Finite Group Databasebeta

← #406783

Subgroups — #406783

Order 512.

1474 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1850C211yesmaximal, characteristic
2C2×#1824C211yesmaximal, characteristic
3C2×#1807C211yesmaximal, characteristic
4C22×#340C211yesmaximal, characteristic
5#14130C211yesmaximal, characteristic
6#13924C211yesmaximal, characteristic
7#11315C212yesmaximal, complemented
8#11231C212yesmaximal, complemented
9#11227C212yesmaximal, complemented
10#11220C212yesmaximal, complemented
11#10730C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#106C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C2×#389C2212yes—
20C2×#385—21no—
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24#1850—24no—
25#1824—24no—
26#1807—24no—
27C2×#340—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1445—21no—
38#1445—21no—
39#1444—21no—
40#1442—21no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C22⋊#95—21no—
45C22⋊#95—21no—
46C22⋊#95—21no—
47C24×C4—21no—
48C24×C4—21no—
49C24×C4C2311yescharacteristic
50C23×C8—22no—
51C23×C8—22no—
52C23×C8—24no—
53C23×C8—42no—
54C23×C8D414yes—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—42no—
71C22×C4⋊C4Q814yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4D411yescharacteristic
79C22×C22⋊C4D411yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×#106—24no—
83C2×#106—24no—
84C2×#106—42no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—24no—
88C2×C4⋊C8—42no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92#389—24no—
93#389—42no—
94#385—44no—
95#380—22no—
96#380—22no—
97#380—22no—
98#380D414yes—
99C4×C22⋊C4—22no—
100C4×C22⋊C4—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95D411yescharacteristic
111C2×#95D411yescharacteristic
112#340—24no—
113#340—24no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—42no—
119#336—22no—
120#336—22no—
121C25C2×D411yescharacteristic
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—24no—
130C23×C4—22no—
131C23×C4—44no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4—82no—
138C23×C4—82no—
139C23×C4—82no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×Q812yes—
148C23×C4#4112yes—
149C23×C4#4112yes—
150C23×C4#4112yes—
151C23×C4C2411yescharacteristic
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—28no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—28no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—44no—
172C22×C8—48no—
173C22×C8—48no—
174C22×C8C2×D412yes—
175C22×C8C2×D412yes—
176C22×C8#4114yes—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
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—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—48no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×Q814yes—
215C2×C4⋊C4#4114yes—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4SD1614yes—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236#106—48no—
237C4⋊C8—44no—
238C4⋊C8—44no—
239C22⋊C8—44no—
240#95—24no—
241#95—22no—
242#95—22no—
243#95—44no—
244#95—42no—
245#95—44no—
246#95—82no—
247#95—82no—
248#95—82no—
249#95—82no—
250#95—82no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24#12112yes—
256C24#12112yes—
257C24C22⋊Q812yes—
258C24C2×SD1612yes—
259C24#13512yes—
260C24#13712yes—
261C24#13712yes—
262C24C22×D411yescharacteristic
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—48no—
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—48no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—88no—
328C22×C4—88no—
329C22×C4—88no—
330C22×C4—88no—
331C22×C4—84no—
332C22×C4—84no—
333C22×C4—84no—
334C22×C4—84no—
335C22×C4#12114yes—
336C22×C4#12112yes—
337C22×C4#12112yes—
338C22×C4C22⋊Q814yes—
339C22×C4C22⋊Q814yes—
340C22×C4C22⋊Q812yes—
341C22×C4#12614yes—
