Finite Group Databasebeta

← #38741985

Subgroups — #38741985

Order 25216.

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#27161799C211yesmaximal, characteristic
2#27161477C211yesmaximal, characteristic, complemented
3#27161359C211yesmaximal, characteristic
4#27161290C211yesmaximal, characteristic, complemented
5#27160824C211yesmaximal, characteristic, complemented
6#27160811C211yesmaximal, characteristic, complemented
7D4⋊C8×C197C211yesmaximal, characteristic
8C2×Dic788—21no—
9#20263326C2211yescharacteristic
10#20263326C2211yescharacteristic
11#20263319—22no—
12#20263313—21no—
13#20263305—21no—
14C4×Dic394C2211yescharacteristic
15#20263291—21no—
16#20263285—21no—
17#20263259—21no—
18#20263258—22no—
19C4⋊#13096802C2211yescharacteristic
20#20263251—21no—
21#20263247—21no—
22C4×D4×C197C2211yescharacteristic
23C4⋊C8×C197C2211yescharacteristic
24C4×C8×C197C2211yescharacteristic
25C2×Dic394—21no—
26C2×Dic394—21no—
27C2×Dic394D411yescharacteristic
28Dic788—42no—
29C2×#13096802—21no—
30C4⋊Dic197—21no—
31C4⋊Dic197—21no—
32C4⋊Dic197—21no—
33C4⋊Dic197D411yescharacteristic
34C4⋊Dic197D412yes—
35#15691337—22no—
36#15691337—21no—
37C22⋊Dic197—22no—
38C22⋊Dic197—22no—
39C4×Dic197—22no—
40C4×Dic197—21no—
41C2×D4×C197D411yescharacteristic
42C22×C4×C197—21no—
43C2×C8×C197—21no—
44C2×C8×C197—21no—
45C4⋊C4×C197D411yescharacteristic
46C22⋊C4×C197—21no—
47C42×C197C2311yescharacteristic
48Dic394—21no—
49Dic394—21no—
50Dic394—41no—
51Dic394—41no—
52Dic394—41no—
53Dic394—41no—
54C2×Dic197—21no—
55C2×Dic197—21no—
56C2×Dic197—22no—
57C2×Dic197—41no—
58#13096802—41no—
59C23×C197—21no—
60D4×C197—21no—
61D4×C197—21no—
62C2×C4×C197—21no—
63C2×C4×C197—41no—
64C2×C4×C197—41no—
65C2×C4×C197C2×D411yescharacteristic
66C2×C4×C197C2×D411yescharacteristic
67C2×C4×C197C2×D411yescharacteristic
68C8×C197—22no—
69C8×C197—41no—
70C22×C197—41no—
71C22×C197—41no—
72C22×C197—41no—
73C22×C197#12011yescharacteristic
74C4×C197—21no—
75C4×C197—41no—
76C4×C197#13511yescharacteristic
77C4×C197#13511yescharacteristic
78C4×C197AGL(1,ℤ/8)11yescharacteristic
79C4×C197#13711yescharacteristic
80Dic197—41no—
81Dic197—42no—
82Dic197—41no—
83Dic197—81no—
84C2×C197—81no—
85C2×C197#44911yescharacteristic
86C2×C197#45211yescharacteristic
87C2×C197#45611yescharacteristic
88C197#165211yescharacteristic, complemented, Sylow
89#1652—1971nomaximal, Sylow
90#556—1971no—
91#498—1971no—
92#474—1971no—
93#462—1971no—
94#332—1971no—
95Q8⋊C8—1971no—
96D4⋊C8D19711yescharacteristic
97C2×Q16—3941no—
98C4⋊Q8—1971no—
99C4⋊Q8—1971no—
100#126—3942no—
101#124—3941no—
102C22⋊Q8—3941no—
103C4×Q8—3941no—
104C4×Q8—1971no—
105C4×D4C2×D19711yescharacteristic
106#117—3941no—
107#107—3941no—
108#106—3942no—
109C4⋊C8—1971no—
110C4⋊C8C2×D19711yescharacteristic
111#103—3941no—
112#102—3941no—
113C4×C8C2×D19711yescharacteristic
114C2×Q8—3941no—
115C2×Q8—3941no—
116C2×Q8—1971no—
117C2×D4D78811yescharacteristic
118C22×C4—21no—
119Q16—7882no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—3941no—
123C4⋊C4—3942no—
124C4⋊C4—3941no—
125C4⋊C4—3941no—
126C4⋊C4—3941no—
127C4⋊C4—3941no—
128C4⋊C4—1971no—
129C4⋊C4—1972no—
130C4⋊C4D78811yescharacteristic
131C22⋊C4—21no—
132C22⋊C4—3942no—
133C22⋊C4—3942no—
134C42—3942no—
135C42—3941no—
136C42C22×D19711yescharacteristic
137C23—21no—
138Q8—7881no—
139Q8—7881no—
140Q8—7881no—
141Q8—7881no—
142Q8—3941no—
143Q8—3941no—
144D4—21no—
145D4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—7881no—
150C2×C4—3941no—
151C2×C4—3941no—
152C2×C4—3942no—
153C2×C4D4×D19711yescharacteristic
154C2×C4D4×D19711yescharacteristic
155C2×C4C2×D78811yescharacteristic
156C8—22no—
157C8—41no—
158C8—7881no—
159C22—41no—
160C22—41no—
161C22—41no—
162C22#2026329611yescharacteristic
163C4—21no—
164C4—41no—
165C4—7881no—
166C4—7882no—
167C4—7881no—
168C4—15761no—
169C4#2026334711yescharacteristic
170C4#2026334811yescharacteristic
171C4#2026335011yescharacteristic
172C4#2026335911yescharacteristic
173C2—81no—
174C2#2716122511yescharacteristic
175C2#2716124311yescharacteristic
176C2#2716126311yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
FittingD4⋊C8×C197C2
socleC22×C197#120
prime radicalC2×D4×C197D4
hypercenterD4⋊C8D197
hypocenterC197#1652
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #27161477C2semidirect
4. #27161290C2semidirect
5. #27160824C2semidirect
6. #27160811C2semidirect
88. C197#1652semidirect

Maximal subgroups

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

subgroupindex
#271617992
#271614772
#271613592
#271612902
#271608242
#271608112
D4⋊C8×C1972
#1652197