Finite Group Databasebeta

← #404509

Subgroups — #404509

Order 512.

1394 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2345C211yesmaximal, characteristic
2C2×#2099C211yesmaximal, characteristic, complemented
3C2×#1948C211yesmaximal, characteristic, complemented
4#15135C211yesmaximal, characteristic
5#11327C212yesmaximal, complemented
6#11228C211yesmaximal, characteristic, complemented
7#11095C212yesmaximal
8#11042C211yesmaximal, characteristic
9#10884C212yesmaximal, complemented
10#10807C211yesmaximal, characteristic, complemented
11#10730C212yesmaximal
12C23×C4⋊C4C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C2×#399—21no—
17#2345—24no—
18#2345—24no—
19C2×#389C2211yescharacteristic
20C2×#389C2212yes—
21C2×#389C2211yescharacteristic
22C2×#387—21no—
23C2×#387—21no—
24C2×#384C2211yescharacteristic
25C2×#380—21no—
26C2×#380C2211yescharacteristic
27C2×#380C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4—21no—
31C2×C4×C22⋊C4C2211yescharacteristic
32#2099—24no—
33#1948—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43#1445—21no—
44#1444—21no—
45#1439—21no—
46C4⋊#95C2211yescharacteristic
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4Q814yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4D411yescharacteristic
72C22×C22⋊C4D411yescharacteristic
73C22×C42—21no—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×#106—24no—
78C2×#106—42no—
79C2×#106—42no—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—22no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—42no—
85#399—44no—
86#389—22no—
87#389—22no—
88#389—22no—
89#389—24no—
90#389—42no—
91#389D414yes—
92#387—28no—
93#387—28no—
94#384—22no—
95#384—22no—
96#380—28no—
97#380—22no—
98#380—22no—
99#380—24no—
100#380—42no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4D414yes—
105C4×C22⋊C4—28no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95D411yescharacteristic
118C2×#95D411yescharacteristic
119#340—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—24no—
125#340—24no—
126#336—22no—
127#336—42no—
128C25C2×D411yescharacteristic
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—82no—
147C23×C4—82no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×Q812yes—
154C23×C4#4112yes—
155C23×C4C2411yescharacteristic
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8#4114yes—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
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—21no—
192C2×C4⋊C4—21no—
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—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×Q812yes—
217C2×C4⋊C4C2×Q812yes—
218C2×C4⋊C4#4114yes—
219C2×C4⋊C4#4112yes—
220C2×C4⋊C4#4112yes—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—82no—
232C2×C22⋊C4SD1614yes—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—82no—
242C2×C42—82no—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245#106—48no—
246C4⋊C8—28no—
247C22⋊C8—48no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—22no—
254#95—22no—
255#95—44no—
256#95—44no—
257#95—42no—
258#95—42no—
259C24—21no—
260C24—21no—
261C24—22no—
262C24—22no—
263C24—21no—
264C24—21no—
265C24—22no—
266C24—22no—
267C24C22⋊Q812yes—
268C24C2×SD1612yes—
269C24#13712yes—
270C24C22×D411yescharacteristic
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—216no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—41no—
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—88no—
344C22×C4—88no—
345C22×C4—84no—
346C22×C4—84no—
347C22×C4#12012yes—
348C22×C4#12012yes—
349C22×C4C22⋊Q812yes—
350C22×C4C2×SD1612yes—
351C22×C4C2×SD1612yes—
352C22×C4#13712yes—
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×Q811yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4112yes—
360C22×C4C2×#4112yes—
361C22×C4H(5,𝔽2)12yes—
362C22×C42−1+412yes—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C4⋊C4—216no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—84no—
399C4⋊C4—88no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C4⋊C4—84no—
403C4⋊C4C22⋊Q814yes—
404C4⋊C4C22⋊Q814yes—
405C22⋊C4—28no—
406C22⋊C4—44no—
407C22⋊C4—48no—
408C22⋊C4—88no—
409C42—22no—
410C42—22no—
411C42—24no—
412C42—41no—
413C42—41no—
414C42—42no—
415C42—84no—
416C42—84no—
417C23—22no—
418C23—24no—
419C23—24no—
420C23—24no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—24no—
425C23—22no—
426C23—22no—
427C23—24no—
428C23—24no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—41no—
434C23—41no—
435C23—42no—
436C23—41no—
437C23—41no—
438C23—42no—
439C23#47514yes—
440C23C2×#12011yescharacteristic
441C23C2×C22⋊Q811yescharacteristic
442C23#52511yescharacteristic
443C23#54612yes—
444C23#54712yes—
445C23C22×SD1611yescharacteristic
446C23C2×#13711yescharacteristic
447C23C2×#13711yescharacteristic
448C23#57511yescharacteristic
449C23#57612yes—
450C23#57712yes—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—816no—
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#44814yes—
517C2×C4C2×C22⋊Q811yescharacteristic
518C2×C4C2×C22⋊Q811yescharacteristic
519C2×C4#52511yescharacteristic
520C2×C4#52511yescharacteristic
521C2×C4#52511yescharacteristic
522C2×C4#52912yes—
523C2×C4#53211yescharacteristic
524C2×C4#53312yes—
525C2×C4#53511yescharacteristic
526C2×C4#53611yescharacteristic
527C2×C4#54212yes—
528C2×C4#54412yes—
529C2×C4#54712yes—
530C2×C4#55012yes—
531C2×C4#55112yes—
532C2×C4#55212yes—
533C2×C4#55212yes—
534C2×C4#55312yes—
535C8—44no—
536C8—82no—
537C22—28no—
538C22—24no—
539C22—44no—
540C22—42no—
541C22—42no—
542C22—44no—
543C22—42no—
544C22—44no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22#240611yescharacteristic
549C22#243111yescharacteristic
550C22#243812yes—
551C22#247012yes—
552C22#284811yescharacteristic
553C22C2×#44811yescharacteristic
554C22#300711yescharacteristic
555C22#300811yescharacteristic
556C22#300911yescharacteristic
557C22C2×#47511yescharacteristic
558C22#307811yescharacteristic
559C22#308112yes—
560C22#310311yescharacteristic
561C22#311311yescharacteristic
562C22#311812yes—
563C22#329512yes—
564C22#330712yes—
565C22#332012yes—
566C22#333312yes—
567C22#333512yes—
568C22#334412yes—
569C22#334612yes—
570C22#335512yes—
571C4—24no—
572C4—42no—
573C4—42no—
574C4—44no—
575C4—44no—
576C4—42no—
577C4—84no—
578C4—82no—
579C4—81no—
580C4—82no—
581C4—162no—
582C4#239611yescharacteristic
583C4#244112yes—
584C4#247912yes—
585C4#259811yescharacteristic
586C4#261111yescharacteristic
587C4#262811yescharacteristic
588C2—44no—
589C2#1489311yescharacteristic
590C2#2409711yescharacteristic
591C2#2410211yescharacteristic
592C2#2556911yescharacteristic
593C2#2558211yescharacteristic
594C2#2599212yes—
595C2#2601612yes—
596C2#2716212yes—
597C2#2717712yes—
598C2#3295411yescharacteristic
599C2#3296211yescharacteristic
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×#2099C2semidirect
3. C2×#1948C2semidirect
5. #11327C2semidirect
6. #11228C2semidirect
9. #10884C2semidirect
10. #10807C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23452
C2×#20992
C2×#19482
#151352
#113272
#112282
#110952
#110422
#108842
#108072
#107302