Finite Group Databasebeta

← #949922

Subgroups — #949922

Order 512.

1964 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17305C212yesmaximal, complemented
2#17209C212yesmaximal, complemented
3#16899C214yesmaximal
4#16331C214yesmaximal
5#16240C212yesmaximal, complemented
6#15363C212yesmaximal, complemented
7#15146C212yesmaximal, complemented
8#15135C211yesmaximal, characteristic
9#15135C211yesmaximal, characteristic
10#15133C212yesmaximal, complemented
11#14264C212yesmaximal
12#14151C212yesmaximal, complemented
13#14005C212yesmaximal, complemented
14#13876C212yesmaximal
15#13844C211yesmaximal, characteristic
16C22×#117C2211yescharacteristic
17C2×#400C2214yes—
18C2×#399C2212yes—
19C2×#399C2212yes—
20C2×#398C2214yes—
21C2×#389C2212yes—
22C2×#389C2214yes—
23C2×#389C2214yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#388C2214yes—
28C2×#384C2212yes—
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C22⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2211yescharacteristic
37#1445C2212yes—
38#1445C2212yes—
39#1444C2211yescharacteristic
40#1444C2212yes—
41#1444C2212yes—
42#1444C2214yes—
43#1444C2211yescharacteristic
44#1444C2214yes—
45#1442C2211yescharacteristic
46#1442C2211yescharacteristic
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1439C2212yes—
59#1439C2212yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2212yes—
64#1439C2212yes—
65#1436C2214yes—
66#1436C2214yes—
67C4⋊#95C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C4×#95C2212yes—
75C24×C4C2311yescharacteristic
76C2×#117—44no—
77C2×#117—42no—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C22⋊C4C2311yescharacteristic
97C22×C22⋊C4C2311yescharacteristic
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C42C2312yes—
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2312yes—
106C22×C42C2314yes—
107#400—48no—
108#399—44no—
109#399—44no—
110#398—48no—
111#389—28no—
112#389—48no—
113#389—48no—
114#389—44no—
115#389—44no—
116#389—44no—
117#388—48no—
118#384—28no—
119#384—42no—
120#384—42no—
121#380—28no—
122#380—44no—
123C4×C4⋊C4—44no—
124C4×C4⋊C4—44no—
125C4×C22⋊C4—42no—
126C4×C22⋊C4Q818yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2312yes—
130C2×#95C2312yes—
131C2×#95C2312yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2311yescharacteristic
153C2×#95C2311yescharacteristic
154C2×#95C2311yescharacteristic
155C2×#95C2311yescharacteristic
156C25C2411yescharacteristic
157C23×C4—24no—
158C23×C4—22no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4C2×Q814yes—
162C23×C4C2414yes—
163C23×C4C2412yes—
164C23×C4C2414yes—
165C23×C4C2414yes—
166C23×C4C2412yes—
167C23×C4C2412yes—
168C23×C4C2411yescharacteristic
169C23×C4C2414yes—
170C23×C4C2411yescharacteristic
171C23×C4C2411yescharacteristic
172C23×C4C2411yescharacteristic
173C23×C4C2411yescharacteristic
174C23×C4C2412yes—
175C23×C4C2411yescharacteristic
176#117—416no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4#4118yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4C2×Q818yes—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—28no—
255C2×C42—44no—
256C2×C42—44no—
257C2×C42C2×Q818yes—
258#95—28no—
259#95—28no—
260#95—28no—
261#95—24no—
262#95—24no—
263#95—28no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—41no—
269#95—41no—
270#95—42no—
271#95—44no—
272#95—42no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—42no—
281#95—42no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95C2×Q814yes—
286#95C2×Q814yes—
287C24—22no—
288C24—22no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—22no—
293C24—22no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24C22×Q812yes—
