Finite Group Databasebeta

← #500276

Subgroups — #500276

Order 512.

956 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3169C211yesmaximal, characteristic
2C2×#2234C212yesmaximal, complemented
3#21151C211yesmaximal, characteristic, complemented
4#13452C211yesmaximal, characteristic, complemented
5#13451C211yesmaximal, characteristic, complemented
6#13407C211yesmaximal, characteristic, complemented
7#13407C211yesmaximal, characteristic, complemented
8#12992C212yesmaximal, complemented
9#12937C212yesmaximal, complemented
10#12935C212yesmaximal, complemented
11#12844C211yesmaximal, characteristic, complemented
12C2×#532—21no—
13#3169—28no—
14C2×#500C2211yescharacteristic
15C2×#493C2211yescharacteristic
16C2×#486—21no—
17C2×#466—21no—
18C2×#466C2212yes—
19C2×C4×D8—21no—
20C2×C4×D8C2212yes—
21C2×#427C2211yescharacteristic
22C2×#427C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24#2296—21no—
25#2234—28no—
26C2×#367C2212yes—
27C2×#366C2212yes—
28C2×#357C2212yes—
29C2×#357C2212yes—
30C2×#357C2212yes—
31C2×#357C2212yes—
32C2×C22⋊C16C2212yes—
33#1942C2211yescharacteristic
34#1942C2211yescharacteristic
35#1942C2211yescharacteristic
36#1942C2211yescharacteristic
37#1941—21no—
38#1941—21no—
39#1735—21no—
40#1380C2212yes—
41#1380C2212yes—
42#1374C2211yescharacteristic
43#1374C2211yescharacteristic
44#1374C2212yes—
45C22×#41—41no—
46C22×D8—21no—
47C22×D8C2312yes—
48C23×C8D411yescharacteristic
49C23×C8C2311yescharacteristic
50#532—28no—
51#532—44no—
52C2×C4⋊Q8—21no—
53C2×#127—21no—
54C2×#124—41no—
55C2×#121—22no—
56C2×#121—41no—
57C2×C4×D4—22no—
58C2×C4×D4—41no—
59C2×#117—21no—
60C2×#117—21no—
61C22×C42D411yescharacteristic
62C22×C16C2312yes—
63C22×C16C2312yes—
64#500—24no—
65#493—24no—
66#486—28no—
67#466—28no—
68#466—24no—
69#466D418yes—
70C4×D8—28no—
71C4×D8—24no—
72C4×D8—24no—
73#427—22no—
74#427—22no—
75#427—24no—
76C2×#107—21no—
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2312yes—
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#106—21no—
83C2×#106—21no—
84C2×#102—22no—
85C2×#102—22no—
86C2×C4×C8—24no—
87C2×C4×C8D411yescharacteristic
88C2×C4×C8D411yescharacteristic
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91#389—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#385—41no—
96#385—41no—
97C4×C4⋊C4—41no—
98C4×C4⋊C4—41no—
99#367—24no—
100#366—24no—
101#357—24no—
102#357—24no—
103#357—24no—
104#357—24no—
105C22⋊C16—24no—
106#340—21no—
107#340—21no—
108#340—21no—
109#340—21no—
110C2×#41—48no—
111C2×#41—84no—
112C2×#41—84no—
113C22×Q8—41no—
114C22×D4—22no—
115C22×D4—42no—
116C22×D4—41no—
117C22×D4—41no—
118C23×C4—42no—
119C23×C4D811yescharacteristic
120C23×C4D811yescharacteristic
121C23×C4C2×D411yescharacteristic
122C2×D8—24no—
123C2×D8—22no—
124C2×D8—22no—
125C2×D8—22no—
126C2×D8—22no—
127C2×D8—22no—
128C2×D8—42no—
129C2×D8—42no—
130C2×D8C2×D414yes—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—41no—
137C22×C8C2×D411yescharacteristic
138C22×C8C2×D411yescharacteristic
139C22×C8C2×D414yes—
140C22×C8C2×D411yescharacteristic
141C22×C8C2411yescharacteristic
142C4⋊Q8—24no—
143C4⋊Q8—42no—
144#127—24no—
145#127—42no—
146#124—48no—
147#121—28no—
148#121—48no—
149#121—44no—
150C4×D4—28no—
151C4×D4—44no—
152C4×D4—48no—
153#117—24no—
154#117—42no—
155#117—42no—
156#117—42no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—42no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—41no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—41no—
174C2×C42—41no—
175C2×C42D811yescharacteristic
176C2×C42D811yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C16—22no—
180C2×C16—22no—
181C2×C16—22no—
182C2×C16—22no—
183C2×C16—22no—
184C2×C16—22no—
185#107—24no—
186#107—22no—
187#107—21no—
188#107—21no—
189#107—22no—
190#107—22no—
191#107C2×D414yes—
192#107#4112yescomplemented