342C22×C4C2×SD1612yes—
343C22×C4C2×SD1612yes—
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×Q811yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4112yes—
351C22×C4C2×#4112yes—
352C22×C4H(5,𝔽2)12yes—
353C22×C42−1+412yes—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—416no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8#12314yes—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—48no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—48no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—88no—
394C22⋊C4—28no—
395C22⋊C4—48no—
396C22⋊C4—44no—
397C22⋊C4—84no—
398C42—41no—
399C42—41no—
400C42—41no—
401C42—41no—
402C42—42no—
403C42—42no—
404C23—24no—
405C23—24no—
406C23—24no—
407C23—24no—
408C23—24no—
409C23—24no—
410C23—24no—
411C23—22no—
412C23—24no—
413C23—24no—
414C23—24no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—24no—
419C23—24no—
420C23#39614yes—
421C23#46114yes—
422C23#46414yes—
423C23#46514yes—
424C23#47014yes—
425C23#47514yes—
426C23#47914yes—
427C23C2×#12111yescharacteristic
428C23C2×#12111yescharacteristic
429C23C2×C22⋊Q811yescharacteristic
430C23#54612yes—
431C23#54712yes—
432C23C22×SD1611yescharacteristic
433C23C2×#13511yescharacteristic
434C23C2×#13711yescharacteristic
435C23C2×#13711yescharacteristic
436C23#57612yes—
437C23#57712yes—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—28no—
447C2×C4—28no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—41no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—41no—
470C2×C4—44no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—816no—
478C2×C4—816no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4#39614yes—
499C2×C4C2×#12111yescharacteristic
500C2×C4C2×C22⋊Q811yescharacteristic
501C2×C4C2×C22⋊Q811yescharacteristic
502C2×C4C2×#12311yescharacteristic
503C2×C4#52511yescharacteristic
504C2×C4C2×#12611yescharacteristic
505C2×C4#52912yes—
506C2×C4#53312yes—
507C2×C4#53411yescharacteristic
508C2×C4#53611yescharacteristic
509C2×C4#53912yes—
510C2×C4#54412yes—
511C2×C4#54612yes—
512C2×C4#54712yes—
513C2×C4#55112yes—
514C2×C4#55112yes—
515C2×C4#55212yes—
516C2×C4#55212yes—
517C8—24no—
518C8—42no—
519C8—81no—
520C8—81no—
521C22—24no—
522C22—24no—
523C22—24no—
524C22—24no—
525C22—28no—
526C22—28no—
527C22—28no—
528C22—28no—
529C22—24no—
530C22—24no—
531C22#206618yes—
532C22C2×#39611yescharacteristic
533C22#242512yes—
534C22#243812yes—
535C22#248611yescharacteristic
536C22#253411yescharacteristic
537C22C2×#46111yescharacteristic
538C22C2×#46411yescharacteristic
539C22#304512yes—
540C22C2×#46511yescharacteristic
541C22C2×#47011yescharacteristic
542C22#305711yescharacteristic
543C22#305811yescharacteristic
544C22C2×#47511yescharacteristic
545C22C2×#47911yescharacteristic
546C22#308112yes—
547C22#329512yes—
548C22#329712yes—
549C22#331612yes—
550C22#332012yes—
551C22#333512yes—
552C22#333712yes—
553C22#334612yes—
554C22#334612yes—
555C4—24no—
556C4—42no—
557C4—42no—
558C4—81no—
559C4—84no—
560C4—84no—
561C4—84no—
562C4—81no—
563C4—162no—
564C4#241211yescharacteristic
565C4#241311yescharacteristic
566C4#243812yes—
567C4#249312yes—
568C4#262711yescharacteristic
569C4#285511yescharacteristic
570C2—28no—
571C2#1493511yescharacteristic
572C2C2×#206611yescharacteristic
573C2#2361211yescharacteristic
574C2#2520212yes—
575C2#2521012yes—
576C2#2599212yes—
577C2#2600012yes—
578C2#2765811yescharacteristic
579C2#2766611yescharacteristic
580C2#2809711yescharacteristic
581C2#2810811yescharacteristic
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. #11227C2semidirect
10. #11220C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18502
C2×#18242
C2×#18072
C22×#3402
#141302
#139242
#113152
#112312
#112272
#112202
#107302