Finite Group Databasebeta

← #24917586

Subgroups — #24917586

Order 10624.

424 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 129 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#19128251C211yesmaximal, characteristic
2C2×C4×Dic166C211yesmaximal, characteristic, complemented
3#19128015C212yesmaximal, complemented
4#19127997C212yesmaximal, complemented
5#19127902C212yesmaximal, complemented
6#19127842C212yesmaximal, complemented
7#19127803C214yesmaximal, complemented
8#432×C83C211yesmaximal, characteristic, complemented
9C2×#13050590—21no—
10C22×Dic166C2211yescharacteristic
11C2×#13050579C2214yes—
12#14581792C2211yescharacteristic
13#14581789C2211yescharacteristic
14#14581780—21no—
15#14581766—21no—
16#14581766C2212yes—
17C4×Dic166—21no—
18C4×Dic166—21no—
19C4×Dic166C2212yes—
20C4×Dic166C2212yes—
21C4×D332—21no—
22C4×D332C2212yes—
23C2×C4⋊Dic83C2211yescharacteristic
24C2×Dic83⋊C4—21no—
25C2×C4×Dic83—21no—
26#14581726C2214yes—
27#14581717C2214yes—
28#14581714C2214yes—
29C2×C42×C83C2211yescharacteristic
30C4⋊C8×C83C2212yes—
31C22⋊C8×C83C2212yes—
32#97×C83C2212yes—
33#13050590—44no—
34C2×Dic166—21no—
35C2×Dic166—24no—
36C2×Dic166—21no—
37C2×Dic166C2312yes—
38C2×D332—21no—
39C2×D332—21no—
40C2×D332C2312yes—
41C2×C22⋊D83—22no—
42C22×Dic83—21no—
43C22×Dic83—21no—
44C2×C4×D83—22no—
45#13050579—28no—
46C4⋊Dic83—21no—
47C4⋊Dic83—21no—
48C4⋊Dic83C2312yes—
49C4⋊Dic83C2312yes—
50Dic83⋊C4—22no—
51Dic83⋊C4—22no—
52(C2×D83)⋊C4—22no—
53(C2×D83)⋊C4—22no—
54C4×Dic83—22no—
55C4×Dic83—22no—
56C22×C4×C83D411yescharacteristic
57C22×C4×C83D411yescharacteristic
58C22×C4×C83C2311yescharacteristic
59C2×C8×C83C2314yes—
60C42×C83D411yescharacteristic
61C42×C83D411yescharacteristic
62C42×C83C2311yescharacteristic
63C42×C83C2311yescharacteristic
64C23×C83C2×D411yescharacteristic
65C22×D83—22no—
66C2×C4×C83—22no—
67C2×C4×C83—22no—
68C2×C4×C83—22no—
69C2×C4×C83C2×D411yescharacteristic
70C2×C4×C83C2×D411yescharacteristic
71C2×C4×C83C2×D411yescharacteristic
72C2×C4×C83C2×D411yescharacteristic
73C2×C4×C83C2×D411yescharacteristic
74C2×C4×C83C2411yescharacteristic
75C22⋊D83—44no—
76C2×Dic83—24no—
77C2×Dic83—22no—
78C2×Dic83—22no—
79C2×Dic83—22no—
80C2×Dic83—42no—
81D332—24no—
82D332—42no—
83D332—42no—
84C4×D83—44no—
85Dic166—24no—
86Dic166—22no—
87Dic166—42no—
88Dic166#4114yescomplemented
89C8×C83—24no—
90C22×C83—21no—
91C22×C83—21no—
92C22×C83AGL(1,ℤ/8)12yes—
93C22×C83C22×D411yescharacteristic
94C2×D83—42no—
95C2×D83—42no—
96C2×D83—42no—
97C4×C83—21no—
98C4×C83—21no—
99C4×C83—21no—
100C4×C83—21no—
101C4×C83#13712yes—
102C4×C83C2×#4112yes—
103Dic83—24no—
104Dic83—42no—
105Dic83—42no—
106C2×C83—22no—
107C2×C83#52511yescharacteristic
108C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
109C2×C83C2×#13711yescharacteristic
110D83—82no—
111#3112—831nomaximal, Sylow
