Finite Group Databasebeta

← #404965

Subgroups — #404965

Order 512.

1428 conjugacy classes of subgroups.

The subgroup lattice has 609 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2357C211yesmaximal, characteristic
2C2×#2071C211yesmaximal, characteristic
3C2×#1948C211yesmaximal, characteristic, complemented
4C2×#1864C212yesmaximal
5#15151C211yesmaximal, characteristic, complemented
6#11208C212yesmaximal, complemented
7#11202C211yesmaximal, characteristic, complemented
8#11189C212yesmaximal
9#11076C211yesmaximal, characteristic
10#10908C212yesmaximal, complemented
11#10807C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2212yes—
17C2×#398—21no—
18C2×#398—21no—
19#2357—24no—
20#2357—24no—
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#389C2212yes—
24C2×#388—21no—
25C2×#388—21no—
26C2×#384C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×#381—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4—21no—
33#2071—24no—
34#1948—24no—
35#1864—28no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1445—21no—
45#1445—21no—
46C4⋊#95C2211yescharacteristic
47C4⋊#95C2212yes—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4D414yes—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C42—21no—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×#106—24no—
79C2×#106—24no—
80C2×#106—24no—
81C2×C4⋊C8—24no—
82C2×C4⋊C8—22no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—24no—
87C2×#102—24no—
88C2×#102—24no—
89#398—44no—
90#398—44no—
91#390—22no—
92#390—22no—
93#390—22no—
94#390D414yes—
95#389—24no—
96#389—24no—
97#388—28no—
98#388—28no—
99#384—22no—
100#384—22no—
101#382—22no—
102#382—22no—
103#381—28no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4D414yes—
110C4×C22⋊C4—28no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95D411yescharacteristic
121C2×#95D411yescharacteristic
122#340—24no—
123#340—24no—
124#340—24no—
125#340—24no—
126#340—24no—
127#336—24no—
128#336—22no—
129#336—24no—
130C25—21no—
131C22×D4—21no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4C2×D412yes—
139C22×D4C2×D412yes—
140C22×D4#4112yes—
141C23×C4—21no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—82no—
150C23×C4—82no—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2411yescharacteristic
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—22no—
170C22×C8—22no—
171C22×C8#4114yes—
172C4×D4—28no—
173C4×D4—28no—
174C4×D4—28no—
175C4×D4—28no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
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—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
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—41no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4#4114yes—
216C2×C4⋊C4#4112yes—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—82no—
229C2×C22⋊C4—82no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—24no—
241C2×C42—24no—
242C2×C42—82no—
243C2×C42—82no—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42#4112yes—
248#106—48no—
249C4⋊C8—28no—
250#102—216no—
251#102—48no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—22no—
257#95—22no—
258#95—42no—
259#95—44no—
260#95—44no—
261#95—42no—
262#95—44no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—42no—
267C24—42no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24—42no—
273C24—42no—
274C24C22×D411yescharacteristic
275C2×D4—22no—
276C2×D4—24no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—41no—
285C2×D4—42no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4#12114yes—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—216no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
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—88no—
349C22×C4—88no—
350C22×C4—84no—
351C22×C4—84no—
352C22×C4#12012yes—
353C22×C4#12012yes—
354C22×C4#12112yes—
355C22×C4#12112yes—
356C22×C4#13512yes—
357C22×C4AGL(1,ℤ/8)12yes—
358C22×C4AGL(1,ℤ/8)12yes—
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4112yes—
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4112yes—
367C22×C42−1+412yes—
368C22×C42−1+412yes—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—44no—
392C4⋊C4—41no—
393C4⋊C4—42no—
394C4⋊C4—41no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—44no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—84no—
403C4⋊C4—88no—
404C4⋊C4—84no—
405C4⋊C4#12114yes—
406C22⋊C4—216no—
407C22⋊C4—48no—
408C22⋊C4—48no—
409C22⋊C4—84no—
410C22⋊C4—84no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—24no—
418C42—41no—
419C42—41no—
420C42—42no—
421C42—84no—
422C42—84no—
423C23—24no—
424C23—42no—
425C23—42no—
426C23—44no—
427C23—42no—
428C23—44no—
429C23—42no—
430C23—44no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—44no—
438C23—82no—
439C23—82no—
440C23—82no—
441C23—82no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23C2×#12011yescharacteristic
447C23C2×#12111yescharacteristic
448C23#52511yescharacteristic
449C23#54712yes—
450C23#54812yes—
451C23C2×#13511yescharacteristic
452C23C2×AGL(1,ℤ/8)11yescharacteristic
453C23C2×AGL(1,ℤ/8)11yescharacteristic
454C23#57511yescharacteristic
455C23#57712yes—
456C23#57712yes—
457D4—28no—
458D4—42no—
459D4—42no—
460D4—42no—
461D4—42no—
462D4—44no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—41no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—816no—
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—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4#44914yes—
525C2×C4C2×#12111yescharacteristic
526C2×C4C2×#12111yescharacteristic
527C2×C4#52511yescharacteristic
528C2×C4#52511yescharacteristic
529C2×C4#52512yes—
530C2×C4#52511yescharacteristic
531C2×C4#52912yes—
532C2×C4#53211yescharacteristic
533C2×C4#53611yescharacteristic
534C2×C4#53611yescharacteristic
535C2×C4#53612yes—
536C2×C4#54112yes—
537C2×C4#54712yes—
538C2×C4#54712yes—
539C2×C4#55212yes—
540C2×C4#55212yes—
541C2×C4#55212yes—
542C2×C4#55612yes—
543C8—44no—
544C8—82no—
545C22—44no—
546C22—44no—
547C22—44no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—84no—
553C22—84no—
554C22—84no—
555C22—84no—
556C22#240611yescharacteristic
557C22#241512yes—
558C22#243211yescharacteristic
559C22#247512yes—
560C22#284711yescharacteristic
561C22C2×#44911yescharacteristic
562C22#300611yescharacteristic
563C22#300911yescharacteristic
564C22#300911yescharacteristic
565C22#303511yescharacteristic
566C22#303811yescharacteristic
567C22#304212yes—
568C22#310611yescharacteristic
569C22#311511yescharacteristic
570C22#312312yes—
571C22#333712yes—
572C22#334512yes—
573C22#334512yes—
574C22#335112yes—
575C22#336512yes—
576C22#337112yes—
577C22#337312yes—
578C22#337312yes—
579C4—24no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—42no—
584C4—44no—
585C4—81no—
586C4—82no—
587C4—82no—
588C4—84no—
589C4—162no—
590C4#239211yescharacteristic
591C4#241312yes—
592C4#247812yes—
593C4#259911yescharacteristic
594C4#261111yescharacteristic
595C4#262111yescharacteristic
596C2—84no—
597C2#1490711yescharacteristic
598C2#2409111yescharacteristic
599C2#2410111yescharacteristic
600C2#2468612yes—
601C2#2469212yes—
602C2#2565211yescharacteristic
603C2#2565911yescharacteristic
604C2#2736712yes—
605C2#2738612yes—
606C2#3290411yescharacteristic
607C2#3291411yescharacteristic
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
5. #15151C2semidirect
6. #11208C2semidirect
7. #11202C2semidirect
10. #10908C2semidirect
11. #10807C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23572
C2×#20712
C2×#19482
C2×#18642
#151512
#112082
#112022
#111892
#110762
#109082
#108072