Finite Group Databasebeta

← #416675

Subgroups — #416675

Order 512.

1112 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic
2#21171C211yesmaximal, characteristic
3#20800C211yesmaximal, characteristic
4#14687C211yesmaximal, characteristic
5#11342C214yesmaximal
6#11217C214yesmaximal
7#10857C211yesmaximal, characteristic
8#10839C211yesmaximal, characteristic
9C4×#339C211yesmaximal, characteristic
10C2×#428C2214yes—
11C2×#428C2212yes—
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2859—21no—
15#2859—21no—
16#2647—22no—
17#2571—21no—
18#2571—21no—
19#2552—21no—
20#2552—21no—
21#2442—22no—
22#2442—22no—
23#2317C2212yes—
24#2316C2212yes—
25#2285C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#382—22no—
29C4×#117C2211yescharacteristic
30C2×C43C2211yescharacteristic
31#1948—21no—
32#1948—21no—
33#1947—21no—
34#1947—21no—
35#1847—22no—
36#1845—21no—
37#1844—21no—
38C2×#343C2214yes—
39#1736C2212yes—
40C2×#339C2212yes—
41C2×#339C2212yes—
42C2×#339C2214yes—
43C4⋊#95—22no—
44C4⋊#95C2214yes—
45C22×M16C2312yes—
46C22×M16C2312yes—
47C2×#125—22no—
48C2×#125—22no—
49C2×#117C2311yescharacteristic
50C2×#117C2312yes—
51C2×#117C2311yescharacteristic
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4D412yes—
55C22×C4⋊C4D412yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C42—21no—
59C22×C42—21no—
60C22×C42C2312yes—
61C22×C42C2311yescharacteristic
62#428—28no—
63#428—44no—
64#428—44no—
65C2×#107—24no—
66C2×#107—22no—
67C2×#106—24no—
68C2×#106—22no—
69#423—22no—
70#423Q814yes—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C4×M16—22no—
74C4×M16Q814yes—
75C2×#97—21no—
76C2×C4×C8—21no—
77#400—24no—
78#400—24no—
79#398—24no—
80#398—24no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389—22no—
85#389—22no—
86#389—22no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389—22no—
91#385—24no—
92#384—22no—
93#384—21no—
94#384—21no—
95#384—21no—
96#384—21no—
97#384—21no—
98#384—21no—
99#384—21no—
100#384—21no—
101#384D411yescharacteristic
102#384D411yescharacteristic
103#382—22no—
104#382—22no—
105#382—22no—
106#382—48no—
107#382D412yes—
108#380—24no—
109#380—21no—
110#380—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4D411yescharacteristic
114C4×C4⋊C4D411yescharacteristic
115C4×C4⋊C4D411yescharacteristic
116C4×C4⋊C4D411yescharacteristic
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119#376—21no—
120C2×#95—28no—
121C43—21no—
122C43—22no—
123C43Q814yes—
124#343—28no—
125#340—22no—
126#340—22no—
127#339—24no—
128#339—24no—
129#339—44no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D414yes—
135C23×C4C2411yescharacteristic
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—24no—
141C2×M16—24no—
142C2×M16C2×Q814yes—
143C2×M16C2×Q814yes—
144C2×M16#4114yes—
145C2×M16#4114yes—
146C22×C8—22no—
147C22×C8—22no—
148#125—84no—
149#125—84no—
150#117—22no—
151#117—22no—
152#117—24no—
153#117—24no—
154#117—22no—
155#117—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C42—24no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—24no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×Q814yes—
216C2×C42C2×Q812yes—
217C2×C42C2×Q812yes—
218C2×C42C2×Q812yes—
219C2×C42#4112yes—
220#107—48no—
221#107—44no—
222#106—48no—
223#106—44no—
224C4⋊C8—24no—
225C4⋊C8—24no—
226#97—24no—
227C4×C8—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—22no—
237#95—22no—
238#95—416no—
239C24C22×D411yescharacteristic
