Finite Group Databasebeta

← #950057

Subgroups — #950057

Order 512.

2258 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17266C214yesmaximal, complemented
2#16986C214yesmaximal, complemented
3#16708C214yesmaximal, complemented
4#15958C214yesmaximal
5#15235C211yesmaximal, characteristic, complemented
6#15220C212yesmaximal, complemented
7#15218C211yesmaximal, characteristic, complemented
8#15216C212yesmaximal, complemented
9#14266C212yesmaximal, complemented
10#14130C212yesmaximal, complemented
11#13907C212yesmaximal, complemented
12#13872C212yesmaximal
13#13847C211yesmaximal, characteristic
14C22×#125C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C23×C4⋊C4C2212yes—
17C2×#389C2212yes—
18C2×#389C2212yes—
19C2×#389C2212yes—
20C2×#386C2214yes—
21C2×#385C2214yes—
22C2×#382C2211yescharacteristic
23C2×#380C2212yes—
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×#380C2214yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2212yes—
29C2×C4×C22⋊C4C2212yes—
30C2×#376C2211yescharacteristic
31#1445C2212yes—
32#1445C2214yes—
33#1444C2214yes—
34#1444C2214yes—
35#1443C2214yes—
36#1443C2214yes—
37#1443C2214yes—
38#1442C2214yes—
39#1442C2214yes—
40#1442C2214yes—
41#1442C2214yes—
42#1442C2214yes—
43#1441C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1439C2212yes—
50#1439C2212yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1438C2212yes—
57C4⋊#95C2212yes—
58C4⋊#95C2212yes—
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2212yes—
61C4⋊#95C2211yescharacteristic
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C23×D4C2311yescharacteristic
68C24×C4C2312yes—
69C24×C4C2314yes—
70C2×#125—44no—
71C2×#125—42no—
72C2×C4×D4—42no—
73C2×C4×D4—42no—
74C2×C4×D4Q818yes—
75C22×C4⋊C4—28no—
76C22×C4⋊C4—44no—
77C22×C4⋊C4—44no—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2312yes—
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C42C2311yescharacteristic
98C22×C42C2312yes—
99C22×C42C2311yescharacteristic
100C22×C42C2311yescharacteristic
101#389—28no—
102#389—44no—
103#389—44no—
104#386—48no—
105#385—48no—
106#382—42no—
107#380—28no—
108#380—48no—
109#380—44no—
110#380—44no—
111C4×C4⋊C4—42no—
112C4×C22⋊C4—44no—
113C4×C22⋊C4—44no—
114#376—42no—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2311yescharacteristic
135C2×#95C2314yes—
136C2×#95C2311yescharacteristic
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C25C2412yes—
143C22×D4—24no—
144C22×D4—24no—
145C22×D4—41no—
146C22×D4—41no—
147C22×D4C2×Q814yes—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—28no—
155C23×C4—28no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4C2×Q818yes—
162C23×C4C2414yes—
163C23×C4C2412yes—
164C23×C4C2412yes—
165C23×C4C2412yes—
166C23×C4C2414yes—
167C23×C4C2412yes—
168C23×C4C2414yes—
169C23×C4C2411yescharacteristic
170C23×C4C2412yes—
171C23×C4C2411yescharacteristic
172C23×C4C2414yes—
173C23×C4C2411yescharacteristic
174#125—416no—
175C4×D4—416no—
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—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—432no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4C2×Q814yes—
209C2×C4⋊C4#4118yes—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4C2×Q818yes—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—22no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—24no—
242C2×C42—22no—
243C2×C42C2×Q814yes—
244#95—28no—
245#95—28no—
246#95—44no—
247#95—44no—
248#95—44no—
249#95—44no—
250#95—44no—
251#95—44no—
252#95—41no—
253#95—41no—
