Finite Group Databasebeta

← #24917080

Subgroups — #24917080

Order 10624.

548 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581818C211yesmaximal, characteristic, complemented
2C2×M16×D83C211yesmaximal, characteristic, complemented
3#19128213C211yesmaximal, characteristic, complemented
4C2×#14581763C211yesmaximal, characteristic, complemented
5#19127953C212yesmaximal, complemented
6#19127946C212yesmaximal, complemented
7#19127936C212yesmaximal, complemented
8#19127929C212yesmaximal, complemented
9#19127744C211yesmaximal, characteristic, complemented
10#19127703C211yesmaximal, characteristic, complemented
11#418×C83C211yesmaximal, characteristic, complemented
12C2×#13050590C2211yescharacteristic
13C22×C22⋊D83—21no—
14C22×C4×D83C2211yescharacteristic
15#14581818—24no—
16C2×#13050578C2212yescomplemented
17C2×#13050575C2212yescomplemented
18C2×#13050574C2211yescharacteristic
19C2×#13050573C2212yescomplemented
20M16×D83—24no—
21C2×C8×D83C2212yescomplemented
22#14581776—21no—
23#14581772C2211yescharacteristic, complemented
24#14581772C2211yescharacteristic, complemented
25#14581767C2211yescharacteristic, complemented
26#14581767C2211yescharacteristic, complemented
27#14581763—21no—
28#14581763—21no—
29#14581763C2212yes—
30#14581763C2212yes—
31#14581762—21no—
32C4×D332—21no—
33C4×D332—21no—
34C2×C4⋊Dic83C2211yescharacteristic
35C2×C22⋊Dic83—21no—
36#14581725C2212yescomplemented
37#14581723C2212yescomplemented
38#14581717C2212yescomplemented
39#14581714C2212yescomplemented
40C22×D4×C83C2211yescharacteristic
41C2×M16×C83C2211yescharacteristic
42#117×C83C2211yescharacteristic
43#102×C83C2212yescomplemented
44#102×C83C2212yescomplemented
45C23×D83D411yescharacteristic
46#13050590—22no—
47#13050590—22no—
48C2×Dic166C2311yescharacteristic
49C2×D332—21no—
50C2×D332—21no—
51C2×D332C2311yescharacteristic
52C2×C22⋊D83—21no—
53C2×C22⋊D83—22no—
54C2×C22⋊D83—22no—
55C2×C22⋊D83—44no—
56C22×Dic83—21no—
57C22×Dic83D411yescharacteristic
58C2×C4×D83—22no—
59C2×C4×D83—22no—
60C2×C4×D83—21no—
61C2×C4×D83D411yescharacteristic
62C2×C4×D83D411yescharacteristic
63C2×C4×D83C2311yescharacteristic
64C2×C4×D83C2311yescharacteristic
65#13050578—24no—
66#13050575—24no—
67#13050574D414yescomplemented
68#13050573—24no—
69C2×C83⋊C8C2312yes—
70C8×D83—24no—
71C4⋊Dic83—21no—
72C4⋊Dic83—21no—
73C4⋊Dic83—21no—
74C4⋊Dic83C2312yes—
75C22⋊Dic83—22no—
76C22⋊Dic83—22no—
77(C2×D83)⋊C4—21no—
78(C2×D83)⋊C4—21no—
79(C2×D83)⋊C4—21no—
80(C2×D83)⋊C4—21no—
81C24×C83—21no—
82C2×D4×C83—22no—
83C2×D4×C83—21no—
84C2×D4×C83—22no—
85C2×D4×C83C2312yes—
86C22×C4×C83C2311yescharacteristic
87M16×C83—22no—
88C2×C8×C83C2312yes—
89C4⋊C4×C83C2311yescharacteristic
90C4⋊C4×C83C2311yescharacteristic
91C22⋊C4×C83—21no—
92C42×C83—21no—
93C23×C83—22no—
94C23×C83—42no—
95C23×C83—42no—
96C23×C83—42no—
97C23×C83C2×D411yescharacteristic
98C22×D83—22no—
99C22×D83—22no—
100C22×D83—21no—
