Finite Group Databasebeta

← #24917950

Subgroups — #24917950

Order 10624.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14581817C212yesmaximal, complemented
2C22×D664C211yesmaximal, characteristic, complemented
3#19128265C212yesmaximal, complemented
4#19128087C214yesmaximal, complemented
5#19128086C212yesmaximal, complemented
6#19128076C212yesmaximal, complemented
7#19127656C211yesmaximal, characteristic, complemented
8C2×#135×C83C211yesmaximal, characteristic, complemented
9C22×D332C2212yes—
10#14581817—28no—
11C2×#13050577C2214yescomplemented
12C2×D664—24no—
13C2×D664C2212yes—
14C2×D664C2212yes—
15C2×#13050575C2214yescomplemented
16C2×#13050574C2211yescharacteristic
17#14581788C2214yescomplemented
18#14581781C2214yescomplemented
19#14581766—24no—
20#14581760—24no—
21#14581751C2211yescharacteristic
22#14581702C2212yescomplemented
23C8×Dic83C2212yescomplemented
24C2×#41×C83C2212yes—
25#135×C83—24no—
26C2×Q16×C83C2211yescharacteristic, complemented
27C2×SD16×C83C2212yescomplemented
28C2×D8×C83C2211yescharacteristic, complemented
29C22×C8×C83C2211yescharacteristic
30C23×D83—22no—
31C2×D332—24no—
32C2×D332—24no—
33C2×D332—22no—
34C2×D332C2314yes—
35C2×C22⋊D83—28no—
36#13050577—28no—
37D664—24no—
38D664—24no—
39#13050575—28no—
40#13050574D414yescomplemented
41C2×C83⋊C8C2312yes—
42C4⋊Dic83D412yes—
43C22⋊Dic83D412yes—
44(C2×D83)⋊C4—28no—
45C4×Dic83C2312yes—
46#41×C83—24no—
47#41×C83—24no—
48C2×Q8×C83C2312yes—
49C2×D4×C83—22no—
50C2×D4×C83C2312yes—
51C22×C4×C83—22no—
52C22×C4×C83C2311yescharacteristic
53Q16×C83—22no—
54SD16×C83—24no—
55D8×C83—22no—
56C2×C8×C83D414yescomplemented
57C2×C8×C83C2311yescharacteristic
58C2×C8×C83C2311yescharacteristic
59C23×C83—22no—
60C23×C83C2×D411yescharacteristic
61C22×D83—24no—
62C22×D83—44no—
63C22×D83—84no—
64Q8×C83—22no—
65Q8×C83—22no—
66D4×C83—22no—
67D4×C83—22no—
68D4×C83—44no—
69C2×C4×C83—22no—
70C2×C4×C83—44no—
71C2×C4×C83C2×D411yescharacteristic
72C2×C4×C83C2×D412yes—
73C2×C4×C83C2×D412yes—
74C2×C4×C83C2411yescharacteristic
75C22⋊D83—88no—
76C2×Dic83C2×D414yes—
77D332—24no—
78D332—24no—
79D332—44no—
80C8×C83C2×D412yes—
81C8×C83C2×D412yes—
82C83⋊C8C2×D414yes—
83C22×C83—22no—
84C22×C83—42no—
85C22×C83—44no—
86C22×C83#12712yes—
87C22×C83C22×D411yescharacteristic
88C2×D83—44no—
89C2×D83—48no—
90C2×D83—88no—
91C4×C83—42no—
92C4×C83#12712yes—
93C4×C83C22×D411yescharacteristic
94C4×C83C22×D411yescharacteristic
95Dic83—24no—
96C2×C83—22no—
97C2×C83—82no—
98C2×C83C2×#12711yescharacteristic
99C2×C83#57612yes—
100D83—84no—
101#3155—831nomaximal, Sylow
102C83#315511yescharacteristic, complemented, Sylow
103C2×AGL(1,ℤ/8)—832no—
104C2×#135D8311yescharacteristic, complemented
105C22×D8—831no—
106#535—832no—
107#496—834no—
108#495—832no—
109#493—832no—
