Finite Group Databasebeta

← #4463919

Subgroups — #4463919

Order 512.

1906 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17252C214yesmaximal, complemented
2#17207C214yesmaximal, complemented
3#17181C214yesmaximal
4#17165C214yesmaximal, complemented
5#15218C211yesmaximal, characteristic, complemented
6#15216C212yesmaximal, complemented
7#15183C212yesmaximal, complemented
8#15181C212yesmaximal
9#15151C211yesmaximal, characteristic, complemented
10#15132C212yesmaximal, complemented
11#15131C212yesmaximal, complemented
12#14266C212yesmaximal, complemented
13#13847C211yesmaximal, characteristic
14C22×#125C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C2×#398C2214yes—
17C2×#397C2214yes—
18C2×#396C2214yes—
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#384C2211yescharacteristic
23C2×#380C2212yes—
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2212yes—
29C2×C4×C22⋊C4C2212yes—
30C2×#376C2211yescharacteristic
31#1445C2214yes—
32#1445C2212yes—
33#1445C2214yes—
34#1445C2214yes—
35#1444C2214yes—
36#1443C2214yes—
37#1443C2214yes—
38#1443C2214yes—
39#1442C2214yes—
40#1442C2214yes—
41#1442C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1440C2214yes—
49#1440C2214yes—
50#1439C2212yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2212yes—
56#1439C2214yes—
57#1439C2214yes—
58#1438C2212yes—
59#1438C2214yes—
60C4⋊#95C2212yes—
61C4⋊#95C2211yescharacteristic
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2211yescharacteristic
67C23×D4C2311yescharacteristic
68C24×C4C2312yes—
69C2×#125—44no—
70C2×#125—42no—
71C2×C4×D4—24no—
72C2×C4×D4—42no—
73C2×C4×D4—42no—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C22⋊C4C2312yes—
86C22×C22⋊C4C2312yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2312yes—
89C22×C22⋊C4C2312yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2312yes—
94C22×C42C2311yescharacteristic
95C22×C42C2312yes—
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98#398—48no—
99#397—48no—
100#396—48no—
101#389—28no—
102#389—44no—
103#389—44no—
104#384—42no—
105#380—44no—
106#380—44no—
107#380—44no—
108#380—44no—
109C4×C4⋊C4—42no—
110C4×C22⋊C4—44no—
111C4×C22⋊C4—44no—
112#376—42no—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2312yes—
127C2×#95C2312yes—
128C2×#95C2311yescharacteristic
129C2×#95C2311yescharacteristic
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C25C2412yes—
141C22×D4—24no—
142C22×D4—22no—
143C22×D4—41no—
144C22×D4—41no—
145C22×D4—42no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4C2414yes—
154C23×C4C2414yes—
155C23×C4C2414yes—
156C23×C4C2412yes—
157C23×C4C2414yes—
158C23×C4C2412yes—
159C23×C4C2412yes—
160C23×C4C2412yes—
161C23×C4C2411yescharacteristic
162C23×C4C2411yescharacteristic
163C23×C4C2412yes—
164C23×C4C2411yescharacteristic
165#125—416no—
166C4×D4—416no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—24no—
230C2×C42—22no—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42#4114yes—
237#95—28no—
238#95—28no—
239#95—44no—
240#95—44no—
241#95—44no—
242#95—44no—
243#95—44no—
244#95—44no—
245#95—41no—
246#95—41no—
247#95—42no—
248#95—42no—
249#95—42no—
250#95—42no—
251#95—42no—
252#95—42no—
253#95—44no—
254#95—44no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—44no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C242−1+414yes—
279C24C2511yescharacteristic
280C2×D4—48no—
281C2×D4—44no—
282C2×D4—44no—
283C2×D4—48no—
284C2×D4—44no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4C2×#4112yes—
363C22×C4C2×#4112yes—
364C22×C4C2×#4112yes—
365C22×C4H(5,𝔽2)14yes—
366C22×C4H(5,𝔽2)12yes—
367C22×C4H(5,𝔽2)12yes—
368C22×C4H(5,𝔽2)14yes—
369C22×C42−1+414yes—
370C4⋊C4—48no—
371C4⋊C4—44no—
372C4⋊C4—416no—
373C4⋊C4—48no—
374C4⋊C4—48no—
375C4⋊C4—48no—
376C4⋊C4—48no—
377C4⋊C4—88no—
378C4⋊C4—88no—
379C4⋊C4—88no—
380C4⋊C4—88no—
381C22⋊C4—48no—
382C22⋊C4—48no—
383C22⋊C4—48no—
384C22⋊C4—48no—
385C22⋊C4—48no—
386C22⋊C4—88no—
387C22⋊C4—88no—
388C22⋊C4—88no—
389C22⋊C4—88no—
390C42—28no—
391C42—44no—
392C42—44no—
393C42—48no—
394C23—24no—
395C23—24no—
396C23—24no—
397C23—24no—
398C23—24no—
399C23—24no—
400C23—24no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23C22×#4111yescharacteristic
430C23C2×H(5,𝔽2)11yescharacteristic
431C23C2×H(5,𝔽2)11yescharacteristic
432C23C2×H(5,𝔽2)12yes—
433C23C2×H(5,𝔽2)12yes—
434C23C2×2−1+412yes—
435C23C2×2−1+412yes—
436C23#58512yes—
437C23#58512yes—
438D4—416no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—816no—
513C2×C4#53814yes—
514C2×C4#54218yes—
515C2×C4#56014yes—
516C2×C4#56218yes—
517C2×C4#56314yes—
518C22—44no—
519C22—44no—
520C22—44no—
521C22—44no—
522C22—44no—
523C22—44no—
524C22—44no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22C2×#53811yescharacteristic
534C22C2×#54212yes—
535C22#349512yes—
536C22#350811yescharacteristic
537C22#351211yescharacteristic
538C22C2×#56011yescharacteristic
539C22C2×#56212yes—
540C22C2×#56311yescharacteristic
541C22#353612yes—
542C22#353811yescharacteristic
543C22#353911yescharacteristic
544C22#353912yes—
545C22#354112yes—
546C22#354112yes—
547C22#354212yes—
548C22#354212yes—
549C22#354612yes—
550C22#354712yes—
551C22#355212yes—
552C22#356312yes—
553C22#356512yes—
554C4—42no—
555C4—44no—
556C4—42no—
557C4—82no—
558C4—82no—
559C4—84no—
560C4—82no—
561C4—84no—
562C4—84no—
563C4—84no—
564C4—82no—
565C4#283018yes—
566C2—84no—
567C2C2×#283011yescharacteristic
568C2#3731711yescharacteristic
569C2#3733312yes—
570C2#3738011yescharacteristic
571C2#3741612yes—
572C2#5298212yes—
573C2#5306212yes—
574C2#5309812yes—
575C2#5311912yes—
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. #17252C2semidirect
2. #17207C2semidirect
4. #17165C2semidirect
5. #15218C2semidirect
6. #15216C2semidirect
7. #15183C2semidirect
9. #15151C2semidirect
10. #15132C2semidirect
11. #15131C2semidirect
12. #14266C2semidirect

Maximal subgroups

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

subgroupindex
#172522
#172072
#171812
#171652
#152182
#152162
#151832
#151812
#151512
#151322
#151312
#142662
#138472