Finite Group Databasebeta

← #38741846

Subgroups — #38741846

Order 25216.

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

normal subgroups only

This group has 214 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#27161458C211yesmaximal, characteristic, complemented
2#27161374C211yesmaximal, characteristic, complemented
3#27161369C211yesmaximal, characteristic
4#27160828C211yesmaximal, characteristic, complemented
5Q8⋊#13096802C211yesmaximal, characteristic, complemented
6D4⋊#13096802C211yesmaximal, characteristic
7#554×C197C211yesmaximal, characteristic
8C2×#15691348—22no—
9#20263326C2211yescharacteristic
10#20263259—21no—
11#20263257—21no—
12C4⋊#13096802C2211yescharacteristic
13C4⋊#13096802C2211yescharacteristic
14#20263250—21no—
15#20263247—21no—
16C4×#13096802C2211yescharacteristic
17C4⋊Q8×C197C2211yescharacteristic
18#125×C197—21no—
19C22⋊Q8×C197—21no—
20C22⋊Q8×C197—22no—
21C4×Q8×C197C2211yescharacteristic
22C4×D4×C197—21no—
23C4×D4×C197C2211yescharacteristic
24C2×C4⋊C4×C197—21no—
25C2×Dic394—21no—
26#15691348—44no—
27C2×#13096802—21no—
28C2×#13096802—21no—
29C2×#13096802—21no—
30C4⋊Dic197—21no—
31C4⋊Dic197—21no—
32C2×Q8×C197D411yescharacteristic
33C2×Q8×C197D412yes—
34C2×D4×C197D411yescharacteristic
35C22×C4×C197—22no—
36C22×C4×C197—21no—
37C4⋊C4×C197—22no—
38C4⋊C4×C197—22no—
39C4⋊C4×C197—21no—
40C4⋊C4×C197—21no—
41C4⋊C4×C197—21no—
42C4⋊C4×C197D411yescharacteristic
43C4⋊C4×C197D411yescharacteristic
44C22⋊C4×C197—22no—
45C22⋊C4×C197—21no—
46C42×C197—21no—
47C42×C197C2311yescharacteristic
48Dic394—42no—
49C2×Dic197—41no—
50#13096802—22no—
51#13096802—41no—
52#13096802—41no—
53C23×C197—21no—
54Q8×C197—21no—
55Q8×C197—21no—
56Q8×C197—42no—
57D4×C197—21no—
58D4×C197Q1612yescomplemented
59C2×C4×C197—21no—
60C2×C4×C197—21no—
61C2×C4×C197—21no—
62C2×C4×C197—22no—
63C2×C4×C197—41no—
64C2×C4×C197—41no—
65C2×C4×C197—82no—
66C2×C4×C197C2×D411yescharacteristic
67C2×C4×C197C2×D411yescharacteristic
68C2×C4×C197C2×D411yescharacteristic
69C22×C197—22no—
70C22×C197—41no—
71C22×C197—41no—
72C22×C197#12011yescharacteristic
73C4×C197—21no—
74C4×C197—41no—
75C4×C197—41no—
76C4×C197—41no—
77C4×C197—42no—
78C4×C197C2×Q1611yescharacteristic
79C4×C197#13511yescharacteristic
80C4×C197#13711yescharacteristic
81C4×C197#13711yescharacteristic
82Dic197—81no—
83C2×C197—42no—
84C2×C197#45111yescharacteristic
85C2×C197#45211yescharacteristic
86C2×C197#45311yescharacteristic
87C197#163211yescharacteristic, complemented, Sylow
88#1632—1971nomaximal, Sylow
89#554D19711yescharacteristic
90#494—1971no—
91#477—1971no—
92#476—1971no—
93#333—1971no—
94Q8⋊C8—1971no—
95D4⋊C8—1971no—
96C2×Q16—3942no—
97C4⋊Q8—1971no—
98C4⋊Q8C2×D19711yescharacteristic
99#125—21no—
100C22⋊Q8—21no—
101C22⋊Q8—22no—
102C4×Q8C2×D19711yescharacteristic
103C4×D4—21no—
104C4×D4C2×D19711yescharacteristic
105C2×C4⋊C4—21no—
106#107—3941no—
107#106—3941no—
108C4⋊C8—1971no—
109C4⋊C8—1971no—
110#103—3941no—
111#102—3941no—
112C4×C8—1971no—
113C2×Q8—3941no—
114C2×Q8C22⋊D19711yescharacteristic
115C2×Q8C22⋊D19712yes—
116C2×D4C22⋊D19711yescharacteristic
117C22×C4—22no—
118C22×C4—21no—
119Q16—7884no—
120C2×C8—3941no—
121C2×C8—3941no—
122C2×C8—3941no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—3941no—
129C4⋊C4—3941no—
130C4⋊C4C22⋊D19711yescharacteristic
131C4⋊C4C22⋊D19711yescharacteristic
132C22⋊C4—22no—
133C22⋊C4—21no—
134C42—21no—
135C42C22×D19711yescharacteristic
136C23—21no—
137Q8—21no—
138Q8—21no—
139Q8—42no—
140Q8—7882no—
141D4—21no—
142D4#1569134812yescomplemented
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—82no—
150C2×C4—7881no—
151C2×C4C2×C22⋊D19711yescharacteristic
152C2×C4C2×C22⋊D19711yescharacteristic
153C2×C4C2×C22⋊D19711yescharacteristic
154C8—7881no—
155C8—7881no—
156C8—3942no—
157C22—22no—
158C22—41no—
159C22—41no—
160C22#2026329511yescharacteristic
161C4—21no—
162C4—41no—
163C4—41no—
164C4—41no—
165C4—42no—
166C4—15761no—
167C4C2×#1569134811yescharacteristic
168C4#2026334611yescharacteristic
169C4#2026335611yescharacteristic
170C4#2026335711yescharacteristic
171C2—42no—
172C2#2716123511yescharacteristic
173C2#2716124111yescharacteristic
174C2#2716124711yescharacteristic
subgroupG/H
centerC22#20263295
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#554×C197C2
socleC22×C197#120
prime radicalC2×D4×C197D4
hypercenter#554D197
hypocenterC197#1632
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #554

Complements

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

NH ≅ G/Ntype
1. #27161458C2semidirect
2. #27161374C2semidirect
4. #27160828C2semidirect
5. Q8⋊#13096802C2semidirect
58. D4×C197Q16semidirect
87. C197#1632semidirect
142. D4#15691348semidirect

Maximal subgroups

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

subgroupindex
#271614582
#271613742
#271613692
#271608282
Q8⋊#130968022
D4⋊#130968022
#554×C1972
#1632197