Finite Group Databasebeta

← #951780

Subgroups — #951780

Order 512.

1998 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#17303C214yesmaximal, complemented
2#17028C214yesmaximal
3#16972C214yesmaximal, complemented
4#15977C214yesmaximal, complemented
5#15233C212yesmaximal, complemented
6#15216C212yesmaximal, complemented
7#15181C212yesmaximal
8#14283C211yesmaximal, characteristic, complemented
9#14277C211yesmaximal, characteristic, complemented
10#14042C212yesmaximal, complemented
11#13876C212yesmaximal, complemented
12#13876C212yesmaximal, complemented
13#13847C211yesmaximal, characteristic
14C22×#125C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C2×#389C2212yes—
17C2×#388C2214yes—
18C2×#388C2214yes—
19C2×#387C2214yes—
20C2×#384C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×#380C2212yes—
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C22⋊C4C2212yes—
30C2×C4×C22⋊C4C2212yes—
31C2×#376C2211yescharacteristic
32#1445C2212yes—
33#1444C2214yes—
34#1443C2214yes—
35#1443C2214yes—
36#1443C2214yes—
37#1442C2214yes—
38#1442C2214yes—
39#1442C2214yes—
40#1442C2214yes—
41#1442C2214yes—
42#1442C2214yes—
43#1441C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1440C2214yes—
47#1440C2214yes—
48#1439C2212yes—
49#1439C2212yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1438C2212yes—
57#1438C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66#1434C2214yes—
67C23×D4C2311yescharacteristic
68C24×C4C2312yes—
69C2×#125—24no—
70C2×#125—42no—
71C2×#125—42no—
72C2×C4×D4—24no—
73C2×C4×D4—42no—
74C2×C4×D4—42no—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C22⋊C4C2312yes—
87C22×C22⋊C4C2312yes—
88C22×C22⋊C4C2312yes—
89C22×C22⋊C4C2312yes—
90C22×C22⋊C4C2312yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C42C2312yes—
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99C22×C42C2314yes—
100#389—28no—
101#388—48no—
102#388—48no—
103#387—48no—
104#384—42no—
105#384—42no—
106#382—24no—
107#380—44no—
108#380—44no—
109#380—44no—
110#380—44no—
111C4×C4⋊C4—42no—
112C4×C4⋊C4—44no—
113C4×C22⋊C4—28no—
114C4×C22⋊C4—28no—
115#376—42no—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2312yes—
120C2×#95C2312yes—
121C2×#95C2312yes—
122C2×#95C2312yes—
123C2×#95C2314yes—
124C2×#95C2312yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C25C2412yes—
143C22×D4—24no—
144C22×D4—22no—
145C22×D4—41no—
146C22×D4—41no—
147C22×D4—42no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4C2414yes—
156C23×C4C2414yes—
157C23×C4C2414yes—
158C23×C4C2414yes—
159C23×C4C2412yes—
160C23×C4C2412yes—
161C23×C4C2411yescharacteristic
162C23×C4C2412yes—
163C23×C4C2412yes—
164C23×C4C2411yescharacteristic
165C23×C4C2412yes—
166C23×C4C2411yescharacteristic
167#125—416no—
168C4×D4—416no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—28no—
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—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
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—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—28no—
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—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—28no—
237C2×C42—44no—
238C2×C42—44no—
239C2×C42#4114yes—
240C2×C42#4114yes—
241C2×C42#4114yes—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—28no—
247#95—28no—
248#95—28no—
249#95—28no—
250#95—44no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—44no—
255#95—44no—
256#95—42no—
