Finite Group Databasebeta

← #24906731

Subgroups — #24906731

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#19128424C211yesmaximal, characteristic
2#19128097C211yesmaximal, characteristic, complemented
3#19127978C211yesmaximal, characteristic
4#19127446C211yesmaximal, characteristic, complemented
5Q8⋊(C83⋊C8)C211yesmaximal, characteristic, complemented
6#19127428C211yesmaximal, characteristic
7#462×C83C211yesmaximal, characteristic, complemented
8#14581792C2211yescharacteristic
9#14581787—22no—
10#14581780—21no—
11#14581768—21no—
12C4×Dic166—21no—
13C4×D332C2211yescharacteristic
14#14581750—21no—
15#14581725—21no—
16#14581723—22no—
17C4⋊(C83⋊C8)C2211yescharacteristic
18#14581712—21no—
19C4×C83⋊C8C2211yescharacteristic
20C2×Q16×C83—21no—
21C4⋊Q8×C83C2211yescharacteristic
22C4×Q8×C83C2211yescharacteristic
23C4⋊C8×C83C2211yescharacteristic
24#103×C83—21no—
25C2×Dic166—21no—
26C2×D332D411yescharacteristic
27C2×C4×D83—21no—
28C2×C83⋊C8—21no—
29C2×C83⋊C8—21no—
30C4⋊Dic83—21no—
31C4⋊Dic83D412yes—
32C4⋊Dic83D411yescharacteristic
33Dic83⋊C4—22no—
34(C2×D83)⋊C4—22no—
35(C2×D83)⋊C4—22no—
36(C2×D83)⋊C4—21no—
37C4×Dic83—22no—
38C2×Q8×C83—21no—
39C2×Q8×C83D411yescharacteristic
40Q16×C83—42no—
41C2×C8×C83—21no—
42C4⋊C4×C83—21no—
43C4⋊C4×C83—21no—
44C4⋊C4×C83—21no—
45C4⋊C4×C83D411yescharacteristic
46C42×C83—21no—
47C42×C83C2311yescharacteristic
48C22×D83—21no—
49Q8×C83—21no—
50Q8×C83—21no—
51Q8×C83—41no—
52Q8×C83—41no—
53C2×C4×C83—21no—
54C2×C4×C83—21no—
55C2×C4×C83—41no—
56C2×C4×C83C2×D411yescharacteristic
57C2×C4×C83C2×D411yescharacteristic
58C2×C4×C83C2×D411yescharacteristic
59C2×Dic83—21no—
60C2×Dic83—22no—
61D332—21no—
62D332—21no—
63C4×D83—41no—
64C4×D83—41no—
65Dic166—41no—
66Dic166—41no—
67C8×C83—41no—
68C83⋊C8—22no—
69C83⋊C8—41no—
70C22×C83#12011yescharacteristic
71C2×D83—41no—
72C2×D83—41no—
73C2×D83—41no—
74C4×C83—21no—
75C4×C83—41no—
76C4×C83—41no—
77C4×C83—81no—
78C4×C83#13511yescharacteristic
79C4×C83#13511yescharacteristic
80C4×C83AGL(1,ℤ/8)11yescharacteristic
81C4×C83#13711yescharacteristic
82Dic83—41no—
83Dic83—42no—
84C2×C83#44911yescharacteristic
85C2×C83#45211yescharacteristic
86C2×C83#45611yescharacteristic
87D83—81no—
88#1652—831nomaximal, Sylow
89C83#165211yescharacteristic, complemented, Sylow
90#556—831no—
91#498—831no—
92#474—831no—
93#462D8311yescharacteristic, complemented
94#332—831no—
95Q8⋊C8—831no—
96D4⋊C8—831no—
97C2×Q16—21no—
98C4⋊Q8—831no—
99C4⋊Q8C2×D8311yescharacteristic
100#126—1662no—
101#124—1661no—
102C22⋊Q8—1661no—
103C4×Q8—1661no—
104C4×Q8C2×D8311yescharacteristic
105C4×D4—831no—
106#117—1661no—
107#107—1661no—
108#106—1662no—
109C4⋊C8—831no—
110C4⋊C8C2×D8311yescharacteristic
111#103—21no—
112#102—1661no—
113C4×C8—831no—
114C2×Q8—21no—
115C2×Q8—1661no—
116C2×Q8C22⋊D8311yescharacteristic
117C2×D4—831no—
118C22×C4—1661no—
119Q16—42no—
120C2×C8—21no—
121C2×C8—1661no—
122C2×C8—1661no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—832no—
127C4⋊C4—831no—
128C4⋊C4—1662no—
129C4⋊C4—1661no—
130C4⋊C4C22⋊D8311yescharacteristic
131C22⋊C4—1662no—
132C22⋊C4—1662no—
133C22⋊C4—1661no—
134C42—21no—
135C42—1662no—
136C42C22×D8311yescharacteristic
137C23—1661no—
138Q8—21no—
139Q8—21no—
140Q8—41no—
141Q8—41no—
142Q8—3321no—
143Q8—3321no—
144D4—1661no—
145D4—1661no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—41no—
149C2×C4—3321no—
150C2×C4—3321no—
151C2×C4—1661no—
152C2×C4—1662no—
153C2×C4D4×D8311yescharacteristic
154C2×C4D4×D8311yescharacteristic
155C2×C4C2×C22⋊D8311yescharacteristic
156C8—41no—
157C8—3321no—
158C8—1662no—
159C22—3321no—
160C22—3321no—
161C22—3321no—
162C22#1458176011yescharacteristic
163C4—21no—
164C4—41no—
165C4—41no—
166C4—81no—
167C4—3321no—
168C4—3322no—
169C4#1458181211yescharacteristic
170C4#1458181511yescharacteristic
171C4#1458181811yescharacteristic
172C4#1458182411yescharacteristic
173C2—6641no—
174C2#1912784411yescharacteristic
175C2#1912786211yescharacteristic
176C2#1912788211yescharacteristic
subgroupG/H
centerC22#14581760
commutator, cosocleC42×C83C23
FrattiniC42C22×D83
Fitting#462×C83C2
socleC22×C83#120
prime radicalC2×D332D4
hypercenter#462D83
hypocenterC83#1652
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
2. #19128097C2semidirect
4. #19127446C2semidirect
5. Q8⋊(C83⋊C8)C2semidirect
7. #462×C83C2semidirect
89. C83#1652semidirect
93. #462D83semidirect

Maximal subgroups

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

subgroupindex
#191284242
#191280972
#191279782
#191274462
Q8⋊(C83⋊C8)2
#191274282
#462×C832
#165283