Finite Group Databasebeta

← #24908808

Subgroups — #24908808

Order 10624.

246 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 73 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#19128332C211yesmaximal, characteristic
2#19127741C212yesmaximal
3#19127470C211yesmaximal, characteristic
4#19127461C212yesmaximal
5#384×C83C211yesmaximal, characteristic
6C2×#13050574C2212yes—
7#14581791—22no—
8#14581783—22no—
9#14581771—22no—
10#14581751C2212yes—
11#14581725—22no—
12#14581723—22no—
13C2×C4⋊C4×C83—22no—
14C2×C4⋊C4×C83C2211yescharacteristic
15C2×C4⋊C4×C83C2211yescharacteristic
16C2×C42×C83C2211yescharacteristic
17C2×Dic166—22no—
18#13050574—22no—
19#13050574Q814yescomplemented
20C2×C83⋊C8—22no—
21C4⋊Dic83—22no—
22C4⋊Dic83D412yes—
23C4⋊Dic83D412yes—
24Dic83⋊C4—24no—
25C22⋊Dic83—22no—
26C4×Dic83—22no—
27C22×C4×C83—21no—
28C22×C4×C83—21no—
29C22×C4×C83D412yes—
30C22×C4×C83C2311yescharacteristic
31C4⋊C4×C83—22no—
32C4⋊C4×C83—22no—
33C4⋊C4×C83—44no—
34C42×C83—21no—
35C42×C83Q812yescomplemented
36C23×C83C2×D411yescharacteristic
37C2×C4×C83—22no—
38C2×C4×C83—22no—
39C2×C4×C83—22no—
40C2×C4×C83—22no—
41C2×C4×C83—42no—
42C2×C4×C83—42no—
43C2×C4×C83—41no—
44C2×C4×C83C2×D412yes—
45C2×C4×C83C2×Q812yes—
46C2×C4×C83C2×Q811yescharacteristic
47C2×C4×C83#4111yescharacteristic
48C2×Dic83—22no—
49C2×Dic83—22no—
50Dic166—82no—
51C83⋊C8—24no—
52C22×C83—21no—
53C22×C83—21no—
54C22×C83#12011yescharacteristic
55C22×C83C22⋊Q811yescharacteristic
56C22×C83C4⋊Q811yescharacteristic
57C4×C83—24no—
58C4×C83—42no—
59C4×C83—81no—
60C4×C83C22⋊Q812yes—
61C4×C83C4⋊Q812yes—
62Dic83—42no—
63Dic83—42no—
64C2×C83—21no—
65C2×C83—21no—
66C2×C83#39511yescharacteristic
67C2×C83#45311yescharacteristic
68C2×C83#45611yescharacteristic
69#2060—831nomaximal, Sylow
70C83#206011yescharacteristic, complemented, Sylow
71#544—831no—
72#428—832no—
73#384D8311yescharacteristic
74#343—831no—
75#339—832no—
76C2×M16—832no—
77C4⋊Q8—1662no—
78#125—1662no—
79C22⋊Q8—1662no—
80#117—832no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4C2×D8311yescharacteristic
83C2×C4⋊C4C2×D8311yescharacteristic
84C2×C42C2×D8311yescharacteristic
85#107—1662no—
86#106—1662no—
87C2×Q8—1662no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4C22⋊D8312yes—
91C22×C4C22×D8311yescharacteristic
92M16—834no—
93M16—1662no—
94C2×C8—1662no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—44no—
98C4⋊C4—832no—
99C4⋊C4—832no—
100C4⋊C4—1662no—
101C4⋊C4—1664no—
102C22⋊C4—1662no—
103C42—21no—
104C42—1662no—
105C42Dic16612yescomplemented
106C23C2×C22⋊D8311yescharacteristic
107Q8—6642no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—41no—
115C2×C4—1662no—
116C2×C4—1662no—
117C2×C4D4×D8312yes—
118C2×C4Q8×D8312yes—
119C2×C4C2×Dic16611yescharacteristic
120C2×C4#1305059011yescharacteristic
121C8—1664no—
122C22—21no—
123C22—21no—
124C22#1458176011yescharacteristic
125C22#1458177111yescharacteristic
126C22#1458179011yescharacteristic
127C4—24no—
128C4—42no—
129C4—81no—
130C4—3322no—
131C4—3322no—
132C4#1458177012yes—
133C4#1458179112yes—
134C2—21no—
135C2—21no—
136C2#1912762111yescharacteristic
137C2#1912786811yescharacteristic
138C2#1912788311yescharacteristic
subgroupG/H
centerC22#14581760
commutator, cosocleC22×C4×C83C23
FrattiniC22×C4C22×D83
Fitting#384×C83C2
socleC22×C83#120
prime radicalC23×C83C2×D4
hypercenter#384D83
hypocenterC83#2060
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
19. #13050574Q8semidirect
35. C42×C83Q8semidirect
70. C83#2060semidirect
105. C42Dic166semidirect

Maximal subgroups

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

subgroupindex
#191283322
#191277412
#191274702
#191274612
#384×C832
#206083