2311yescharacteristic
176#384C2312yes—
177#384C2312yes—
178#382C2311yescharacteristic
179#382C2314yes—
180#380—22no—
181#380—22no—
182#380—22no—
183#380—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4D412yes—
189C4×C4⋊C4C2312yes—
190C4×C4⋊C4C2311yescharacteristic
191C4×C22⋊C4—22no—
192C4×C22⋊C4D412yes—
193C4×C22⋊C4D412yes—
194C4×C22⋊C4C2312yes—
195C43C2311yescharacteristic
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#336C2314yes—
203#336C2312yes—
204C22×Q8—22no—
205C22×Q8C2412yes—
206C22×Q8C2412yes—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4C2×D411yescharacteristic
211C22×D4C2412yes—
212C23×C4—22no—
213C23×C4C2×D412yes—
214C23×C4C2×D412yes—
215C23×C4C2412yes—
216C2×Q16—44no—
217C2×Q16—44no—
218C2×Q16—44no—
219C2×SD16—44no—
220C2×SD16—44no—
221C2×SD16—44no—
222C22×C8C2414yes—
223C22×C8C2412yes—
224C22×C8C2412yes—
225C4⋊Q8—22no—
226C4⋊Q8—28no—
227C4⋊Q8—22no—
228C4⋊Q8—48no—
229C4⋊Q8—42no—
230#127—42no—
231#126—48no—
232#126—416no—
233#125—22no—
234#125#4114yes—
235#124—24no—
236#124—24no—
237#124—28no—
238#124—48no—
239#123—48no—
240C22⋊Q8—24no—
241C22⋊Q8—42no—
242C22⋊Q8—44no—
243C22⋊Q8—44no—
244C22⋊Q8—44no—
245#121—24no—
246#121—48no—
247#121—42no—
248#121—44no—
249C4×Q8—24no—
250C4×Q8—28no—
251C4×Q8—42no—
252C4×Q8—48no—
253C4×D4—24no—
254C4×D4—24no—
255C4×D4—44no—
256C4×D4—42no—
257C4×D4—48no—
258#117—48no—
259#117—44no—
260#117—44no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
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—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4C2×D411yescharacteristic
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2414yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
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—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4C2×D412yes—
295C2×C22⋊C4C2×D412yes—
296C2×C22⋊C4C2×D412yes—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2412yes—
308C2×C42C2411yescharacteristic
309C2×C42C2411yescharacteristic
310C2×C42C2411yescharacteristic
311C4⋊C8—24no—
312C4⋊C8—44no—
313#103—28no—
314#103—44no—
315#103—44no—
316#103—44no—
317#103—44no—
318#102—44no—
319#102—44no—
320#102—44no—
321C22⋊C8—44no—
322C22⋊C8—44no—
323#97—22no—
324#97—22no—
325#97—24no—
326#97—24no—
327C4×C8—42no—
328#95—22no—
329#95—24no—
330#95—24no—
331#95—22no—
332#95—22no—
333#95—24no—
334#95—22no—
335#95—24no—
336#95—24no—
337#95—22no—
338#95C2×D411yescharacteristic
339#95C2×D411yescharacteristic
340#95C2×D412yes—
341#95C2×D412yes—
342C24—22no—
343C24C22×D412yes—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—48no—
353C2×Q8—44no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—24no—
357C2×D4—22no—
358C2×D4—24no—
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—82no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—82no—
388C22×C4C22×D412yes—
389C22×C4C22×D412yes—
390C22×C4C22×D412yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D412yes—
394C22×C4C22×D412yes—
395C22×C4C2511yescharacteristic
396Q16—88no—
397SD16—88no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—24no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—24no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—24no—
407C4⋊C4—22no—
408C4⋊C4—28no—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—42no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—48no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—48no—
426C4⋊C4—82no—
427C4⋊C4—82no—
428C4⋊C4C2×#4114yes—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—48no—
437C22⋊C4—48no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—82no—
443C22⋊C4—82no—
444C22⋊C4—82no—
445C42—24no—
446C42—24no—
447C42—24no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—42no—
452C42—44no—
453C42—48no—
454C42—44no—
455C42—82no—
456C42C2×#4112yes—
457C42H(5,𝔽2)14yes—
458C42H(5,𝔽2)12yes—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23C23×D411yescharacteristic
474Q8—44no—
475Q8—44no—
476Q8—44no—
477Q8—44no—
478Q8—88no—
479D4—44no—
480D4—44no—
481D4—44no—
482D4—88no—
483D4—164no—
484D4—164no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—84no—
520C2×C4—84no—
521C2×C4—88no—
522C2×C4#53512yes—
523C2×C4#53512yes—
524C2×C4#54614yes—
525C2×C4#57514yes—
526C2×C4C22×#4111yescharacteristic
527C2×C4C2×H(5,𝔽2)12yes—
528C2×C4C2×H(5,𝔽2)11yescharacteristic
529C2×C4#58512yes—
530C8—42no—
531C8—42no—
532C8—44no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22C2×#53511yescharacteristic
548C22C2×#54611yescharacteristic
549C22#350011yescharacteristic
550C22C2×#57512yes—
551C22#358612yes—
552C4—24no—
553C4—22no—
554C4—22no—
555C4—24no—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—44no—
560C4—82no—
561C4—82no—
562C4—82no—
563C4—84no—
564C4—82no—
565C4#351712yes—
566C4#354612yes—
567C2—82no—
568C2—162no—
569C2#3895611yescharacteristic
570C2#6080111yescharacteristic
571C2#6083111yescharacteristic
572C2#6116512yes—
573C2#6258512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36136C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36188C2semidirect
3. #33230C2semidirect
4. #31841C2semidirect
5. #31839C2semidirect
6. #29815C2semidirect
7. #29226C2semidirect
8. #28447C2semidirect
9. #27979C2semidirect
10. #25604C2semidirect
11. #25580C2semidirect
12. #25337C2semidirect
13. #21868C2semidirect
14. #21657C2semidirect
15. #21271C2semidirect

Maximal subgroups

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

subgroupindex
#361882
#361362
#332302
#318412
#318392
#298152
#292262
#284472
#279792
#256042
#255802
#253372
#218682
#216572
#212712