257#95—42no—
258#95—42no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C242−1+414yes—
283C24C2511yescharacteristic
284C2×D4—48no—
285C2×D4—44no—
286C2×D4—44no—
287C2×D4—48no—
288C2×D4—44no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
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—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—48no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—48no—
364C22×C4C2×#4114yes—
365C22×C4C2×#4112yes—
366C22×C4C2×#4114yes—
367C22×C4C2×#4112yes—
368C22×C4C2×#4112yes—
369C22×C4C2×#4112yes—
370C22×C4C2×#4112yes—
371C22×C4H(5,𝔽2)14yes—
372C22×C4H(5,𝔽2)12yes—
373C22×C4H(5,𝔽2)12yes—
374C22×C4H(5,𝔽2)12yes—
375C22×C42−1+412yes—
376C4⋊C4—44no—
377C4⋊C4—416no—
378C4⋊C4—44no—
379C4⋊C4—416no—
380C4⋊C4—44no—
381C4⋊C4—48no—
382C4⋊C4—48no—
383C4⋊C4—48no—
384C4⋊C4—48no—
385C4⋊C4—88no—
386C4⋊C4—88no—
387C22⋊C4—216no—
388C22⋊C4—216no—
389C22⋊C4—48no—
390C22⋊C4—48no—
391C22⋊C4—48no—
392C22⋊C4—88no—
393C22⋊C4—88no—
394C22⋊C4—88no—
395C22⋊C4—88no—
396C42—28no—
397C42—28no—
398C42—28no—
399C42—48no—
400C42—88no—
401C23—24no—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—24no—
406C23—24no—
407C23—24no—
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—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23C22×#4111yescharacteristic
437C23C22×#4111yescharacteristic
438C23C22×#4111yescharacteristic
439C23C2×H(5,𝔽2)11yescharacteristic
440C23C2×H(5,𝔽2)11yescharacteristic
441C23C2×H(5,𝔽2)11yescharacteristic
442C23C2×H(5,𝔽2)12yes—
443C23C2×2−1+411yescharacteristic
444C23C2×2−1+412yes—
445C23#58512yes—
446C23#58512yes—
447D4—416no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—816no—
522C2×C4#53814yes—
523C2×C4#53814yes—
524C2×C4#53914yes—
525C2×C4#55514yes—
526C2×C4#55614yes—
527C2×C4#55914yes—
528C2×C4#56214yes—
529C22—44no—
530C22—44no—
531C22—44no—
532C22—44no—
533C22—44no—
534C22—44no—
535C22—44no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22C2×#53811yescharacteristic
545C22C2×#53811yescharacteristic
546C22C2×#53911yescharacteristic
547C22#346012yes—
548C22C2×#55511yescharacteristic
549C22C2×#55611yescharacteristic
550C22C2×#55911yescharacteristic
551C22#349212yes—
552C22#349212yes—
553C22#349312yes—
554C22#349312yes—
555C22#349512yes—
556C22C2×#56211yescharacteristic
557C22#353711yescharacteristic
558C22#353811yescharacteristic
559C22#353911yescharacteristic
560C22#353911yescharacteristic
561C22#354112yes—
562C22#354612yes—
563C22#354612yes—
564C22#354712yes—
565C22#356112yes—
566C22#356312yes—
567C4—42no—
568C4—44no—
569C4—44no—
570C4—44no—
571C4—42no—
572C4—84no—
573C4—82no—
574C4—84no—
575C4—84no—
576C4—82no—
577C4—84no—
578C4#280818yes—
579C2—84no—
580C2#3738011yescharacteristic
581C2#3751511yescharacteristic
582C2#3795312yes—
583C2#3796112yes—
584C2#4250512yes—
585C2#4250712yes—
586C2#4537511yescharacteristic
587C2#4540011yescharacteristic
588C2C2×#280811yescharacteristic
589C2#5303011yescharacteristic
590C2#5307611yescharacteristic
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. #17303C2semidirect
3. #16972C2semidirect
4. #15977C2semidirect
5. #15233C2semidirect
6. #15216C2semidirect
8. #14283C2semidirect
9. #14277C2semidirect
10. #14042C2semidirect
11. #13876C2semidirect
12. #13876C2semidirect

Maximal subgroups

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

subgroupindex
#173032
#170282
#169722
#159772
#152332
#152162
#151812
#142832
#142772
#140422
#138762
#138472