Finite Group Databasebeta

← #469412

Subgroups — #469412

Order 512.

1214 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2494C211yesmaximal, characteristic
2C2×#2058C211yesmaximal, characteristic, complemented
3C2×#1987C211yesmaximal, characteristic, complemented
4#15125C211yesmaximal, characteristic
5#11268C212yesmaximal, complemented
6#11227C211yesmaximal, characteristic, complemented
7#11220C212yesmaximal, complemented
8#11199C212yesmaximal
9#11095C212yesmaximal
10#11042C211yesmaximal, characteristic
11#10807C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16#2494—24no—
17#2494—24no—
18C2×#401—21no—
19C2×#397—21no—
20C2×#393—21no—
21C2×#389C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#388—21no—
24C2×#387—21no—
25C2×#382C2211yescharacteristic
26C2×#380C2211yescharacteristic
27C2×#380C2211yescharacteristic
28C2×#380C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#2058—24no—
32#1987—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42#1446—21no—
43#1442—21no—
44#1439—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C23×Q8—21no—
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—42no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4D411yescharacteristic
68C22×C22⋊C4D411yescharacteristic
69C22×C42—21no—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C2×#106—24no—
73C2×#106—24no—
74C2×#106—42no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—42no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—42no—
79C2×C22⋊C8—42no—
80#401—28no—
81#397—28no—
82#393—44no—
83#389—22no—
84#389—22no—
85#389—22no—
86#389D414yes—
87#388—28no—
88#387—28no—
89#382—22no—
90#382—22no—
91#380—22no—
92#380—22no—
93#380—22no—
94#380—22no—
95#380—24no—
96#380—42no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4D414yes—
99C4×C22⋊C4—22no—
100C4×C22⋊C4—22no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95D411yescharacteristic
114C2×#95D411yescharacteristic
115#340—24no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—42no—
120#340—42no—
121#340—42no—
122#336—24no—
123#336—22no—
124C25C2×D411yescharacteristic
125C22×Q8—44no—
126C22×Q8—42no—
127C22×Q8—42no—
128C22×Q8—42no—
129C22×Q8—82no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2411yescharacteristic
152C22×C8—22no—
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—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8#4114yes—
166C4⋊Q8—28no—
167C4⋊Q8—48no—
168C4⋊Q8—44no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
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—24no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4#4114yes—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—82no—
221C2×C22⋊C4—82no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—24no—
229C2×C42—82no—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232#106—48no—
233C4⋊C8—44no—
234C22⋊C8—48no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—22no—
243#95—22no—
244#95—44no—
245#95—42no—
246#95—44no—
247#95—42no—
248#95—82no—
249C24—21no—
250C24—21no—
251C24—21no—
252C24—22no—
253C24—22no—
254C24—21no—
255C24—21no—
256C24—21no—
257C24—22no—
258C24—22no—
259C24AGL(1,ℤ/8)12yes—
260C24C22×D411yescharacteristic
261C2×Q8—48no—
262C2×Q8—84no—
263C2×Q8—84no—
264C2×Q8—88no—
265C2×Q8—88no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
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#12012yes—
342C22×C4#12012yes—
343C22×C4#13512yes—
344C22×C4#13712yes—
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C2×#4112yes—
348C22×C4C2×#4112yes—
349C22×C4H(5,𝔽2)12yes—
350C22×C4H(5,𝔽2)11yescharacteristic
351C22×C42−1+411yescharacteristic
352C22×C42−1+411yescharacteristic
353C22×C42−1+411yescharacteristic
354C22×C42−1+412yes—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—48no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—42no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—48no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—84no—
394C22⋊C4—44no—
395C22⋊C4—48no—
396C22⋊C4—44no—
397C22⋊C4—84no—
398C22⋊C4—88no—
399C42—22no—
400C42—22no—
401C42—24no—
402C42—41no—
403C42—41no—
404C42—42no—
405C42—84no—
406C23—24no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—24no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—42no—
418C23—42no—
419C23—41no—
420C23—42no—
421C23—41no—
422C23—41no—
423C23—42no—
424C23—41no—
425C23—41no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23C2×#12011yescharacteristic
430C23#53611yescharacteristic
431C23#53711yescharacteristic
432C23#54612yes—
433C23#54712yes—
434C23C2×#13511yescharacteristic
435C23C2×AGL(1,ℤ/8)11yescharacteristic
436C23C2×#13711yescharacteristic
437C23#57511yescharacteristic
438C23#57712yes—
439C23#57712yes—
440Q8—816no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—82no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—82no—
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#45214yes—
508C2×C4#52511yescharacteristic
509C2×C4#53211yescharacteristic
510C2×C4#53511yescharacteristic
511C2×C4#53611yescharacteristic
512C2×C4#53611yescharacteristic
513C2×C4#53611yescharacteristic
514C2×C4#53711yescharacteristic
515C2×C4#53711yescharacteristic
516C2×C4#53812yes—
517C2×C4#54112yes—
518C2×C4#54212yes—
519C2×C4#54212yes—
520C2×C4#54612yes—
521C2×C4#54812yes—
522C2×C4#55212yes—
523C2×C4#55612yes—
524C2×C4#56312yes—
525C2×C4#56312yes—
526C8—44no—
527C8—82no—
528C22—42no—
529C22—42no—
530C22—44no—
531C22—44no—
532C22—42no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—82no—
538C22—82no—
539C22#240411yescharacteristic
540C22#243211yescharacteristic
541C22#261611yescharacteristic
542C22#269812yes—
543C22#271512yes—
544C22C2×#45211yescharacteristic
545C22#300611yescharacteristic
546C22#300911yescharacteristic
547C22#301011yescharacteristic
548C22#323311yescharacteristic
549C22#323811yescharacteristic
550C22#324412yes—
551C22#324711yescharacteristic
552C22#325211yescharacteristic
553C22#326312yes—
554C22#330112yes—
555C22#331012yes—
556C22#332012yes—
557C22#332312yes—
558C22#334012yes—
559C22#334512yes—
560C22#335012yes—
561C22#335612yes—
562C4—24no—
563C4—42no—
564C4—82no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—82no—
569C4—82no—
570C4—84no—
571C4—82no—
572C4—162no—
573C4#260411yescharacteristic
574C4#263711yescharacteristic
575C4#266211yescharacteristic
576C4#270212yes—
577C4#271612yes—
578C4#282111yescharacteristic
579C2—82no—
580C2#1665611yescharacteristic
581C2#2396411yescharacteristic
582C2#2398211yescharacteristic
583C2#2570111yescharacteristic
584C2#2571411yescharacteristic
585C2#2969211yescharacteristic
586C2#2969711yescharacteristic
587C2#3169912yes—
588C2#3170612yes—
589C2#3220812yes—
590C2#3221412yes—
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×#2058C2semidirect
3. C2×#1987C2semidirect
5. #11268C2semidirect
6. #11227C2semidirect
7. #11220C2semidirect
11. #10807C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24942
C2×#20582
C2×#19872
#151252
#112682
#112272
#112202
#111992
#110952
#110422
#108072