Finite Group Databasebeta

← #440947

Subgroups — #440947

Order 512.

1466 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2454C211yesmaximal, characteristic
2C2×#2058C211yesmaximal, characteristic, complemented
3C2×#2023C211yesmaximal, characteristic
4#14155C211yesmaximal, characteristic, complemented
5#11385C212yesmaximal, complemented
6#11346C211yesmaximal, characteristic, complemented
7#11220C212yesmaximal, complemented
8#11121C212yesmaximal
9#10931C211yesmaximal, characteristic
10#10885C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#106C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18#2454—24no—
19#2454—24no—
20C2×#399—21no—
21C2×#396—21no—
22C2×#391—21no—
23C2×#391—21no—
24C2×#390C2211yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#385—21no—
29C2×#380—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#2058—24no—
33#2023—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1445—21no—
44#1445—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—42no—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125—24no—
59C2×#125—42no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—24no—
63C2×C4×D4—42no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×#106—24no—
86C2×#106—24no—
87C2×#106—42no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—42no—
91C2×#102—42no—
92C2×#102—42no—
93C2×#102Q818yescomplemented
94#399—28no—
95#396—28no—
96#391—44no—
97#391—44no—
98#390—22no—
99#390—22no—
100#390—22no—
101#390D414yes—
102#389—22no—
103#389—22no—
104#389—22no—
105#389D414yes—
106#385—28no—
107#380—28no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—42no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95D411yescharacteristic
119C2×#95D411yescharacteristic
120#340—24no—
121#340—24no—
122#340—24no—
123#340—42no—
124#340—42no—
125#336—22no—
126#336—22no—
127#336—42no—
128#336—42no—
129C25—21no—
130C22×Q8—24no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×D4—21no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4C2×D412yes—
144C22×D4C2×Q814yes—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—44no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—82no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8C2×Q814yes—
176C4⋊Q8—48no—
177C4⋊Q8—48no—
178#125—48no—
179#125—44no—
180#125—44no—
181C4×D4—48no—
182C4×D4—44no—
183C4×D4—44no—
184C2×C4⋊C4—21no—
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—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×Q814yes—
223C2×C4⋊C4#4114yes—
224C2×C4⋊C4#4114yes—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—82no—
234C2×C22⋊C4—82no—
235C2×C22⋊C4—82no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251#106—48no—
252#102—48no—
253#102—48no—
254#95—24no—
255#95—24no—
256#95—22no—
257#95—22no—
258#95—42no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—44no—
263C24—22no—
264C24—24no—
265C24—41no—
266C24—41no—
267C24—42no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24—42no—
273C24—42no—
274C24C22×D411yescharacteristic
275C2×Q8—48no—
276C2×Q8—48no—
277C2×Q8—48no—
278C2×Q8—84no—
279C2×Q8—84no—
280C2×Q8—84no—
281C2×Q8—84no—
282C2×D4—24no—
283C2×D4—22no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4C4⋊Q814yes—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—48no—
332C22×C4—48no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—88no—
357C22×C4—84no—
358C22×C4—88no—
359C22×C4#12012yes—
360C22×C4#12012yes—
361C22×C4#13512yes—
362C22×C4AGL(1,ℤ/8)12yes—
363C22×C4AGL(1,ℤ/8)12yes—
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×Q812yes—
367C22×C4C2×#4112yes—
368C22×C4C2×#4112yes—
369C22×C4H(5,𝔽2)12yes—
370C22×C4H(5,𝔽2)12yes—
371C22×C42−1+412yes—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C4⋊C4—22no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—44no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—48no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—44no—
406C4⋊C4—84no—
407C4⋊C4—84no—
408C4⋊C4#12414yes—
409C22⋊C4—48no—
410C22⋊C4—48no—
411C22⋊C4—84no—
412C22⋊C4—84no—
413C22⋊C4—84no—
414C42—24no—
415C42—24no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—42no—
421C42—42no—
422C42—84no—
423C23—24no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—42no—
429C23—42no—
430C23—44no—
431C23—44no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—82no—
437C23—81no—
438C23—81no—
439C23—82no—
440C23—81no—
441C23—81no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23C2×#12011yescharacteristic
447C23#52511yescharacteristic
448C23#53511yescharacteristic
449C23#54612yes—
450C23D4×Q812yes—
451C23C2×#13511yescharacteristic
452C23C2×AGL(1,ℤ/8)11yescharacteristic
453C23C2×AGL(1,ℤ/8)11yescharacteristic
454C23#57511yescharacteristic
455C23#57712yes—
456C23#57712yes—
457Q8—816no—
458D4—28no—
459D4—44no—
460D4—44no—
461D4—44no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—816no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4#44914yes—
524C2×C4C2×#12411yescharacteristic
525C2×C4C2×C4⋊Q811yescharacteristic
526C2×C4#53312yes—
527C2×C4#53511yescharacteristic
528C2×C4#53611yescharacteristic
529C2×C4#54212yes—
530C2×C4#54212yes—
531C2×C4#54312yes—
532C2×C4#54612yes—
533C2×C4#54612yes—
534C2×C4#54612yes—
535C2×C4D4×Q812yes—
536C2×C4#55412yes—
537C2×C4#55412yes—
538C2×C4#55612yes—
539C2×C4#56212yes—
540C8—44no—
541C8—82no—
542C22—44no—
543C22—44no—
544C22—44no—
545C22—82no—
546C22—82no—
547C22—84no—
548C22—84no—
549C22—84no—
550C22—82no—
551C22—82no—
552C22—84no—
553C22#240411yescharacteristic
554C22#243411yescharacteristic
555C22#243511yescharacteristic
556C22#246512yes—
557C22#269212yes—
558C22C2×#44911yescharacteristic
559C22#300611yescharacteristic
560C22#300911yescharacteristic
561C22#300911yescharacteristic
562C22#310511yescharacteristic
563C22#311411yescharacteristic
564C22#312612yes—
565C22#321111yescharacteristic
566C22#321711yescharacteristic
567C22#322112yes—
568C22#330412yes—
569C22#330612yes—
570C22#331112yes—
571C22#333212yes—
572C22#338112yes—
573C22#338412yes—
574C22#338412yes—
575C22#338812yes—
576C4—24no—
577C4—42no—
578C4—44no—
579C4—42no—
580C4—81no—
581C4—81no—
582C4—82no—
583C4—81no—
584C4—84no—
585C4—84no—
586C4—82no—
587C4#249212yes—
588C4#254912yes—
589C4#269212yes—
590C4#270012yes—
591C2—84no—
592C2#1567111yescharacteristic
593C2#2393611yescharacteristic
594C2#2394011yescharacteristic
595C2#2583511yescharacteristic
596C2#2585111yescharacteristic
597C2#2588711yescharacteristic
598C2#2589811yescharacteristic
599C2#2699912yes—
600C2#2701412yes—
601C2#3153112yes—
602C2#3153812yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2058C2semidirect
4. #14155C2semidirect
5. #11385C2semidirect
6. #11346C2semidirect
7. #11220C2semidirect
10. #10885C2semidirect
93. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24542
C2×#20582
C2×#20232
#141552
#113852
#113462
#112202
#111212
#109312
#108852
#107882