Finite Group Databasebeta

← #196860

Subgroups — #196860

Order 512.

2109 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36621C211yesmaximal, characteristic
2C22×C4×C4⋊C4C211yesmaximal, characteristic, complemented
3#20715C211yesmaximal, characteristic, complemented
4C2×#1509C212yesmaximal, complemented
5#9148C212yesmaximal, complemented
6#9147C212yesmaximal
7#9144C212yesmaximal, complemented
8#9143C212yesmaximal, complemented
9#8021C212yesmaximal, complemented
10C2×#529C2211yescharacteristic
11C23×C4⋊C4—21no—
12C23×C42—21no—
13C23×C42C2211yescharacteristic
14C2×#432C2212yes—
15#2855C2211yescharacteristic
16#2854C2211yescharacteristic
17#2853—21no—
18#2852—22no—
19#2851C412yescomplemented
20#2850C412yescomplemented
21#2849—21no—
22#2848C2211yescharacteristic
23#2847C2211yescharacteristic
24C4×C22⋊Q8C412yescomplemented
25C4×#123—21no—
26C4×#121C412yescomplemented
27C4×#120—21no—
28#2290—22no—
29#2283C2211yescharacteristic
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4—28no—
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C4⋊C4C2212yes—
35C22×#95C2212yes—
36#1817C2212yes—
37#1797—22no—
38#1779C2212yes—
39#1749—22no—
40#1509—28no—
41C2×#328C2214yes—
42C2×#327C2214yes—
43#1294C2214yes—
44#1294C2214yes—
45C24×C4—22no—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49#529—24no—
50#529—24no—
51C2×#126—21no—
52C2×#125C2×C411yescharacteristic, complemented
53C2×#124C2×C411yescharacteristic, complemented
54C2×#123—21no—
55C2×#123—21no—
56C2×C22⋊Q8C2×C412yes—
57C2×C22⋊Q8C2311yescharacteristic
58C2×#121C2×C412yes—
59C2×#121C2311yescharacteristic
60C2×#120—21no—
61C2×C4×Q8C2×C411yescharacteristic, complemented
62C2×C4×D4—22no—
63C2×C4×D4—21no—
64C2×C4×D4C2×C411yescharacteristic, complemented
65C2×#117—21no—
66C22×C4⋊C4—216no—
67C22×C4⋊C4—28no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C42—24no—
71C22×C42—24no—
72C22×C42—28no—
73C22×C42—22no—
74C22×C42—22no—
75C22×C42—44no—
76C22×C42—44no—
77C22×C42C2×C412yes—
78C22×C42C2×C412yes—
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D412yes—
82C22×C42D412yes—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85#432—24no—
86#432—24no—
87C2×C4⋊C8—22no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8C2314yes—
90C2×#97—22no—
91#391C2×C412yescomplemented
92#390C2×C412yescomplemented
93#389C2×C412yescomplemented
94#389C2×C412yescomplemented
95#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#386—22no—
104#386—22no—
105#386—22no—
106#386—22no—
107#386—21no—
108#386—21no—
109#385—22no—
110#385—22no—
111#385—22no—
112#385—22no—
113#385—21no—
114#385—21no—
115#383—21no—
116#382C2×C412yescomplemented
117#381—21no—
118#381—21no—
119#380C2×C412yes—
120#380C2×C412yes—
121#380C2311yescharacteristic
122#380C2311yescharacteristic
123#379—21no—
124C4×C4⋊C4—28no—
125C4×C4⋊C4—28no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—24no—
128C4×C4⋊C4—28no—
129C4×C4⋊C4C2×C412yescomplemented
130C4×C4⋊C4D418yescomplemented
131C4×C22⋊C4—22no—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—22no—
134#376—21no—
135C2×#95—22no—
136C2×#95—28no—
137C2×#95C2314yes—
138#336—24no—
139#328—28no—
140#327—28no—
141C25C2×D411yescharacteristic
142C22×Q8C22×C411yescharacteristic
143C22×D4—22no—
144C22×D4—21no—
145C22×D4C22×C411yescharacteristic
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—28no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—216no—
156C23×C4—28no—
157C23×C4—24no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—44no—
162C23×C4—48no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4C22⋊C412yes—
166C23×C4C22⋊C412yes—
167C23×C4C22⋊C412yes—
168C23×C4C22⋊C412yes—
169C23×C4C22×C412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2411yescharacteristic
177C22×C8—24no—
178#126—44no—
179#125—24no—
180#124—24no—
181#123—42no—
182#123—42no—
183#123—42no—
184#123—42no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C22⋊Q8—44no—
188#121—22no—
189#121—22no—
190#121—44no—
191#120—42no—
192#120—42no—
193C4×Q8—24no—
194C4×D4—24no—
195C4×D4—48no—
196C4×D4—42no—
197C4×D4—42no—
198#117—44no—
199C2×C4⋊C4—264no—
200C2×C4⋊C4—216no—
201C2×C4⋊C4—216no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C412yes—
214C2×C4⋊C4C22×C412yes—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—21no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—28no—
235C2×C42—28no—
236C2×C42—28no—
237C2×C42—28no—
238C2×C42—28no—
239C2×C42—21no—
240C2×C42—22no—
241C2×C42—28no—
242C2×C42—432no—
243C2×C42—44no—
244C2×C42—44no—
245C2×C42—44no—
246C2×C42—44no—
247C2×C42C22⋊C412yes—
248C2×C42C22⋊C412yes—
249C2×C42C22⋊C412yes—
250C2×C42C22⋊C412yes—
251C2×C42C22×C412yes—
252C2×C42C22×C411yescharacteristic
253C2×C42C22×C411yescharacteristic
254C2×C42C2×D412yes—
255C2×C42C2×D414yes—
