Finite Group Databasebeta

← #496809

Subgroups — #496809

Order 512.

904 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2184C211yesmaximal, characteristic
2C4×#466C211yesmaximal, characteristic, complemented
3C2×#1970C211yesmaximal, characteristic, complemented
4#13235C212yesmaximal, complemented
5#13147C212yesmaximal, complemented
6#13130C212yesmaximal, complemented
7#13012C212yesmaximal, complemented
8#12991C211yesmaximal, characteristic, complemented
9#12919C211yesmaximal, characteristic, complemented
10#12731C211yesmaximal, characteristic, complemented
11C4×C22⋊C16C211yesmaximal, characteristic, complemented
12C23×D8C2211yescharacteristic
13C22×D16C2212yes—
14C2×#466C2211yescharacteristic
15C2×C4×D8C2211yescharacteristic, complemented
16C22×#107C2211yescharacteristic
17C22×#102—21no—
18C22×C4×C8C2211yescharacteristic
19C4×#121—21no—
20C2×#390—21no—
21#2184C418yescomplemented
22C2×#366C2212yescomplemented
23C2×#357C2212yes—
24C2×#357C2212yes—
25C2×#357C2212yes—
26C2×#357C2212yescomplemented
27C2×C22⋊C16C2211yescharacteristic
28C2×C22⋊C16C2211yescharacteristic
29C2×C4×C16C2212yescomplemented
30#1970—21no—
31#1970—21no—
32#1970C2212yes—
33#1970C2212yes—
34#1970C2211yescharacteristic, complemented
35#1942C2211yescharacteristic, complemented
36#1934—21no—
37#1893—21no—
38#1808C2211yescharacteristic
39C4×#107C2211yescharacteristic, complemented
40C4×#102—21no—
41#1380C2212yescomplemented
42#1380C2212yescomplemented
43#1374C2212yescomplemented
44C23×D4—21no—
45C22×D8—21no—
46C22×D8—22no—
47C22×D8—42no—
48C22×D8—42no—
49C22×D8C2×C414yescomplemented
50C22×D8C2311yescharacteristic
51C22×D8C2312yes—
52C23×C8D411yescharacteristic
53C23×C8C2311yescharacteristic
54C2×#121—21no—
55C2×C4×D4—21no—
56C2×C4×D4—41no—
57C22×C4⋊C4—21no—
58C22×C22⋊C4—41no—
59C22×C22⋊C4—41no—
60C22×C42D411yescharacteristic
61C2×D16—42no—
62C2×D16—42no—
63C2×D16C2×C418yescomplemented
64C22×C16C2312yes—
65C22×C16C2312yes—
66#466—22no—
67#466C2×C414yes—
68C4×D8—42no—
69C2×#107—21no—
70C2×#107—42no—
71C2×#107C2311yescharacteristic
72C2×#107C2311yescharacteristic
73C2×#107C2312yes—
74C2×#107C2311yescharacteristic
75C2×#102—21no—
76C2×#102—22no—
77C2×#102—22no—
78C2×#102—21no—
79C2×#102—21no—
80C2×#102—44no—
81C2×C4×C8—42no—
82C2×C4×C8D411yescharacteristic
83C2×C4×C8D411yescharacteristic
84C2×C4×C8C2311yescharacteristic
85C2×C4×C8C2311yescharacteristic
86#390—21no—
87#390—22no—
88#390—22no—
89#390—41no—
90#390—41no—
91#389—21no—
92#388—41no—
93#386—41no—
94#380—21no—
95C4×C4⋊C4—21no—
96C4×C22⋊C4—41no—
97#366—42no—
98#357—42no—
99#357—42no—
100#357—42no—
101#357C2×C418yescomplemented
102C22⋊C16—22no—
103C22⋊C16C2×C414yescomplemented
104C4×C16—42no—
105#340—21no—
106#340—21no—
107#340—21no—
108C25—41no—
109C22×D4—24no—
110C22×D4—21no—
111C22×D4—22no—
112C22×D4—42no—
113C22×D4—42no—
114C22×D4—42no—
115C22×D4—42no—
116C22×D4—41no—
117C22×D4—41no—
118C22×D4—42no—
119C23×C4—41no—
120C23×C4—41no—
121C23×C4D811yescharacteristic
122C23×C4D811yescharacteristic
123C23×C4C2×D411yescharacteristic
124C2×D8—24no—
125C2×D8—22no—
126C2×D8—21no—
127C2×D8—22no—
128C2×D8—22no—
129C2×D8—22no—
130C2×D8—22no—
131C2×D8—22no—
132C2×D8—22no—
133C2×D8—48no—
134C2×D8—42no—
135C2×D8—42no—
136C2×D8—48no—
137C2×D8C22×C414yes—
138C2×D8C22×C412yes—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—41no—
144C22×C8C22×C412yes—
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8#4112yes—
149C22×C8C2411yescharacteristic
150#121—24no—
151#121—42no—
152C4×D4—42no—
153C4×D4—42no—
154C4×D4—84no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—42no—
166C2×C22⋊C4—42no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—42no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—84no—
173C2×C22⋊C4—84no—
174C2×C42—41no—
175C2×C42—41no—
176C2×C42—41no—
177C2×C42D811yescharacteristic
178C2×C42D811yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181D16—48no—
182C2×C16—22no—
183C2×C16—22no—
184C2×C16—22no—
185C2×C16—22no—
186C2×C16—22no—
187C2×C16C22×C414yes—
188#107—21no—
189#107—22no—
190#107—22no—
191#107—22no—
192#107—22no—
193#107—42no—
194#107C22×C412yes—
195#102—24no—
196#102—44no—
197#102—42no—
198#102—42no—
199#102—44no—
200C4×C8—22no—
201C4×C8—22no—
202C4×C8—41no—
203C4×C8—41no—
204#95—41no—
205#95—41no—
206#95—41no—
207C24—41no—
