Finite Group Databasebeta

← #196281

Subgroups — #196281

Order 512.

1414 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic, complemented
2C2×#2326C211yesmaximal, characteristic
3#17448C212yesmaximal
4#17365C211yesmaximal, characteristic, complemented
5#9106C214yesmaximal, complemented
6#9102C214yesmaximal, complemented
7#7855C212yesmaximal, complemented
8C22×#124C411yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10C22×C4×D4C2211yescharacteristic
11#2984—24no—
12#2983—22no—
13#2983C2212yescomplemented
14#2982C2212yescomplemented
15Q8×M16—24no—
16#2920—22no—
17#2920C2212yes—
18#2920C2212yes—
19#2326—22no—
20#2326—22no—
21#2326C412yes—
22#2326C412yes—
23#2326C2212yes—
24#2326C2212yes—
25C2×#391C411yescharacteristic
26C2×#390C411yescharacteristic
27C2×#388—22no—
28C2×#385—22no—
29C2×#384C411yescharacteristic
30C2×#376C2211yescharacteristic
31#1490—24no—
32#1486—24no—
33#1478C2218yes—
34#1467—22no—
35#1463—22no—
36#1461C2212yescomplemented
37#1461C2212yescomplemented
38#1274C2218yescomplemented
39C23×Q8C2×C411yescharacteristic
40C23×D4C2×C411yescharacteristic
41C24×C4—21no—
42C2×#124—22no—
43C2×#124—28no—
44C2×#124C2×C412yes—
45C2×#124C2×C412yes—
46C2×C4×Q8—22no—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8D412yes—
53C2×C4×Q8D412yes—
54C2×C4×Q8C2311yescharacteristic
55C2×C4×Q8C2311yescharacteristic
56C2×C4×D4—21no—
57C2×C4×D4—21no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4C2×C412yes—
63C2×C4×D4C2312yes—
64C22×C4⋊C4—22no—
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4D412yes—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42D412yes—
74C22×C42C2311yescharacteristic
75#446—216no—
76#446—28no—
77C8×Q8—28no—
78#432—24no—
79#432—24no—
80#432—22no—
81#432—22no—
82#432—22no—
83#432—22no—
84#431—24no—
85#431—24no—
86#431—22no—
87#431D414yes—
88#431C2314yes—
89#431C2314yes—
90#423—24no—
91#423—22no—
92#423—22no—
93#407—22no—
94#407—22no—
95C4×M16—24no—
96C4×M16—22no—
97C4×M16D414yes—
98#391—22no—
99#391C2×C412yes—
100#391C2×C412yes—
101#390—22no—
102#390C2×C412yes—
103#390C2×C412yes—
104#388—24no—
105#388—24no—
106#388—24no—
107#388—24no—
108#385—24no—
109#385—24no—
110#385—24no—
111#385—24no—
112#384—22no—
113#384C2×C412yes—
114#384C2×C412yes—
115#376—21no—
116#376—21no—
117#376C2×C412yes—
118#376C2312yes—
119C2×#95—22no—
120C2×#95—21no—
121C2×#95—21no—
122D4⋊C8—216no—
123#324—24no—
124#324—24no—
125C25—21no—
126C22×Q8—24no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—42no—
130C22×Q8C22⋊C412yes—
131C22×Q8C22×C411yescharacteristic
132C22×Q8C22×C411yescharacteristic
133C22×D4—24no—
134C22×D4—21no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—42no—
138C22×D4C22×C412yes—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—84no—
147C23×C4C22⋊C412yes—
148C23×C4C22⋊C412yes—
149C23×C4C22×C411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C2×M16—24no—
153C2×M16—42no—
154#124—216no—
155#124—24no—
156#124—24no—
157#124—24no—
158#124M1618yescomplemented
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—24no—
164C4×Q8—24no—
165C4×Q8—24no—
166C4×Q8—24no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×D4—24no—
172C4×D4—24no—
173C4×D4—24no—
174C4×D4—24no—
175C4×D4—42no—
176C4×D4—42no—
177C4×D4—42no—
178C4×D4—42no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—84no—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C22⋊C4—216no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—84no—
209C2×C42—21no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—24no—
215C2×C42—21no—
216C2×C42—42no—
217C2×C42C22⋊C412yes—
218C2×C42C22⋊C412yes—
219C2×C42C22×C412yes—
220C2×C42C22×C412yes—
221C2×C42C22×C411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42#4112yes—
229C2×C42#4112yes—
230C2×C42C2411yescharacteristic