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42#4114yes—
259C4⋊C8—44no—
260C22⋊C8—24no—
261C22⋊C8—24no—
262C22⋊C8—48no—
263#97—44no—
264#95—28no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—24no—
275#95—21no—
276#95—21no—
277#95C2×D418yes—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—24no—
283C24—24no—
284C24—21no—
285C24C2×C22⋊C412yes—
286C24AGL(1,ℤ/8)12yes—
287C24#13712yes—
288C24C22×D411yescharacteristic
289C2×Q8—22no—
290C2×Q8—42no—
291C2×D4—22no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—44no—
302C22×C4—28no—
303C22×C4—216no—
304C22×C4—28no—
305C22×C4—28no—
306C22×C4—28no—
307C22×C4—216no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—232no—
332C22×C4—216no—
333C22×C4—21no—
334C22×C4—28no—
335C22×C4—21no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—416no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—416no—
353C22×C4—432no—
354C22×C4—48no—
355C22×C4—432no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—42no—
359C22×C4—44no—
360C22×C4#10418yes—
361C22×C4C2×C22⋊C412yes—
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C412yes—
366C22×C4C2×C22⋊C412yes—
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4#12012yes—
369C22×C4#12012yes—
370C22×C4#12312yes—
371C22×C4#12312yes—
372C22×C4AGL(1,ℤ/8)12yes—
373C22×C4#13712yes—
374C22×C4C23×C411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D412yes—
377C22×C4C2×#4112yes—
378C2×C8—48no—
379C2×C8—44no—
380C4⋊C4—232no—
381C4⋊C4—232no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C42—24no—
410C42—24no—
411C42—24no—
412C42—24no—
413C42—24no—
414C42—24no—
415C42—416no—
416C42—416no—
417C42—42no—
418C42—44no—
419C42C4×D414yes—
420C42C4×D414yes—
421C42#12114yes—
422C42#12114yes—
423C23—24no—
424C23—24no—
425C23—24no—
426C23—28no—
427C23—24no—
428C23—28no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—28no—
439C23—28no—
440C23—22no—
441C23—42no—
442C23—41no—
443C23—41no—
444C23—42no—
445C23—41no—
446C23#41014yes—
447C23#41714yes—
448C23C2×#10414yes—
449C23C22×C22⋊C411yescharacteristic
450C23C2×#12011yescharacteristic
451C23C2×#12311yescharacteristic
452C23C2×AGL(1,ℤ/8)11yescharacteristic
453C23C2×AGL(1,ℤ/8)11yescharacteristic
454C23C2×#13711yescharacteristic
455C23C2×#13711yescharacteristic
456Q8—82no—
457D4—82no—
458D4—81no—
459D4—81no—
460D4—81no—
461D4—81no—
462D4—88no—
463C2×C4—24no—
464C2×C4—216no—
465C2×C4—216no—
466C2×C4—216no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—216no—
489C2×C4—24no—
490C2×C4—216no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—48no—
497C2×C4—48no—
498C2×C4—464no—
499C2×C4—416no—
500C2×C4—416no—
501C2×C4—48no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—48no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—84no—
519C2×C4#38612yes—
520C2×C4#38612yes—
521C2×C4#38612yes—
522C2×C4#38612yes—
523C2×C4#41714yes—
524C2×C4C2×#10414yes—
525C2×C4C2×#10414yes—
526C2×C4C2×C4×D412yes—
527C2×C4C2×#12112yes—
528C8—84no—
529C22—28no—
530C22—28no—
531C22—28no—
532C22—28no—
533C22—22no—
534C22—22no—
535C22—24no—
536C22—22no—
537C22—22no—
538C22—24no—
539C22—24no—
540C22—24no—
541C22—21no—
542C22—21no—
543C22—24no—
544C22—21no—
545C22—21no—
546C22—24no—
547C22—82no—
548C22—82no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22#150918yescomplemented
553C22C2×#38611yescharacteristic
554C22C2×#41012yes—
555C22C2×#41711yescharacteristic
556C22C2×#41711yescharacteristic
557C22C22×#10412yes—
558C22#300311yescharacteristic
559C22#300411yescharacteristic
560C22#309211yescharacteristic
561C22#309211yescharacteristic
562C4—22no—
563C4—22no—
564C4—24no—
565C4—22no—
566C4—24no—
567C4—22no—
568C4—22no—
569C4—416no—
570C4—42no—
571C4—82no—
572C4—82no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4#189614yes—
577C2—21no—
578C2—21no—
579C2—21no—
580C2—21no—
581C2—28no—
582C2—161no—
583C2C2×#150912yes—
584C2#1754612yes—
585C2#2090911yescharacteristic
586C2C2×#189611yescharacteristic
587C2#2098311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#2847C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×C4×C4⋊C4C2semidirect
3. #20715C2semidirect
4. C2×#1509C2semidirect
5. #9148C2semidirect
7. #9144C2semidirect
8. #9143C2semidirect
9. #8021C2semidirect
19. #2851C4semidirect
20. #2850C4semidirect
24. C4×C22⋊Q8C4semidirect
26. C4×#121C4semidirect
52. C2×#125C2×C4semidirect
53. C2×#124C2×C4semidirect
61. C2×C4×Q8C2×C4semidirect
64. C2×C4×D4C2×C4semidirect
91. #391C2×C4semidirect
92. #390C2×C4semidirect
93. #389C2×C4semidirect
94. #389C2×C4semidirect
116. #382C2×C4semidirect
129. C4×C4⋊C4C2×C4semidirect
130. C4×C4⋊C4D4semidirect
552. C22#1509semidirect

Maximal subgroups

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

subgroupindex
#366212
C22×C4×C4⋊C42
#207152
C2×#15092
#91482
#91472
#91442
#91432
#80212