101C22×D83—41no—
102C22×D83C2×D411yescharacteristic
103C22×D83C2×D411yescharacteristic
104D4×C83—22no—
105D4×C83—22no—
106D4×C83—42no—
107C2×C4×C83—21no—
108C2×C4×C83—21no—
109C2×C4×C83C2×D411yescharacteristic
110C2×C4×C83C2×D412yes—
111C2×C4×C83#4112yes—
112C2×C4×C83C2411yescharacteristic
113C22⋊D83—44no—
114C22⋊D83—42no—
115C22⋊D83—44no—
116C2×Dic83—22no—
117C2×Dic83—21no—
118C2×Dic83—22no—
119C2×Dic83—42no—
120C2×Dic83C2×D411yescharacteristic
121C2×Dic83C2×D411yescharacteristic
122D332—21no—
123D332—21no—
124D332—42no—
125D332—42no—
126C4×D83—22no—
127C4×D83—22no—
128C4×D83—21no—
129C4×D83—21no—
130C4×D83—21no—
131C4×D83—21no—
132C4×D83—42no—
133Dic166—21no—
134Dic166—21no—
135C8×C83—22no—
136C83⋊C8C2×D414yescomplemented
137C22×C83—21no—
138C22×C83—21no—
139C22×C83—42no—
140C22×C83—42no—
141C22×C83—42no—
142C22×C83—84no—
143C22×C83#12112yes—
144C22×C83C22×D411yescharacteristic
145C2×D83—21no—
146C2×D83—21no—
147C2×D83—21no—
148C2×D83—21no—
149C2×D83—21no—
150C2×D83—44no—
151C2×D83—41no—
152C2×D83—41no—
153C2×D83—41no—
154C2×D83AGL(1,ℤ/8)12yescomplemented
155C4×C83—41no—
156C4×C83—41no—
157C4×C83#12112yes—
158C4×C83C22×D411yescharacteristic
159C4×C83C2×#4111yescharacteristic
160Dic83—21no—
161Dic83—21no—
162Dic83—41no—
163Dic83—42no—
164C2×C83—22no—
165C2×C83—82no—
166C2×C83C2×#12111yescharacteristic
167C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C83#57511yescharacteristic
169D83—22no—
170D83—41no—
171D83—81no—
172#3055—831nomaximal, Sylow
173C83#305511yescharacteristic, complemented, Sylow
174C2×AGL(1,ℤ/8)—831no—
175C22×M16—831no—
176#525—831no—
177C2×#121—831no—
178#469—832no—
179#468—832no—
180#466—832no—
181#465—832no—
182#428—831no—
183#418D8311yescharacteristic, complemented
184#417—831no—
185C2×#41—831no—
186C22×D4—1661no—
187C22×D4C2×D8311yescharacteristic
188C23×C4—831no—
189AGL(1,ℤ/8)—1664no—
190C2×SD16—832no—
191C2×D8—832no—
192C2×M16—831no—
193C2×M16—832no—
194C2×M16—1664no—
195C2×M16C2×D8311yescharacteristic
196C22×C8—832no—
197#123—1661no—
198C22⋊Q8—831no—
199C22⋊Q8—831no—
200#121—832no—
201#121—832no—
202#121—831no—
203#121—831no—
204#121—1661no—
205#121—1661no—
206#120—1661no—
207C4×D4—1661no—
208C4×D4—1661no—
209#117C2×D8311yescharacteristic
210C2×C4⋊C4—831no—
211C2×C22⋊C4—1661no—
212#107—832no—
213#106—832no—
214#103—832no—
215#102—832no—
216#102C2×D8312yescomplemented
217#102C2×D8312yescomplemented
218C24—21no—
219C24—831no—
220#41—1662no—
221#41—1662no—
222C2×Q8—831no—
223C2×D4—22no—
224C2×D4—21no—
225C2×D4—22no—
226C2×D4—3324no—
227C2×D4—831no—
228C2×D4—1661no—
229C2×D4—1661no—
230C2×D4—1662no—
231C2×D4—1662no—
232C2×D4—1661no—
233C2×D4C22×D8312yes—
234C22×C4—831no—
235C22×C4—831no—
236C22×C4—831no—
237C22×C4—831no—
238C22×C4—831no—
239C22×C4—1661no—
240C22×C4—1661no—
241C22×C4—1662no—
242C22×C4—1662no—
243C22×C4C22×D8311yescharacteristic
244SD16—1664no—
245D8—1664no—
246M16—22no—
247M16—834no—
