Finite Group Databasebeta

← #461439

Subgroups — #461439

Order 512.

1188 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1880C211yesmaximal, characteristic
2C2×#1803C211yesmaximal, characteristic, complemented
3#15244C211yesmaximal, characteristic
4#13931C211yesmaximal, characteristic, complemented
5#11357C211yesmaximal, characteristic, complemented
6#11268C212yesmaximal, complemented
7#11255C211yesmaximal, characteristic, complemented
8#11189C212yesmaximal
9#11076C211yesmaximal, characteristic
10#11052C212yesmaximal, complemented
11#11023C212yesmaximal
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2211yescharacteristic
19C2×#391C2212yes—
20C2×#390C2211yescharacteristic
21C2×#390C2212yes—
22C2×#384C2211yescharacteristic
23C2×#382C2212yes—
24#1880—24no—
25#1803—24no—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34#1445—22no—
35#1439—21no—
36#1439—21no—
37#1439—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40#1434—21no—
41#1434—21no—
42C4×#95C2211yescharacteristic
43C23×Q8C2311yescharacteristic
44C23×D4C2311yescharacteristic
45C22×SD16—22no—
46C22×SD16—22no—
47C22×SD16—24no—
48C22×SD16—22no—
49C22×SD16—42no—
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×C4×Q8—22no—
54C2×C4×Q8—24no—
55C2×C4×Q8—42no—
56C2×C4×Q8D414yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C42—21no—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71C2×C4⋊C8—22no—
72C2×C4⋊C8—42no—
73C2×#103—24no—
74C2×#103—22no—
75C2×#103—42no—
76C2×#103—42no—
77C2×#103—42no—
78C2×#102—24no—
79C2×#102—22no—
80C2×#102—42no—
81C2×#102—42no—
82C2×#102—42no—
83#391—24no—
84#391—42no—
85#390—22no—
86#390—22no—
87#390—42no—
88#390—42no—
89#384—22no—
90#384—22no—
91#382—42no—
92#382—42no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95D411yescharacteristic
102C2×#95D411yescharacteristic
103C2×#95D412yes—
104#340—24no—
105#340—22no—
106#340—42no—
107#340—42no—
108#340—42no—
109#336—24no—
110#336—22no—
111#336—42no—
112C25—21no—
113C22×Q8—21no—
114C22×Q8—22no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8C2×D412yes—
122C22×Q8#4114yes—
123C22×D4—21no—
124C22×D4—22no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4#4112yes—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2411yescharacteristic
143C2×SD16—28no—
144C2×SD16—48no—
145C2×SD16—48no—
146C2×SD16—44no—
147C2×SD16—44no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8#4112yes—
165C4×Q8—28no—
166C4×Q8—44no—
167C4×Q8—48no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
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—44no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—81no—
193C2×C4⋊C4—81no—
194C2×C4⋊C4C2×D412yes—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—82no—
203C2×C22⋊C4—82no—
204C2×C22⋊C4—82no—
205C2×C22⋊C4—82no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—24no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—44no—
216C2×C42C2×D412yes—
217C2×C42#4112yes—
218C4⋊C8—44no—
219#103—48no—
220#103—44no—
221#102—44no—
222#102—48no—
223#95—22no—
224#95—22no—
225#95—42no—
226#95—44no—
227#95—44no—
228#95—44no—
229#95—44no—
230#95—44no—
231#95—42no—
232#95—81no—
233#95—82no—
234C24—22no—
235C24—41no—
236C24—42no—
237C24—42no—
238C24—41no—
239C24—41no—
240C24—41no—
241C24—42no—
242C24—41no—
243C24—41no—
244C24—42no—
245C24C22×D411yescharacteristic
246C2×Q8—22no—
247C2×Q8—24no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8#12314yes—
264C2×D4—22no—
265C2×D4—24no—
266C2×D4—24no—
267C2×D4—22no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4—42no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
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—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
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—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4#12112yes—
