Finite Group Databasebeta

← #393502

Subgroups — #393502

Order 512.

1834 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2500C211yesmaximal, characteristic
2#33996C211yesmaximal, characteristic, complemented
3#20912C212yesmaximal
4C2×#1822C211yesmaximal, characteristic, complemented
5#10958C212yesmaximal, complemented
6#10958C212yesmaximal, complemented
7#10892C212yesmaximal, complemented
8#10891C212yesmaximal, complemented
9#10733C212yesmaximal, complemented
10C22×#121C2211yescharacteristic
11C22×#117C2212yes—
12C22×#117C2211yescharacteristic
13C2×#418C2214yes—
14C2×#407C2212yes—
15#2500—24no—
16#2500C418yescomplemented
17C2×#394—22no—
18C2×#390C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#388—22no—
21C2×#384C2211yescharacteristic
22C2×#382C2211yescharacteristic
23#2274—24no—
24#2274C2212yes—
25#2274C2212yes—
26#2273—22no—
27C4×#117—22no—
28C4×#117—24no—
29C2×C4×C22⋊C4C2211yescharacteristic
30#1875—24no—
31#1874—24no—
32#1822—24no—
33C2×#343C2212yes—
34C2×#343C2212yes—
35#1736C2214yes—
36C2×#339C2214yes—
37C2×#339C2214yes—
38C23×D4—21no—
39C23×D4C2311yescharacteristic
40C24×C4C2311yescharacteristic
41C22×M16C2314yes—
42C2×#121—22no—
43C2×#121—22no—
44C2×#121—22no—
45C2×#121—24no—
46C2×#121C2×C414yes—
47C2×#117—22no—
48C2×#117—22no—
49C2×#117—22no—
50C2×#117—22no—
51C2×#117—24no—
52C2×#117—28no—
53C2×#117—28no—
54C2×#117C2×C414yes—
55C2×#117D414yes—
56C2×#117D414yes—
57C2×#117C2314yes—
58C22×C4⋊C4—22no—
59C22×C4⋊C4D414yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C22⋊C4—22no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4D412yes—
69C22×C42—22no—
70C22×C42D412yes—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73#418—28no—
74#418—28no—
75C2×#102—28no—
76#407—24no—
77#407—24no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80#394—216no—
81#390—22no—
82#390—22no—
83#390C2×C414yescomplemented
84#390C2×C414yescomplemented
85#388—216no—
86#384—22no—
87#384C2×C414yescomplemented
88#382—22no—
89#382C2×C414yescomplemented
90C4×C4⋊C4—24no—
91C4×C4⋊C4—24no—
92C4×C4⋊C4—28no—
93C4×C4⋊C4—28no—
94C4×C22⋊C4—216no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—24no—
97C4×C22⋊C4—24no—
98C4×C22⋊C4C2×C414yes—
99#376—24no—
100C2×#95—22no—
101C2×#95—21no—
102C43—24no—
103#343—24no—
104#343—24no—
105#340—28no—
106#339—28no—
107#339—28no—
108C25—21no—
109C25C2×D411yescharacteristic
110C22×D4—24no—
111C22×D4—22no—
112C22×D4—21no—
113C22×D4—22no—
114C22×D4—24no—
115C22×D4—24no—
116C22×D4—42no—
117C22×D4—42no—
118C22×D4—84no—
119C22×D4C22×C412yes—
120C23×C4—24no—
121C23×C4—24no—
122C23×C4—21no—
123C23×C4—24no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4C22×C412yes—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D412yes—
129C23×C4C2×D412yes—
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C23×C4#4114yes—
134C23×C4C2411yescharacteristic
135C2×M16—24no—
136C2×M16—24no—
137C2×M16—24no—
138C2×M16—24no—
139C2×M16—24no—
140C2×M16—24no—
141C22×C8—24no—
142#121—28no—
143#121—28no—
144#121—44no—
145#121—44no—
146#117—28no—
147#117—28no—
148#117—216no—
149#117—28no—
150#117—28no—
151#117—28no—
152#117—28no—
153#117—24no—
154#117—24no—
155#117—216no—
156#117#4118yes—
157#117#4118yes—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—216no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—84no—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4C2×D418yes—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—48no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—84no—
191C2×C22⋊C4C22⋊C418yes—
192C2×C42—216no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—28no—
203C2×C42—28no—
204C2×C42C22⋊C418yes—
205C2×C42C22×C412yes—
206C2×C42C22×C412yes—
207#102—416no—
208C22⋊C8—44no—
209C22⋊C8—44no—
210#95—28no—
211#95—28no—
212#95—24no—
213#95—28no—
214C24—22no—
215C24—22no—
216C24—22no—
217C24—42no—
218C24—42no—
219C24—44no—
220C24—44no—
221C24—41no—
222C24—41no—
