Finite Group Databasebeta

← #480388

Subgroups — #480388

Order 512.

1734 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#127C211yesmaximal, characteristic
2#17362C212yesmaximal, complemented
3#11529C214yesmaximal, complemented
4#11512C212yesmaximal, complemented
5#11505C212yesmaximal, complemented
6#11475C214yesmaximal, complemented
7C22×#127C411yescharacteristic
8C22×C4×D4—22no—
9C22×C4×D4C411yescharacteristic
10C22×C4×D4C2211yescharacteristic
11C2×#407—22no—
12C4×#127—22no—
13C4×#127—22no—
14C4×#127C412yes—
15C4×#127C412yes—
16C4×#127C2212yes—
17C4×#127C2212yes—
18C2×#390—22no—
19C2×#390C411yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×C43C411yescharacteristic
23#1465—22no—
24#1465C2214yescomplemented
25#1463C2214yes—
26#1462—22no—
27#1459—22no—
28#1459C2214yescomplemented
29#1325C2218yescomplemented
30#1318C2218yescomplemented
31C23×D4—22no—
32C23×D4C2×C411yescharacteristic
33C23×D4C2×C411yescharacteristic
34C24×C4—21no—
35C24×C4D412yes—
36C22×M16—42no—
37C2×#127—22no—
38C2×#127—28no—
39C2×#127C2×C412yes—
40C2×#127C2×C412yes—
41C2×C4×D4—24no—
42C2×C4×D4—24no—
43C2×C4×D4—24no—
44C2×C4×D4—24no—
45C2×C4×D4—21no—
46C2×C4×D4—21no—
47C2×C4×D4—41no—
48C2×C4×D4—41no—
49C2×C4×D4—41no—
50C2×C4×D4—41no—
51C2×C4×D4—416no—
52C2×C4×D4—41no—
53C2×C4×D4—41no—
54C2×C4×D4—41no—
55C2×C4×D4—41no—
56C2×C4×D4C2×C412yes—
57C2×C4×D4C2×C412yes—
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4D412yes—
61C2×C4×D4D412yes—
62C2×C4×D4C2311yescharacteristic
63C2×C4×D4C2311yescharacteristic
64C22×C4⋊C4—22no—
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4—22no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4D412yes—
71C22×C42—21no—
72C22×C42—21no—
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76#407—24no—
77#407—24no—
78#407—42no—
79#407—42no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—42no—
82#390—22no—
83#390—22no—
84#390—24no—
85#390—24no—
86#390—24no—
87#390—24no—
88#390C2×C412yes—
89#390C2×C412yes—
90#390C2×C412yes—
91#390C2312yes—
92#384—22no—
93#384C2×C412yes—
94#384C2312yes—
95C43—21no—
96C43—21no—
97C43C2×C412yes—
98C43C2×C412yes—
99#324—24no—
100#324C2318yes—
101#323—24no—
102#323—24no—
103C25—21no—
104C25C22⋊C412yes—
105C22×D4—216no—
106C22×D4—24no—
107C22×D4—24no—
108C22×D4—24no—
109C22×D4—24no—
110C22×D4—21no—
111C22×D4—42no—
112C22×D4—41no—
113C22×D4—41no—
114C22×D4—41no—
115C22×D4—41no—
116C22×D4—48no—
117C22×D4—48no—
118C22×D4—42no—
119C22×D4C22⋊C412yes—
120C22×D4C22×C411yescharacteristic
121C22×D4C22×C412yes—
122C22×D4C22×C411yescharacteristic
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—48no—
133C23×C4—48no—
134C23×C4—41no—
135C23×C4—41no—
136C23×C4—41no—
137C23×C4—41no—
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C412yes—
140C23×C4C22⋊C412yes—
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C2×M16—44no—
147C2×M16—44no—
148C2×M16—88no—
149C22×C8—44no—
150#127—216no—
151#127—24no—
152#127—24no—
153#127—24no—
154#127M1618yescomplemented
155C4×D4—41no—
156C4×D4—41no—
157C4×D4—41no—
158C4×D4—41no—
159C4×D4—42no—
160C4×D4—42no—
161C4×D4—42no—
162C4×D4—42no—
163C4×D4—41no—
164C4×D4—41no—
165C4×D4—41no—
166C4×D4—41no—
167C4×D4—416no—
168C4×D4—416no—
169C4×D4—416no—
170C4×D4—416no—
171C4×D4—42no—
172C4×D4—42no—
173C4×D4—42no—
174C4×D4—42no—
175C4×D4—42no—
176C4×D4—42no—
177C4×D4—42no—
178C4×D4—42no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—48no—
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2411yescharacteristic
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—48no—
200C2×C22⋊C4—48no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4C22⋊C412yes—
204C2×C22⋊C4C22⋊C412yes—
205C2×C22⋊C4C2×D412yes—
206C2×C22⋊C4C2×D412yes—
207C2×C42—28no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—21no—
215C2×C42—24no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—44no—
220C2×C42—42no—
221C2×C42C22⋊C411yescharacteristic
222C2×C42C22⋊C411yescharacteristic
223C2×C42C22×C411yescharacteristic
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C412yes—
226C2×C42C22×C412yes—
227C22⋊C8—28no—
228C22⋊C8—44no—
229C22⋊C8—44no—
230C22⋊C8—44no—
231C24—24no—
232C24—22no—
233C24—24no—
