Finite Group Databasebeta

← #949917

Subgroups — #949917

Order 512.

1844 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#17243C212yesmaximal, complemented
2#17221C212yesmaximal, complemented
3#16953C214yesmaximal
4#16531C214yesmaximal
5#16236C212yesmaximal, complemented
6#16002C212yesmaximal, complemented
7#15232C212yesmaximal, complemented
8#15135C211yesmaximal, characteristic
9#15135C211yesmaximal, characteristic
10#15133C212yesmaximal, complemented
11#14264C212yesmaximal
12#14150C212yesmaximal, complemented
13#14005C212yesmaximal, complemented
14#13875C212yesmaximal
15#13844C211yesmaximal, characteristic
16C22×#117C2211yescharacteristic
17C2×#399C2212yes—
18C2×#399C2212yes—
19C2×#398C2214yes—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#389C2214yes—
25C2×#384C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#382C2212yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×C4×C4⋊C4C2212yes—
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C22⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34#1445C2212yes—
35#1445C2212yes—
36#1445C2214yes—
37#1444C2211yescharacteristic
38#1444C2211yescharacteristic
39#1444C2212yes—
40#1444C2212yes—
41#1444C2214yes—
42#1444C2214yes—
43#1444C2214yes—
44#1443C2214yes—
45#1443C2214yes—
46#1443C2214yes—
47#1442C2211yescharacteristic
48#1442C2211yescharacteristic
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1440C2214yes—
57#1439C2212yes—
58#1439C2212yes—
59#1439C2212yes—
60#1439C2212yes—
61#1439C2214yes—
62#1439C2214yes—
63#1437C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2212yes—
73#1434C2214yes—
74C4×#95C2212yes—
75C24×C4C2311yescharacteristic
76C2×#117—44no—
77C2×#117—42no—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C22⋊C4C2311yescharacteristic
95C22×C22⋊C4C2311yescharacteristic
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C42C2312yes—
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103C22×C42C2312yes—
104C22×C42C2314yes—
105#399—44no—
106#399—44no—
107#398—48no—
108#389—28no—
109#389—48no—
110#389—44no—
111#389—44no—
112#389—44no—
113#384—42no—
114#384—42no—
115#382—28no—
116#380—28no—
117#380—44no—
118C4×C4⋊C4—44no—
119C4×C4⋊C4—44no—
120C4×C22⋊C4—42no—
121C4×C22⋊C4Q818yes—
122C2×#95C2312yes—
123C2×#95C2312yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2312yes—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2311yescharacteristic
150C2×#95C2311yescharacteristic
151C2×#95C2311yescharacteristic
152C2×#95C2311yescharacteristic
153C25C2411yescharacteristic
154C23×C4—24no—
155C23×C4—22no—
156C23×C4—24no—
157C23×C4—22no—
158C23×C4C2×Q814yes—
159C23×C4C2412yes—
160C23×C4C2414yes—
161C23×C4C2414yes—
162C23×C4C2414yes—
163C23×C4C2412yes—
164C23×C4C2412yes—
165C23×C4C2412yes—
166C23×C4C2411yescharacteristic
167C23×C4C2414yes—
168C23×C4C2411yescharacteristic
169C23×C4C2411yescharacteristic
170C23×C4C2411yescharacteristic
171C23×C4C2411yescharacteristic
172C23×C4C2411yescharacteristic
173#117—416no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—28no—
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—44no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4#4118yes—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4C2×Q818yes—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—24no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—24no—
245C2×C42—44no—
246C2×C42—44no—
247C2×C42—44no—
248C2×C42C2×Q818yes—
249#95—28no—
250#95—28no—
251#95—28no—
252#95—28no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—44no—
258#95—44no—
259#95—42no—
260#95—42no—
261#95—44no—
262#95—44no—
263#95—41no—
264#95—41no—
