Finite Group Databasebeta

← #404556

Subgroups — #404556

Order 512.

1376 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×#2357C211yesmaximal, characteristic
2C2×#2093C211yesmaximal, characteristic
3C2×#1948C211yesmaximal, characteristic, complemented
4#15152C211yesmaximal, characteristic, complemented
5#11325C212yesmaximal, complemented
6#11202C211yesmaximal, characteristic, complemented
7#11121C212yesmaximal
8#11076C211yesmaximal, characteristic
9#10907C212yesmaximal, complemented
10#10807C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×C4×D4C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2212yes—
18C2×#400—21no—
19#2357—24no—
20#2357—24no—
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#388—21no—
24C2×#388—21no—
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#381—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4—21no—
32#2093—24no—
33#1948—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1445—21no—
42#1442—21no—
43#1441—21no—
44C4⋊#95C2212yes—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2212yes—
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4Q814yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C42—21no—
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×#107—24no—
77C2×#107—24no—
78C2×#107—42no—
79C2×#106—24no—
80C2×#106—42no—
81C2×#106—42no—
82C2×C4⋊C8—24no—
83C2×C4⋊C8—22no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—24no—
87C2×#102—24no—
88C2×#102—42no—
89C2×#102—42no—
90#400—44no—
91#390—22no—
92#390—22no—
93#390—22no—
94#390D414yes—
95#388—28no—
96#388—28no—
97#384—22no—
98#384—22no—
99#382—22no—
100#382—22no—
101#381—28no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4D414yes—
108C4×C22⋊C4—28no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95D411yescharacteristic
120C2×#95D411yescharacteristic
121#340—24no—
122#340—24no—
123#340—24no—
124#340—42no—
125#336—24no—
126#336—22no—
127#336—24no—
128C25—21no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4C2×D412yes—
137C22×D4C2×Q812yes—
138C22×D4#4112yes—
139C23×C4—21no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—82no—
148C23×C4—82no—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
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#4114yes—
169C22×C8#4114yes—
170C4×D4—28no—
171C4×D4—28no—
172C4×D4—28no—
173C4×D4—28no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
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—21no—
187C2×C4⋊C4—21no—
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—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×Q812yes—
210C2×C4⋊C4#4112yes—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—82no—
223C2×C22⋊C4—82no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—82no—
237C2×C42—82no—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2×Q812yes—
241C2×C42#4112yes—
242#107—48no—
243#106—48no—
244C4⋊C8—28no—
245#102—48no—
246#102—48no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—22no—
252#95—22no—
253#95—42no—
254#95—44no—
255#95—44no—
256#95—42no—
257#95—82no—
258#95—82no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—42no—
263C24—42no—
264C24—41no—
265C24—41no—
266C24—41no—
267C24—41no—
268C24—42no—
269C24—42no—
270C24C22×D411yescharacteristic
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—41no—
281C2×D4—42no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4C22⋊Q814yes—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—216no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
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—88no—
345C22×C4—88no—
346C22×C4—84no—
347C22×C4—84no—
348C22×C4#12012yes—
349C22×C4#12012yes—
350C22×C4C22⋊Q812yes—
351C22×C4C22⋊Q812yes—
352C22×C4#13512yes—
353C22×C4AGL(1,ℤ/8)12yes—
354C22×C4#13712yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×Q811yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4112yes—
362C22×C4C2×#4112yes—
363C22×C4H(5,𝔽2)12yes—
364C22×C42−1+412yes—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—44no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—42no—
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—84no—
400C4⋊C4C22⋊Q814yes—
401C22⋊C4—216no—
402C22⋊C4—48no—
403C22⋊C4—48no—
404C22⋊C4—84no—
405C22⋊C4—84no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—24no—
413C42—41no—
414C42—41no—
415C42—42no—
416C42—84no—
417C42—84no—
418C23—24no—
419C23—42no—
420C23—42no—
421C23—44no—
422C23—42no—
423C23—42no—
424C23—44no—
425C23—42no—
426C23—42no—
427C23—44no—
428C23—44no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—82no—
434C23—82no—
435C23—82no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—82no—
440C23—82no—
441C23C2×#12011yescharacteristic
442C23C2×C22⋊Q811yescharacteristic
443C23#52511yescharacteristic
444C23#54712yes—
445C23#55012yes—
446C23C2×#13511yescharacteristic
447C23C2×AGL(1,ℤ/8)11yescharacteristic
448C23C2×#13711yescharacteristic
449C23#57511yescharacteristic
450C23#57612yes—
451C23#57712yes—
452D4—28no—
453D4—42no—
454D4—42no—
455D4—42no—
456D4—42no—
457D4—44no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
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—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4#44914yes—
520C2×C4C2×C22⋊Q811yescharacteristic
521C2×C4C2×C22⋊Q811yescharacteristic
522C2×C4#52511yescharacteristic
523C2×C4#52511yescharacteristic
524C2×C4#52511yescharacteristic
525C2×C4#52912yes—
526C2×C4#53211yescharacteristic
527C2×C4#53312yes—
528C2×C4#53511yescharacteristic
529C2×C4#53611yescharacteristic
530C2×C4#54212yes—
531C2×C4#54412yes—
532C2×C4#54612yes—
533C2×C4#54712yes—
534C2×C4#55112yes—
535C2×C4#55212yes—
536C2×C4#55212yes—
537C2×C4#55312yes—
538C8—44no—
539C8—44no—
540C22—44no—
541C22—44no—
542C22—44no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—84no—
548C22—84no—
549C22—84no—
550C22—84no—
551C22#240611yescharacteristic
552C22#243111yescharacteristic
553C22#244112yes—
554C22#247912yes—
555C22#284811yescharacteristic
556C22C2×#44911yescharacteristic
557C22#300711yescharacteristic
558C22#300811yescharacteristic
559C22#300911yescharacteristic
560C22#307611yescharacteristic
561C22#307911yescharacteristic
562C22#308112yes—
563C22#310711yescharacteristic
564C22#311511yescharacteristic
565C22#311812yes—
566C22#333712yes—
567C22#334512yes—
568C22#334612yes—
569C22#335512yes—
570C22#339612yes—
571C22#339912yes—
572C22#340012yes—
573C22#340112yes—
574C4—24no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—44no—
580C4—81no—
581C4—82no—
582C4—82no—
583C4—162no—
584C4—162no—
585C4#239611yescharacteristic
586C4#243812yes—
587C4#247012yes—
588C4#259811yescharacteristic
589C4#261111yescharacteristic
590C4#262811yescharacteristic
591C2—84no—
592C2#1489311yescharacteristic
593C2#2409111yescharacteristic
594C2#2410211yescharacteristic
595C2#2563311yescharacteristic
596C2#2564111yescharacteristic
597C2#2609412yes—
598C2#2610212yes—
599C2#2749712yes—
600C2#2751412yes—
601C2#3295511yescharacteristic
602C2#3296211yescharacteristic
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
3. C2×#1948C2semidirect
4. #15152C2semidirect
5. #11325C2semidirect
6. #11202C2semidirect
9. #10907C2semidirect
10. #10807C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23572
C2×#20932
C2×#19482
#151522
#113252
#112022
#111212
#110762
#109072
#108072
#107882