Finite Group Databasebeta

← #38768095

Subgroups — #38768095

Order 25472.

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

normal subgroups only

This group has 194 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#27164747C211yesmaximal, characteristic
2#27164423C211yesmaximal, characteristic
3#27164303C211yesmaximal, characteristic
4#27164229C211yesmaximal, characteristic
5#27163765C211yesmaximal, characteristic
6#27163750C211yesmaximal, characteristic
7Q8⋊C8×C199C211yesmaximal, characteristic
8C2×Dic796—21no—
9#20263955—21no—
10#20263955C2211yescharacteristic
11#20263955C2211yescharacteristic
12#20263946—22no—
13#20263946—21no—
14Q8×Dic199—21no—
15C4×Dic398—21no—
16C4×Dic398C2211yescharacteristic
17#20263889—22no—
18#20263886—21no—
19C4⋊(C199⋊C8)C2211yescharacteristic
20#20263880—21no—
21#20263879—21no—
22C4×Q8×C199C2211yescharacteristic
23C4⋊C8×C199C2211yescharacteristic
24C4×C8×C199C2211yescharacteristic
25C2×Dic398—21no—
26C2×Dic398—21no—
27C2×Dic398D411yescharacteristic
28Dic796—42no—
29C2×C199⋊C8—21no—
30C4⋊Dic199—22no—
31C4⋊Dic199—22no—
32C4⋊Dic199—21no—
33C4⋊Dic199—21no—
34C4⋊Dic199—21no—
35C4⋊Dic199D412yes—
36C4⋊Dic199D411yescharacteristic
37Dic199⋊C4—22no—
38Dic199⋊C4—21no—
39C4×Dic199—22no—
40C4×Dic199—21no—
41C2×Q8×C199D411yescharacteristic
42C2×C8×C199—21no—
43C2×C8×C199—21no—
44C4⋊C4×C199—21no—
45C4⋊C4×C199D411yescharacteristic
46C42×C199—21no—
47C42×C199C2311yescharacteristic
48Dic398—21no—
49Dic398—21no—
50Dic398—41no—
51Dic398—41no—
52Dic398—41no—
53Dic398—41no—
54C2×Dic199—21no—
55C2×Dic199—21no—
56C2×Dic199—22no—
57C2×Dic199—41no—
58C199⋊C8—41no—
59Q8×C199—21no—
60Q8×C199Q1612yescomplemented
61C2×C4×C199—21no—
62C2×C4×C199—21no—
63C2×C4×C199C2×D411yescharacteristic
64C2×C4×C199C2×D411yescharacteristic
65C2×C4×C199C2×D411yescharacteristic
66C8×C199—22no—
67C8×C199—41no—
68C22×C199#12011yescharacteristic
69C4×C199—22no—
70C4×C199—21no—
71C4×C199—41no—
72C4×C199C2×Q1611yescharacteristic
73C4×C199#13511yescharacteristic
74C4×C199AGL(1,ℤ/8)11yescharacteristic
75C4×C199#13711yescharacteristic
76Dic199—41no—
77Dic199—42no—
78Dic199—41no—
79Dic199—81no—
80C2×C199#44911yescharacteristic
81C2×C199#45111yescharacteristic
82C2×C199#45611yescharacteristic
83C199#164811yescharacteristic, complemented, Sylow
84#1648—1991nomaximal, Sylow
85#557—1991no—
86#500—1991no—
87#475—1991no—
88#462—1991no—
89#332—1991no—
90Q8⋊C8—1991no—
91Q8⋊C8D19911yescharacteristic
92C2×Q16—3981no—
93C4⋊Q8—3981no—
94C4⋊Q8—1991no—
95C4⋊Q8—1991no—
96#125—3982no—
97#125—3981no—
98C4×Q8—3981no—
99C4×Q8—3981no—
100C4×Q8—1991no—
101C4×Q8C2×D19911yescharacteristic
102#107—3982no—
103#106—3981no—
104C4⋊C8—1991no—
105C4⋊C8C2×D19911yescharacteristic
106#103—3981no—
107#103—3981no—
108C4×C8C2×D19911yescharacteristic
109C2×Q8—3981no—
110C2×Q8—3981no—
111C2×Q8—1991no—
112C2×Q8D79611yescharacteristic
113Q16—7962no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—3981no—
117C4⋊C4—21no—
118C4⋊C4—3982no—
119C4⋊C4—3982no—
120C4⋊C4—3982no—
121C4⋊C4—3981no—
122C4⋊C4—3981no—
123C4⋊C4—3981no—
124C4⋊C4—3981no—
125C4⋊C4—1991no—
126C4⋊C4—1992no—
127C4⋊C4D79611yescharacteristic
128C42—21no—
129C42—3982no—
130C42—3981no—
131C42C22×D19911yescharacteristic
132Q8—21no—
133Q8—7961no—
134Q8—7961no—
135Q8—7961no—
136Q8—7961no—
137Q8—3981no—
138Q8—3981no—
139Q8Dic79612yescomplemented
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—7961no—
143C2×C4—3981no—
144C2×C4—3981no—
145C2×C4—3982no—
146C2×C4D4×D19911yescharacteristic
147C2×C4D4×D19911yescharacteristic
148C2×C4C2×D79611yescharacteristic
149C8—22no—
150C8—41no—
151C8—7961no—
152C22#2026392511yescharacteristic
153C4—22no—
154C4—21no—
155C4—41no—
156C4—7962no—
157C4—7961no—
158C4—7961no—
159C4—15921no—
160C4C2×Dic79611yescharacteristic
161C4#2026397811yescharacteristic
162C4#2026398211yescharacteristic
163C4#2026398811yescharacteristic
164C2#2716416611yescharacteristic
165C2#2716417611yescharacteristic
166C2#2716420211yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
FittingQ8⋊C8×C199C2
socle, prime radicalC22×C199#120
hypercenterQ8⋊C8D199
hypocenterC199#1648
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
60. Q8×C199Q16semidirect
83. C199#1648semidirect
139. Q8Dic796semidirect

Maximal subgroups

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

subgroupindex
#271647472
#271644232
#271643032
#271642292
#271637652
#271637502
Q8⋊C8×C1992
#1648199