223C24—42no—
224C24—42no—
225C24C2×C22⋊C412yes—
226C24#12012yes—
227C24#12312yes—
228C24C22×D411yescharacteristic
229C2×D4—24no—
230C2×D4—24no—
231C2×D4—24no—
232C2×D4—48no—
233C2×D4—42no—
234C2×D4—44no—
235C2×D4—44no—
236C2×D4—44no—
237C2×D4—44no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—88no—
241C2×D4—88no—
242C2×D4—88no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—28no—
252C22×C4—28no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—28no—
260C22×C4—216no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—84no—
276C22×C4—84no—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C414yes—
279C22×C4C2×C22⋊C414yes—
280C22×C4C4×D414yes—
281C22×C4C4×D414yes—
282C22×C4#12011yescharacteristic
283C22×C4#12011yescharacteristic
284C22×C4#12114yes—
285C22×C4#12114yes—
286C22×C4#12312yes—
287C22×C4C23×C411yescharacteristic
288C22×C4C22×D412yes—
289C22×C4C22×D411yescharacteristic
290C22×C4C2×#4112yes—
291C22×C4C2×#4114yes—
292C22×C4C2×#4114yes—
293M16—44no—
294M16—44no—
295M16—44no—
296M16—44no—
297C2×C8—44no—
298C2×C8—44no—
299C2×C8—44no—
300C4⋊C4—216no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—28no—
306C4⋊C4—24no—
307C4⋊C4—42no—
308C4⋊C4—88no—
309C4⋊C4#123116yes—
310C22⋊C4—216no—
311C22⋊C4—28no—
312C22⋊C4—28no—
313C22⋊C4—216no—
314C22⋊C4—416no—
315C22⋊C4—48no—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C22⋊C4—88no—
319C42—216no—
320C42—28no—
321C42—28no—
322C42—28no—
323C42—28no—
324C42—24no—
325C42—24no—
326C42—42no—
327C42—42no—
328C42—88no—
329C42—88no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—24no—
335C23—24no—
336C23—24no—
337C23—22no—
338C23—22no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—44no—
344C23—44no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—84no—
349C23—84no—
350C23—82no—
351C23—82no—
352C23—88no—
353C23—88no—
354C23—82no—
355C23—82no—
356C23—82no—
357C23—82no—
358C23#38614yes—
359C23C22×C22⋊C411yescharacteristic
360C23C2×#12011yescharacteristic
361C23C2×#12311yescharacteristic
362C23#52512yes—
363C23#52512yes—
364D4—48no—
365D4—816no—
366D4—84no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—28no—
373C2×C4—28no—
374C2×C4—28no—
375C2×C4—28no—
376C2×C4—24no—
377C2×C4—216no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—42no—
381C2×C4—48no—
382C2×C4—44no—
383C2×C4—48no—
384C2×C4—44no—
385C2×C4—48no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—816no—
397C2×C4—84no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4#38612yes—
401C2×C4#38612yes—
402C2×C4C2×C4×D412yes—
403C2×C4C4×#4112yes—
404C2×C4C4×#4112yes—
405C2×C4C2×#12112yes—
406C2×C4C2×#12314yes—
407C2×C4#52514yes—
408C2×C4#52914yes—
409C2×C4#52914yes—
410C2×C4#53212yes—
411C2×C4#53212yes—
412C8—84no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—24no—
418C22—24no—
419C22—24no—
420C22—24no—
421C22—24no—
422C22—44no—
423C22—44no—
424C22—82no—
425C22—82no—
426C22—84no—
427C22—84no—
428C22—84no—
429C22—82no—
430C22—82no—
431C22—82no—
432C22—164no—
433C22—164no—
434C22C2×#38611yescharacteristic
435C22#229711yescharacteristic
436C22#229711yescharacteristic
437C22#284912yes—
438C22#285312yes—
439C22#301812yes—
440C22#302112yes—
441C4—22no—
442C4—22no—
443C4—44no—
444C4—44no—
445C4—44no—
446C4—416no—
447C4—82no—
448C4—82no—
449C4C4×#12314yes—
450C4#285414yes—
451C2—21no—
452C2—21no—
453C2—22no—
454C2—44no—
455C2—162no—
456C2#1470611yescharacteristic
457C2#2095211yescharacteristic
458C2#2095611yescharacteristic
459C2#3299814yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33996C2semidirect
4. C2×#1822C2semidirect
5. #10958C2semidirect
6. #10958C2semidirect
7. #10892C2semidirect
8. #10891C2semidirect
9. #10733C2semidirect
16. #2500C4semidirect
83. #390C2×C4semidirect
84. #390C2×C4semidirect
87. #384C2×C4semidirect
89. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25002
#339962
#209122
C2×#18222
#109582
#108922
#108912
#107332