Finite Group Databasebeta

← #24918516

Subgroups — #24918516

Order 10624.

420 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 109 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
1#19128307C211yesmaximal, characteristic, complemented
2#19128269C211yesmaximal, characteristic, complemented
3#19128091C211yesmaximal, characteristic, complemented
4#19128078C211yesmaximal, characteristic, complemented
5#19128067C211yesmaximal, characteristic, complemented
6#19127919C212yesmaximal, complemented
7#19127895C212yesmaximal, complemented
8#19127849C212yesmaximal, complemented
9#19127836C212yesmaximal, complemented
10#19127760C211yesmaximal, characteristic, complemented
11#488×C83C211yesmaximal, characteristic, complemented
12C2×#13050591C2211yescharacteristic
13C2×D4×D83C2211yescharacteristic
14C2×#13050577C2212yescomplemented
15C2×D664C2212yes—
16C2×D664C2212yescomplemented
17C2×#13050575C2212yes—
18#14581789C2211yescharacteristic
19#14581789C2211yescharacteristic, complemented
20#14581788C2211yescharacteristic
21#14581781C2211yescharacteristic
22#14581777—22no—
23#14581767—22no—
24#14581767C2212yes—
25#14581765—22no—
26#14581762—22no—
27#14581754C2212yescomplemented
28#14581749C2211yescharacteristic
29Dic83⋊C8C2212yescomplemented
30#14581712C2212yes—
31#14581712C2212yescomplemented
32(C2×D83)⋊C8C2212yes—
33#14581703C2211yescharacteristic, complemented
34#125×C83C2211yescharacteristic, complemented
35#124×C83C2211yescharacteristic
36#103×C83C2212yescomplemented
37#102×C83C2212yes—
38C4×C8×C83C2211yescharacteristic, complemented
39C23×D83—21no—
40#13050591—42no—
41C2×D332—21no—
42C2×D332—22no—
43C2×D332—22no—
44C2×D332—22no—
45C2×D332C2312yes—
46C2×D332C2311yescharacteristic
47C2×D332C2311yescharacteristic
48C2×C22⋊D83—21no—
49C2×C22⋊D83—22no—
50D4×D83—42no—
51C2×C4×D83—21no—
52C2×C4×D83—22no—
53C2×C4×D83C2311yescharacteristic
54#13050577—42no—
55D664—42no—
56D664—42no—
57#13050575—42no—
58C2×C83⋊C8C2312yes—
59Dic83⋊C4D412yes—
60(C2×D83)⋊C4—22no—
61(C2×D83)⋊C4—22no—
62(C2×D83)⋊C4—22no—
63(C2×D83)⋊C4D412yes—
64C4×Dic83C2311yescharacteristic
65C2×Q8×C83C2311yescharacteristic
66C2×D4×C83C2311yescharacteristic
67C2×C8×C83C2312yes—
68C4⋊C4×C83—22no—
69C4⋊C4×C83C2312yes—
70C22⋊C4×C83—22no—
71C42×C83C2311yescharacteristic
72C23×C83—21no—
73C22×D83—22no—
74C22×D83—21no—
75C22×D83—21no—
76C22×D83—42no—
77C22×D83—82no—
78C22×D83C2×D411yescharacteristic
79Q8×C83—22no—
80D4×C83—22no—
81C2×C4×C83—21no—
82C2×C4×C83—22no—
83C2×C4×C83C2×D412yes—
84C2×C4×C83C2411yescharacteristic
85C22⋊D83—42no—
86C22⋊D83—82no—
87C2×Dic83C2×D412yes—
88C2×Dic83C2×D411yescharacteristic
89D332—22no—
90D332—22no—
91D332—22no—
92D332—22no—
93D332—42no—
94D332—82no—
95D332—82no—
96D332—82no—
97C4×D83—22no—
98C4×D83—42no—
99C4×D83—82no—
100C8×C83—22no—
101C83⋊C8—22no—
102C22×C83—41no—
103C22×C83—42no—
104C22×C83C22×D411yescharacteristic
105C2×D83—21no—
106C2×D83—22no—
107C2×D83—41no—
108C2×D83—41no—
109C2×D83—42no—
110C2×D83—42no—
111C2×D83—42no—
112C2×D83—42no—
113C2×D83—42no—
114C2×D83—84no—
115C4×C83—22no—
116C4×C83—41no—
117C4×C83—42no—
118C4×C83H(5,𝔽2)12yes—
119Dic83—21no—
120Dic83—22no—
121C2×C83—81no—
122C2×C83#53511yescharacteristic
123C2×C83#57612yes—
124D83—41no—
125D83—81no—
126D83—82no—
127D83—81no—
128#3220—831nomaximal, Sylow