112C83#311211yescharacteristic, complemented, Sylow
113#532—831no—
114C2×C4×Q8—831no—
115#480—832no—
116#477—832no—
117#460—832no—
118#448—832no—
119#440—834no—
120#432D8311yescharacteristic, complemented
121C2×#41—1661no—
122C22×Q8—831no—
123C2×SD16—834no—
124C4⋊Q8—831no—
125#127—831no—
126#124—1661no—
127#121—832no—
128#121—1661no—
129C4×Q8—832no—
130C4×Q8—832no—
131C4×Q8—1661no—
132C4×Q8—1661no—
133C4×D4—832no—
134C4×D4—1661no—
135C2×C4⋊C4—831no—
136C2×C4⋊C4—1661no—
137C2×C42—1661no—
138C2×C42C2×D8311yescharacteristic
139#107—834no—
140C4⋊C8C2×D8312yes—
141#103—834no—
142#102—834no—
143C22⋊C8C2×D8312yes—
144#97C2×D8312yes—
145#41—3324no—
146C2×Q8—832no—
147C2×Q8—1664no—
148C2×Q8—1661no—
149C2×Q8—1661no—
150C2×D4—832no—
151C2×D4—1662no—
152C2×D4—1661no—
153C2×D4—1661no—
154C22×C4—1661no—
155C22×C4—1662no—
156C22×C4—1661no—
157C22×C4D33211yescharacteristic
158C22×C4D33211yescharacteristic
159C22×C4C22×D8311yescharacteristic
160SD16—1668no—
161C2×C8C22×D8314yes—
162C4⋊C4—832no—
163C4⋊C4—832no—
164C4⋊C4—1661no—
165C4⋊C4—1662no—
166C4⋊C4—1662no—
167C4⋊C4—1661no—
168C22⋊C4—1662no—
169C22⋊C4—1662no—
170C42—1662no—
171C42—1662no—
172C42D33211yescharacteristic
173C42D33211yescharacteristic
174C42C22×D8311yescharacteristic
175C42C22×D8311yescharacteristic
176C23—1662no—
177C23C2×D33211yescharacteristic
178Q8—3322no—
179Q8—834no—
180Q8—1664no—
181Q8—1662no—
182D4—3322no—
183D4—3324no—
184D4—3322no—
185D4—1664no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—3324no—
190C2×C4—3322no—
191C2×C4—1662no—
192C2×C4—1662no—
193C2×C4—1664no—
194C2×C4—1662no—
195C2×C4C2×D33211yescharacteristic
196C2×C4C2×D33211yescharacteristic
197C2×C4C2×D33211yescharacteristic
198C2×C4C2×D33211yescharacteristic
199C2×C4C2×D33211yescharacteristic
200C2×C4C23×D8311yescharacteristic
201C8—24no—
202C22—21no—
203C22—21no—
204C22—3322no—
205C22—3322no—
206C22—3322no—
207C22#1458182012yes—
208C22C22×D33211yescharacteristic
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—3322no—
214C4—3322no—
215C4—1664no—
216C4#1458182512yes—
217C4#41×D8312yes—
218C2—22no—
219C2—6642no—
220C2#1912821311yescharacteristic
221C2C2×#1458182011yescharacteristic
222C2C2×#1458182511yescharacteristic
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#432×C83C2
socleC22×C83C22×D4
prime radical#19128251C2
hypercenter#432D83
hypocenterC83#3112
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic166C2semidirect
3. #19128015C2semidirect
4. #19127997C2semidirect
5. #19127902C2semidirect
6. #19127842C2semidirect
7. #19127803C2semidirect
8. #432×C83C2semidirect
88. Dic166#41semidirect
112. C83#3112semidirect
120. #432D83semidirect

Maximal subgroups

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

subgroupindex
#191282512
C2×C4×Dic1662
#191280152
#191279972
#191279022
#191278422
#191278032
#432×C832
#311283