Finite Group Databasebeta

← #24906736

Subgroups — #24906736

Order 10624.

200 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 53 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#19128421C211yesmaximal, characteristic
2#19128099C211yesmaximal, characteristic, complemented
3#19127981C211yesmaximal, characteristic
4#19127912C211yesmaximal, characteristic, complemented
5#19127446C211yesmaximal, characteristic, complemented
6#19127433C211yesmaximal, characteristic, complemented
7D4⋊C8×C83C211yesmaximal, characteristic
8C2×Dic332—21no—
9#14581792C2211yescharacteristic
10#14581792C2211yescharacteristic
11#14581785—22no—
12#14581779—21no—
13#14581771—21no—
14C4×Dic166C2211yescharacteristic
15#14581757—21no—
16#14581751—21no—
17#14581725—21no—
18#14581724—22no—
19C4⋊(C83⋊C8)C2211yescharacteristic
20#14581717—21no—
21#14581713—21no—
22C4×D4×C83C2211yescharacteristic
23C4⋊C8×C83C2211yescharacteristic
24C4×C8×C83C2211yescharacteristic
25C2×Dic166—21no—
26C2×Dic166—21no—
27C2×Dic166D411yescharacteristic
28Dic332—42no—
29C2×C83⋊C8—21no—
30C4⋊Dic83—21no—
31C4⋊Dic83—21no—
32C4⋊Dic83—21no—
33C4⋊Dic83D412yes—
34C4⋊Dic83D411yescharacteristic
35Dic83⋊C4—22no—
36Dic83⋊C4—21no—
37C22⋊Dic83—22no—
38C22⋊Dic83—22no—
39C4×Dic83—22no—
40C4×Dic83—21no—
41C2×D4×C83D411yescharacteristic
42C22×C4×C83—21no—
43C2×C8×C83—21no—
44C2×C8×C83—21no—
45C4⋊C4×C83D411yescharacteristic
46C22⋊C4×C83—21no—
47C42×C83C2311yescharacteristic
48C23×C83—21no—
49D4×C83—21no—
50D4×C83—21no—
51C2×C4×C83—21no—
52C2×C4×C83—41no—
53C2×C4×C83—41no—
54C2×C4×C83C2×D411yescharacteristic
55C2×C4×C83C2×D411yescharacteristic
56C2×C4×C83C2×D411yescharacteristic
57C2×Dic83—21no—
58C2×Dic83—21no—
59C2×Dic83—22no—
60C2×Dic83—41no—
61Dic166—21no—
62Dic166—21no—
63Dic166—41no—
64Dic166—41no—
65Dic166—41no—
66Dic166—41no—
67C8×C83—22no—
68C8×C83—41no—
69C83⋊C8—41no—
70C22×C83—41no—
71C22×C83—41no—
72C22×C83—41no—
73C22×C83#12011yescharacteristic
74C4×C83—21no—
75C4×C83—41no—
76C4×C83#13511yescharacteristic
77C4×C83#13511yescharacteristic
78C4×C83AGL(1,ℤ/8)11yescharacteristic
79C4×C83#13711yescharacteristic
80Dic83—41no—
81Dic83—42no—
82Dic83—41no—
83Dic83—81no—
84C2×C83—81no—
85C2×C83#44911yescharacteristic
86C2×C83#45211yescharacteristic
87C2×C83#45611yescharacteristic
88#1652—831nomaximal, Sylow
89C83#165211yescharacteristic, complemented, Sylow
90#556—831no—
91#498—831no—
92#474—831no—
93#462—831no—
94#332—831no—
95Q8⋊C8—831no—
96D4⋊C8D8311yescharacteristic
97C2×Q16—1661no—
98C4⋊Q8—831no—
99C4⋊Q8—831no—
100#126—1662no—
101#124—1661no—
102C22⋊Q8—1661no—
103C4×Q8—831no—
104C4×Q8—1661no—
105C4×D4C2×D8311yescharacteristic
106#117—1661no—
107#107—1661no—
108#106—1662no—
109C4⋊C8—831no—
110C4⋊C8C2×D8311yescharacteristic
111#103—1661no—
112#102—1661no—
113C4×C8C2×D8311yescharacteristic
114C2×Q8—831no—
115C2×Q8—1661no—
116C2×Q8—1661no—
117C2×D4D33211yescharacteristic
118C22×C4—21no—
119Q16—3322no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—1661no—
123C4⋊C4—831no—
124C4⋊C4—832no—
125C4⋊C4—1662no—
126C4⋊C4—1661no—
127C4⋊C4—1661no—
128C4⋊C4—1661no—
129C4⋊C4—1661no—
130C4⋊C4D33211yescharacteristic
131C22⋊C4—21no—
132C22⋊C4—1662no—
133C22⋊C4—1662no—
134C42—1662no—
135C42—1661no—
136C42C22×D8311yescharacteristic
137C23—21no—
138Q8—3321no—
139Q8—3321no—
140Q8—3321no—
141Q8—3321no—
142Q8—1661no—
143Q8—1661no—
144D4—21no—
145D4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—3321no—
150C2×C4—1661no—
151C2×C4—1662no—
152C2×C4—1661no—
153C2×C4D4×D8311yescharacteristic
154C2×C4D4×D8311yescharacteristic
155C2×C4C2×D33211yescharacteristic
156C8—22no—
157C8—41no—
158C8—3321no—
159C22—41no—
160C22—41no—
161C22—41no—
162C22#1458176211yescharacteristic
163C4—21no—
164C4—41no—
165C4—3321no—
166C4—3322no—
167C4—3321no—
168C4—6641no—
169C4#1458181311yescharacteristic
170C4#1458181411yescharacteristic
171C4#1458181611yescharacteristic
172C4#1458182511yescharacteristic
173C2—81no—
174C2#1912784711yescharacteristic
175C2#1912786511yescharacteristic
176C2#1912788511yescharacteristic
subgroupG/H
centerC22#14581762
commutator, cosocleC42×C83C23
FrattiniC42C22×D83
FittingD4⋊C8×C83C2
socleC22×C83#120
prime radicalC2×D4×C83D4
hypercenterD4⋊C8D83
hypocenterC83#1652
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #19128099C2semidirect
4. #19127912C2semidirect
5. #19127446C2semidirect
6. #19127433C2semidirect
89. C83#1652semidirect

Maximal subgroups

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

subgroupindex
#191284212
#191280992
#191279812
#191279122
#191274462
#191274332
D4⋊C8×C832
#165283