129C83#322011yescharacteristic, complemented, Sylow
130#540—831no—
131#535—831no—
132#496—831no—
133#493—831no—
134#490—831no—
135#488D8311yescharacteristic, complemented
136#463—832no—
137#459—832no—
138#449—832no—
139#447—832no—
140#433—831no—
141C2×#41—831no—
142C22×D4—831no—
143C2×SD16—832no—
144C2×D8—832no—
145C2×D8—832no—
146C2×D8—832no—
147#127—831no—
148#127—831no—
149#127—831no—
150#125C2×D8311yescharacteristic, complemented
151#124—831no—
152#124C2×D8311yescharacteristic
153#123—1662no—
154#121—832no—
155#121—1662no—
156#121—1662no—
157#120—1662no—
158C4×D4—832no—
159#117—831no—
160C4⋊C8—832no—
161#103C2×D8312yescomplemented
162#102—832no—
163#102—832no—
164#102C2×D8312yes—
165C22⋊C8—832no—
166#97—831no—
167C4×C8C2×D8311yescharacteristic, complemented
168C24—1661no—
169#41—3322no—
170C2×Q8C22×D8311yescharacteristic
171C2×D4—3322no—
172C2×D4—831no—
173C2×D4—831no—
174C2×D4—832no—
175C2×D4—1661no—
176C2×D4—1662no—
177C2×D4—1661no—
178C2×D4—1662no—
179C2×D4—1662no—
180C2×D4—1662no—
181C2×D4C22×D8311yescharacteristic
182C22×C4—831no—
183C22×C4—1661no—
184C22×C4—1662no—
185SD16—3322no—
186D8—3322no—
187D8—3322no—
188D8—3322no—
189C2×C8—832no—
190C2×C8C22×D8312yes—
191C4⋊C4—22no—
192C4⋊C4—832no—
193C4⋊C4C22×D8312yes—
194C22⋊C4—22no—
195C22⋊C4—832no—
196C22⋊C4—1662no—
197C22⋊C4—1662no—
198C22⋊C4—1662no—
199C42—831no—
200C42C22×D8311yescharacteristic
201C23—21no—
202C23—3322no—
203C23—831no—
204C23—6642no—
205C23—1662no—
206C23—1661no—
207C23—1661no—
208Q8—22no—
209D4—22no—
210D4—3322no—
211D4—3322no—
212D4—6642no—
213D4—6642no—
214D4—6642no—
215D4—6642no—
216D4—1662no—
217D4—1662no—
218D4—1662no—
219D4—1662no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—3322no—
223C2×C4—832no—
224C2×C4—831no—
225C2×C4—6642no—
226C2×C4—1662no—
227C2×C4D4×D8312yes—
228C2×C4C23×D8311yescharacteristic
229C8—22no—
230C8—1662no—
231C22—42no—
232C22—41no—
233C22—3322no—
234C22—3322no—
235C22—3322no—
236C22—3322no—
237C22—3321no—
238C22—3321no—
239C22—3322no—
240C22—6644no—
241C22—1662no—
242C22—1661no—
243C22C2×D4×D8311yescharacteristic
244C4—22no—
245C4—42no—
246C4—41no—
247C4—1662no—
248C4—1661no—
249C4#1458183812yes—
250C2—81no—
251C2—3321no—
252C2—6641no—
253C2—6642no—
254C2—6641no—
255C2#1912826911yescharacteristic
256C2#1912850312yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#488×C83C2
socleC22×C83C22×D4
hypercenter#488D83
hypocenterC83#3220
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #19128307C2semidirect
2. #19128269C2semidirect
3. #19128091C2semidirect
4. #19128078C2semidirect
5. #19128067C2semidirect
6. #19127919C2semidirect
7. #19127895C2semidirect
8. #19127849C2semidirect
9. #19127836C2semidirect
10. #19127760C2semidirect
11. #488×C83C2semidirect
14. C2×#13050577C22semidirect
16. C2×D664C22semidirect
19. #14581789C22semidirect
27. #14581754C22semidirect
29. Dic83⋊C8C22semidirect
31. #14581712C22semidirect
33. #14581703C22semidirect
34. #125×C83C22semidirect
36. #103×C83C22semidirect
38. C4×C8×C83C22semidirect
129. C83#3220semidirect
135. #488D83semidirect
150. #125C2×D83semidirect
161. #103C2×D83semidirect
167. C4×C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191283072
#191282692
#191280912
#191280782
#191280672
#191279192
#191278952
#191278492
#191278362
#191277602
#488×C832
#322083