110#405—831no—
111C2×#41C2×D8312yes—
112C22×D4—832no—
113AGL(1,ℤ/8)—1668no—
114#135—24no—
115C2×Q16C2×D8311yescharacteristic, complemented
116C2×SD16—834no—
117C2×SD16C2×D8312yescomplemented
118C2×D8—834no—
119C2×D8—832no—
120C2×D8—832no—
121C2×D8—1664no—
122C2×D8C2×D8311yescharacteristic, complemented
123C2×M16—831no—
124C22×C8C2×D8311yescharacteristic
125#127—834no—
126#124—834no—
127#121—1664no—
128#120—1664no—
129#117—831no—
130#97—832no—
131C4×C8—832no—
132C24—1662no—
133#41—24no—
134#41—24no—
135C2×Q8C22×D8312yes—
136C2×D4—22no—
137C2×D4—834no—
138C2×D4—1668no—
139C2×D4—1662no—
140C2×D4—1664no—
141C2×D4—1664no—
142C2×D4C22×D8312yes—
143C22×C4—22no—
144C22×C4C22×D8311yescharacteristic
145Q16—22no—
146SD16—24no—
147SD16—1668no—
148D8—22no—
149D8—1668no—
150D8—1664no—
151D8—1664no—
152M16—834no—
153C2×C8—832no—
154C2×C8C22⋊D8314yescomplemented
155C2×C8C22×D8311yescharacteristic
156C2×C8C22×D8311yescharacteristic
157C4⋊C4—832no—
158C22⋊C4—832no—
159C22⋊C4—1668no—
160C42—832no—
161C23—22no—
162C23—3324no—
163C23—6644no—
164C23—1664no—
165C23D4×D8311yescharacteristic
166Q8—22no—
167Q8—22no—
168D4—22no—
169D4—22no—
170D4—44no—
171D4—3324no—
172D4—6648no—
173D4—1664no—
174D4—1664no—
175C2×C4—22no—
176C2×C4—44no—
177C2×C4—834no—
178C2×C4D4×D8311yescharacteristic
179C2×C4D4×D8312yes—
180C2×C4C2×C22⋊D8312yes—
181C2×C4C23×D8311yescharacteristic
182C8—834no—
183C8C2×C22⋊D8312yes—
184C8C2×C22⋊D8312yes—
185C22—22no—
186C22—42no—
187C22—44no—
188C22—3324no—
189C22—3328no—
190C22—6648no—
191C22#1458178812yes—
192C22C2×D4×D8311yescharacteristic
193C4—42no—
194C4—1664no—
195C4#1458178812yes—
196C4C2×D4×D8311yescharacteristic
197C4C22×C22⋊D8311yescharacteristic
198C2—22no—
199C2—82no—
200C2—6644no—
201C2C2×#1458178811yescharacteristic
202C2#1912850312yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×#135×C83C2
socleC22×C83C22×D4
hypercenterC2×#135D83
hypocenterC83#3155
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×#14581817C2semidirect
2. C22×D664C2semidirect
3. #19128265C2semidirect
4. #19128087C2semidirect
5. #19128086C2semidirect
6. #19128076C2semidirect
7. #19127656C2semidirect
8. C2×#135×C83C2semidirect
11. C2×#13050577C22semidirect
15. C2×#13050575C22semidirect
17. #14581788C22semidirect
18. #14581781C22semidirect
22. #14581702C22semidirect
23. C8×Dic83C22semidirect
26. C2×Q16×C83C22semidirect
27. C2×SD16×C83C22semidirect
28. C2×D8×C83C22semidirect
40. #13050574D4semidirect
56. C2×C8×C83D4semidirect
102. C83#3155semidirect
104. C2×#135D83semidirect
115. C2×Q16C2×D83semidirect
117. C2×SD16C2×D83semidirect
122. C2×D8C2×D83semidirect
154. C2×C8C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818172
C22×D6642
#191282652
#191280872
#191280862
#191280762
#191276562
C2×#135×C832
#315583