Finite Group Databasebeta

← #196874

Subgroups — #196874

Order 512.

1633 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36621C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic, complemented
3#20715C211yesmaximal, characteristic, complemented
4C2×#1509C212yesmaximal, complemented
5#9156C212yesmaximal, complemented
6#9155C212yesmaximal
7#9152C212yesmaximal, complemented
8#9151C212yesmaximal, complemented
9#8021C212yesmaximal, complemented
10C2×#529C2211yescharacteristic
11C22×#117—21no—
12C23×C42C2211yescharacteristic
13C2×#432C2212yes—
14#2855C2211yescharacteristic
15#2854C2211yescharacteristic
16#2853—21no—
17#2852—22no—
18#2851C412yescomplemented
19#2850C412yescomplemented
20#2849—21no—
21#2848C2211yescharacteristic
22#2847C2211yescharacteristic
23C4×C22⋊Q8C412yescomplemented
24C4×#123—21no—
25C4×#121C412yescomplemented
26C4×#120—21no—
27#2290—22no—
28#2285—28no—
29C2×#384—21no—
30#2283C2211yescharacteristic
31C2×#382C2212yes—
32C2×#380—21no—
33C2×#380C2212yes—
34C2×C4×C4⋊C4—21no—
35C2×C4×C4⋊C4C2212yes—
36#1817C2212yes—
37#1797—22no—
38#1779C2212yes—
39#1749—22no—
40#1509—28no—
41C2×#328C2214yes—
42C2×#327C2214yes—
43#1294C2214yes—
44#1294C2214yes—
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48#529—24no—
49#529—24no—
50C2×#126—21no—
51C2×#125C2×C411yescharacteristic
52C2×#124C2×C411yescharacteristic
53C2×#123—21no—
54C2×#123—21no—
55C2×C22⋊Q8C2×C412yes—
56C2×C22⋊Q8C2311yescharacteristic
57C2×#121C2×C412yes—
58C2×#121C2311yescharacteristic
59C2×#120—21no—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×D4—22no—
62C2×C4×D4—21no—
63C2×C4×D4C2×C411yescharacteristic
64C2×#117—28no—
65C2×#117—21no—
66C2×#117—44no—
67C2×#117—44no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C42—24no—
73C22×C42—24no—
74C22×C42—22no—
75C22×C42C2×C412yes—
76C22×C42C2×C412yes—
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D412yes—
80C22×C42D412yes—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#432—24no—
84#432—24no—
85C2×C4⋊C8—22no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8C2314yes—
88C2×#97—22no—
89#391C2×C412yes—
90#390C2×C412yes—
91#389C2×C412yes—
92#389C2×C412yes—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#386—22no—
102#386—22no—
103#386—22no—
104#386—22no—
105#386—21no—
106#386—21no—
107#385—22no—
108#385—22no—
109#385—22no—
110#385—22no—
111#385—21no—
112#385—21no—
113#384—28no—
114#383—21no—
115#382—28no—
116#382C2×C412yes—
117#381—21no—
118#381—21no—
119#380—28no—
120#380—28no—
121#380C2×C412yes—
122#380C2×C412yes—
123#380C2311yescharacteristic
124#380C2311yescharacteristic
125#379—21no—
126C4×C4⋊C4—28no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—24no—
129C4×C4⋊C4C2×C412yes—
130C4×C4⋊C4D418yescomplemented
131C4×C22⋊C4—22no—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—22no—
134#376—21no—
135C2×#95—22no—
136C2×#95C2314yes—
137#336—24no—
138#328—28no—
139#327—28no—
140C25C2×D411yescharacteristic
141C22×Q8C22×C411yescharacteristic
142C22×D4—22no—
143C22×D4—21no—
144C22×D4C22×C411yescharacteristic
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—21no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4C22⋊C412yes—
160C23×C4C22⋊C412yes—
161C23×C4C22⋊C412yes—
162C23×C4C22⋊C412yes—
163C23×C4C22×C412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2411yescharacteristic
171C22×C8—24no—
172#126—44no—
173#125—24no—
174#124—24no—
175#123—42no—
176#123—42no—
177#123—42no—
178#123—42no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8—44no—
182#121—22no—
183#121—22no—
184#121—44no—
185#120—42no—
186#120—42no—
187C4×Q8—24no—
188C4×D4—24no—
189C4×D4—48no—
190C4×D4—42no—
191C4×D4—42no—
192#117—432no—
193#117—44no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—48no—
205C2×C4⋊C4—48no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C412yes—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
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—21no—
239C2×C42—22no—
240C2×C42—28no—
241C2×C42—44no—
242C2×C42—44no—
243C2×C42C22⋊C412yes—
244C2×C42C22⋊C412yes—
245C2×C42C22⋊C412yes—
246C2×C42C22⋊C412yes—
247C2×C42C22×C412yes—
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C411yescharacteristic
250C2×C42C2×D412yes—
251C2×C42C2×D414yes—
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42#4114yes—
255C4⋊C8—44no—
