Finite Group Databasebeta

← #408596

Subgroups — #408596

Order 512.

1518 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#467C211yesmaximal, characteristic
2C2×#1935C212yesmaximal, complemented
3C2×#1752C211yesmaximal, characteristic
4#15179C212yesmaximal
5#11328C211yesmaximal, characteristic, complemented
6#11256C211yesmaximal, characteristic, complemented
7#11024C211yesmaximal, characteristic, complemented
8#11024C211yesmaximal, characteristic, complemented
9#10970C212yesmaximal
10#10928C212yesmaximal, complemented
11#10721C211yesmaximal, characteristic
12C23×Q16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C22⋊Q8C2212yes—
15C2×#467—28no—
16C2×#467—24no—
17C22×#107C2211yescharacteristic
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×C22⋊C8C2211yescharacteristic
22C22×#97C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#389C2212yes—
27C2×#380C2212yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29#1935—28no—
30#1752—24no—
31C2×#340C2211yescharacteristic
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1442—22no—
39#1441—22no—
40#1440—22no—
41#1439—22no—
42C4⋊#95C2212yes—
43C23×Q8C2312yes—
44C24×C4C2311yescharacteristic
45C22×Q16—24no—
46C22×Q16—22no—
47C22×Q16—22no—
48C22×Q16—22no—
49C22×Q16—42no—
50C23×C8—24no—
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—42no—
54C23×C8D414yes—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8—24no—
63C2×C22⋊Q8—42no—
64C2×C22⋊Q8—42no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4D411yescharacteristic
73C22×C22⋊C4D411yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76#467—232no—
77C2×#107—22no—
78C2×#107—24no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#103D418yes—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×#97—24no—
92C2×#97—22no—
93C2×#97—22no—
94C2×#97—42no—
95#391—24no—
96#391—24no—
97#391—42no—
98#391—42no—
99#389—24no—
100#389—42no—
101#380—24no—
102#380—42no—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95D411yescharacteristic
114C2×#95D411yescharacteristic
115#340—24no—
116#340—22no—
117#340—22no—
118#340—24no—
119#340—42no—
120#336—22no—
121#336—22no—
122C25C2×D411yescharacteristic
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8C2×D414yes—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4#4112yes—
156C23×C4C2411yescharacteristic
157C2×Q16—28no—
158C2×Q16—48no—
159C2×Q16—48no—
160C2×Q16—44no—
161C2×Q16—44no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—28no—
176C22×C8—28no—
177C22×C8—44no—
178C22×C8—48no—
179C22×C8—48no—
180C22×C8C2×D414yes—
181C22×C8C2×D412yes—
182C22×C8C2×D412yes—
183C22×C8#4112yes—
184C22×C8#4112yes—
185C22×C8#4112yes—
186C22×C8#4112yes—
187C22⋊Q8—216no—
188C22⋊Q8—48no—
189C22⋊Q8—48no—
190C22⋊Q8—48no—
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—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4#4114yes—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—82no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233#107—28no—
234#103—216no—
235#103—216no—
236#103—48no—
237C22⋊C8—28no—
238#97—44no—
239#97—44no—
240#95—22no—
241#95—22no—
242#95—44no—
243#95—44no—
244#95—44no—
245#95—44no—
246#95—44no—
247#95—44no—
248#95—44no—
249#95—82no—
250C24—21no—
251C24—21no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—21no—
256C24—21no—
257C24—21no—
258C24—21no—
259C24#12112yes—
260C24#13714yes—
261C24C22×D411yescharacteristic
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C22×C4—21no—
291C22×C4—21no—
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—24no—
303C22×C4—24no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4#12112yes—
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D412yes—
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4112yes—
368C22×C4C2×#4111yescharacteristic
369C22×C4H(5,𝔽2)12yes—
370C22×C4H(5,𝔽2)12yes—
371Q16—416no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—416no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—42no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—42no—
391C2×C8#12014yes—
392C2×C8#12314yes—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—416no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—88no—
407C22⋊C4—44no—
408C22⋊C4—416no—
409C22⋊C4—44no—
410C22⋊C4—88no—
411C42—41no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42—42no—
416C42—42no—
417C23—22no—
418C23—22no—
419C23—24no—
420C23—24no—
421C23—24no—
422C23—24no—
423C23—22no—
424C23—22no—
425C23—24no—
426C23—24no—
427C23—22no—
428C23—22no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23#47014yes—
440C23C2×#12111yescharacteristic
441C23#53211yescharacteristic
442C23#53211yescharacteristic
443C23D4212yes—
444C23#54612yes—
445C23C2×#13712yes—
446C23#57512yes—
447C23#57611yescharacteristic
448C23#57811yescharacteristic
449C23#57811yescharacteristic
450C23#57811yescharacteristic
451Q8—44no—
452Q8—44no—
453Q8—44no—
454Q8—44no—
455Q8—82no—
456Q8—82no—
457Q8—82no—
458Q8—82no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—44no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4C2×#12011yescharacteristic
519C2×C4C2×#12311yescharacteristic
520C2×C4#52511yescharacteristic
521C2×C4#52512yes—
522C2×C4#52911yescharacteristic
523C2×C4#52911yescharacteristic
524C2×C4#53211yescharacteristic
525C2×C4#53212yes—
526C2×C4#53411yescharacteristic
527C2×C4#53511yescharacteristic
528C2×C4#53812yes—
529C2×C4#54012yes—
530C2×C4#54612yes—
531C2×C4#54612yes—
532C2×C4#54612yes—
533C2×C4#55312yes—
534C2×C4#55312yes—
535C2×C4#55512yes—
536C8—24no—
537C8—24no—
538C8—42no—
539C8—42no—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—44no—
544C22—44no—
545C22—44no—
546C22—44no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22#240811yescharacteristic
552C22#246412yes—
553C22#254311yescharacteristic
554C22#255512yes—
555C22#285411yescharacteristic
556C22C2×#47011yescharacteristic
557C22#305411yescharacteristic
558C22#305911yescharacteristic
559C22#306111yescharacteristic
560C22#317511yescharacteristic
561C22#317511yescharacteristic
562C22#317711yescharacteristic
563C22#317711yescharacteristic
564C22#318112yes—
565C22#318112yes—
566C22#326812yes—
567C22#326912yes—
568C22#327012yes—
569C22#327012yes—
570C22#331212yes—
571C22#331312yes—
572C22#331412yes—
573C22#331512yes—
574C4—42no—
575C4—84no—
576C4—81no—
577C4—84no—
578C4—81no—
579C4—81no—
580C4—84no—
581C4—81no—
582C4—162no—
583C4#238811yescharacteristic
584C4#240511yescharacteristic
585C4#247612yes—
586C4#250011yescharacteristic
587C4#255912yes—
588C4#263511yescharacteristic
589C2—44no—
590C2#1499611yescharacteristic
591C2#2427011yescharacteristic
592C2#2427311yescharacteristic
593C2#2697912yes—
594C2#2698112yes—
595C2#2856311yescharacteristic
596C2#2856411yescharacteristic
597C2#2893412yes—
598C2#2893512yes—
599C2#3313611yescharacteristic
600C2#3313911yescharacteristic
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
2. C2×#1935C2semidirect
5. #11328C2semidirect
6. #11256C2semidirect
7. #11024C2semidirect
8. #11024C2semidirect
10. #10928C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4672
C2×#19352
C2×#17522
#151792
#113282
#112562
#110242
#109702
#109282
#107212