Finite Group Databasebeta

← #24919253

Subgroups — #24919253

Order 10624.

460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 117 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#19128510C212yesmaximal, complemented
2#19128509C212yesmaximal, complemented
3#19128075C211yesmaximal, characteristic, complemented
4#19127967C212yesmaximal, complemented
5#19127966C212yesmaximal, complemented
6#19127883C211yesmaximal, characteristic, complemented
7#19127870C212yesmaximal, complemented
8#19127815C212yesmaximal, complemented
9#456×C83C211yesmaximal, characteristic, complemented
10#14581840C2211yescharacteristic
11#14581838C2212yescomplemented
12#14581824—22no—
13#14581823—22no—
14#14581823C2212yescomplemented
15#14581820C2212yes—
16#14581819—22no—
17#14581817—22no—
18#14581817C2212yes—
19#14581815—22no—
20#14581812—22no—
21#14581812C2212yes—
22C2×#13050577—22no—
23C2×#13050577C2212yescomplemented
24C2×#13050577C2212yescomplemented
25SD16×D83—22no—
26#14581800C2212yescomplemented
27#14581799C2212yescomplemented
28#14581789C2211yescharacteristic
29#14581727C2212yescomplemented
30#14581720C2212yescomplemented
31#14581711C2211yescharacteristic, complemented
32#14581708C2212yescomplemented
33C4×C83⋊C8C2211yescharacteristic, complemented
342−1+4×C83C2211yescharacteristic
35#137×C83C2212yescomplemented
36C4⋊Q8×C83C2211yescharacteristic, complemented
37#104×C83C2212yescomplemented
38#101×C83C2211yescharacteristic, complemented
39#13050591—22no—
40#13050591—22no—
41#13050590—22no—
42#13050590—41no—
43#13050590C2312yes—
44Q8×D83—22no—
45C2×D332—22no—
46C2×D332—41no—
47C2×D332C2312yes—
48D4×D83—22no—
49D4×D83—42no—
50#13050580—22no—
51#13050580D412yescomplemented
52#13050579—22no—
53#13050578D412yescomplemented
54#13050577—22no—
55#13050577—22no—
56#13050577—22no—
57#13050577—22no—
58#13050577—22no—
59#13050577—22no—
60#13050577D412yescomplemented
61D664—22no—
62#13050575—22no—
63#13050575D412yescomplemented
64#13050574—22no—
65#13050574C2312yes—
66#13050573—22no—
67C2×C83⋊C8—22no—
68C2×C83⋊C8C2312yes—
69C8×D83—22no—
70#41×C83—22no—
71#41×C83—41no—
72#41×C83C2312yes—
73C2×Q8×C83—22no—
74C2×Q8×C83C2311yescharacteristic
75C2×Q8×C83C2311yescharacteristic
76Q16×C83—22no—
77SD16×C83—22no—
78M16×C83C2312yes—
79C4⋊C4×C83—41no—
80C42×C83C2311yescharacteristic
81C22×D83—22no—
82C22×D83—42no—
83Q8×C83—22no—
84Q8×C83—22no—
85Q8×C83—41no—
86Q8×C83C2×D412yes—
87D4×C83—22no—
88D4×C83—41no—
89D4×C83C2×D412yes—
90C2×C4×C83—21no—
91C2×C4×C83—22no—
92C2×C4×C83—21no—
93C2×C4×C83—21no—
94C2×C4×C83—42no—
95C2×C4×C83C2411yescharacteristic
96C22⋊D83—22no—
97C22⋊D83—42no—
98D332—22no—
99D332—22no—
100D332—42no—
101D332—41no—
102D332—82no—
103D332C2×D412yes—
104C4×D83—22no—
105C4×D83—42no—
106C4×D83—42no—
107Dic166—41no—
108Dic166C2×D412yes—
109C8×C83—22no—
110C83⋊C8—22no—
111C83⋊C8C2×D412yes—
112C83⋊C8C2×D412yes—
113C22×C83—22no—
114C22×C83H(5,𝔽2)11yescharacteristic
115C2×D83—22no—
116C2×D83—42no—
117C2×D83—82no—
118C2×D83—162no—
119C4×C83—22no—
120C4×C83—22no—
121C4×C83—41no—
122C4×C83—41no—
123C4×C83C22×D412yes—
124Dic83—22no—
125C2×C83—21no—
126C2×C83—42no—
127C2×C83D4211yescharacteristic
128D83—42no—
129D83—82no—
130#3291—831nomaximal, Sylow
131C83#329111yescharacteristic, complemented, Sylow
132#577—832no—
133#577—832no—
134#492—831no—
135#471—832no—
136#471—832no—
137#456—831no—
