Finite Group Databasebeta

← #24906708

Subgroups — #24906708

Order 10624.

194 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 57 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#19128430C211yesmaximal, characteristic
2#19128106C211yesmaximal, characteristic
3#19127986C211yesmaximal, characteristic
4#19127912C211yesmaximal, characteristic
5#19127448C211yesmaximal, characteristic
6#19127433C211yesmaximal, characteristic
7Q8⋊C8×C83C211yesmaximal, characteristic
8C2×Dic332—21no—
9#14581792—21no—
10#14581792C2211yescharacteristic
11#14581792C2211yescharacteristic
12#14581783—22no—
13#14581783—21no—
14Q8×Dic83—21no—
15C4×Dic166—21no—
16C4×Dic166C2211yescharacteristic
17#14581726—22no—
18#14581723—21no—
19C4⋊(C83⋊C8)C2211yescharacteristic
20#14581717—21no—
21#14581716—21no—
22C4×Q8×C83C2211yescharacteristic
23C4⋊C8×C83C2211yescharacteristic
24C4×C8×C83C2211yescharacteristic
25C2×Dic166—21no—
26C2×Dic166—21no—
27C2×Dic166D411yescharacteristic
28Dic332—42no—
29C2×C83⋊C8—21no—
30C4⋊Dic83—22no—
31C4⋊Dic83—22no—
32C4⋊Dic83—21no—
33C4⋊Dic83—21no—
34C4⋊Dic83—21no—
35C4⋊Dic83D412yes—
36C4⋊Dic83D411yescharacteristic
37Dic83⋊C4—22no—
38Dic83⋊C4—21no—
39C4×Dic83—22no—
40C4×Dic83—21no—
41C2×Q8×C83D411yescharacteristic
42C2×C8×C83—21no—
43C2×C8×C83—21no—
44C4⋊C4×C83—21no—
45C4⋊C4×C83D411yescharacteristic
46C42×C83—21no—
47C42×C83C2311yescharacteristic
48Q8×C83—21no—
49Q8×C83Q1612yescomplemented
50C2×C4×C83—21no—
51C2×C4×C83—21no—
52C2×C4×C83C2×D411yescharacteristic
53C2×C4×C83C2×D411yescharacteristic
54C2×C4×C83C2×D411yescharacteristic
55C2×Dic83—21no—
56C2×Dic83—21no—
57C2×Dic83—22no—
58C2×Dic83—41no—
59Dic166—21no—
60Dic166—21no—
61Dic166—41no—
62Dic166—41no—
63Dic166—41no—
64Dic166—41no—
65C8×C83—22no—
66C8×C83—41no—
67C83⋊C8—41no—
68C22×C83#12011yescharacteristic
69C4×C83—22no—
70C4×C83—21no—
71C4×C83—41no—
72C4×C83C2×Q1611yescharacteristic
73C4×C83#13511yescharacteristic
74C4×C83AGL(1,ℤ/8)11yescharacteristic
75C4×C83#13711yescharacteristic
76Dic83—41no—
77Dic83—42no—
78Dic83—41no—
79Dic83—81no—
80C2×C83#44911yescharacteristic
81C2×C83#45111yescharacteristic
82C2×C83#45611yescharacteristic
83#1648—831nomaximal, Sylow
84C83#164811yescharacteristic, complemented, Sylow
85#557—831no—
86#500—831no—
87#475—831no—
88#462—831no—
89#332—831no—
90Q8⋊C8—831no—
91Q8⋊C8D8311yescharacteristic
92C2×Q16—1661no—
93C4⋊Q8—831no—
94C4⋊Q8—831no—
95C4⋊Q8—1661no—
96#125—1662no—
97#125—1661no—
98C4×Q8—831no—
99C4×Q8—1661no—
100C4×Q8—1661no—
101C4×Q8C2×D8311yescharacteristic
102#107—1662no—
103#106—1661no—
104C4⋊C8—831no—
105C4⋊C8C2×D8311yescharacteristic
106#103—1661no—
107#103—1661no—
108C4×C8C2×D8311yescharacteristic
109C2×Q8—831no—
110C2×Q8—1661no—
111C2×Q8—1661no—
112C2×Q8D33211yescharacteristic
113Q16—3322no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—1661no—
117C4⋊C4—21no—
118C4⋊C4—831no—
119C4⋊C4—832no—
120C4⋊C4—1662no—
121C4⋊C4—1662no—
122C4⋊C4—1662no—
123C4⋊C4—1661no—
124C4⋊C4—1661no—
125C4⋊C4—1661no—
126C4⋊C4—1661no—
127C4⋊C4D33211yescharacteristic
128C42—21no—
129C42—1662no—
130C42—1661no—
131C42C22×D8311yescharacteristic
132Q8—21no—
133Q8—3321no—
134Q8—3321no—
135Q8—3321no—
136Q8—3321no—
137Q8—1661no—
138Q8—1661no—
139Q8Dic33212yescomplemented
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—3321no—
143C2×C4—1661no—
144C2×C4—1662no—
145C2×C4—1661no—
146C2×C4D4×D8311yescharacteristic
147C2×C4D4×D8311yescharacteristic
148C2×C4C2×D33211yescharacteristic
149C8—22no—
150C8—41no—
151C8—3321no—
152C22#1458176211yescharacteristic
153C4—22no—
154C4—21no—
155C4—41no—
156C4—3322no—
157C4—3321no—
158C4—3321no—
159C4—6641no—
160C4C2×Dic33211yescharacteristic
161C4#1458181511yescharacteristic
162C4#1458181911yescharacteristic
163C4#1458182511yescharacteristic
164C2#1912784911yescharacteristic
165C2#1912785911yescharacteristic
166C2#1912788511yescharacteristic
subgroupG/H
centerC22#14581762
commutator, cosocleC42×C83C23
FrattiniC42C22×D83
FittingQ8⋊C8×C83C2
socle, prime radicalC22×C83#120
hypercenterQ8⋊C8D83
hypocenterC83#1648
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
49. Q8×C83Q16semidirect
84. C83#1648semidirect
139. Q8Dic332semidirect

Maximal subgroups

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

subgroupindex
#191284302
#191281062
#191279862
#191279122
#191274482
#191274332
Q8⋊C8×C832
#164883