332C22×C4#12112yes—
333C22×C4#13512yes—
334C22×C4#13712yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C2×#4112yes—
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4H(5,𝔽2)12yes—
341C22×C4H(5,𝔽2)12yes—
342C22×C4H(5,𝔽2)12yes—
343C22×C42−1+411yescharacteristic
344C22×C42−1+411yescharacteristic
345SD16—416no—
346C2×C8—22no—
347C2×C8—24no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—41no—
353C2×C8—42no—
354C2×C8—41no—
355C2×C8—42no—
356C2×C8—41no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—48no—
366C4⋊C4—42no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—84no—
379C4⋊C4—84no—
380C22⋊C4—84no—
381C22⋊C4—88no—
382C22⋊C4—88no—
383C22⋊C4—84no—
384C42—22no—
385C42—22no—
386C42—24no—
387C42—48no—
388C42—42no—
389C42—42no—
390C42—42no—
391C42—42no—
392C23—42no—
393C23—42no—
394C23—44no—
395C23—42no—
396C23—44no—
397C23—81no—
398C23—82no—
399C23—81no—
400C23—81no—
401C23—81no—
402C23—82no—
403C23—82no—
404C23—82no—
405C23—82no—
406C23—82no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23C2×#12111yescharacteristic
416C23#53512yes—
417C23#54612yes—
418C23#54711yescharacteristic
419C23#54811yescharacteristic
420C23C2×#13511yescharacteristic
421C23C2×#13711yescharacteristic
422C23#57512yes—
423C23#57611yescharacteristic
424C23#57711yescharacteristic
425C23#57712yes—
426Q8—24no—
427Q8—24no—
428Q8—42no—
429Q8—42no—
430Q8—44no—
431Q8—44no—
432D4—42no—
433D4—42no—
434D4—44no—
435D4—81no—
436D4—81no—
437D4—82no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—81no—
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—81no—
485C2×C4—81no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4#46414yes—
491C2×C4C2×#12311yescharacteristic
492C2×C4#52511yescharacteristic
493C2×C4#53411yescharacteristic
494C2×C4#53511yescharacteristic
495C2×C4#53511yescharacteristic
496C2×C4#53611yescharacteristic
497C2×C4#53812yes—
498C2×C4#53912yes—
499C2×C4#53912yes—
500C2×C4#54012yes—
501C2×C4#54012yes—
502C2×C4#55112yes—
503C2×C4#55112yes—
504C2×C4#55112yes—
505C2×C4#55211yescharacteristic
506C2×C4#55611yescharacteristic
507C2×C4#56212yes—
508C2×C4#56212yes—
509C8—42no—
510C8—81no—
511C8—82no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—84no—
516C22—84no—
517C22—161no—
518C22—161no—
519C22—162no—
520C22—162no—
521C22—161no—
522C22—161no—
523C22#241211yescharacteristic
524C22#241511yescharacteristic
525C22#261911yescharacteristic
526C22#269612yes—
527C22#270612yes—
528C22C2×#46411yescharacteristic
529C22#304011yescharacteristic
530C22#304311yescharacteristic
531C22#304511yescharacteristic
532C22#321212yes—
533C22#321612yes—
534C22#322212yes—
535C22#322412yes—
536C22#330112yes—
537C22#330912yes—
538C22#332012yes—
539C22#332612yes—
540C22#334011yescharacteristic
541C22#334812yes—
542C22#335711yescharacteristic
543C22#336111yescharacteristic
544C22#337211yescharacteristic
545C22#337612yes—
546C4—24no—
547C4—44no—
548C4—42no—
549C4—82no—
550C4—82no—
551C4—82no—
552C4—161no—
553C4—162no—
554C4—162no—
555C4#248211yescharacteristic
556C4#262311yescharacteristic
557C4#263511yescharacteristic
558C4#274012yes—
559C4#274312yes—
560C4#281611yescharacteristic
561C2—162no—
562C2#1639211yescharacteristic
563C2#2453311yescharacteristic
564C2#2454711yescharacteristic
565C2#2467911yescharacteristic
566C2#2469511yescharacteristic
567C2#2979711yescharacteristic
568C2#2980311yescharacteristic
569C2#3164812yes—
570C2#3166012yes—
571C2#3194412yes—
572C2#3194812yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1803C2semidirect
4. #13931C2semidirect
5. #11357C2semidirect
6. #11268C2semidirect
7. #11255C2semidirect
10. #11052C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18802
C2×#18032
#152442
#139312
#113572
#112682
#112552
#111892
#110762
#110522
#110232