208C24—42no—
209C24—44no—
210C24—82no—
211C24—82no—
212C24—82no—
213C24—82no—
214C24—82no—
215C24—82no—
216C24C2×D811yescharacteristic
217C2×D4—22no—
218C2×D4—24no—
219C2×D4—48no—
220C2×D4—48no—
221C2×D4—44no—
222C2×D4—41no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—42no—
232C2×D4—44no—
233C2×D4—82no—
234C2×D4—82no—
235C2×D4—84no—
236C2×D4—84no—
237C2×D4—82no—
238C2×D4—82no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4D1614yescomplemented
259C22×C4C4×D412yes—
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4#13512yes—
266C22×C4C22×D411yescharacteristic
267D8—24no—
268D8—24no—
269D8—24no—
270D8—48no—
271D8—42no—
272D8—42no—
273D8—42no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—44no—
284C2×C8C4×D412yes—
285C2×C8AGL(1,ℤ/8)12yes—
286C2×C8C23×C411yescharacteristic
287C2×C8C2×#4111yescharacteristic
288C2×C8H(5,𝔽2)12yes—
289C2×C8H(5,𝔽2)12yes—
290C4⋊C4—22no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—42no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C22⋊C4—44no—
301C22⋊C4—84no—
302C22⋊C4—82no—
303C22⋊C4—82no—
304C22⋊C4—84no—
305C22⋊C4—84no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C22⋊C4—84no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—41no—
314C42—41no—
315C42—82no—
316C16—24no—
317C16—42no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—42no—
323C23—44no—
324C23—88no—
325C23—88no—
326C23—82no—
327C23—81no—
328C23—81no—
329C23—82no—
330C23—82no—
331C23—81no—
332C23—81no—
333C23—82no—
334C23—81no—
335C23—82no—
336C23—88no—
337C23—82no—
338C23—81no—
339C23—164no—
340C23—164no—
341C23—164no—
342C23—164no—
343C23C4×D812yes—
344C23C2×D1612yes—
345C23#50812yes—
346C23C22×D811yescharacteristic
347D4—44no—
348D4—42no—
349D4—42no—
350D4—44no—
351D4—88no—
352D4—88no—
353D4—81no—
354D4—82no—
355D4—81no—
356D4—82no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—82no—
379C2×C4—81no—
380C2×C4—82no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—82no—
385C2×C4—81no—
386C2×C4—164no—
387C2×C4—164no—
388C2×C4C4×D812yes—
389C2×C4C2×D1612yes—
390C2×C4C2×D1612yes—
391C2×C4#50912yes—
392C2×C4C2×C4×D411yescharacteristic
393C2×C4#53411yescharacteristic
394C2×C4#53511yescharacteristic
395C2×C4C2×#13511yescharacteristic
396C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
397C2×C4#57511yescharacteristic
398C8—22no—
399C8—22no—
400C8—41no—
401C8#51812yes—
402C8#53812yes—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—84no—
410C22—82no—
411C22—82no—
412C22—84no—
413C22—82no—
414C22—84no—
415C22—1616no—
416C22—161no—
417C22—162no—
418C22—161no—
419C22—162no—
420C22—161no—
421C22—162no—
422C22—161no—
423C22—162no—
424C22C4×D1614yes—
425C22C2×C4×D811yescharacteristic
426C22#318611yescharacteristic
427C22#320511yescharacteristic
428C22C22×D1611yescharacteristic
429C22C2×#50811yescharacteristic
430C22C2×#50911yescharacteristic
431C22#341411yescharacteristic
432C4—22no—
433C4—22no—
434C4—22no—
435C4—22no—
436C4—41no—
437C4—82no—
438C4—161no—
439C4—161no—
440C4—162no—
441C4#232111yescharacteristic
442C4C4×AGL(1,ℤ/8)11yescharacteristic
443C4#318811yescharacteristic
444C4#321111yescharacteristic
445C2—24no—
446C2—162no—
447C2—164no—
448C2#2204011yescharacteristic
449C2C2×C4×D1611yescharacteristic
450C2C4×#50911yescharacteristic
451C2#3375211yescharacteristic
452C2#3375411yescharacteristic
453C2#3377111yescharacteristic
454C2#3377711yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#2184C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#466C2semidirect
3. C2×#1970C2semidirect
4. #13235C2semidirect
5. #13147C2semidirect
6. #13130C2semidirect
7. #13012C2semidirect
8. #12991C2semidirect
9. #12919C2semidirect
10. #12731C2semidirect
11. C4×C22⋊C16C2semidirect
15. C2×C4×D8C22semidirect
21. #2184C4semidirect
22. C2×#366C22semidirect
26. C2×#357C22semidirect
29. C2×C4×C16C22semidirect
34. #1970C22semidirect
35. #1942C22semidirect
39. C4×#107C22semidirect
41. #1380C22semidirect
42. #1380C22semidirect
43. #1374C22semidirect
49. C22×D8C2×C4semidirect
63. C2×D16C2×C4semidirect
101. #357C2×C4semidirect
103. C22⋊C16C2×C4semidirect
258. C22×C4D16semidirect

Maximal subgroups

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

subgroupindex
C2×#21842
C4×#4662
C2×#19702
#132352
#131472
#131302
#130122
#129912
#129192
#127312
C4×C22⋊C162