Finite Group Databasebeta

← #406757

Subgroups — #406757

Order 512.

1346 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1854C211yesmaximal, characteristic
2C2×#1851C211yesmaximal, characteristic
3C2×#1823C211yesmaximal, characteristic
4C2×#1736C211yesmaximal, characteristic
5#13951C211yesmaximal, characteristic
6#13924C211yesmaximal, characteristic
7#11315C212yesmaximal, complemented
8#11231C212yesmaximal, complemented
9#11228C212yesmaximal, complemented
10#11219C212yesmaximal, complemented
11#10722C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C22×C4⋊C8C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C2×#389C2212yes—
20C2×#380C2211yescharacteristic
21C2×#380C2211yescharacteristic
22C2×C4×C22⋊C4C2211yescharacteristic
23#1854—24no—
24#1851—24no—
25#1823—24no—
26#1736—24no—
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36#1445—21no—
37#1445—21no—
38#1439—22no—
39C4⋊#95—21no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43#1434—21no—
44C22⋊#95—22no—
45C24×C4—21no—
46C24×C4C2311yescharacteristic
47C22×M16—22no—
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—24no—
51C22×M16—42no—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2312yes—
55C2×#117—22no—
56C2×#117—24no—
57C2×#117—42no—
58C2×#117D414yes—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4Q814yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4D412yes—
77C22×C42—21no—
78C22×C42C2312yes—
79C2×#107—22no—
80C2×#107—24no—
81C2×#106—22no—
82C2×#106—42no—
83C2×C4⋊C8—24no—
84C2×C4⋊C8—24no—
85C2×C4⋊C8—42no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8D414yes—
89#389—24no—
90#389—42no—
91#380—22no—
92#380—22no—
93#380—22no—
94#380—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95D411yescharacteristic
107C2×#95D411yescharacteristic
108#340—24no—
109#340—24no—
110#340—24no—
111#340—22no—
112#340—22no—
113#340—24no—
114#340—24no—
115#340—42no—
116#336—24no—
117C25C2×D411yescharacteristic
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—24no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4—81no—
132C23×C4—81no—
133C23×C4—81no—
134C23×C4—81no—
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2×Q812yes—
141C23×C4#4112yes—
142C23×C4#4112yes—
143C23×C4#4112yes—
144C23×C4C2411yescharacteristic
145C2×M16—28no—
146C2×M16—48no—
147C2×M16—48no—
148C2×M16—44no—
149C2×M16—44no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8C2×D412yes—
165C22×C8C2×D412yes—
166C22×C8#4114yes—
167#117—28no—
168#117—44no—
169#117—48no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—28no—
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—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×Q814yes—
207C2×C4⋊C4#4114yes—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—81no—
217C2×C22⋊C4—81no—
218C2×C42—24no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—44no—
226#107—44no—
227#106—44no—
228C4⋊C8—48no—
229C22⋊C8—28no—
230#95—22no—
231#95—22no—
232#95—44no—
233#95—42no—
234#95—42no—
235#95—44no—
236#95—82no—
237#95—82no—
238#95—82no—
239#95—82no—
240#95—82no—
241C24—22no—
242C24—22no—
243C24—22no—
244C24—22no—
245C24—21no—
246C24—21no—
247C24—21no—
248C24—21no—
249C24#12112yes—
250C24#12112yes—
251C24C22⋊Q812yes—
252C24C22×D411yescharacteristic
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—48no—
300C22×C4—41no—
301C22×C4—41no—
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—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—84no—
318C22×C4—84no—
319C22×C4—84no—
320C22×C4—84no—
321C22×C4—84no—
322C22×C4—84no—
323C22×C4#12114yes—
324C22×C4#12112yes—
325C22×C4#12112yes—
326C22×C4C22⋊Q814yes—
327C22×C4C22⋊Q814yes—
