Finite Group Databasebeta

← #24920606

Subgroups — #24920606

Order 10624.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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#19128515C211yesmaximal, characteristic, complemented
2#19128503C211yesmaximal, characteristic, complemented
3#19128143C212yesmaximal, complemented
4#19128141C211yesmaximal, characteristic, complemented
5#19128136C211yesmaximal, characteristic, complemented
6#19128135C212yesmaximal, complemented
7#19128132C212yesmaximal, complemented
8C2×#14581737C211yesmaximal, characteristic, complemented
9SD32×D83C212yesmaximal, complemented
10#19128117C211yesmaximal, characteristic, complemented
11#508×C83C211yesmaximal, characteristic, complemented
12#14581839—21no—
13#14581838—21no—
14#14581822—21no—
15#14581821—22no—
16#14581819—22no—
17#14581817—21no—
18#14581815C2212yes—
19#14581811C2212yescomplemented
20C2×Dic332C2211yescharacteristic, complemented
21Q16×D83C2212yescomplemented
22C2×D664C2211yescharacteristic
23D8×D83C2212yes—
24#14581798C2211yescharacteristic
25#14581738C2212yescomplemented
26#14581737C2212yescomplemented
27#14581737C2212yescomplemented
28#14581736C2212yescomplemented
29#14581735C2212yescomplemented
30#14581733C2212yescomplemented
31#14581732C2211yescharacteristic, complemented
32#14581731C2212yescomplemented
33C16×D83C2212yescomplemented
34#135×C83C2211yescharacteristic
35#135×C83C2211yescharacteristic, complemented
36Q32×C83C2211yescharacteristic, complemented
37SD32×C83C2211yescharacteristic, complemented
38SD32×C83C2211yescharacteristic, complemented
39D16×C83C2211yescharacteristic, complemented
40C2×C16×C83C2211yescharacteristic, complemented
41#13050591—22no—
42#13050590—41no—
43#13050590—41no—
44#13050590D411yescharacteristic
45#13050589—22no—
46#13050589—42no—
47C2×Dic166—21no—
48C2×Dic166—41no—
49Q8×D83—22no—
50C2×D332—21no—
51C2×D332—41no—
52D4×D83—22no—
53D4×D83—42no—
54Dic332—21no—
55Dic332C2312yes—
56#13050580—22no—
57#13050578—22no—
58#13050577—22no—
59D664—21no—
60D664C2312yes—
61#13050575—22no—
62#13050574D411yescharacteristic
63#13050573D412yes—
64C8×D83C2312yes—
65C83⋊C16C2312yes—
66#41×C83—21no—
67#41×C83—21no—
68Q16×C83C2311yescharacteristic
69Q16×C83C2311yescharacteristic
70SD16×C83—21no—
71SD16×C83—21no—
72D8×C83C2311yescharacteristic
73D8×C83C2311yescharacteristic
74C2×C8×C83C2311yescharacteristic
75C16×C83C2311yescharacteristic
76C16×C83C2311yescharacteristic
77C22×D83—41no—
78C22×D83—42no—
79Q8×C83—21no—
80Q8×C83—21no—
81D4×C83—21no—
82D4×C83—21no—
83D4×C83—41no—
84D4×C83—41no—
85C2×C4×C83—41no—
86C2×C4×C83—41no—
87C2×C4×C83C2×D411yescharacteristic
88C22⋊D83—42no—
89C22⋊D83—42no—
90C22⋊D83D812yes—
91C2×Dic83—41no—
92C2×Dic83—42no—
93D332—22no—
94D332—42no—
95D332—41no—
96D332D811yescharacteristic
97C4×D83—42no—
98C4×D83—42no—
99C4×D83C2×D412yes—
100Dic166—22no—
101Dic166—41no—
102Dic166—42no—
103Dic166D811yescharacteristic
104C8×C83C2×D411yescharacteristic
105C8×C83C2411yescharacteristic
106C83⋊C8C2×D412yes—
107C22×C83—41no—
108C22×C83—41no—
109C22×C83C2×D811yescharacteristic
110C2×D83—42no—
111C2×D83—162no—
112C2×D83—161no—
113C2×D83C2×D812yes—
114C4×C83—41no—
115C4×C83—41no—
116C4×C83C2×D811yescharacteristic
117C4×C83C22×D411yescharacteristic
118Dic83—42no—
119Dic83C2×D812yes—
120C2×C83—21no—
121C2×C83—81no—
122C2×C83—81no—
123C2×C83C22×D811yescharacteristic
124D83—22no—
125D83—82no—
126#3416—831nomaximal, Sylow
127C83#341611yescharacteristic, complemented, Sylow
128#578—831no—
129#576—831no—
130#510—831no—
131#510—832no—
132#509—831no—
133#509—832no—
134#508—832no—
135#508D8311yescharacteristic, complemented