265#95—42no—
266#95—42no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—42no—
277#95—42no—
278#95C2×Q814yes—
279#95C2×Q814yes—
280C24—22no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—22no—
285C24—21no—
286C24—21no—
287C24—21no—
288C24—22no—
289C24—22no—
290C24C22×Q812yes—
291C24C2511yescharacteristic
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—28no—
331C22×C4—216no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—48no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4C22×Q812yes—
371C22×C4C22×Q814yes—
372C22×C4C22×Q814yes—
373C22×C4C22×Q812yes—
374C22×C4C2×#4118yes—
375C22×C4C2×#4114yes—
376C22×C4H(5,𝔽2)12yes—
377C22×C4H(5,𝔽2)12yes—
378C22×C4H(5,𝔽2)14yes—
379C22×C4H(5,𝔽2)12yes—
380C22×C42−1+412yes—
381C22×C42−1+412yes—
382C22×C42−1+412yes—
383C4⋊C4—216no—
384C4⋊C4—416no—
385C4⋊C4—48no—
386C4⋊C4—48no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C4⋊C4—84no—
393C4⋊C4—84no—
394C4⋊C4—88no—
395C4⋊C4—88no—
396C4⋊C4—84no—
397C4⋊C4—84no—
398C4⋊C4—88no—
399C22⋊C4—48no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—48no—
403C22⋊C4—88no—
404C22⋊C4—88no—
405C42—216no—
406C42—44no—
407C42—44no—
408C42—48no—
409C42—88no—
410C23—22no—
411C23—24no—
412C23—24no—
413C23—22no—
414C23—22no—
415C23—42no—
416C23—41no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—42no—
433C23C23×Q811yescharacteristic
434C23C22×#4112yes—
435C23C2×H(5,𝔽2)12yes—
436C23C2×H(5,𝔽2)11yescharacteristic
437C23C2×H(5,𝔽2)11yescharacteristic
438C23C2×H(5,𝔽2)11yescharacteristic
439C23C2×2−1+411yescharacteristic
440C23C2×2−1+411yescharacteristic
441C23C2×2−1+411yescharacteristic
442C23#58512yes—
443C23#58512yes—
444C2×C4—28no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—28no—
454C2×C4—24no—
455C2×C4—28no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—44no—
490C2×C4—416no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4#53918yes—
527C2×C4#54414yes—
528C2×C4#55418yes—
529C2×C4#55914yes—
530C2×C4#56014yes—
531C22—42no—
532C22—44no—
533C22—42no—
534C22—42no—
535C22—44no—
536C22—81no—
537C22—81no—
538C22—82no—
539C22—82no—
540C22—81no—
541C22—81no—
542C22C2×#53912yes—
543C22C2×#54411yescharacteristic
544C22#345812yes—
545C22C2×#55412yes—
546C22C2×#55911yescharacteristic
547C22#350311yescharacteristic
548C22#350311yescharacteristic
549C22#351312yes—
550C22#351512yes—
551C22#351612yes—
552C22#351712yes—
553C22C2×#56011yescharacteristic
554C22#353711yescharacteristic
555C22#353811yescharacteristic
556C22#354112yes—
557C22#354212yes—
558C22#354612yes—
559C22#354712yes—
560C22#355011yescharacteristic
561C22#355011yescharacteristic
562C22#355111yescharacteristic
563C22#355111yescharacteristic
564C22#355712yes—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—44no—
569C4—48no—
570C4—42no—
571C4—44no—
572C4—42no—
573C4—84no—
574C4—82no—
575C4—84no—
576C4—82no—
577C4—84no—
578C4#280718yes—
579C2—82no—
580C2#3730811yescharacteristic
581C2#3756511yescharacteristic
582C2#3810412yes—
583C2#3896012yes—
584C2#4257212yes—
585C2#4284612yes—
586C2#4522611yescharacteristic
587C2#4531511yescharacteristic
588C2C2×#280711yescharacteristic
589C2#5296611yescharacteristic
590C2#5300611yescharacteristic
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. #17243C2semidirect
2. #17221C2semidirect
5. #16236C2semidirect
6. #16002C2semidirect
7. #15232C2semidirect
10. #15133C2semidirect
12. #14150C2semidirect
13. #14005C2semidirect

Maximal subgroups

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

subgroupindex
#172432
#172212
#169532
#165312
#162362
#160022
#152322
#151352
#151332
#142642
#141502
#140052
#138752
#138442