328C22×C4C22⋊Q812yes—
329C22×C4#12614yes—
330C22×C4#13714yes—
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×Q811yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4112yes—
338C22×C4C2×#4112yes—
339C22×C4H(5,𝔽2)12yes—
340C22×C42−1+412yes—
341M16—416no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—42no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—42no—
354C2×C8—41no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—48no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—42no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—88no—
380C4⋊C4#12314yes—
381C22⋊C4—48no—
382C22⋊C4—48no—
383C22⋊C4—84no—
384C22⋊C4—84no—
385C42—48no—
386C42—42no—
387C42—42no—
388C42—42no—
389C42—42no—
390C23—24no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—24no—
396C23—24no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—24no—
402C23—22no—
403C23—22no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23#39614yes—
413C23C2×#12111yescharacteristic
414C23C2×#12111yescharacteristic
415C23C2×C22⋊Q811yescharacteristic
416C23#54612yes—
417C23#54712yes—
418C23C2×#13712yes—
419C23#57512yes—
420C23#57611yescharacteristic
421C23#57611yescharacteristic
422C23#57711yescharacteristic
423C23#57711yescharacteristic
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—28no—
431C2×C4—24no—
432C2×C4—28no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—816no—
460C2×C4—82no—
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—81no—
472C2×C4—81no—
473C2×C4—82no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4#39614yes—
484C2×C4C2×#12111yescharacteristic
485C2×C4C2×C22⋊Q811yescharacteristic
486C2×C4C2×C22⋊Q811yescharacteristic
487C2×C4C2×#12311yescharacteristic
488C2×C4#52511yescharacteristic
489C2×C4C2×#12611yescharacteristic
490C2×C4#52912yes—
491C2×C4#53312yes—
492C2×C4#53411yescharacteristic
493C2×C4#53611yescharacteristic
494C2×C4#53912yes—
495C2×C4#54412yes—
496C2×C4#54612yes—
497C2×C4#54712yes—
498C2×C4#55112yes—
499C2×C4#55112yes—
500C2×C4#55212yes—
501C2×C4#55212yes—
502C8—42no—
503C8—42no—
504C8—44no—
505C22—24no—
506C22—24no—
507C22—24no—
508C22—42no—
509C22—42no—
510C22—44no—
511C22—44no—
512C22—44no—
513C22—44no—
514C22—42no—
515C22—42no—
516C22C2×#39611yescharacteristic
517C22#241211yescharacteristic
518C22#241311yescharacteristic
519C22#242512yes—
520C22#243812yes—
521C22#303911yescharacteristic
522C22#304011yescharacteristic
523C22#304511yescharacteristic
524C22#304511yescharacteristic
525C22#305612yes—
526C22#305711yescharacteristic
527C22#305811yescharacteristic
528C22#307812yes—
529C22#308111yescharacteristic
530C22#308111yescharacteristic
531C22#330712yes—
532C22#331612yes—
533C22#332212yes—
534C22#333312yes—
535C22#334412yes—
536C22#335012yes—
537C22#335512yes—
538C22#335712yes—
539C4—24no—
540C4—44no—
541C4—44no—
542C4—84no—
543C4—82no—
544C4—82no—
545C4—161no—
546C4—162no—
547C4—161no—
548C4#243812yes—
549C4#248611yescharacteristic
550C4#249312yes—
551C4#253411yescharacteristic
552C4#262711yescharacteristic
553C4#285511yescharacteristic
554C2—44no—
555C2#1493511yescharacteristic
556C2#2361211yescharacteristic
557C2#2361911yescharacteristic
558C2#2454611yescharacteristic
559C2#2454911yescharacteristic
560C2#2459711yescharacteristic
561C2#2459811yescharacteristic
562C2#2524412yes—
563C2#2525812yes—
564C2#2601012yes—
565C2#2601612yes—
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
7. #11315C2semidirect
8. #11231C2semidirect
9. #11228C2semidirect
10. #11219C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18542
C2×#18512
C2×#18232
C2×#17362
#139512
#139242
#113152
#112312
#112282
#112192
#107222