Finite Group Databasebeta

← #412579

Subgroups — #412579

Order 512.

1142 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1943C211yesmaximal, characteristic, complemented
2C2×#1809C211yesmaximal, characteristic
3C2×#1809C211yesmaximal, characteristic
4C2×#1753C211yesmaximal, characteristic
5#15183C212yesmaximal
6#11316C212yesmaximal, complemented
7#11292C212yesmaximal, complemented
8#11199C212yesmaximal
9#10863C211yesmaximal, characteristic, complemented
10#10862C211yesmaximal, characteristic, complemented
11#10861C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C22×#97C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C2×#389C2212yes—
21C2×#380C2212yes—
22C2×#380C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24#1943—24no—
25#1809—24no—
26#1809—24no—
27#1753—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36#1445—22no—
37#1443—22no—
38#1440—22no—
39#1439—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C24×C4C2311yescharacteristic
44C22×M16—22no—
45C22×M16—22no—
46C22×M16—22no—
47C22×M16—22no—
48C22×M16—22no—
49C22×M16—22no—
50C22×M16D414yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C22×C4⋊C4—22no—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4D412yes—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×#107—22no—
66C2×#107—22no—
67C2×#107—22no—
68C2×#107—22no—
69C2×#107—22no—
70C2×#107D414yes—
71C2×#106—22no—
72C2×#106—22no—
73C2×#106—22no—
74C2×#106—22no—
75C2×#106—22no—
76C2×#106—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×#97—22no—
81C2×#97—22no—
82C2×#97—22no—
83C2×C4×C8—22no—
84C2×C4×C8—22no—
85C2×C4×C8—22no—
86#389—24no—
87#389—42no—
88#380—24no—
89#380—24no—
90#380—24no—
91#380—42no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95D412yes—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—42no—
110#336—22no—
111#336—22no—
112C25C2×D411yescharacteristic
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4C2×D412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C23×C4#4112yes—
134C23×C4C2411yescharacteristic
135C2×M16—28no—
136C2×M16—28no—
137C2×M16—28no—
138C2×M16—44no—
139C2×M16—44no—
140C2×M16—44no—
141C2×M16—44no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8C2×D412yes—
161C22×C8C2×D412yes—
162C22×C8C2×D412yes—
163C22×C8#4112yes—
164C22×C8#4112yes—
165C22×C8#4112yes—
166C22×C8#4112yes—
167C22×C8#4112yes—
168C22×C8#4112yes—
169C22×C8#4112yes—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4C2×D414yes—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—82no—
204C2×C22⋊C4—82no—
205C2×C22⋊C4—82no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218#107—28no—
219#107—44no—
220#106—44no—
221#106—44no—
222C22⋊C8—28no—
223#97—44no—
224C4×C8—44no—
225#95—24no—
226#95—44no—
227#95—44no—
228#95—44no—
229#95—44no—
230#95—82no—
231#95—82no—
232#95—82no—
233#95—82no—
234#95—82no—
235C24—21no—
236C24—21no—
237C24—21no—
238C24—21no—
239C24—21no—
240C24—21no—
241C24—22no—
242C24—22no—
243C24—22no—
244C24—22no—
245C24#12112yes—
246C24C22×D411yescharacteristic
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4#12112yes—
323C22×C4#13714yes—
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4H(5,𝔽2)11yescharacteristic
335C22×C42−1+411yescharacteristic
336C22×C42−1+411yescharacteristic
337C22×C42−1+411yescharacteristic
338M16—416no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8#12114yes—
361C2×C8#12314yes—
362C2×C8#12314yes—
363C2×C8#12314yes—
364C2×C8#12414yes—
365C2×C8#12614yes—
366C2×C8#12614yes—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—88no—
384C22⋊C4—48no—
385C22⋊C4—88no—
386C22⋊C4—88no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—41no—
391C42—41no—
392C42—41no—
393C42—41no—
394C42—41no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—24no—
399C23—24no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—41no—
408C23—41no—
409C23—42no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—42no—
417C23—42no—
418C23C2×#12111yescharacteristic
419C23#53211yescharacteristic
420C23#53211yescharacteristic
421C23D4211yescharacteristic
422C23#54711yescharacteristic
423C23#54711yescharacteristic
424C23#54711yescharacteristic
425C23C2×#13712yes—
426C23#57512yes—
427C23#57612yes—
428C23#57712yes—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4C2×#12111yescharacteristic
487C2×C4C2×#12311yescharacteristic
488C2×C4C2×#12311yescharacteristic
489C2×C4C2×#12311yescharacteristic
490C2×C4#52511yescharacteristic
491C2×C4C2×#12411yescharacteristic
492C2×C4C2×#12611yescharacteristic
493C2×C4C2×#12611yescharacteristic
494C2×C4#52911yescharacteristic
495C2×C4#52911yescharacteristic
496C2×C4#53611yescharacteristic
497C2×C4#53711yescharacteristic
498C2×C4#54111yescharacteristic
499C2×C4#54111yescharacteristic
500C2×C4#54211yescharacteristic
501C2×C4#54211yescharacteristic
502C2×C4#54711yescharacteristic
503C2×C4#54811yescharacteristic
504C2×C4#54811yescharacteristic
505C2×C4#55211yescharacteristic
506C2×C4#55211yescharacteristic
507C2×C4#55211yescharacteristic
508C2×C4#55211yescharacteristic
509C2×C4#55311yescharacteristic
510C2×C4#55311yescharacteristic
511C2×C4#55511yescharacteristic
512C2×C4#55611yescharacteristic
513C2×C4#55611yescharacteristic
514C8—42no—
515C8—42no—
516C8—42no—
517C8#39718yes—
518C22—44no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—44no—
523C22—82no—
524C22—82no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22#241611yescharacteristic
530C22#242211yescharacteristic
531C22#248611yescharacteristic
532C22#249111yescharacteristic
533C22#249611yescharacteristic
534C22#249611yescharacteristic
535C22#285411yescharacteristic
536C22#305612yes—
537C22#306212yes—
538C22#317212yes—
539C22#317311yescharacteristic
540C22#317611yescharacteristic
541C22#317912yes—
542C22#318311yescharacteristic
543C22#318511yescharacteristic
544C22#327312yes—
545C22#327812yes—
546C22#334412yes—
547C22#334612yes—
548C22#334712yes—
549C22#335012yes—
550C22#335112yes—
551C22#335712yes—
552C4—44no—
553C4—42no—
554C4—84no—
555C4—82no—
556C4—162no—
557C4—162no—
558C4—162no—
559C4C2×#39711yescharacteristic
560C4#241511yescharacteristic
561C4#242111yescharacteristic
562C4#248511yescharacteristic
563C4#248911yescharacteristic
564C4#249411yescharacteristic
565C4#249511yescharacteristic
566C4#262511yescharacteristic
567C2—82no—
568C2#1513211yescharacteristic
569C2#2479512yes—
570C2#2504711yescharacteristic
571C2#2504811yescharacteristic
572C2#2766812yes—
573C2#2779512yes—
574C2#2784112yes—
575C2#2785011yescharacteristic
576C2#2785111yescharacteristic
577C2#3314011yescharacteristic
578C2#3314411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1943C2semidirect
6. #11316C2semidirect
7. #11292C2semidirect
9. #10863C2semidirect
10. #10862C2semidirect
11. #10861C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19432
C2×#18092
C2×#17532
#151832
#113162
#112922
#111992
#108632
#108622
#108612