231C4⋊C8—28no—
232C4⋊C8—28no—
233C4⋊C8—24no—
234C4⋊C8—24no—
235C4⋊C8—24no—
236C4⋊C8—42no—
237C22⋊C8—24no—
238C22⋊C8—24no—
239C22⋊C8—24no—
240C22⋊C8—42no—
241#97—28no—
242#97—24no—
243#97—42no—
244C4×C8—28no—
245C4×C8—24no—
246C4×C8C2×D418yes—
247#95—22no—
248#95—22no—
249#95—24no—
250#95—24no—
251#95—22no—
252#95—22no—
253C24—24no—
254C24—22no—
255C24—44no—
256C24—42no—
257C24—44no—
258C24—41no—
259C24—41no—
260C24C2×C22⋊C411yescharacteristic
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—42no—
269C2×Q8—48no—
270C2×Q8—48no—
271C2×Q8C2×M1614yes—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—22no—
275C2×D4—24no—
276C2×D4—48no—
277C2×D4—48no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4C2×M1614yes—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—44no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—44no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—88no—
307C22×C4—84no—
308C22×C4—84no—
309C22×C4—88no—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C4×D412yes—
318C22×C4C4×D412yes—
319C22×C4#12011yescharacteristic
320C22×C4#12011yescharacteristic
321C22×C4#12112yes—
322C22×C4#12112yes—
323C22×C4#12311yescharacteristic
324C22×C4#12311yescharacteristic
325C22×C4C23×C411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328M16—48no—
329M16—84no—
330C2×C8—28no—
331C2×C8—44no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—42no—
341C4⋊C4—84no—
342C4⋊C4—84no—
343C22⋊C4—216no—
344C22⋊C4—216no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—42no—
349C22⋊C4—84no—
350C22⋊C4—84no—
351C42—24no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—24no—
362C42—42no—
363C42C4×D412yes—
364C42C4×D412yes—
365C42#12012yes—
366C42#12112yes—
367C42#12112yes—
368C42#12312yes—
369C42C2×M1614yes—
370C42C22×D412yes—
371C42C2×#4112yes—
372C23—21no—
373C23—21no—
374C23—22no—
375C23—22no—
376C23—24no—
377C23—44no—
378C23—44no—
379C23—44no—
380C23—416no—
381C23—44no—
382C23—42no—
383C23—42no—
384C23—82no—
385C23—88no—
386C23—82no—
387C23—82no—
388C23—82no—
389C23#38611yescharacteristic
390C23#38611yescharacteristic
391C23C22×C22⋊C411yescharacteristic
392Q8—28no—
393Q8—48no—
394Q8—44no—
395D4—28no—
396D4—48no—
397D4—44no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—48no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—88no—
434C2×C4—88no—
435C2×C4—82no—
436C2×C4—84no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4#38611yescharacteristic
440C2×C4#38611yescharacteristic
441C2×C4#38612yes—
442C2×C4#38612yes—
443C2×C4#40714yes—
444C2×C4C2×C4×D412yes—
445C2×C4C2×#12011yescharacteristic
446C2×C4C2×#12112yes—
447C2×C4C2×#12311yescharacteristic
448C2×C4C22×M1612yes—
449C2×C4#56812yes—
450C8—48no—
451C8—84no—
452C22—24no—
453C22—24no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—48no—
461C22—44no—
462C22—84no—
463C22—88no—
464C22—88no—
465C22—82no—
466C22—82no—
467C22#191012yes—
468C22#191212yes—
469C22C2×#38611yescharacteristic
470C22C2×#40711yescharacteristic
471C22#287911yescharacteristic
472C4—22no—
473C4—22no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—22no—
479C4—24no—
480C4—44no—
481C4—42no—
482C4—42no—
483C4—42no—
484C4—82no—
485C4—82no—
486C4D4×M1614yes—
487C2—21no—
488C2—21no—
489C2—22no—
490C2—22no—
491C2—84no—
492C2#2089811yescharacteristic
493C2C2×#191011yescharacteristic
494C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60248C2semidirect
4. #17365C2semidirect
5. #9106C2semidirect
6. #9102C2semidirect
7. #7855C2semidirect
13. #2983C22semidirect
14. #2982C22semidirect
36. #1461C22semidirect
37. #1461C22semidirect
38. #1274C22semidirect
158. #124M16semidirect

Maximal subgroups

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

subgroupindex
#602482
C2×#23262
#174482
#173652
#91062
#91022
#78552