Finite Group Databasebeta

← #24917121

Subgroups — #24917121

Order 10624.

484 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14581798C211yesmaximal, characteristic, complemented
2#19128279C212yesmaximal, complemented
3#19127956C212yesmaximal, complemented
4#19127951C212yesmaximal, complemented
5#19127927C214yesmaximal, complemented
6C2×#14581724C211yesmaximal, characteristic, complemented
7#19127702C212yesmaximal, complemented
8C2×#135×C83C211yesmaximal, characteristic, complemented
9C2×#13050590C2211yescharacteristic
10#14581798—24no—
11C2×#13050574C2211yescharacteristic
12C2×#13050573C2212yes—
13C2×C8×D83C2212yes—
14#14581771—22no—
15#14581768C2214yes—
16#14581766—22no—
17#14581764—24no—
18#14581763C2214yes—
19C2×C4⋊Dic83C2212yes—
20#14581724C2212yes—
21#14581724C2212yes—
22#14581716C2214yes—
23#14581713C2214yes—
24C2×#41×C83C2212yes—
25#135×C83—24no—
26C2×Q16×C83C2211yescharacteristic
27C2×SD16×C83C2212yes—
28C2×D8×C83C2211yescharacteristic
29C22×C8×C83C2211yescharacteristic
30#13050590—22no—
31#13050590—22no—
32C2×Dic166D411yescharacteristic
33C2×D332D411yescharacteristic
34C2×C22⋊D83—24no—
35C2×C22⋊D83D412yes—
36C22×Dic83—22no—
37C2×C4×D83C2312yes—
38#13050574—22no—
39#13050573—24no—
40C2×C83⋊C8C2312yes—
41C8×D83—24no—
42C4⋊Dic83—22no—
43C4⋊Dic83C2314yes—
44Dic83⋊C4—24no—
45C22⋊Dic83—24no—
46(C2×D83)⋊C4—24no—
47#41×C83—24no—
48#41×C83—24no—
49C2×Q8×C83C2312yes—
50C2×D4×C83—22no—
51C2×D4×C83C2312yes—
52C22×C4×C83—22no—
53C22×C4×C83C2311yescharacteristic
54Q16×C83—22no—
55SD16×C83—24no—
56D8×C83—22no—
57C2×C8×C83D414yescomplemented
58C2×C8×C83C2311yescharacteristic
59C2×C8×C83C2311yescharacteristic
60C23×C83—22no—
61C23×C83C2×D411yescharacteristic
62C22×D83C2×D412yes—
63Q8×C83—22no—
64Q8×C83—22no—
65D4×C83—22no—
66D4×C83—22no—
67D4×C83—44no—
68C2×C4×C83—22no—
69C2×C4×C83—44no—
70C2×C4×C83C2×D411yescharacteristic
71C2×C4×C83C2×D412yes—
72C2×C4×C83#4112yes—
73C2×C4×C83C2411yescharacteristic
74C22⋊D83—24no—
75C22⋊D83—84no—
76C2×Dic83—24no—
77C2×Dic83—82no—
78C2×Dic83C2×D412yes—
79D332—22no—
80C4×D83—22no—
81C4×D83—22no—
82Dic166—22no—
83C8×C83C2×D412yes—
84C8×C83C2×D412yes—
85C83⋊C8—22no—
86C22×C83—22no—
87C22×C83—42no—
88C22×C83—42no—
89C22×C83—42no—
90C22×C83#12112yes—
91C22×C83C22×D411yescharacteristic
92C2×D83—22no—
93C2×D83—24no—
94C4×C83—42no—
95C4×C83#12112yes—
96C4×C83C22×D411yescharacteristic
97C4×C83C2×#4111yescharacteristic
98Dic83—22no—
99Dic83—44no—
100C2×C83—22no—
101C2×C83—82no—
102C2×C83C2×#12111yescharacteristic
103C2×C83#57712yes—
104D83—42no—
105#3060—831nomaximal, Sylow
106C83#306011yescharacteristic, complemented, Sylow