248M16—1664no—
249C2×C8—832no—
250C2×C8—1664no—
251C2×C8C22×D8312yes—
252C4⋊C4—832no—
253C4⋊C4—1661no—
254C4⋊C4—1661no—
255C4⋊C4—1661no—
256C4⋊C4C22×D8311yescharacteristic
257C4⋊C4C22×D8311yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—1661no—
260C22⋊C4—1661no—
261C22⋊C4—1661no—
262C22⋊C4—1661no—
263C22⋊C4—1662no—
264C22⋊C4—1662no—
265C42—21no—
266C23—22no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—3321no—
271C23—831no—
272C23—831no—
273C23—1662no—
274C23—1661no—
275C23—1662no—
276C23D4×D8311yescharacteristic
277Q8—1661no—
278Q8—1661no—
279D4—22no—
280D4—22no—
281D4—42no—
282D4—3322no—
283D4—3322no—
284D4—3324no—
285D4—3324no—
286D4—3322no—
287D4—1661no—
288D4—1661no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—3322no—
292C2×C4—3322no—
293C2×C4—831no—
294C2×C4—831no—
295C2×C4—1662no—
296C2×C4—1661no—
297C2×C4—1662no—
298C2×C4—1662no—
299C2×C4—1662no—
300C2×C4—1661no—
301C2×C4—1661no—
302C2×C4—1661no—
303C2×C4—1661no—
304C2×C4D4×D8311yescharacteristic
305C2×C4D4×D8312yes—
306C2×C4#1305059012yes—
307C2×C4C23×D8311yescharacteristic
308C8—22no—
309C8—834no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—84no—
316C22—3324no—
317C22—3321no—
318C22—3321no—
319C22—3321no—
320C22—832no—
321C22—1661no—
322C22—1661no—
323C22—1661no—
324C22—1661no—
325C22—1661no—
326C22#1458176512yes—
327C22C2×D4×D8311yescharacteristic
328C4—41no—
329C4—41no—
330C4—3321no—
331C4—3322no—
332C4—1661no—
333C4—1661no—
334C4#1458176512yes—
335C4C2×D4×D8311yescharacteristic
336C4C2×#1305059011yescharacteristic
337C2—22no—
338C2—82no—
339C2—3321no—
340C2—6641no—
341C2—1662no—
342C2C2×#1458176511yescharacteristic
343C2AGL(1,ℤ/8)×D8311yescharacteristic
344C2#1912849711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#418×C83C2
socleC22×C83C22×D4
hypercenter#418D83
hypocenterC83#3055
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#14581818C2semidirect
2. C2×M16×D83C2semidirect
3. #19128213C2semidirect
4. C2×#14581763C2semidirect
5. #19127953C2semidirect
6. #19127946C2semidirect
7. #19127936C2semidirect
8. #19127929C2semidirect
9. #19127744C2semidirect
10. #19127703C2semidirect
11. #418×C83C2semidirect
16. C2×#13050578C22semidirect
17. C2×#13050575C22semidirect
19. C2×#13050573C22semidirect
21. C2×C8×D83C22semidirect
23. #14581772C22semidirect
24. #14581772C22semidirect
25. #14581767C22semidirect
26. #14581767C22semidirect
36. #14581725C22semidirect
37. #14581723C22semidirect
38. #14581717C22semidirect
39. #14581714C22semidirect
43. #102×C83C22semidirect
44. #102×C83C22semidirect
67. #13050574D4semidirect
136. C83⋊C8C2×D4semidirect
154. C2×D83AGL(1,ℤ/8)semidirect
173. C83#3055semidirect
183. #418D83semidirect
216. #102C2×D83semidirect
217. #102C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818182
C2×M16×D832
#191282132
C2×#145817632
#191279532
#191279462
#191279362
#191279292
#191277442
#191277032
#418×C832
#305583