Finite Group Databasebeta

← #38767943

Subgroups — #38767943

Order 25472.

210 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 55 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#27164743C211yesmaximal, characteristic, complemented
2#27164215C211yesmaximal, characteristic
3#27164215C211yesmaximal, characteristic
4#27163764C211yesmaximal, characteristic, complemented
5#27163750C211yesmaximal, characteristic, complemented
6#27163750C211yesmaximal, characteristic, complemented
7#486×C199C211yesmaximal, characteristic
8C2×#15691874—21no—
9C2×#15691874—21no—
10#20263953—21no—
11#20263946—21no—
12#20263945—22no—
13Q8×Dic199—21no—
14C4×Dic398C2211yescharacteristic
15C4×Dic398C2211yescharacteristic
16#20263920—21no—
17C4⋊(C199⋊C8)C2211yescharacteristic
18C4⋊(C199⋊C8)C2211yescharacteristic
19#20263876—21no—
20#20263876—21no—
21C4⋊Q8×C199C2211yescharacteristic
22#127×C199C2211yescharacteristic
23#102×C199—22no—
24C4×C8×C199C2211yescharacteristic
25C2×Dic398D411yescharacteristic
26C2×Dic398D411yescharacteristic
27#15691874—42no—
28#15691874—42no—
29C2×C199⋊C8—21no—
30C2×C199⋊C8—21no—
31C4⋊Dic199—22no—
32C4⋊Dic199D411yescharacteristic
33C4⋊Dic199D411yescharacteristic
34Dic199⋊C4—22no—
35Dic199⋊C4—22no—
36Dic199⋊C4—21no—
37Dic199⋊C4—21no—
38C4×Dic199—22no—
39C4×Dic199—21no—
40C4×Dic199—21no—
41C2×Q8×C199—21no—
42C2×D4×C199—21no—
43C2×D4×C199—21no—
44C2×D4×C199—21no—
45C2×C8×C199—21no—
46C4⋊C4×C199—21no—
47C4⋊C4×C199D412yes—
48C42×C199C2311yescharacteristic
49Dic398—21no—
50Dic398—21no—
51Dic398—21no—
52Dic398SD1612yescomplemented
53C2×Dic199—21no—
54C2×Dic199—21no—
55C2×Dic199—21no—
56C2×Dic199—21no—
57C199⋊C8—41no—
58C199⋊C8—41no—
59C23×C199—41no—
60Q8×C199—41no—
61Q8×C199—41no—
62D4×C199—42no—
63D4×C199—41no—
64D4×C199—41no—
65D4×C199—41no—
66D4×C199—41no—
67C2×C4×C199—22no—
68C2×C4×C199C2×D411yescharacteristic
69C2×C4×C199C2×D411yescharacteristic
70C2×C4×C199C2×D411yescharacteristic
71C8×C199—41no—
72C22×C199—81no—
73C22×C199—81no—
74C22×C199—81no—
75C22×C199#12011yescharacteristic
76C4×C199—21no—
77C4×C199—42no—
78C4×C199C2×SD1611yescharacteristic
79C4×C199#13511yescharacteristic
80C4×C199AGL(1,ℤ/8)11yescharacteristic
81C4×C199#13711yescharacteristic
82Dic199—22no—
83Dic199—41no—
84Dic199—41no—
85Dic199—41no—
86C2×C199—161no—
87C2×C199#44811yescharacteristic
88C2×C199#44911yescharacteristic
89C2×C199#45411yescharacteristic
90C199#162611yescharacteristic, complemented, Sylow
91#1626—1991nomaximal, Sylow
92#557—1991no—
93#486D19911yescharacteristic
94#460—1991no—
95#460—1991no—
96#332—1991no—
97Q8⋊C8—1991no—
98Q8⋊C8—1991no—
99C2×SD16—3981no—
100C2×SD16—3981no—
101C4⋊Q8—3981no—
102C4⋊Q8C2×D19911yescharacteristic
103#127C2×D19911yescharacteristic
104#125—3981no—
105#125—3982no—
106C4×Q8—3981no—
107C4×Q8—3981no—
108C4×Q8—1991no—
109C4×Q8—1991no—
110C4⋊C8—1991no—
111C4⋊C8—1991no—
112#102—22no—
113#102—3981no—
114#102—3981no—
115C4×C8C2×D19911yescharacteristic
116C2×Q8—21no—
117C2×Q8—1991no—
118C2×Q8—1991no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—21no—
122SD16—7962no—
123SD16—7962no—
124C2×C8—21no—
125C2×C8—3981no—
126C2×C8—3981no—
127C4⋊C4—21no—
128C4⋊C4—3982no—
129C4⋊C4—3982no—
130C4⋊C4—3982no—
131C4⋊C4—3981no—
132C4⋊C4—3981no—
133C4⋊C4—1991no—
134C4⋊C4—1991no—
135C4⋊C4C22⋊D19912yes—
136C42—3982no—
137C42—3981no—
138C42—3981no—
139C42C22×D19911yescharacteristic
140C23—41no—
141Q8—41no—
142Q8—41no—
143Q8—3981no—
144Q8—3981no—
145Q8—3981no—
146Q8—1992no—
147D4—42no—
148D4—41no—
149D4—41no—
150D4—41no—
151D4—41no—
152C2×C4—22no—
153C2×C4—3981no—
154C2×C4—3981no—
155C2×C4—3981no—
156C2×C4—3981no—
157C2×C4D4×D19911yescharacteristic
158C2×C4D4×D19911yescharacteristic
159C2×C4C2×C22⋊D19911yescharacteristic
160C8—41no—
161C8—7961no—
162C8—7961no—
163C22—81no—
164C22—81no—
165C22—81no—
166C22#2026392311yescharacteristic
167C4—21no—
168C4—42no—
169C4—7961no—
170C4—7961no—
171C4—7961no—
172C4—3982no—
173C4SD16×D19911yescharacteristic
174C4#2026397411yescharacteristic
175C4#2026397911yescharacteristic
176C4#2026398411yescharacteristic
177C2—161no—
178C2#2716415511yescharacteristic
179C2#2716416011yescharacteristic
180C2#2716419011yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#486×C199C2
socleC22×C199#120
prime radical#127×C199C22
hypercenter#486D199
hypocenterC199#1626
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #27164743C2semidirect
4. #27163764C2semidirect
5. #27163750C2semidirect
6. #27163750C2semidirect
52. Dic398SD16semidirect
90. C199#1626semidirect

Maximal subgroups

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

subgroupindex
#271647432
#271642152
#271637642
#271637502
#486×C1992
#1626199