Finite Group Databasebeta

← #38768108

Subgroups — #38768108

Order 25472.

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#27164738C211yesmaximal, characteristic
2#27164423C211yesmaximal, characteristic, complemented
3#27164304C211yesmaximal, characteristic, complemented
4#27164223C211yesmaximal, characteristic
5#27163763C211yesmaximal, characteristic, complemented
6#27163750C211yesmaximal, characteristic, complemented
7D4⋊C8×C199C211yesmaximal, characteristic
8C2×#15691874—21no—
9#20263955C2211yescharacteristic
10#20263955C2211yescharacteristic
11#20263948—22no—
12#20263942—21no—
13#20263934—21no—
14C4×Dic398C2211yescharacteristic
15#20263920—21no—
16#20263914—21no—
17#20263889—22no—
18#20263887—21no—
19C4⋊(C199⋊C8)C2211yescharacteristic
20#20263880—21no—
21#20263878—21no—
22C4×D4×C199C2211yescharacteristic
23C4⋊C8×C199C2211yescharacteristic
24C4×C8×C199C2211yescharacteristic
25C2×Dic398—21no—
26C2×Dic398—21no—
27C2×Dic398D411yescharacteristic
28#15691874—42no—
29C2×C199⋊C8—21no—
30C4⋊Dic199—21no—
31C4⋊Dic199—21no—
32C4⋊Dic199—21no—
33C4⋊Dic199D412yes—
34C4⋊Dic199D411yescharacteristic
35Dic199⋊C4—22no—
36Dic199⋊C4—21no—
37C22⋊Dic199—22no—
38C22⋊Dic199—22no—
39C4×Dic199—22no—
40C4×Dic199—21no—
41C2×D4×C199D411yescharacteristic
42C22×C4×C199—21no—
43C2×C8×C199—21no—
44C2×C8×C199—21no—
45C4⋊C4×C199D411yescharacteristic
46C22⋊C4×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—
59C23×C199—21no—
60D4×C199—21no—
61D4×C199—21no—
62C2×C4×C199—21no—
63C2×C4×C199—41no—
64C2×C4×C199—41no—
65C2×C4×C199C2×D411yescharacteristic
66C2×C4×C199C2×D411yescharacteristic
67C2×C4×C199C2×D411yescharacteristic
68C8×C199—22no—
69C8×C199—41no—
70C22×C199—41no—
71C22×C199—41no—
72C22×C199—41no—
73C22×C199#12011yescharacteristic
74C4×C199—21no—
75C4×C199—41no—
76C4×C199#13511yescharacteristic
77C4×C199#13511yescharacteristic
78C4×C199AGL(1,ℤ/8)11yescharacteristic
79C4×C199#13711yescharacteristic
80Dic199—42no—
81Dic199—41no—
82Dic199—41no—
83Dic199—81no—
84C2×C199—81no—
85C2×C199#44911yescharacteristic
86C2×C199#45211yescharacteristic
87C2×C199#45611yescharacteristic
88C199#165011yescharacteristic, complemented, Sylow
89#1650—1991nomaximal, Sylow
90#556—1991no—
91#500—1991no—
92#475—1991no—
93#461—1991no—
94#332—1991no—
95Q8⋊C8—1991no—
96D4⋊C8D19911yescharacteristic
97C2×SD16—3981no—
98C4⋊Q8—1991no—
99C4⋊Q8—1991no—
100#126—3982no—
101#124—3981no—
102C22⋊Q8—3981no—
103C4×Q8—3981no—
104C4×Q8—1991no—
105C4×D4C2×D19911yescharacteristic
106#117—3981no—
107#107—3982no—
108#106—3981no—
109C4⋊C8—1991no—
110C4⋊C8C2×D19911yescharacteristic
111#103—3981no—
112#103—3981no—
113C4×C8C2×D19911yescharacteristic
114C2×Q8—3981no—
115C2×Q8—3981no—
116C2×Q8—1991no—
117C2×D4D79611yescharacteristic
118C22×C4—21no—
119SD16—7962no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—3981no—
123C4⋊C4—3982no—
124C4⋊C4—3981no—
125C4⋊C4—3981no—
126C4⋊C4—3981no—
127C4⋊C4—3981no—
128C4⋊C4—1991no—
129C4⋊C4—1992no—
130C4⋊C4D79611yescharacteristic
131C22⋊C4—21no—
132C22⋊C4—3982no—
133C22⋊C4—3982no—
134C42—3982no—
135C42—3981no—
136C42C22×D19911yescharacteristic
137C23—21no—
138Q8—7961no—
139Q8—7961no—
140Q8—7961no—
141Q8—7961no—
142Q8—3981no—
143Q8—3981no—
144D4—21no—
145D4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—7961no—
150C2×C4—3981no—
151C2×C4—3981no—
152C2×C4—3982no—
153C2×C4D4×D19911yescharacteristic
154C2×C4D4×D19911yescharacteristic
155C2×C4C2×D79611yescharacteristic
156C8—22no—
157C8—41no—
158C8—7961no—
159C22—41no—
160C22—41no—
161C22—41no—
162C22#2026392511yescharacteristic
163C4—21no—
164C4—41no—
165C4—7962no—
166C4—7961no—
167C4—7961no—
168C4—15921no—
169C4#2026397411yescharacteristic
170C4#2026397711yescharacteristic
171C4#2026398311yescharacteristic
172C4#2026398411yescharacteristic
173C2—81no—
174C2#2716416511yescharacteristic
175C2#2716418111yescharacteristic
176C2#2716420211yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
FittingD4⋊C8×C199C2
socleC22×C199#120
prime radicalC2×D4×C199D4
hypercenterD4⋊C8D199
hypocenterC199#1650
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #27164423C2semidirect
3. #27164304C2semidirect
5. #27163763C2semidirect
6. #27163750C2semidirect
88. C199#1650semidirect

Maximal subgroups

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

subgroupindex
#271647382
#271644232
#271643042
#271642232
#271637632
#271637502
D4⋊C8×C1992
#1650199