Finite Group Databasebeta

← #497130

Subgroups — #497130

Order 512.

778 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2250C211yesmaximal, characteristic
2C4×#495C211yesmaximal, characteristic, complemented
3C4×#493C211yesmaximal, characteristic, complemented
4#13206C211yesmaximal, characteristic, complemented
5#13205C211yesmaximal, characteristic, complemented
6#13014C212yesmaximal, complemented
7#13013C212yesmaximal, complemented
8#13012C212yesmaximal, complemented
9#13012C212yesmaximal, complemented
10#12890C211yesmaximal, characteristic, complemented
11#12591C211yesmaximal, characteristic, complemented
12C22×SD32C2212yes—
13C22×D16C2212yes—
14C2×#495C2211yescharacteristic
15C2×#493C2211yescharacteristic
16C2×C4×Q16C2211yescharacteristic
17C2×C4×SD16—21no—
18C2×C4×D8—21no—
19C2×C4×D8C2212yes—
20C2×C4×D8C2211yescharacteristic
21C4×#127—21no—
22C4×#124—21no—
23#2250C418yescomplemented
24C2×#358C2212yes—
25C2×#357C2212yes—
26C2×#357C2212yes—
27C2×#357C2212yes—
28C2×#346C2211yescharacteristic
29C2×#346C2211yescharacteristic
30#1970—21no—
31#1970C2212yes—
32#1970C2211yescharacteristic
33#1969C2211yescharacteristic
34#1968—21no—
35#1958—21no—
36#1957—21no—
37#1846C2211yescharacteristic
38#1845C2211yescharacteristic
39C42×C8C2211yescharacteristic
40#1380C2212yes—
41#1380C2212yes—
42#1374C2212yes—
43#1374C2212yes—
44C22×Q16C2311yescharacteristic
45C22×SD16—21no—
46C22×D8—21no—
47C22×D8C2312yes—
48C22×D8C2311yescharacteristic
49C2×#127—21no—
50C2×#124—21no—
51C2×C4×Q8—21no—
52C2×C4×D4—22no—
53C2×C4×D4—21no—
54C2×C4×D4—41no—
55C2×C4×D4—41no—
56C2×SD32—42no—
57C2×SD32—42no—
58C2×SD32C2×C418yescomplemented
59C2×D16—42no—
60C2×D16—42no—
61C2×D16C2×C418yescomplemented
62C22×C16C2312yes—
63C22×C16C2312yes—
64#495—22no—
65#495C2×C414yescomplemented
66#493—22no—
67#493C2×C414yescomplemented
68C4×Q16—42no—
69C4×SD16—44no—
70C4×D8—42no—
71C4×D8—42no—
72C4×D8—42no—
73C4×D8—44no—
74C2×#107—21no—
75C2×#107C2312yes—
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#106—21no—
79C2×#103—21no—
80C2×#103—21no—
81C2×#102—22no—
82C2×#102—22no—
83C2×#102—21no—
84C2×#102—21no—
85C2×C4×C8D411yescharacteristic
86C2×C4×C8D411yescharacteristic
87C2×C4×C8D411yescharacteristic
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91#391—21no—
92#390—22no—
93#390—21no—
94#390—41no—
95#390—41no—
96#388—41no—
97#388—41no—
98#384—21no—
99#382—21no—
100C4×C22⋊C4—41no—
101C4×C22⋊C4—41no—
102C43D411yescharacteristic
103#358—42no—
104#357—42no—
105#357—42no—
106#357—42no—
107#346—22no—
108#346C2×C414yescomplemented
109C22×Q8—21no—
110C22×D4—21no—
111C22×D4—22no—
112C22×D4—41no—
113C22×D4—41no—
114C23×C4—42no—
115C23×C4—41no—
116C2×Q16—21no—
117C2×Q16—22no—
118C2×Q16—22no—
119C2×Q16C22×C412yes—
120C2×SD16—24no—
121C2×SD16—42no—
122C2×SD16—42no—
123C2×D8—24no—
124C2×D8—22no—
125C2×D8—21no—
126C2×D8—22no—
127C2×D8—22no—
128C2×D8—22no—
129C2×D8—22no—
130C2×D8—22no—
131C2×D8—22no—
132C2×D8—42no—
133C2×D8—42no—
134C2×D8C22×C414yes—
135C2×D8C22×C412yes—
136C22×C8C2×D411yescharacteristic
137C22×C8C2×D411yescharacteristic
138C22×C8C2×D411yescharacteristic
139C22×C8C2411yescharacteristic
140#127—24no—
141#127—42no—
142#124—24no—
143#124—42no—
144C4×Q8—42no—
145C4×Q8—42no—
146C4×D4—44no—
147C4×D4—42no—
148C4×D4—44no—
149C4×D4—42no—
150C4×D4—84no—
151C4×D4—84no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C42—41no—
167C2×C42D811yescharacteristic
168C2×C42D811yescharacteristic
169C2×C42D811yescharacteristic
170C2×C42D811yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174SD32—48no—
175D16—48no—
176C2×C16—22no—
177C2×C16—22no—
178C2×C16—22no—
179C2×C16—22no—
180C2×C16—22no—
181C2×C16C22×C414yes—
182#107—22no—
183#107—22no—
184#107—22no—
185#107—22no—
186#107—42no—
187#106—42no—
188#103—42no—
189#103—42no—
190#102—44no—
191#102—42no—
192#102—44no—
193#102—42no—
194C4×C8—22no—
195C4×C8—22no—
196C4×C8—22no—
197C4×C8—22no—
198C4×C8—22no—
199C4×C8C22×C412yes—
200C4×C8#4112yes—
201#95—41no—
