Finite Group Databasebeta

← #165429

Subgroups — #165429

Order 512.

1073 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36289C211yesmaximal, characteristic
2#21998C212yesmaximal, complemented
3#7719C212yesmaximal, complemented
4#7710C212yesmaximal, complemented
5#7670C212yesmaximal, complemented
6#7649C211yesmaximal, characteristic, complemented
7#7606C212yesmaximal, complemented
8#7601C211yesmaximal, characteristic, complemented
9#7572C212yesmaximal, complemented
10C2×#554C411yescharacteristic
11C2×D42C411yescharacteristic
12#2858C2211yescharacteristic
13#2850C2212yes—
14#2563C411yescharacteristic
15#2548C411yescharacteristic
16#2504C412yes—
17#2490—22no—
18#2472—22no—
19#2472—22no—
20#2435C412yes—
21D4×C22⋊C4—22no—
22C4×#124C2211yescharacteristic
23C2×#390C2212yes—
24C42×D4C2211yescharacteristic
25#1923—22no—
26#1916—22no—
27#1870—22no—
28#1851—22no—
29#1839C2212yescomplemented
30#1831C2212yescomplemented
31#1817C2212yescomplemented
32#1797C2212yescomplemented
33C4×C22⋊C8C2212yescomplemented
34#1728C2212yescomplemented
35#1272C2211yescharacteristic, complemented
36#1272C2212yescomplemented
37#1272C2211yescharacteristic, complemented
38#1270C2211yescharacteristic
39#1270C2212yes—
40#1270C2211yescharacteristic
41#1270C2212yescomplemented
42#1270C2212yescomplemented
43#1270C2212yescomplemented
44#1270C2212yescomplemented
45C23×D4—21no—
46C23×D4C2×C412yes—
47C22×M16—22no—
48#554—44no—
49D42—44no—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#127C2×C411yescharacteristic
52C2×#125—21no—
53C2×#124—22no—
54C2×#124C2×C411yescharacteristic
55C2×#124C2×C412yes—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8C2×C412yes—
58C2×#121—21no—
59C2×#121C2×C412yes—
60C2×#120—22no—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×D4—21no—
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4D412yes—
66C22×C4⋊C4C2×C412yes—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C42C2312yes—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C4×C8—22no—
76#397—22no—
77#397—22no—
78#394—22no—
79#394—22no—
80#394—21no—
81#394—21no—
82#393—21no—
83#393—21no—
84#391C2311yescharacteristic
85#390—22no—
86#390—22no—
87#390—22no—
88#390—22no—
89#390C2×C412yes—
90#390C2×C412yes—
91#390C2×C412yes—
92#390C2312yes—
93#390C2311yescharacteristic
94#388—22no—
95#388—22no—
96#388—22no—
97#386—22no—
98#386—22no—
99#386—22no—
100#386—22no—
101#382C2×C412yes—
102#382C2×C411yescharacteristic
103#380—22no—
104#379—22no—
105#379—22no—
106C4×C4⋊C4C2×C411yescharacteristic
107C4×C4⋊C4D412yes—
108C4×C22⋊C4—22no—
109C4×C22⋊C4C2×C412yes—
110C4×C22⋊C4D412yes—
111C4×C22⋊C4D412yes—
112C43C2311yescharacteristic
113#336—22no—
114#336—22no—
115#336—22no—
116#336—22no—
117#336—22no—
118#336C2312yes—
119#336C2312yes—
120#336C2312yes—
121#336C2312yes—
122C25—22no—
123C22×Q8—21no—
124C22×Q8C22×C411yescharacteristic
125C22×D4—24no—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4—42no—
131C22×D4—44no—
132C22×D4—44no—
133C22×D4—48no—
134C22×D4—82no—
135C22×D4C22⋊C411yescharacteristic
136C22×D4C22×C412yes—
137C22×D4C22×C411yescharacteristic
138C23×C4—22no—
139C23×C4—21no—
140C23×C4C22×C412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C2×M16—44no—
144C2×M16—82no—
145C2×M16—82no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C4⋊Q8—42no—
151#127—42no—
152#125—44no—
153#124—22no—
154#124—41no—
155#124—82no—
156#124—82no—
157#124—82no—
158C22⋊Q8—48no—
159C22⋊Q8—44no—
160#121—44no—
161#121—44no—
162#120—48no—
163C4×Q8—42no—
164C4×D4—24no—
165C4×D4—42no—
166C4×D4—44no—
167C4×D4—44no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22⋊C411yescharacteristic
175C2×C4⋊C4C22×C411yescharacteristic
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C411yescharacteristic
178C2×C22⋊C4—22no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—82no—
194C2×C22⋊C4—82no—
195C2×C22⋊C4—82no—
196C2×C22⋊C4—82no—
197C2×C22⋊C4C22⋊C412yes—
