Finite Group Databasebeta

← #480406

Subgroups — #480406

Order 512.

1211 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#535C211yesmaximal, characteristic
2#17362C212yesmaximal, complemented
3#11531C212yesmaximal, complemented
4#11529C212yesmaximal, complemented
5#11514C212yesmaximal, complemented
6#11508C212yesmaximal, complemented
7#11481C212yesmaximal, complemented
8#11479C212yesmaximal, complemented
9C2×#535C411yescharacteristic
10C2×C4×#41C411yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C2×#407—22no—
13#2328C411yescharacteristic
14#2328C2211yescharacteristic
15#2326C411yescharacteristic
16#2326C2211yescharacteristic
17#2324—22no—
18#2321—22no—
19#2317C2211yescharacteristic
20#2311C2211yescharacteristic
21#2309C411yescharacteristic
22C4×#127C411yescharacteristic
23C4×#127C2211yescharacteristic
24C4×#124C411yescharacteristic
25C4×#124C2211yescharacteristic
26C4×#121—22no—
27C4×#120—22no—
28C4×#117C411yescharacteristic
29#1465—22no—
30#1465C2214yes—
31#1463C2214yes—
32#1462—22no—
33#1459—22no—
34#1459C2214yes—
35#1325C2214yescomplemented
36#1324C2214yescomplemented
37#1319C2214yescomplemented
38#1318C2214yescomplemented
39C22×#41C2×C411yescharacteristic
40C23×D4C2×C411yescharacteristic
41C24×C4D412yes—
42C22×M16—42no—
43#535—24no—
44#535—24no—
45C2×#127C2×C411yescharacteristic
46C2×#127C2×C411yescharacteristic
47C2×#124C2×C411yescharacteristic
48C2×#124C2×C411yescharacteristic
49C2×#121—22no—
50C2×#120—22no—
51C4×#41—42no—
52C4×#41—42no—
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—42no—
60C2×C4×D4—42no—
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4D412yes—
66C2×C4×D4D412yes—
67C2×C4×D4C2311yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117—21no—
70C2×#117C2×C411yescharacteristic
71C2×#117C2×C411yescharacteristic
72C2×#117C2×C411yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C22⋊C4D412yes—
75C22×C42—21no—
76C22×C42C2×C411yescharacteristic
77#407—24no—
78#407—24no—
79#407—42no—
80#407—42no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—42no—
83#391C2×C411yescharacteristic
84#391C2311yescharacteristic
85#390—22no—
86#390—22no—
87#390—22no—
88#390C2×C411yescharacteristic
89#390C2×C411yescharacteristic
90#390C2×C411yescharacteristic
91#390C2×C411yescharacteristic
92#390C2×C411yescharacteristic
93#390C2311yescharacteristic
94#389—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#386—22no—
100#386—22no—
101#386—22no—
102#386—22no—
103#386—22no—
104#385—22no—
105#385—22no—
106#385—22no—
107#384C2×C411yescharacteristic
108#384C2311yescharacteristic
109#382C2×C411yescharacteristic
110#382C2311yescharacteristic
111#380—22no—
112#379—22no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C22⋊C4—21no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—22no—
119#376C2×C411yescharacteristic
120#376C2×C411yescharacteristic
121C43C2×C411yescharacteristic
122C43C2×C411yescharacteristic
123#324—24no—
124#324C2318yes—
125#323—24no—
126#323—24no—
127C25C22⋊C412yes—
128C2×#41—22no—
129C2×#41—22no—
130C2×#41—42no—
131C2×#41—42no—
132C22×Q8C22×C411yescharacteristic
133C22×D4—21no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—42no—
139C22×D4—42no—
140C22×D4C22⋊C412yes—
141C22×D4C22×C411yescharacteristic
142C22×D4C22×C411yescharacteristic
143C22×D4C22×C411yescharacteristic
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—84no—
152C23×C4C22⋊C412yes—
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C412yes—
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C2×M16—44no—
161C2×M16—44no—
162C2×M16—88no—
163C22×C8—44no—
164#127—22no—
165#127—22no—
166#127—22no—
167#127M1614yescomplemented
168#124—22no—
169#124—22no—
170#124—22no—
171#124M1614yescomplemented
172#121—216no—
173#120—216no—
174C4×Q8—42no—
175C4×Q8—41no—
176C4×Q8—41no—
177C4×D4—41no—
178C4×D4—42no—
179C4×D4—41no—
180C4×D4—42no—
181C4×D4—42no—
182C4×D4—42no—
183C4×D4—42no—
184C4×D4—42no—
185C4×D4—41no—
186C4×D4—41no—
187C4×D4—41no—
188C4×D4—41no—
189C4×D4—42no—
190C4×D4—44no—
191C4×D4—42no—
192C4×D4—42no—
193C4×D4—84no—
194C4×D4—84no—
195#117—22no—
196#117—22no—
197#117—44no—
198#117—42no—
199#117—42no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2411yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4C22⋊C412yes—
228C2×C22⋊C4C22⋊C412yes—
229C2×C22⋊C4C2×D412yes—