202#95—41no—
203#95—41no—
204#95—41no—
205C24—41no—
206C24—42no—
207C2×Q8—22no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×D4—22no—
214C2×D4—24no—
215C2×D4—41no—
216C2×D4—42no—
217C2×D4—41no—
218C2×D4—42no—
219C2×D4—41no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—41no—
223C2×D4—42no—
224C2×D4—41no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4—82no—
228C2×D4—82no—
229C2×D4—82no—
230C2×D4—82no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—82no—
236C22×C4—84no—
237C22×C4—84no—
238C22×C4—82no—
239C22×C4—84no—
240C22×C4—82no—
241C22×C4C2×D811yescharacteristic
242C22×C4C2×D811yescharacteristic
243C22×C4C2×D811yescharacteristic
244C22×C4C2×D811yescharacteristic
245C22×C4C2×D811yescharacteristic
246C22×C4C2×D811yescharacteristic
247C22×C4C22×D411yescharacteristic
248Q16—24no—
249Q16—42no—
250SD16—48no—
251D8—24no—
252D8—24no—
253D8—24no—
254D8—42no—
255D8—42no—
256D8—42no—
257D8—48no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8C4×D412yes—
265C2×C8AGL(1,ℤ/8)12yes—
266C2×C8AGL(1,ℤ/8)12yes—
267C2×C8C23×C411yescharacteristic
268C2×C8C2×#4111yescharacteristic
269C2×C8H(5,𝔽2)12yes—
270C2×C8H(5,𝔽2)12yes—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C22⋊C4—44no—
281C22⋊C4—44no—
282C22⋊C4—82no—
283C22⋊C4—84no—
284C22⋊C4—82no—
285C22⋊C4—84no—
286C22⋊C4—82no—
287C22⋊C4—84no—
288C42—21no—
289C42—21no—
290C42—21no—
291C42—21no—
292C42—22no—
293C42—22no—
294C42—41no—
295C42—41no—
296C42—82no—
297C42C4×D412yes—
298C42#13512yes—
299C16—24no—
300C16—42no—
301C23—42no—
302C23—44no—
303C23—82no—
304C23—82no—
305C23—81no—
306C23—81no—
307C23—81no—
308C23—81no—
309C23—82no—
310C23—82no—
311C23—81no—
312C23—82no—
313C23—82no—
314C23—81no—
315C23C22×D811yescharacteristic
316Q8—42no—
317Q8—42no—
318Q8—81no—
319Q8—81no—
320D4—42no—
321D4—44no—
322D4—44no—
323D4—42no—
324D4—88no—
325D4—88no—
326D4—81no—
327D4—82no—
328D4—81no—
329D4—82no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—42no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—82no—
356C2×C4—164no—
357C2×C4—168no—
358C2×C4C4×D812yes—
359C2×C4C4×D812yes—
360C2×C4#50912yes—
361C2×C4#50912yes—
362C2×C4C2×C4×D411yescharacteristic
363C2×C4#53511yescharacteristic
364C2×C4#53511yescharacteristic
365C2×C4C2×#13511yescharacteristic
366C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
367C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
368C8—41no—
369C8—41no—
370C8#44214yes—
371C8#51812yes—
372C8#54012yes—
373C22—84no—
374C22—84no—
375C22—82no—
376C22—82no—
377C22—84no—
378C22—82no—
379C22—161no—
380C22—162no—
381C22—162no—
382C22—161no—
383C22—162no—
384C22—161no—
385C22—161no—
386C22—162no—
387C22C2×C4×D811yescharacteristic
388C22#320511yescharacteristic
389C22#320511yescharacteristic
390C22C2×#50911yescharacteristic
391C22C2×#50911yescharacteristic
392C22#341411yescharacteristic
393C22#341411yescharacteristic
394C4—22no—
395C4—22no—
396C4—22no—
397C4—22no—
398C4—41no—
399C4—41no—
400C4—82no—
401C4—161no—
402C4—161no—
403C4—162no—
404C4#232811yescharacteristic
405C4C2×#44211yescharacteristic
406C4#320711yescharacteristic
407C4#320711yescharacteristic
408C2—164no—
409C2—162no—
410C2#2226911yescharacteristic
411C2C4×#50911yescharacteristic
412C2C4×#50911yescharacteristic
413C2#3377911yescharacteristic
414C2#3378011yescharacteristic
415C2#3378311yescharacteristic
416C2#3378511yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#2250C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#495C2semidirect
3. C4×#493C2semidirect
4. #13206C2semidirect
5. #13205C2semidirect
6. #13014C2semidirect
7. #13013C2semidirect
8. #13012C2semidirect
9. #13012C2semidirect
10. #12890C2semidirect
11. #12591C2semidirect
23. #2250C4semidirect
58. C2×SD32C2×C4semidirect
61. C2×D16C2×C4semidirect
65. #495C2×C4semidirect
67. #493C2×C4semidirect
108. #346C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22502
C4×#4952
C4×#4932
#132062
#132052
#130142
#130132
#130122
#128902
#125912