198C2×C22⋊C4C22⋊C412yes—
199C2×C22⋊C4C22⋊C412yes—
200C2×C42—21no—
201C2×C42—82no—
202C2×C42C22×C411yescharacteristic
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2411yescharacteristic
210C4⋊C8—82no—
211C4⋊C8—82no—
212C22⋊C8—44no—
213C22⋊C8—82no—
214C22⋊C8—82no—
215C4×C8—82no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95C22⋊C412yes—
221#95C22⋊C412yes—
222#95C22⋊C411yescharacteristic
223#95C22⋊C411yescharacteristic
224C24—21no—
225C24—22no—
226C24—44no—
227C24—44no—
228C24—44no—
229C24—44no—
230C24—82no—
231C24—82no—
232C24—82no—
233C24C2×C22⋊C412yes—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—42no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—82no—
240C2×D4—24no—
241C2×D4—22no—
242C2×D4—21no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—44no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—44no—
251C2×D4—42no—
252C2×D4—44no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—44no—
256C2×D4—42no—
257C2×D4—82no—
258C2×D4—82no—
259C2×D4—88no—
260C2×D4—88no—
261C2×D4—816no—
262C2×D4#13112yes—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—42no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C23×C411yescharacteristic
291C22×C4C22×D412yes—
292M16—88no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4#10014yes—
314C22⋊C4—24no—
315C22⋊C4—24no—
316C22⋊C4—42no—
317C22⋊C4—44no—
318C22⋊C4—42no—
319C22⋊C4—44no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—44no—
327C22⋊C4—82no—
328C22⋊C4—82no—
329C22⋊C4—82no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C22⋊C4—82no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C42—21no—
336C42—21no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—42no—
341C42—42no—
342C42—41no—
343C42—42no—
344C42—42no—
345C42—42no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—41no—
354C23—41no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—88no—
365C23—88no—
366C23—88no—
367C23—88no—
368C23—164no—
369C23—164no—
370C23—164no—
371C23#42114yes—
372C23C22×C22⋊C411yescharacteristic
373Q8—81no—
374Q8—81no—
375Q8—162no—
376D4—28no—
377D4—42no—
378D4—42no—
379D4—88no—
380D4—82no—
381D4—82no—
382D4—84no—
383D4—88no—
384D4—162no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—84no—
420C2×C4—164no—
421C2×C4C2×#10012yes—
422C2×C4C2×#10014yes—
423C2×C4C2×#13111yescharacteristic
424C2×C4#56811yescharacteristic
425C2×C4#57612yes—
426C2×C4#57712yes—
427C8—82no—
428C8—82no—
429C8—82no—
430C8—82no—
431C22—24no—
432C22—24no—
433C22—24no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—44no—
440C22—81no—
441C22—81no—
442C22—82no—
443C22—82no—
444C22—82no—
445C22—82no—
446C22—82no—
447C22—82no—
448C22—82no—
449C22—82no—
450C22—1616no—
451C22#288011yescharacteristic
452C22#288311yescharacteristic
453C22C22×#10011yescharacteristic
454C22#289712yes—
455C22C2×#42112yes—
456C4—22no—
457C4—21no—
458C4—21no—
459C4—24no—
460C4—22no—
461C4—22no—
462C4—42no—
463C4—41no—
464C4—42no—
465C4—42no—
466C4—81no—
467C4—81no—
468C4—81no—
469C4—82no—
470C2—44no—
471C2—82no—
472C2—162no—
473C2#1742111yescharacteristic
474C2#1752112yes—
475C2#1759712yes—
476C2#1766611yescharacteristic
477C2#1778211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×D42C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21998C2semidirect
3. #7719C2semidirect
4. #7710C2semidirect
5. #7670C2semidirect
6. #7649C2semidirect
7. #7606C2semidirect
8. #7601C2semidirect
9. #7572C2semidirect
29. #1839C22semidirect
30. #1831C22semidirect
31. #1817C22semidirect
32. #1797C22semidirect
33. C4×C22⋊C8C22semidirect
34. #1728C22semidirect
35. #1272C22semidirect
36. #1272C22semidirect
37. #1272C22semidirect
41. #1270C22semidirect
42. #1270C22semidirect
43. #1270C22semidirect
44. #1270C22semidirect

Maximal subgroups

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

subgroupindex
#362892
#219982
#77192
#77102
#76702
#76492
#76062
#76012
#75722