254#95—42no—
255#95—42no—
256#95—42no—
257#95—42no—
258#95—42no—
259#95—42no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24C22×Q814yes—
284C24H(5,𝔽2)14yes—
285C242−1+414yes—
286C24C2511yescharacteristic
287C2×D4—216no—
288C2×D4—44no—
289C2×D4—48no—
290C2×D4—44no—
291C2×D4—44no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—216no—
313C22×C4—216no—
314C22×C4—216no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—416no—
365C22×C4—416no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—88no—
371C22×C4—88no—
372C22×C4—88no—
373C22×C4—88no—
374C22×C4—88no—
375C22×C4C22×Q812yes—
376C22×C4C22×Q812yes—
377C22×C4C22×Q814yes—
378C22×C4C22×Q812yes—
379C22×C4C2×#4118yes—
380C22×C4C2×#4114yes—
381C22×C4H(5,𝔽2)12yes—
382C22×C4H(5,𝔽2)14yes—
383C22×C4H(5,𝔽2)12yes—
384C4⋊C4—216no—
385C4⋊C4—44no—
386C4⋊C4—416no—
387C4⋊C4—48no—
388C4⋊C4—416no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C4⋊C4—88no—
393C4⋊C4—88no—
394C4⋊C4—88no—
395C22⋊C4—48no—
396C22⋊C4—416no—
397C22⋊C4—48no—
398C22⋊C4—48no—
399C22⋊C4—48no—
400C22⋊C4—48no—
401C22⋊C4—88no—
402C22⋊C4—88no—
403C42—28no—
404C42—48no—
405C42—44no—
406C42—44no—
407C23—24no—
408C23—24no—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—24no—
415C23—24no—
416C23—24no—
417C23—24no—
418C23—24no—
419C23—24no—
420C23—24no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—24no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23#54418yes—
442C23C23×Q811yescharacteristic
443C23C22×#4112yes—
444C23C2×H(5,𝔽2)11yescharacteristic
445C23C2×H(5,𝔽2)12yes—
446C23C2×H(5,𝔽2)11yescharacteristic
447C23C2×H(5,𝔽2)12yes—
448C23C2×2−1+412yes—
449C23#58512yes—
450C23#58512yes—
451D4—416no—
452C2×C4—28no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—28no—
462C2×C4—28no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—432no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—832no—
527C2×C4#53818yes—
528C2×C4#54314yes—
529C2×C4#55418yes—
530C2×C4#55514yes—
531C2×C4#56014yes—
532C22—28no—
533C22—28no—
534C22—28no—
535C22—44no—
536C22—44no—
537C22—44no—
538C22—44no—
539C22—44no—
540C22—44no—
541C22—44no—
542C22—44no—
543C22—44no—
544C22—44no—
545C22—44no—
546C22—44no—
547C22C2×#53812yes—
548C22C2×#54311yescharacteristic
549C22C2×#54412yes—
550C22C2×#55412yes—
551C22C2×#55511yescharacteristic
552C22#350411yescharacteristic
553C22#350411yescharacteristic
554C22#351312yes—
555C22#351312yes—
556C22#351612yes—
557C22#351612yes—
558C22C2×#56011yescharacteristic
559C22#353612yes—
560C22#353711yescharacteristic
561C22#353811yescharacteristic
562C22#353912yes—
563C22#354112yes—
564C22#354112yes—
565C22#354212yes—
566C22#354212yes—
567C22#355112yes—
568C4—44no—
569C4—48no—
570C4—42no—
571C4—42no—
572C4—82no—
573C4—84no—
574C4—84no—
575C4—82no—
576C4—82no—
577C4—82no—
578C4—84no—
579C4#278418yes—
580C2—48no—
581C2#3742511yescharacteristic
582C2#3747711yescharacteristic
583C2#3810012yes—
584C2#3810112yes—
585C2#4253912yes—
586C2#4283112yes—
587C2#4538212yes—
588C2C2×#278411yescharacteristic
589C2#5309812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17266C2semidirect
2. #16986C2semidirect
3. #16708C2semidirect
5. #15235C2semidirect
6. #15220C2semidirect
7. #15218C2semidirect
8. #15216C2semidirect
9. #14266C2semidirect
10. #14130C2semidirect
11. #13907C2semidirect

Maximal subgroups

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

subgroupindex
#172662
#169862
#167082
#159582
#152352
#152202
#152182
#152162
#142662
#141302
#139072
#138722
#138472