Finite Group Databasebeta

← #38741863

Subgroups — #38741863

Order 25216.

234 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 61 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#27161431C211yesmaximal, characteristic, complemented
2#27161369C211yesmaximal, characteristic, complemented
3#27161369C211yesmaximal, characteristic, complemented
4#27160825C211yesmaximal, characteristic, complemented
5D4⋊#13096802C211yesmaximal, characteristic, complemented
6D4⋊#13096802C211yesmaximal, characteristic, complemented
7#547×C197C211yesmaximal, characteristic
8#20263326C2211yescharacteristic
9#20263257—21no—
10#20263257—21no—
11C4⋊#13096802C2211yescharacteristic
12C4⋊#13096802C2211yescharacteristic
13#20263249—22no—
14#20263247—21no—
15#20263247—21no—
16C4×#13096802C2211yescharacteristic
17C2×#41×C197—21no—
18C4⋊Q8×C197C2211yescharacteristic
19#123×C197—22no—
20C22⋊Q8×C197—21no—
21#121×C197—21no—
22C4×D4×C197C2211yescharacteristic
23C4×D4×C197C2211yescharacteristic
24C2×C4⋊C4×C197—21no—
25C2×Dic394—21no—
26C2×#13096802—21no—
27C2×#13096802—21no—
28C2×#13096802—21no—
29C4⋊Dic197—21no—
30C4⋊Dic197—21no—
31#41×C197—42no—
32#41×C197—42no—
33C2×Q8×C197—21no—
34C2×D4×C197—22no—
35C2×D4×C197D411yescharacteristic
36C2×D4×C197D411yescharacteristic
37C22×C4×C197—22no—
38C22×C4×C197—21no—
39C22×C4×C197—21no—
40C4⋊C4×C197—22no—
41C4⋊C4×C197—21no—
42C4⋊C4×C197D411yescharacteristic
43C4⋊C4×C197D411yescharacteristic
44C4⋊C4×C197D412yes—
45C22⋊C4×C197—22no—
46C22⋊C4×C197—21no—
47C22⋊C4×C197—21no—
48C42×C197C2311yescharacteristic
49Dic394—42no—
50C2×Dic197—41no—
51#13096802—41no—
52#13096802—41no—
53#13096802—41no—
54C23×C197—21no—
55C23×C197—21no—
56Q8×C197—41no—
57Q8×C197—41no—
58D4×C197—21no—
59D4×C197—21no—
60D4×C197—21no—
61D4×C197—44no—
62D4×C197SD1612yes—
63C2×C4×C197—21no—
64C2×C4×C197—21no—
65C2×C4×C197—22no—
66C2×C4×C197—44no—
67C2×C4×C197—41no—
68C2×C4×C197—41no—
69C2×C4×C197—41no—
70C2×C4×C197—41no—
71C2×C4×C197C2×D411yescharacteristic
72C2×C4×C197C2×D411yescharacteristic
73C2×C4×C197C2×D411yescharacteristic
74C22×C197—22no—
75C22×C197—41no—
76C22×C197—41no—
77C22×C197—41no—
78C22×C197—41no—
79C22×C197—41no—
80C22×C197#12011yescharacteristic
81C4×C197—21no—
82C4×C197—41no—
83C4×C197—42no—
84C4×C197—41no—
85C4×C197C2×SD1611yescharacteristic
86C4×C197#13511yescharacteristic
87C4×C197AGL(1,ℤ/8)11yescharacteristic
88C4×C197#13711yescharacteristic
89Dic197—81no—
90C2×C197—42no—
91C2×C197—81no—
92C2×C197#45011yescharacteristic
93C2×C197#45211yescharacteristic
94C2×C197#45411yescharacteristic
95C197#163511yescharacteristic, complemented, Sylow
96#1635—1971nomaximal, Sylow
97#547D19711yescharacteristic
98#487—1971no—
99#476—1971no—
100#476—1971no—
101#332—1971no—
102D4⋊C8—1971no—
103D4⋊C8—1971no—
104C2×#41—21no—
105C4⋊Q8—1971no—
106C4⋊Q8C2×D19711yescharacteristic
107#123—22no—
108C22⋊Q8—21no—
109#121—21no—
110C4×D4C2×D19711yescharacteristic
111C4×D4C2×D19711yescharacteristic
112C2×C4⋊C4—21no—
113#106—3941no—
114#106—3941no—
115C4⋊C8—1971no—
116C4⋊C8—1971no—
117#103—3942no—
118#102—3941no—
119#102—3941no—
120C4×C8—1971no—
121#41—42no—
122#41—42no—
123C2×Q8—21no—
124C2×Q8—3941no—
125C2×D4—22no—
126C2×D4C22⋊D19711yescharacteristic
127C2×D4C22⋊D19711yescharacteristic
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—21no—
131C2×C8—3941no—
132C2×C8—3941no—
133C2×C8—3941no—
134C4⋊C4—22no—
135C4⋊C4—21no—
136C4⋊C4—3941no—
137C4⋊C4—3941no—
138C4⋊C4C22⋊D19712yes—
139C4⋊C4C22⋊D19711yescharacteristic
140C4⋊C4C22⋊D19711yescharacteristic
141C22⋊C4—22no—
142C22⋊C4—21no—
143C22⋊C4—21no—
144C42C22×D19711yescharacteristic
145C23—21no—
146C23—21no—
147Q8—41no—
148Q8—41no—
149Q8—7882no—
150D4—21no—
151D4—21no—
152D4—21no—
153D4—44no—
154D4#1569134612yes—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—44no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—7881no—
164C2×C4C2×C22⋊D19711yescharacteristic
165C2×C4C2×C22⋊D19711yescharacteristic
166C2×C4C2×C22⋊D19711yescharacteristic
167C8—7881no—
168C8—7881no—
169C8—7881no—
170C22—22no—
171C22—41no—
172C22—41no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22#2026329511yescharacteristic
177C4—21no—
178C4—41no—
179C4—42no—
180C4—41no—
181C4—15761no—
182C4C2×#1569134611yescharacteristic
183C4#2026334611yescharacteristic
184C4#2026335211yescharacteristic
185C4#2026335611yescharacteristic
186C2—42no—
187C2—81no—
188C2#2716123011yescharacteristic
189C2#2716124111yescharacteristic
190C2#2716125211yescharacteristic
subgroupG/H
centerC22#20263295
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting, prime radical#547×C197C2
socleC22×C197#120
hypercenter#547D197
hypocenterC197#1635
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #27161431C2semidirect
2. #27161369C2semidirect
3. #27161369C2semidirect
4. #27160825C2semidirect
5. D4⋊#13096802C2semidirect
6. D4⋊#13096802C2semidirect
95. C197#1635semidirect

Maximal subgroups

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

subgroupindex
#271614312
#271613692
#271608252
D4⋊#130968022
#547×C1972
#1635197