193#107#4112yescomplemented
194#106—24no—
195#106—42no—
196#102—28no—
197#102—28no—
198C4×C8—22no—
199C4×C8—22no—
200C4×C8—22no—
201C4×C8—22no—
202#95—41no—
203#95—41no—
204#95—41no—
205#95—41no—
206C24—42no—
207C24C2×D811yescharacteristic
208#41—832no—
209C2×Q8—44no—
210C2×Q8—82no—
211C2×Q8—82no—
212C2×D4—24no—
213C2×D4—44no—
214C2×D4—44no—
215C2×D4—48no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—82no—
222C2×D4—82no—
223C2×D4—84no—
224C2×D4—84no—
225C2×D4—82no—
226C2×D4—82no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—41no—
237C22×C4—48no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—84no—
244C22×C4—84no—
245C22×C4#12012yes—
246C22×C4C2×D811yescharacteristic
247C22×C4C2×D811yescharacteristic
248C22×C4C2×D811yescharacteristic
249C22×C4C2×D811yescharacteristic
250C22×C4C2×D811yescharacteristic
251C22×C4AGL(1,ℤ/8)12yes—
252C22×C4C22×D411yescharacteristic
253D8—24no—
254D8—24no—
255D8—48no—
256D8—42no—
257D8—42no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—24no—
263C2×C8—24no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—44no—
268C2×C8#12012yes—
269C2×C8#13712yes—
270C2×C8C22×D412yes—
271C2×C8C2×#4111yescharacteristic
272C2×C8C2×#4111yescharacteristic
273C2×C82−1+412yes—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—44no—
278C4⋊C4—42no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C4⋊C4—82no—
289C22⋊C4—48no—
290C22⋊C4—44no—
291C22⋊C4—48no—
292C22⋊C4—82no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—22no—
298C42—22no—
299C42—44no—
300C42—82no—
301C16—42no—
302C16—42no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—44no—
310C23—82no—
311C23—82no—
312C23—82no—
313C23—82no—
314C23—82no—
315C23—82no—
316C23#44712yes—
317C23C22×D811yescharacteristic
318Q8—88no—
319D4—44no—
320D4—44no—
321D4—88no—
322D4—88no—
323D4—816no—
324D4—82no—
325D4—82no—
326C2×C4—28no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—44no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—816no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4#44712yes—
360C2×C4#50812yes—
361C2×C4#50912yes—
362C2×C4#50912yes—
363C2×C4C2×#12011yescharacteristic
364C2×C4#52511yescharacteristic
365C2×C4#53611yescharacteristic
366C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
367C2×C4C2×#13711yescharacteristic
368C2×C4#57511yescharacteristic
369C8—22no—
370C8—22no—
371C8—41no—
372C8#54712yes—
373C8#54712yes—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—84no—
382C22—84no—
383C22—84no—
384C22—162no—
385C22—162no—
386C22—162no—
387C22—162no—
388C22C2×#44711yescharacteristic
389C22#310011yescharacteristic
390C22#322611yescharacteristic
391C22C2×#50811yescharacteristic
392C22C2×#50911yescharacteristic
393C22C2×#50911yescharacteristic
394C22#341411yescharacteristic
395C4—22no—
396C4—22no—
397C4—22no—
398C4—22no—
399C4—41no—
400C4—82no—
401C4—82no—
402C4—84no—
403C4—161no—
404C4—161no—
405C4#240611yescharacteristic
406C4#300711yescharacteristic
407C4#311211yescharacteristic
408C4#323411yescharacteristic
409C2—42no—
410C2—164no—
411C2#2406011yescharacteristic
412C2#3351611yescharacteristic
413C2#3351711yescharacteristic
414C2#3366611yescharacteristic
415C2#3367511yescharacteristic
416C2#3379411yescharacteristic
417C2#3380111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#3169C2
LCSG ▷ C22×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. C2×#2234C2semidirect
3. #21151C2semidirect
4. #13452C2semidirect
5. #13451C2semidirect
6. #13407C2semidirect
7. #13407C2semidirect
8. #12992C2semidirect
9. #12937C2semidirect
10. #12935C2semidirect
11. #12844C2semidirect
192. #107#41semidirect
193. #107#41semidirect

Maximal subgroups

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

subgroupindex
C2×#31692
C2×#22342
#211512
#134522
#134512
#134072
#129922
#129372
#129352
#128442