256C22⋊C8—24no—
257C22⋊C8—24no—
258C22⋊C8—48no—
259#97—44no—
260#95—28no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—24no—
271#95—21no—
272#95—21no—
273#95C2×D418yes—
274C24—21no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—42no—
280C24—42no—
281C24C2×C22⋊C412yes—
282C24AGL(1,ℤ/8)12yes—
283C24#13712yes—
284C24C22×D411yescharacteristic
285C2×Q8—22no—
286C2×Q8—42no—
287C2×D4—22no—
288C2×D4—42no—
289C2×D4—44no—
290C2×D4—44no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—44no—
298C22×C4—28no—
299C22×C4—28no—
300C22×C4—28no—
301C22×C4—28no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—21no—
321C22×C4—21no—
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—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—28no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—48no—
343C22×C4—48no—
344C22×C4—48no—
345C22×C4—48no—
346C22×C4—48no—
347C22×C4—48no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—42no—
351C22×C4—44no—
352C22×C4C2×C22⋊C412yes—
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C412yes—
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4#12012yes—
360C22×C4#12012yes—
361C22×C4#12312yes—
362C22×C4#12312yes—
363C22×C4AGL(1,ℤ/8)12yes—
364C22×C4#13712yes—
365C22×C4C23×C411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D412yes—
368C22×C4C2×#4112yes—
369C2×C8—48no—
370C2×C8—44no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—416no—
376C4⋊C4—416no—
377C4⋊C4—416no—
378C4⋊C4—44no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C22⋊C4—44no—
386C22⋊C4—416no—
387C22⋊C4—44no—
388C22⋊C4—44no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—44no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C42—24no—
403C42—24no—
404C42—24no—
405C42—24no—
406C42—24no—
407C42—24no—
408C42—416no—
409C42—42no—
410C42—44no—
411C42C4×D414yes—
412C42C4×D414yes—
413C42#12114yes—
414C42#12114yes—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—44no—
428C23—44no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—44no—
434C23—44no—
435C23—42no—
436C23—41no—
437C23—41no—
438C23—42no—
439C23—41no—
440C23#41714yes—
441C23C22×C22⋊C411yescharacteristic
442C23C2×#12011yescharacteristic
443C23C2×#12311yescharacteristic
444C23C2×AGL(1,ℤ/8)11yescharacteristic
445C23C2×AGL(1,ℤ/8)11yescharacteristic
446C23C2×#13711yescharacteristic
447C23C2×#13711yescharacteristic
448Q8—82no—
449D4—82no—
450D4—88no—
451D4—81no—
452D4—81no—
453D4—81no—
454D4—81no—
455C2×C4—24no—
456C2×C4—216no—
457C2×C4—216no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—48no—
486C2×C4—48no—
487C2×C4—48no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—48no—
491C2×C4—48no—
492C2×C4—48no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—48no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—88no—
508C2×C4—88no—
509C2×C4—84no—
510C2×C4#38612yes—
511C2×C4#38612yes—
512C2×C4#38612yes—
513C2×C4#38612yes—
514C2×C4#41014yes—
515C2×C4#41714yes—
516C2×C4#42114yes—
517C2×C4C2×C4×D412yes—
518C2×C4C2×#12112yes—
519C8—84no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—24no—
529C22—24no—
530C22—24no—
531C22—44no—
532C22—44no—
533C22—44no—
534C22—44no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—82no—
540C22—81no—
541C22—82no—
542C22—81no—
543C22—81no—
544C22C2×#38611yescharacteristic
545C22C2×#41012yes—
546C22C2×#41711yescharacteristic
547C22C2×#41711yescharacteristic
548C22C2×#42112yes—
549C22#300311yescharacteristic
550C22#300411yescharacteristic
551C22#309211yescharacteristic
552C22#309211yescharacteristic
553C4—22no—
554C4—22no—
555C4—24no—
556C4—22no—
557C4—22no—
558C4—24no—
559C4—22no—
560C4—42no—
561C4—88no—
562C4—82no—
563C4—81no—
564C4—82no—
565C4—81no—
566C4—81no—
567C4#189714yes—
568C2—21no—
569C2—21no—
570C2—21no—
571C2—21no—
572C2—44no—
573C2—161no—
574C2#1758112yes—
575C2#1758512yes—
576C2#2090911yescharacteristic
577C2C2×#189711yescharacteristic
578C2#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. C2×#2285C2semidirect
3. #20715C2semidirect
4. C2×#1509C2semidirect
5. #9156C2semidirect
7. #9152C2semidirect
8. #9151C2semidirect
9. #8021C2semidirect
18. #2851C4semidirect
19. #2850C4semidirect
23. C4×C22⋊Q8C4semidirect
25. C4×#121C4semidirect
130. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
#366212
C2×#22852
#207152
C2×#15092
#91562
#91552
#91522
#91512
#80212