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—21no—
254C22×C4—28no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4#12011yescharacteristic
266C22×C4#12011yescharacteristic
267C22×C4#12011yescharacteristic
268C22×C4#12011yescharacteristic
269C22×C4C22⋊Q812yes—
270C22×C4C22⋊Q812yes—
271C22×C4C4⋊Q812yes—
272C22×C4C4⋊Q812yes—
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×Q811yescharacteristic
276C22×C4C22×Q811yescharacteristic
277C22×C4C22×Q811yescharacteristic
278C22×C4C2×#4111yescharacteristic
279C22×C4C2×#4112yes—
280C22×C4C2×#4112yes—
281C22×C4H(5,𝔽2)12yes—
282C22×C42−1+412yes—
283M16—24no—
284M16—24no—
285M16—24no—
286M16—28no—
287M16—44no—
288M16#12518yescomplemented
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—42no—
294C2×C8—42no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—84no—
322C4⋊C4—84no—
323C4⋊C4—84no—
324C4⋊C4—84no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—44no—
330C42—216no—
331C42—24no—
332C42—24no—
333C42—48no—
334C42—42no—
335C42—42no—
336C42—42no—
337C42—42no—
338C42—42no—
339C42—42no—
340C42—42no—
341C42—42no—
342C42—42no—
343C42—42no—
344C42C22⋊Q812yes—
345C42C22⋊Q812yes—
346C42C22⋊Q812yes—
347C42C22⋊Q812yes—
348C42C4⋊Q812yes—
349C42C4⋊Q812yes—
350C42C4⋊Q812yes—
351C42C4⋊Q812yes—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23C2×#12011yescharacteristic
357C23C2×C22⋊Q811yescharacteristic
358C23C2×C4⋊Q811yescharacteristic
359C23#54712yes—
360C23#55012yes—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—28no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—44no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—88no—
398C2×C4—88no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4#39512yes—
403C2×C4#39512yes—
404C2×C4#39512yes—
405C2×C4#39512yes—
406C2×C4#45312yes—
407C2×C4#45412yes—
408C2×C4#45512yes—
409C2×C4#45612yes—
410C2×C4C2×C22⋊Q811yescharacteristic
411C2×C4C2×C22⋊Q811yescharacteristic
412C2×C4C2×#12512yes—
413C2×C4C2×C4⋊Q811yescharacteristic
414C2×C4C2×C4⋊Q811yescharacteristic
415C2×C4#53211yescharacteristic
416C2×C4#53211yescharacteristic
417C2×C4#53312yes—
418C2×C4#53511yescharacteristic
419C2×C4#53611yescharacteristic
420C2×C4#54412yes—
421C2×C4#54412yes—
422C2×C4#54612yes—
423C2×C4#54812yes—
424C2×C4#55212yes—
425C2×C4#55312yes—
426C2×C4#55412yes—
427C2×C4#55712yes—
428C8—28no—
429C8—44no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22C2×#39511yescharacteristic
439C22#243111yescharacteristic
440C22#243211yescharacteristic
441C22#244112yes—
442C22#255212yes—
443C22C2×#45311yescharacteristic
444C22C2×#45511yescharacteristic
445C22C2×#45411yescharacteristic
446C22C2×#45611yescharacteristic
447C4—22no—
448C4—22no—
449C4—48no—
450C4—81no—
451C4—82no—
452C4—81no—
453C4—82no—
454C4—81no—
455C4—84no—
456C4—81no—
457C4—162no—
458C4#206014yes—
459C4#206114yes—
460C4#244212yes—
461C4#255012yes—
462C4#256511yescharacteristic
463C4#256611yescharacteristic
464C4#266011yescharacteristic
465C4#285911yescharacteristic
466C2—22no—
467C2—22no—
468C2#1522811yescharacteristic
469C2C2×#206011yescharacteristic
470C2C2×#206111yescharacteristic
471C2#2561511yescharacteristic
472C2#2561611yescharacteristic
473C2#2568711yescharacteristic
474C2#2568811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
288. M16#125semidirect

Maximal subgroups

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

subgroupindex
#352862
#211712
#208002
#146872
#113422
#112172
#108572
#108392
C4×#3392