136C2×SD32—832no—
137C2×SD32—831no—
138#504—831no—
1392−1+4—1661no—
140H(5,𝔽2)—1661no—
141#137—1662no—
142#137—1661no—
143AGL(1,ℤ/8)—1662no—
144AGL(1,ℤ/8)—1661no—
145#135—832no—
146#135—832no—
147#135C2×D8311yescharacteristic
148#135C2×D8311yescharacteristic, complemented
149C2×Q16—832no—
150C2×Q16—831no—
151C2×D8—832no—
152C2×D8—831no—
153#131—831no—
154Q32—832no—
155Q32C2×D8311yescharacteristic, complemented
156SD32—832no—
157SD32—832no—
158SD32—832no—
159SD32—832no—
160SD32C2×D8311yescharacteristic, complemented
161SD32C2×D8311yescharacteristic, complemented
162D16—832no—
163D16C2×D8311yescharacteristic, complemented
164M32—831no—
165M32—832no—
166C2×C16—832no—
167C2×C16C2×D8311yescharacteristic, complemented
168#41—21no—
169#41—21no—
170#41—3321no—
171#41—3321no—
172#41—3322no—
173#41—831no—
174#41—1662no—
175#41—1662no—
176C2×Q8—3321no—
177C2×Q8—1662no—
178C2×Q8—1661no—
179C2×D4—3321no—
180C2×D4—3322no—
181C2×D4—1662no—
182C2×D4—1661no—
183Q16—832no—
184Q16—1662no—
185Q16—1661no—
186Q16C22×D8311yescharacteristic
187Q16C22×D8311yescharacteristic
188SD16—21no—
189SD16—21no—
190SD16—1662no—
191SD16—1662no—
192D8—832no—
193D8—1662no—
194D8—1661no—
195D8C22×D8311yescharacteristic
196D8C22×D8311yescharacteristic
197M16—831no—
198M16—832no—
199C2×C8—832no—
200C2×C8C22×D8311yescharacteristic
201C16—832no—
202C16C22×D8311yescharacteristic
203C16C22×D8311yescharacteristic
204C23—3322no—
205C23—3321no—
206Q8—21no—
207Q8—21no—
208Q8—3322no—
209Q8—3321no—
210Q8—831no—
211Q8—1662no—
212D4—21no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—3322no—
217D4—3322no—
218D4—3322no—
219D4—3321no—
220D4—832no—
221D4—831no—
222D4—1662no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—3322no—
226C2×C4—3322no—
227C2×C4—3321no—
228C2×C4—3322no—
229C2×C4—832no—
230C2×C4D4×D8311yescharacteristic
231C8—832no—
232C8D4×D8311yescharacteristic
233C8C23×D8311yescharacteristic
234C22—41no—
235C22—41no—
236C22—13281no—
237C22—13282no—
238C22—3322no—
239C22—832no—
240C22D8×D8311yescharacteristic
241C4—41no—
242C4—41no—
243C4—3322no—
244C4—832no—
245C4D8×D8311yescharacteristic
246C4C2×D4×D8311yescharacteristic
247C2—21no—
248C2—81no—
249C2—81no—
250C2—6642no—
251C2—1662no—
252C2C2×D8×D8311yescharacteristic
subgroupG/H
centerC2C2×D8×D83
commutator, cosocleC8×C83C24
FrattiniC8C23×D83
Fitting#508×C83C2
socleC2×C83C22×D8
hypercenter#508D83
hypocenterC83#3416
LCSG ▷ C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #19128515C2semidirect
2. #19128503C2semidirect
3. #19128143C2semidirect
4. #19128141C2semidirect
5. #19128136C2semidirect
6. #19128135C2semidirect
7. #19128132C2semidirect
8. C2×#14581737C2semidirect
9. SD32×D83C2semidirect
10. #19128117C2semidirect
11. #508×C83C2semidirect
19. #14581811C22semidirect
20. C2×Dic332C22semidirect
21. Q16×D83C22semidirect
25. #14581738C22semidirect
26. #14581737C22semidirect
27. #14581737C22semidirect
28. #14581736C22semidirect
29. #14581735C22semidirect
30. #14581733C22semidirect
31. #14581732C22semidirect
32. #14581731C22semidirect
33. C16×D83C22semidirect
35. #135×C83C22semidirect
36. Q32×C83C22semidirect
37. SD32×C83C22semidirect
38. SD32×C83C22semidirect
39. D16×C83C22semidirect
40. C2×C16×C83C22semidirect
127. C83#3416semidirect
135. #508D83semidirect
148. #135C2×D83semidirect
155. Q32C2×D83semidirect
160. SD32C2×D83semidirect
161. SD32C2×D83semidirect
163. D16C2×D83semidirect
167. C2×C16C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191285152
#191285032
#191281432
#191281412
#191281362
#191281352
#191281322
C2×#145817372
SD32×D832
#191281172
#508×C832
#341683