Finite Group Databasebeta

← #38768069

Subgroups — #38768069

Order 25472.

200 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 59 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
1Q8×Dic398C211yesmaximal, characteristic
2#27164370C211yesmaximal, characteristic
3#27164300C211yesmaximal, characteristic
4#27163766C211yesmaximal, characteristic
5Q8⋊(C199⋊C8)C211yesmaximal, characteristic
6#27163748C211yesmaximal, characteristic
7#462×C199C211yesmaximal, characteristic
8#20263955—21no—
9#20263955C2211yescharacteristic
10#20263954—22no—
11#20263954—21no—
12Q8×Dic199—21no—
13C4×Dic398—21no—
14C4×Dic398C2211yescharacteristic
15#20263886—21no—
16C4⋊(C199⋊C8)C2211yescharacteristic
17#20263878—21no—
18#20263878—22no—
19C4×C199⋊C8C2211yescharacteristic
20C2×Q16×C199—21no—
21C4⋊Q8×C199C2211yescharacteristic
22C4×Q8×C199C2211yescharacteristic
23C4⋊C8×C199C2211yescharacteristic
24#103×C199—21no—
25C2×Dic398—22no—
26C2×Dic398D411yescharacteristic
27C2×Dic398D412yes—
28C2×C199⋊C8—21no—
29C2×C199⋊C8—21no—
30C4⋊Dic199—22no—
31C4⋊Dic199—21no—
32C4⋊Dic199—21no—
33C4⋊Dic199D411yescharacteristic
34Dic199⋊C4—22no—
35Dic199⋊C4—21no—
36C4×Dic199—22no—
37C4×Dic199—21no—
38C2×Q8×C199—21no—
39C2×Q8×C199D411yescharacteristic
40Q16×C199—42no—
41C2×C8×C199—21no—
42C4⋊C4×C199—21no—
43C4⋊C4×C199—21no—
44C4⋊C4×C199—21no—
45C4⋊C4×C199D411yescharacteristic
46C42×C199—21no—
47C42×C199C2311yescharacteristic
48Dic398—21no—
49Dic398—42no—
50Dic398—82no—
51Dic398Q1612yescomplemented
52C2×Dic199—21no—
53C2×Dic199—22no—
54C2×Dic199—21no—
55C199⋊C8—41no—
56C199⋊C8—41no—
57Q8×C199—21no—
58Q8×C199—41no—
59Q8×C199—41no—
60Q8×C199SD1612yes—
61C2×C4×C199—21no—
62C2×C4×C199—21no—
63C2×C4×C199—41no—
64C2×C4×C199C2×D411yescharacteristic
65C2×C4×C199C2×D411yescharacteristic
66C2×C4×C199C2×D411yescharacteristic
67C8×C199—41no—
68C22×C199#12011yescharacteristic
69C4×C199—22no—
70C4×C199—21no—
71C4×C199—41no—
72C4×C199—81no—
73C4×C199C2×SD1611yescharacteristic
74C4×C199C2×Q1611yescharacteristic
75C4×C199#13711yescharacteristic
76C4×C199#13711yescharacteristic
77Dic199—22no—
78Dic199—42no—
79Dic199—41no—
80C2×C199#44811yescharacteristic
81C2×C199#45111yescharacteristic
82C2×C199#45511yescharacteristic
83C199#164511yescharacteristic, complemented, Sylow
84#1645—1991nomaximal, Sylow
85Q82—1991no—
86#487—1991no—
87#475—1991no—
88#462D19911yescharacteristic
89#333—1991no—
90Q8⋊C8—1991no—
91Q8⋊C8—1991no—
92C2×Q16—21no—
93C4⋊Q8—3981no—
94C4⋊Q8—3982no—
95C4⋊Q8—3981no—
96C4⋊Q8—1991no—
97C4⋊Q8C2×D19911yescharacteristic
98C4×Q8—3981no—
99C4×Q8—3981no—
100C4×Q8—1991no—
101C4×Q8C2×D19911yescharacteristic
102#106—3981no—
103C4⋊C8—1991no—
104C4⋊C8C2×D19911yescharacteristic
105#103—21no—
106#103—3981no—
107#103—3982no—
108C4×C8—1991no—
109C2×Q8—21no—
110C2×Q8—3982no—
111C2×Q8—1992no—
112C2×Q8—1991no—
113C2×Q8C22⋊D19911yescharacteristic
114Q16—42no—
115C2×C8—21no—
116C2×C8—3981no—
117C2×C8—3981no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—3982no—
122C4⋊C4—3982no—
123C4⋊C4—3981no—
124C4⋊C4—3981no—
125C4⋊C4—3981no—
126C4⋊C4—1991no—
127C4⋊C4C22⋊D19911yescharacteristic
128C42—21no—
129C42—3982no—
130C42—3981no—
131C42C22×D19911yescharacteristic
132Q8—21no—
133Q8—41no—
134Q8—41no—
135Q8—7962no—
136Q8—15922no—
137Q8—3981no—
138Q8—1992no—
139Q8#1569187412yes—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—41no—
143C2×C4—3982no—
144C2×C4—3981no—
145C2×C4—3981no—
146C2×C4D4×D19911yescharacteristic
147C2×C4D4×D19911yescharacteristic
148C2×C4C2×C22⋊D19911yescharacteristic
149C8—41no—
150C8—7961no—
151C8—7961no—
152C22#2026392311yescharacteristic
153C4—21no—
154C4—22no—
155C4—41no—
156C4—81no—
157C4—7962no—
158C4—7961no—
159C4—3982no—
160C4C2×#1569187411yescharacteristic
161C4Q16×D19911yescharacteristic
162C4#2026398511yescharacteristic
163C4#2026398711yescharacteristic
164C2#2716415611yescharacteristic
165C2#2716417211yescharacteristic
166C2#2716419411yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#462×C199C2
socle, prime radicalC22×C199#120
hypercenter#462D199
hypocenterC199#1645
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
51. Dic398Q16semidirect
83. C199#1645semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic3982
#271643702
#271643002
#271637662
Q8⋊(C199⋊C8)2
#271637482
#462×C1992
#1645199