Finite Group Databasebeta

← #38741848

Subgroups — #38741848

Order 25216.

214 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 214 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#27161459C211yesmaximal, characteristic, complemented
3#27161375C211yesmaximal, characteristic, complemented
4#27161369C211yesmaximal, characteristic
5#27160827C211yesmaximal, characteristic, complemented
6#27160811C211yesmaximal, characteristic, complemented
7D4⋊C8×C197C211yesmaximal, characteristic
8C2×Dic788—22no—
9#20263326C2211yescharacteristic
10#20263326C2211yescharacteristic
11#20263317—21no—
12#20263305—21no—
13#20263303—22no—
14C4×Dic394C2211yescharacteristic
15D4×Dic197—21no—
16C2×C4⋊Dic197—21no—
17#20263260—21no—
18#20263257—21no—
19C4⋊#13096802C2211yescharacteristic
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—44no—
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—42no—
52C2×Dic197—21no—
53C2×Dic197—21no—
54C2×Dic197—22no—
55C2×Dic197—41no—
56C2×Dic197—82no—
57#13096802—41no—
58C23×C197—21no—
59D4×C197—21no—
60D4×C197Q1612yescomplemented
61C2×C4×C197—21no—
62C2×C4×C197—41no—
63C2×C4×C197—41no—
64C2×C4×C197C2×D411yescharacteristic
65C2×C4×C197C2×D411yescharacteristic
66C2×C4×C197C2×D411yescharacteristic
67C8×C197—22no—
68C8×C197—41no—
69C22×C197—22no—
70C22×C197—41no—
71C22×C197—41no—
72C22×C197#12011yescharacteristic
73C4×C197—21no—
74C4×C197—41no—
75C4×C197C2×Q1611yescharacteristic
76C4×C197#13511yescharacteristic
77C4×C197#13711yescharacteristic
78C4×C197#13711yescharacteristic
79Dic197—41no—
80Dic197—41no—
81Dic197—42no—
82Dic197—81no—
83C2×C197—42no—
84C2×C197#45111yescharacteristic
85C2×C197#45211yescharacteristic
86C2×C197#45311yescharacteristic
87C197#163211yescharacteristic, complemented, Sylow
88#1632—1971nomaximal, Sylow
89#554—1971no—
90#494—1971no—
91#477—1971no—
92#476—1971no—
93#333—1971no—
94Q8⋊C8—1971no—
95D4⋊C8D19711yescharacteristic
96C2×Q16—3942no—
97C4⋊Q8—1971no—
98C4⋊Q8—1971no—
99#125—3941no—
100C22⋊Q8—3942no—
101C22⋊Q8—3941no—
102C4×Q8—1971no—
103C4×D4—3941no—
104C4×D4C2×D19711yescharacteristic
105C2×C4⋊C4—3941no—
106#107—3941no—
107#106—3941no—
108C4⋊C8—1971no—
109C4⋊C8C2×D19711yescharacteristic
110#103—3941no—
111#102—3941no—
112C4×C8C2×D19711yescharacteristic
113C2×Q8—3941no—
114C2×Q8—1972no—
115C2×Q8—1971no—
116C2×D4D78811yescharacteristic
117C22×C4—21no—
118C22×C4—3942no—
119Q16—7884no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—3941no—
123C4⋊C4—3942no—
124C4⋊C4—3942no—
125C4⋊C4—3941no—
126C4⋊C4—3941no—
127C4⋊C4—3941no—
128C4⋊C4—3941no—
129C4⋊C4—3941no—
130C4⋊C4—1971no—
131C4⋊C4D78811yescharacteristic
132C22⋊C4—21no—
133C22⋊C4—3942no—
134C42—3941no—
135C42C22×D19711yescharacteristic
136C23—21no—
137Q8—7882no—
138Q8—7882no—
139Q8—3941no—
140Q8—3941no—
141D4—21no—
142D4Dic78812yescomplemented
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—7881no—
147C2×C4—15762no—
148C2×C4—3941no—
149C2×C4—3941no—
150C2×C4—3942no—
151C2×C4D4×D19711yescharacteristic
152C2×C4D4×D19711yescharacteristic
153C2×C4C2×D78811yescharacteristic
154C8—22no—
155C8—41no—
156C8—7881no—
157C22—22no—
158C22—41no—
159C22—41no—
160C22#2026329611yescharacteristic
161C4—21no—
162C4—41no—
163C4—7881no—
164C4—7881no—
165C4—7882no—
166C4—15761no—
167C4C2×Dic78811yescharacteristic
168C4#2026334511yescharacteristic
169C4#2026335511yescharacteristic
170C4#2026335911yescharacteristic
171C2—42no—
172C2#2716123711yescharacteristic
173C2#2716124211yescharacteristic
174C2#2716124811yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
FittingD4⋊C8×C197C2
socleC22×C197#120
prime radicalC2×D4×C197D4
hypercenterD4⋊C8D197
hypocenterC197#1632
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. #27161459C2semidirect
3. #27161375C2semidirect
5. #27160827C2semidirect
6. #27160811C2semidirect
60. D4×C197Q16semidirect
87. C197#1632semidirect
142. D4Dic788semidirect

Maximal subgroups

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

subgroupindex
#271617862
#271614592
#271613752
#271613692
#271608272
#271608112
D4⋊C8×C1972
#1632197