298C24C2511yescharacteristic
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—24no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—28no—
336C22×C4—216no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—48no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4C22×Q812yes—
376C22×C4C22×Q814yes—
377C22×C4C22×Q814yes—
378C22×C4C22×Q812yes—
379C22×C4C2×#4118yes—
380C22×C4C2×#4114yes—
381C22×C4H(5,𝔽2)12yes—
382C22×C4H(5,𝔽2)12yes—
383C22×C4H(5,𝔽2)14yes—
384C22×C4H(5,𝔽2)14yes—
385C22×C4H(5,𝔽2)12yes—
386C22×C42−1+412yes—
387C22×C42−1+412yes—
388C22×C42−1+414yes—
389C22×C42−1+412yes—
390C4⋊C4—216no—
391C4⋊C4—416no—
392C4⋊C4—48no—
393C4⋊C4—48no—
394C4⋊C4—416no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—48no—
398C4⋊C4—48no—
399C4⋊C4—48no—
400C4⋊C4—88no—
401C4⋊C4—84no—
402C4⋊C4—84no—
403C4⋊C4—88no—
404C4⋊C4—88no—
405C4⋊C4—84no—
406C4⋊C4—84no—
407C4⋊C4—88no—
408C22⋊C4—216no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—48no—
412C22⋊C4—88no—
413C22⋊C4—88no—
414C42—216no—
415C42—44no—
416C42—44no—
417C42—48no—
418C42—88no—
419C23—24no—
420C23—22no—
421C23—24no—
422C23—22no—
423C23—22no—
424C23—42no—
425C23—41no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—42no—
441C23—41no—
442C23C23×Q811yescharacteristic
443C23C22×#4112yes—
444C23C2×H(5,𝔽2)12yes—
445C23C2×H(5,𝔽2)11yescharacteristic
446C23C2×H(5,𝔽2)12yes—
447C23C2×H(5,𝔽2)11yescharacteristic
448C23C2×H(5,𝔽2)11yescharacteristic
449C23C2×2−1+411yescharacteristic
450C23C2×2−1+412yes—
451C23C2×2−1+411yescharacteristic
452C23C2×2−1+411yescharacteristic
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—28no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—28no—
466C2×C4—24no—
467C2×C4—28no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—44no—
498C2×C4—416no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4#53918yes—
535C2×C4#54418yes—
536C2×C4#54414yes—
537C2×C4#55418yes—
538C2×C4#55914yes—
539C2×C4#56014yes—
540C22—42no—
541C22—42no—
542C22—44no—
543C22—42no—
544C22—44no—
545C22—81no—
546C22—81no—
547C22—82no—
548C22—82no—
549C22—81no—
550C22—81no—
551C22C2×#53912yes—
552C22C2×#54411yescharacteristic
553C22C2×#54412yes—
554C22C2×#55412yes—
555C22C2×#55911yescharacteristic
556C22#350311yescharacteristic
557C22#350311yescharacteristic
558C22#351912yes—
559C22#352012yes—
560C22#352212yes—
561C22#352312yes—
562C22C2×#56011yescharacteristic
563C22#353611yescharacteristic
564C22#353611yescharacteristic
565C22#353712yes—
566C22#353812yes—
567C22#354812yes—
568C22#354812yes—
569C22#354811yescharacteristic
570C22#354911yescharacteristic
571C22#355011yescharacteristic
572C22#355012yes—
573C22#355611yescharacteristic
574C4—28no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—44no—
579C4—48no—
580C4—42no—
581C4—42no—
582C4—82no—
583C4—84no—
584C4—82no—
585C4—84no—
586C4—84no—
587C4#280718yes—
588C2—82no—
589C2#3730911yescharacteristic
590C2#3764211yescharacteristic
591C2#3809812yes—
592C2#3896512yes—
593C2#4255612yes—
594C2#4284012yes—
595C2#4504711yescharacteristic
596C2#4534511yescharacteristic
597C2C2×#280711yescharacteristic
598C2#5295711yescharacteristic
599C2#5300711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17305C2semidirect
2. #17209C2semidirect
5. #16240C2semidirect
6. #15363C2semidirect
7. #15146C2semidirect
10. #15133C2semidirect
12. #14151C2semidirect
13. #14005C2semidirect

Maximal subgroups

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

subgroupindex
#173052
#172092
#168992
#163312
#162402
#153632
#151462
#151352
#151332
#142642
#141512
#140052
#138762
#138442