230C2×C22⋊C4C2×D412yes—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42C22⋊C411yescharacteristic
243C2×C42C22⋊C411yescharacteristic
244C2×C42C22×C411yescharacteristic
245C2×C42C22×C411yescharacteristic
246C2×C42C22×C411yescharacteristic
247C2×C42C22×C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C411yescharacteristic
250C22⋊C8—28no—
251C22⋊C8—44no—
252C22⋊C8—44no—
253C22⋊C8—44no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262C24—21no—
263C24—22no—
264C24—22no—
265C24—24no—
266C24—42no—
267C24—82no—
268C24—82no—
269C24C2×C22⋊C412yes—
270C24C2×C22⋊C412yes—
271C24C2×C22⋊C411yescharacteristic
272#41—48no—
273#41—48no—
274C2×Q8—21no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8C2×M1612yescomplemented
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—28no—
281C2×D4—28no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—24no—
286C2×D4—42no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—48no—
294C2×D4—48no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—82no—
298C2×D4—82no—
299C2×D4—82no—
300C2×D4—82no—
301C2×D4C2×M1612yes—
302C2×D4C2×M1612yes—
303C2×D4C2×M1612yescomplemented
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—48no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—42no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—84no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4—84no—
342C22×C4—82no—
343C22×C4—84no—
344C22×C4—81no—
345C22×C4#9912yes—
346C22×C4#9912yes—
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C412yes—
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C23×C411yescharacteristic
354C22×C4C22×D412yes—
355M16—88no—
356M16—88no—
357C2×C8—48no—
358C2×C8—88no—
359C4⋊C4—24no—
360C4⋊C4—41no—
361C4⋊C4—42no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C22⋊C4—24no—
371C22⋊C4—28no—
372C22⋊C4—28no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—82no—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—22no—
392C42—22no—
393C42—21no—
394C42—21no—
395C42—41no—
396C42—41no—
397C42—42no—
398C42—42no—
399C42—42no—
400C42—42no—
401C42—41no—
402C42—41no—
403C42—81no—
404C42—81no—
405C42C2×M1612yescomplemented
406C42C2×M1612yescomplemented
407C23—21no—
408C23—22no—
409C23—21no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—41no—
416C23—41no—
417C23—44no—
418C23—44no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—44no—
425C23—42no—
426C23—44no—
427C23—41no—
428C23—41no—
429C23—82no—
430C23—82no—
431C23—88no—
432C23—88no—
433C23#40714yes—
434C23C2×#9912yes—
435C23C22×C22⋊C411yescharacteristic
436Q8—42no—
437Q8—42no—
438D4—42no—
439D4—42no—
440D4—48no—
441D4—42no—
442D4—42no—
443D4—42no—
444D4—42no—
445D4—48no—
446D4—88no—
447D4—88no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—48no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—44no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—84no—
499C2×C4—82no—
500C2×C4C2×#9911yescharacteristic
501C2×C4C2×#9912yes—
502C2×C4C2×#9911yescharacteristic
503C2×C4#41312yes—
504C2×C4C22×M1611yescharacteristic
505C2×C4C22×M1611yescharacteristic
506C8—88no—
507C22—21no—
508C22—21no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—24no—
513C22—44no—
514C22—42no—
515C22—42no—
516C22—44no—
517C22—44no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—88no—
528C22—88no—
529C22—81no—
530C22—81no—
531C22C2×#40711yescharacteristic
532C22C22×#9911yescharacteristic
533C22C2×#41311yescharacteristic
534C4—22no—
535C4—21no—
536C4—22no—
537C4—21no—
538C4—21no—
539C4—42no—
540C4—42no—
541C4—42no—
542C4—42no—
543C4—41no—
544C4—41no—
545C4—42no—
546C4—81no—
547C4—81no—
548C4—81no—
549C4—81no—
550C4#212712yes—
551C2—21no—
552C2—21no—
553C2—42no—
554C2—44no—
555C2—82no—
556C2—82no—
557C2#1736211yescharacteristic
558C2C2×#212711yescharacteristic
559C2#3336911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#535C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17362C2semidirect
3. #11531C2semidirect
4. #11529C2semidirect
5. #11514C2semidirect
6. #11508C2semidirect
7. #11481C2semidirect
8. #11479C2semidirect
35. #1325C22semidirect
36. #1324C22semidirect
37. #1319C22semidirect
38. #1318C22semidirect
167. #127M16semidirect
171. #124M16semidirect
277. C2×Q8C2×M16semidirect
303. C2×D4C2×M16semidirect
405. C42C2×M16semidirect
406. C42C2×M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5352
#173622
#115312
#115292
#115142
#115082
#114812
#114792