138#456D8311yescharacteristic, complemented
139#453—832no—
140#443—832no—
1412−1+4—831no—
1422−1+4C2×D8311yescharacteristic
143H(5,𝔽2)—832no—
144#137—832no—
145#137—1662no—
146#137—1662no—
147#137C2×D8312yescomplemented
148AGL(1,ℤ/8)—832no—
149AGL(1,ℤ/8)—832no—
150AGL(1,ℤ/8)—1662no—
151AGL(1,ℤ/8)—1662no—
152#135—832no—
153#135—1662no—
154#135—1662no—
155C2×SD16—832no—
156C2×SD16—832no—
157C2×SD16—1662no—
158C2×SD16—1662no—
159#131—832no—
160#131—832no—
161C4⋊Q8C2×D8311yescharacteristic, complemented
162#127—831no—
163#108—832no—
164#104—832no—
165#104C2×D8312yescomplemented
166#101—831no—
167#101C2×D8311yescharacteristic, complemented
168#100—832no—
169C4×C8—831no—
170#41—22no—
171#41—41no—
172#41—3321no—
173#41—832no—
174#41—1662no—
175#41—1662no—
176#41—1662no—
177#41C22×D8312yes—
178C2×Q8—22no—
179C2×Q8—1662no—
180C2×Q8C22×D8311yescharacteristic
181C2×Q8C22×D8311yescharacteristic
182C2×D4—3321no—
183C2×D4—3322no—
184C2×D4—832no—
185C2×D4—1662no—
186C2×D4—1662no—
187Q16—22no—
188Q16—832no—
189Q16—1662no—
190SD16—22no—
191SD16—832no—
192SD16—832no—
193SD16—1662no—
194SD16—1662no—
195SD16—1662no—
196SD16—1662no—
197SD16—1662no—
198SD16—1662no—
199SD16—1662no—
200D8—832no—
201D8—1662no—
202D8—1662no—
203M16—832no—
204M16—1662no—
205M16—1662no—
206M16C22×D8312yes—
207C2×C8—832no—
208C2×C8—1662no—
209C2×C8—1662no—
210C4⋊C4—41no—
211C42C22×D8311yescharacteristic
212C23—3322no—
213C23—1662no—
214Q8—22no—
215Q8—22no—
216Q8—41no—
217Q8—3321no—
218Q8—832no—
219Q8D4×D8312yes—
220D4—22no—
221D4—41no—
222D4—3322no—
223D4—3321no—
224D4—3322no—
225D4—832no—
226D4—6642no—
227D4—1662no—
228D4—1662no—
229D4—1662no—
230D4D4×D8312yes—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—3322no—
237C2×C4—3322no—
238C2×C4—1662no—
239C2×C4C23×D8311yescharacteristic
240C8—22no—
241C8—832no—
242C8—832no—
243C8—1662no—
244C22—22no—
245C22—13282no—
246C22—3322no—
247C22—6642no—
248C22—1662no—
249C22#1458183711yescharacteristic
250C4—22no—
251C4—22no—
252C4—41no—
253C4—41no—
254C4—1662no—
255C4C2×D4×D8312yes—
256C2—21no—
257C2—42no—
258C2—3322no—
259C2—6642no—
260C2#1912833311yescharacteristic
subgroupG/H
centerC2#19128333
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#456×C83C2
socleC2×C83D42
hypercenter#456D83
hypocenterC83#3291
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #19128510C2semidirect
2. #19128509C2semidirect
3. #19128075C2semidirect
4. #19127967C2semidirect
5. #19127966C2semidirect
6. #19127883C2semidirect
7. #19127870C2semidirect
8. #19127815C2semidirect
9. #456×C83C2semidirect
11. #14581838C22semidirect
14. #14581823C22semidirect
23. C2×#13050577C22semidirect
24. C2×#13050577C22semidirect
26. #14581800C22semidirect
27. #14581799C22semidirect
29. #14581727C22semidirect
30. #14581720C22semidirect
31. #14581711C22semidirect
32. #14581708C22semidirect
33. C4×C83⋊C8C22semidirect
35. #137×C83C22semidirect
36. C4⋊Q8×C83C22semidirect
37. #104×C83C22semidirect
38. #101×C83C22semidirect
51. #13050580D4semidirect
53. #13050578D4semidirect
60. #13050577D4semidirect
63. #13050575D4semidirect
131. C83#3291semidirect
138. #456D83semidirect
147. #137C2×D83semidirect
161. C4⋊Q8C2×D83semidirect
165. #104C2×D83semidirect
167. #101C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191285102
#191285092
#191280752
#191279672
#191279662
#191278832
#191278702
#191278152
#456×C832
#329183