234C24—22no—
235C24—22no—
236C24—42no—
237C24—44no—
238C24—44no—
239C24—44no—
240C24—42no—
241C24—44no—
242C24—41no—
243C24—41no—
244C24C2×C22⋊C411yescharacteristic
245C24C2×C22⋊C412yes—
246C24C2×C22⋊C412yes—
247C2×D4—24no—
248C2×D4—216no—
249C2×D4—24no—
250C2×D4—216no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—24no—
254C2×D4—42no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—48no—
262C2×D4—48no—
263C2×D4—42no—
264C2×D4—48no—
265C2×D4—48no—
266C2×D4—48no—
267C2×D4—48no—
268C2×D4—48no—
269C2×D4—48no—
270C2×D4—464no—
271C2×D4—42no—
272C2×D4C2×M1614yes—
273C2×D4C2×M1614yescomplemented
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—48no—
303C22×C4—48no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—48no—
307C22×C4—42no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—84no—
317C22×C4—84no—
318C22×C4—84no—
319C22×C4—84no—
320C22×C4—84no—
321C22×C4—84no—
322C22×C4—84no—
323C22×C4#9911yescharacteristic
324C22×C4#9911yescharacteristic
325C22×C4#9911yescharacteristic
326C22×C4#9911yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C412yes—
333C22×C4C23×C411yescharacteristic
334C22×C4C22×D412yes—
335M16—88no—
336M16—88no—
337C2×C8—48no—
338C2×C8—88no—
339C4⋊C4—41no—
340C4⋊C4—42no—
341C4⋊C4—41no—
342C4⋊C4—42no—
343C4⋊C4—48no—
344C4⋊C4—48no—
345C4⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—48no—
351C22⋊C4—48no—
352C22⋊C4—48no—
353C22⋊C4—48no—
354C22⋊C4—44no—
355C22⋊C4—44no—
356C42—28no—
357C42—28no—
358C42—24no—
359C42—22no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—42no—
369C42—44no—
370C42—44no—
371C42—42no—
372C42C2×M1614yescomplemented
373C23—21no—
374C23—21no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—24no—
381C23—22no—
382C23—22no—
383C23—44no—
384C23—44no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—44no—
390C23—416no—
391C23—416no—
392C23—44no—
393C23—44no—
394C23—44no—
395C23—42no—
396C23—42no—
397C23—44no—
398C23—44no—
399C23—82no—
400C23—82no—
401C23—82no—
402C23—88no—
403C23—88no—
404C23—82no—
405C23—82no—
406C23—82no—
407C23#40714yes—
408C23C2×#9911yescharacteristic
409C23C2×#9911yescharacteristic
410C23C22×C22⋊C411yescharacteristic
411D4—44no—
412D4—48no—
413D4—44no—
414D4—432no—
415D4—432no—
416D4—44no—
417D4—44no—
418D4—48no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—216no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—21no—
425C2×C4—24no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—42no—
440C2×C4—48no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—84no—
460C2×C4—84no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—84no—
465C2×C4—84no—
466C2×C4#35314yes—
467C2×C4C2×#9911yescharacteristic
468C2×C4C2×#9911yescharacteristic
469C2×C4C2×#9911yescharacteristic
470C2×C4C2×#9911yescharacteristic
471C2×C4#41312yes—
472C2×C4C22×M1612yes—
473C8—88no—
474C22—24no—
475C22—24no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—24no—
483C22—44no—
484C22—44no—
485C22—48no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—44no—
492C22—82no—
493C22—88no—
494C22—88no—
495C22—82no—
496C22—84no—
497C22—88no—
498C22—88no—
499C22—82no—
500C22—82no—
501C22C2×#35312yes—
502C22#213212yes—
503C22C2×#40711yescharacteristic
504C22C22×#9911yescharacteristic
505C22C2×#41311yescharacteristic
506C4—28no—
507C4—21no—
508C4—21no—
509C4—22no—
510C4—22no—
511C4—44no—
512C4—42no—
513C4—42no—
514C4—41no—
515C4—41no—
516C4—82no—
517C4C2×#35312yes—
518C4C2×#35312yes—
519C2—21no—
520C2—21no—
521C2—22no—
522C2—22no—
523C2—44no—
524C2—82no—
525C2—84no—
526C2#1736211yescharacteristic
527C2C22×#35311yescharacteristic
528C2C2×#213211yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
socleC22C22×#99
prime radicalC22×#127C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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. #11529C2semidirect
4. #11512C2semidirect
5. #11505C2semidirect
6. #11475C2semidirect
24. #1465C22semidirect
28. #1459C22semidirect
29. #1325C22semidirect
30. #1318C22semidirect
154. #127M16semidirect
273. C2×D4C2×M16semidirect
372. C42C2×M16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1272
#173622
#115292
#115122
#115052
#114752