Finite Group Databasebeta

← #38741929

Subgroups — #38741929

Order 25216.

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#27161786C211yesmaximal, characteristic
2#27161430C211yesmaximal, characteristic, complemented
3#27161359C211yesmaximal, characteristic
4#27161290C211yesmaximal, characteristic, complemented
5#27160827C211yesmaximal, characteristic, complemented
6#27160811C211yesmaximal, characteristic, complemented
7D4⋊C8×C197C211yesmaximal, characteristic
8C2×Dic788—21no—
9#20263326C2211yescharacteristic
10#20263326C2211yescharacteristic
11#20263317—21no—
12#20263305—21no—
13#20263303—22no—
14C4×Dic394C2211yescharacteristic
15D4×Dic197—21no—
16C2×C4⋊Dic197—21no—
17#20263259—21no—
18C4⋊#13096802C2211yescharacteristic
19#20263251—22no—
20#20263251—21no—
21#20263247—21no—
22C4×D4×C197C2211yescharacteristic
23C4⋊C8×C197C2211yescharacteristic
24C4×C8×C197C2211yescharacteristic
25C2×Dic394—21no—
26C2×Dic394D411yescharacteristic
27C2×Dic394D412yes—
28C22×Dic197—22no—
29Dic788—42no—
30C2×#13096802—21no—
31C4⋊Dic197—22no—
32C4⋊Dic197—21no—
33C4⋊Dic197—21no—
34C4⋊Dic197—21no—
35C4⋊Dic197—21no—
36C4⋊Dic197D411yescharacteristic
37#15691337—22no—
38#15691337—21no—
39C22⋊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—42no—
51Dic394—41no—
52Dic394—41no—
53C2×Dic197—21no—
54C2×Dic197—21no—
55C2×Dic197—22no—
56C2×Dic197—41no—
57C2×Dic197—82no—
58#13096802—41no—
59C23×C197—21no—
60D4×C197—21no—
61D4×C197Q1612yescomplemented
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—41no—
69C8×C197—41no—
70C22×C197—22no—
71C22×C197—41no—
72C22×C197—41no—
73C22×C197#12011yescharacteristic
74C4×C197—21no—
75C4×C197—41no—
76C4×C197C2×Q1611yescharacteristic
77C4×C197#13511yescharacteristic
78C4×C197AGL(1,ℤ/8)11yescharacteristic
79C4×C197#13711yescharacteristic
80Dic197—41no—
81Dic197—41no—
82Dic197—42no—
83Dic197—81no—
84C2×C197—42no—
85C2×C197#44911yescharacteristic
86C2×C197#45111yescharacteristic
87C2×C197#45511yescharacteristic
88C197#164411yescharacteristic, complemented, Sylow
89#1644—1971nomaximal, Sylow
90#554—1971no—
91#487—1971no—
92#474—1971no—
93#462—1971no—
94#333—1971no—
95Q8⋊C8—1971no—
96D4⋊C8D19711yescharacteristic
97C2×Q16—3941no—
98C4⋊Q8—1971no—
99C4⋊Q8—1971no—
100#125—3941no—
101C22⋊Q8—3942no—
102C22⋊Q8—3941no—
103C4×Q8—1971no—
104C4×D4—3941no—
105C4×D4C2×D19711yescharacteristic
106C2×C4⋊C4—3941no—
107#107—3941no—
108C4⋊C8—1971no—
109C4⋊C8C2×D19711yescharacteristic
110#103—3942no—
111#103—3941no—
112#102—3941no—
113C4×C8C2×D19711yescharacteristic
114C2×Q8—3941no—
115C2×Q8—1972no—
116C2×Q8—1971no—
117C2×D4D78811yescharacteristic
118C22×C4—21no—
119C22×C4—3942no—
120Q16—7882no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—3941no—
124C4⋊C4—3942no—
125C4⋊C4—3942no—
126C4⋊C4—3941no—
127C4⋊C4—3941no—
128C4⋊C4—3941no—
129C4⋊C4—3941no—
130C4⋊C4—3941no—
131C4⋊C4—1971no—
132C4⋊C4D78811yescharacteristic
133C22⋊C4—21no—
134C22⋊C4—3942no—
135C42—3941no—
136C42C22×D19711yescharacteristic
137C23—21no—
138Q8—7882no—
139Q8—7881no—
140Q8—7881no—
141Q8—3941no—
142Q8—3941no—
143D4—21no—
144D4Dic78812yescomplemented
145C2×C4—21no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—7881no—
149C2×C4—15762no—
150C2×C4—3941no—
151C2×C4—3941no—
152C2×C4—3942no—
153C2×C4D4×D19711yescharacteristic
154C2×C4D4×D19711yescharacteristic
155C2×C4C2×D78811yescharacteristic
156C8—41no—
157C8—41no—
158C8—7881no—
159C22—22no—
160C22—41no—
161C22—41no—
162C22#2026329611yescharacteristic
163C4—21no—
164C4—41no—
165C4—7881no—
166C4—7881no—
167C4—7882no—
168C4—15761no—
169C4C2×Dic78811yescharacteristic
170C4#2026334711yescharacteristic
171C4#2026335011yescharacteristic
172C4#2026335911yescharacteristic
173C2—42no—
174C2#2716122511yescharacteristic
175C2#2716123711yescharacteristic
176C2#2716125811yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
FittingD4⋊C8×C197C2
socleC22×C197#120
prime radicalC2×D4×C197D4
hypercenterD4⋊C8D197
hypocenterC197#1644
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. #27161430C2semidirect
4. #27161290C2semidirect
5. #27160827C2semidirect
6. #27160811C2semidirect
61. D4×C197Q16semidirect
88. C197#1644semidirect
144. D4Dic788semidirect

Maximal subgroups

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

subgroupindex
#271617862
#271614302
#271613592
#271612902
#271608272
#271608112
D4⋊C8×C1972
#1644197