107C2×#135D8311yescharacteristic, complemented
108C2×#131—831no—
109#537—832no—
110#470—832no—
111#469—832no—
112#465—834no—
113C2×#106—831no—
114#417—832no—
115C2×#41—831no—
116C2×#41C2×D8312yes—
117#135—24no—
118C2×Q16C2×D8311yescharacteristic
119C2×SD16C2×D8312yes—
120C2×D8C2×D8311yescharacteristic
121#131—1664no—
122C2×M16—831no—
123C2×M16—832no—
124C22×C8—832no—
125C22×C8C2×D8311yescharacteristic
126C22⋊Q8—834no—
127C22⋊Q8—1662no—
128#121—834no—
129#121—1662no—
130#121—1664no—
131C2×C4⋊C4—832no—
132#106—832no—
133#106—832no—
134#103—834no—
135#102—834no—
136#41—24no—
137#41—24no—
138#41—1662no—
139#41—1662no—
140C2×Q8—831no—
141C2×Q8C22×D8312yes—
142C2×D4—22no—
143C2×D4—832no—
144C2×D4—831no—
145C2×D4—1664no—
146C2×D4C22×D8312yes—
147C22×C4—22no—
148C22×C4—832no—
149C22×C4—1662no—
150C22×C4C22×D8311yescharacteristic
151Q16—22no—
152SD16—24no—
153D8—22no—
154M16—1662no—
155M16—1664no—
156C2×C8—832no—
157C2×C8—1664no—
158C2×C8C22⋊D8314yescomplemented
159C2×C8C22×D8311yescharacteristic
160C2×C8C22×D8311yescharacteristic
161C4⋊C4—834no—
162C4⋊C4—1664no—
163C4⋊C4—1662no—
164C22⋊C4—1664no—
165C22⋊C4—1664no—
166C23—22no—
167C23—832no—
168C23D4×D8311yescharacteristic
169Q8—22no—
170Q8—22no—
171Q8—1662no—
172D4—22no—
173D4—22no—
174D4—44no—
175D4—6644no—
176D4—1664no—
177D4—1662no—
178C2×C4—22no—
179C2×C4—44no—
180C2×C4—832no—
181C2×C4—6642no—
182C2×C4—1664no—
183C2×C4—1662no—
184C2×C4—1662no—
185C2×C4D4×D8311yescharacteristic
186C2×C4C2×C22⋊D8312yes—
187C2×C4#1305058912yes—
188C2×C4C23×D8311yescharacteristic
189C8—1662no—
190C8C2×C22⋊D8312yes—
191C8C2×C22⋊D8312yes—
192C22—22no—
193C22—42no—
194C22—42no—
195C22—42no—
196C22—1662no—
197C22—1664no—
198C22#1458176312yes—
199C22C2×D4×D8311yescharacteristic
200C4—42no—
201C4—3324no—
202C4—1662no—
203C4#1458176312yes—
204C4C22×C22⋊D8311yescharacteristic
205C4C2×#1305058911yescharacteristic
206C2—22no—
207C2—82no—
208C2—3322no—
209C2C2×#1458176311yescharacteristic
210C2#1912850812yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×#135×C83C2
socleC22×C83C22×D4
hypercenterC2×#135D83
hypocenterC83#3060
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#14581798C2semidirect
2. #19128279C2semidirect
3. #19127956C2semidirect
4. #19127951C2semidirect
5. #19127927C2semidirect
6. C2×#14581724C2semidirect
7. #19127702C2semidirect
8. C2×#135×C83C2semidirect
57. C2×C8×C83D4semidirect
106. C83#3060semidirect
107. C2×#135D83semidirect
158. C2×C8C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817982
#191282792
#191279562
#191279512
#191279272
C2×#145817242
#191277022
C2×#135×C832
#306083