Finite Group Databasebeta

← #38768063

Subgroups — #38768063

Order 25472.

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

normal subgroups only

This group has 208 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#27164725C211yesmaximal, characteristic
2#27164370C211yesmaximal, characteristic, complemented
3#27164225C211yesmaximal, characteristic, complemented
4#27163766C211yesmaximal, characteristic, complemented
5#27163748C211yesmaximal, characteristic, complemented
6D4⋊(C199⋊C8)C211yesmaximal, characteristic
7#474×C199C211yesmaximal, characteristic
8C2×#15691876—21no—
9#20263955C2211yescharacteristic
10#20263946—21no—
11#20263934—21no—
12#20263932—22no—
13C4×Dic398C2211yescharacteristic
14D4×Dic199—21no—
15C2×C4⋊Dic199—21no—
16C4⋊(C199⋊C8)C2211yescharacteristic
17#20263879—21no—
18#20263878—22no—
19C4×C199⋊C8C2211yescharacteristic
20C4⋊Q8×C199C2211yescharacteristic
21C4×D4×C199C2211yescharacteristic
22#107×C199—21no—
23C4⋊C8×C199C2211yescharacteristic
24#102×C199—21no—
25C2×Dic398D411yescharacteristic
26C2×Dic398D412yes—
27C22×Dic199—22no—
28#15691876—42no—
29C2×C199⋊C8—21no—
30C2×C199⋊C8—21no—
31C4⋊Dic199—22no—
32C4⋊Dic199—21no—
33C4⋊Dic199—21no—
34C4⋊Dic199D411yescharacteristic
35Dic199⋊C4—22no—
36Dic199⋊C4—21no—
37C22⋊Dic199—22no—
38C4×Dic199—21no—
39C2×Q8×C199—21no—
40C2×D4×C199D411yescharacteristic
41C22×C4×C199—21no—
42C2×C8×C199—21no—
43C4⋊C4×C199—21no—
44C4⋊C4×C199—21no—
45C4⋊C4×C199D411yescharacteristic
46C22⋊C4×C199—21no—
47C42×C199C2311yescharacteristic
48Dic398—21no—
49Dic398—21no—
50Dic398—42no—
51C2×Dic199—21no—
52C2×Dic199—21no—
53C2×Dic199—22no—
54C2×Dic199—82no—
55C199⋊C8—41no—
56C199⋊C8—41no—
57C23×C199—21no—
58Q8×C199—41no—
59Q8×C199—41no—
60D4×C199—21no—
61D4×C199Q1612yescomplemented
62C2×C4×C199—21no—
63C2×C4×C199—41no—
64C2×C4×C199—41no—
65C2×C4×C199—41no—
66C2×C4×C199C2×D411yescharacteristic
67C2×C4×C199C2×D411yescharacteristic
68C2×C4×C199C2×D411yescharacteristic
69C8×C199—41no—
70C22×C199—22no—
71C22×C199—41no—
72C22×C199—41no—
73C22×C199#12011yescharacteristic
74C4×C199—21no—
75C4×C199—41no—
76C4×C199—81no—
77C4×C199C2×Q1611yescharacteristic
78C4×C199#13511yescharacteristic
79C4×C199AGL(1,ℤ/8)11yescharacteristic
80C4×C199#13711yescharacteristic
81Dic199—41no—
82Dic199—41no—
83Dic199—42no—
84C2×C199—42no—
85C2×C199#44911yescharacteristic
86C2×C199#45111yescharacteristic
87C2×C199#45511yescharacteristic
88C199#164411yescharacteristic, complemented, Sylow
89#1644—1991nomaximal, Sylow
90#554—1991no—
91#487—1991no—
92#474D19911yescharacteristic
93#462—1991no—
94#333—1991no—
95Q8⋊C8—1991no—
96D4⋊C8—1991no—
97C2×Q16—3981no—
98C4⋊Q8—1991no—
99C4⋊Q8C2×D19911yescharacteristic
100#125—3981no—
101C22⋊Q8—3982no—
102C22⋊Q8—3981no—
103C4×Q8—1991no—
104C4×D4—3981no—
105C4×D4C2×D19911yescharacteristic
106C2×C4⋊C4—3981no—
107#107—21no—
108C4⋊C8—1991no—
109C4⋊C8C2×D19911yescharacteristic
110#103—3982no—
111#103—3981no—
112#102—21no—
113C4×C8—1991no—
114C2×Q8—21no—
115C2×Q8—1992no—
116C2×Q8—1991no—
117C2×D4C22⋊D19911yescharacteristic
118C22×C4—21no—
119C22×C4—3982no—
120Q16—7962no—
121C2×C8—21no—
122C2×C8—3981no—
123C2×C8—3981no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—3982no—
127C4⋊C4—3982no—
128C4⋊C4—3981no—
129C4⋊C4—3981no—
130C4⋊C4—3981no—
131C4⋊C4—1991no—
132C4⋊C4C22⋊D19911yescharacteristic
133C22⋊C4—21no—
134C22⋊C4—3982no—
135C42—3981no—
136C42C22×D19911yescharacteristic
137C23—21no—
138Q8—41no—
139Q8—41no—
140Q8—7962no—
141Q8—3981no—
142Q8—3981no—
143D4—21no—
144D4#1569187612yescomplemented
145C2×C4—21no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—15922no—
150C2×C4—3981no—
151C2×C4—3981no—
152C2×C4—3982no—
153C2×C4D4×D19911yescharacteristic
154C2×C4D4×D19911yescharacteristic
155C2×C4C2×C22⋊D19911yescharacteristic
156C8—41no—
157C8—7961no—
158C8—7961no—
159C22—22no—
160C22—41no—
161C22—41no—
162C22#2026392311yescharacteristic
163C4—21no—
164C4—41no—
165C4—81no—
166C4—7961no—
167C4—7961no—
168C4—7962no—
169C4C2×#1569187611yescharacteristic
170C4#2026397411yescharacteristic
171C4#2026397911yescharacteristic
172C4#2026398511yescharacteristic
173C2—42no—
174C2#2716416011yescharacteristic
175C2#2716417311yescharacteristic
176C2#2716419411yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#474×C199C2
socleC22×C199#120
prime radicalC2×D4×C199D4
hypercenter#474D199
hypocenterC199#1644
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #27164370C2semidirect
3. #27164225C2semidirect
4. #27163766C2semidirect
5. #27163748C2semidirect
61. D4×C199Q16semidirect
88. C199#1644semidirect
144. D4#15691876semidirect

Maximal subgroups

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

subgroupindex
#271647252
#271643702
#271642252
#271637662
#271637482
D4⋊(C199⋊C8)2
#474×C1992
#1644199