Finite Group Databasebeta

← #38741939

Subgroups — #38741939

Order 25216.

212 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 212 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#27161794C211yesmaximal, characteristic
2#27161431C211yesmaximal, characteristic, complemented
3#27161281C211yesmaximal, characteristic, complemented
4#27160827C211yesmaximal, characteristic, complemented
5D4⋊#13096802C211yesmaximal, characteristic, complemented
6#27160806C211yesmaximal, characteristic, complemented
7#474×C197C211yesmaximal, characteristic, complemented
8C2×#15691345—21no—
9#20263326C2211yescharacteristic
10#20263312—22no—
11#20263300—21no—
12#20263297—21no—
13C4×D788C2211yescharacteristic
14#20263285—21no—
15#20263284—21no—
16C4⋊#13096802C2211yescharacteristic
17#20263250—21no—
18#20263249—22no—
19C4×#13096802C2211yescharacteristic
20C4⋊Q8×C197C2211yescharacteristic
21C4×D4×C197C2211yescharacteristic
22#107×C197—21no—
23C4⋊C8×C197C2211yescharacteristic
24#102×C197—21no—
25C2×Dic394D412yes—
26C2×D788D411yescharacteristic
27C2×C22⋊D197—22no—
28C2×C4×D197—21no—
29#15691345—42no—
30C2×#13096802—21no—
31C2×#13096802—21no—
32C4⋊Dic197—21no—
33C4⋊Dic197—21no—
34C4⋊Dic197D411yescharacteristic
35C22⋊Dic197—22no—
36#15691335—22no—
37#15691335—21no—
38C4×Dic197—22no—
39C2×Q8×C197—21no—
40C2×D4×C197D411yescharacteristic
41C22×C4×C197—21no—
42C2×C8×C197—21no—
43C4⋊C4×C197—21no—
44C4⋊C4×C197—21no—
45C4⋊C4×C197D411yescharacteristic
46C22⋊C4×C197—21no—
47C42×C197C2311yescharacteristic
48C22×D197—21no—
49Dic394—42no—
50D788—21no—
51D788—21no—
52C22⋊D197—82no—
53C2×Dic197—21no—
54C2×Dic197—22no—
55C4×D197—41no—
56C4×D197—41no—
57#13096802—41no—
58#13096802—41no—
59C23×C197—21no—
60Q8×C197—41no—
61Q8×C197—41no—
62D4×C197—21no—
63D4×C197—21no—
64C2×C4×C197—21no—
65C2×C4×C197—41no—
66C2×C4×C197—41no—
67C2×C4×C197—41no—
68C2×C4×C197C2×D411yescharacteristic
69C2×C4×C197C2×D411yescharacteristic
70C2×C4×C197C2×D411yescharacteristic
71C8×C197—41no—
72C22×C197—41no—
73C22×C197—41no—
74C22×C197—41no—
75C22×C197#12011yescharacteristic
76C2×D197—41no—
77C2×D197—41no—
78C2×D197—41no—
79C4×C197—21no—
80C4×C197—41no—
81C4×C197—81no—
82C4×C197#13511yescharacteristic
83C4×C197#13511yescharacteristic
84C4×C197AGL(1,ℤ/8)11yescharacteristic
85C4×C197#13711yescharacteristic
86Dic197—41no—
87Dic197—42no—
88C2×C197—81no—
89C2×C197#44911yescharacteristic
90C2×C197#45211yescharacteristic
91C2×C197#45511yescharacteristic
92D197—81no—
93C197#164611yescharacteristic, complemented, Sylow
94#1646—1971nomaximal, Sylow
95#555—1971no—
96#487—1971no—
97#474D19711yescharacteristic, complemented
98#461—1971no—
99#333—1971no—
100D4⋊C8—1971no—
101D4⋊C8—1971no—
102C2×SD16—3941no—
103C4⋊Q8—1971no—
104C4⋊Q8C2×D19711yescharacteristic
105#124—3942no—
106#121—3941no—
107#121—3941no—
108C4×D4—1971no—
109C4×D4C2×D19711yescharacteristic
110#117—3941no—
111#117—3941no—
112#107—21no—
113C4⋊C8—1971no—
114C4⋊C8C2×D19711yescharacteristic
115#103—3942no—
116#103—3941no—
117#102—21no—
118C4×C8—1971no—
119C2×Q8—21no—
120C2×Q8—1972no—
121C2×D4—3942no—
122C2×D4—1971no—
123C2×D4C22⋊D19711yescharacteristic
124C22×C4—21no—
125C22×C4—3941no—
126SD16—7882no—
127C2×C8—21no—
128C2×C8—3941no—
129C2×C8—3941no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—3941no—
133C4⋊C4—3941no—
134C4⋊C4—1971no—
135C4⋊C4C22⋊D19711yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—3942no—
138C22⋊C4—3942no—
139C22⋊C4—3941no—
140C42—3942no—
141C42C22×D19711yescharacteristic
142C23—21no—
143C23—3941no—
144Q8—41no—
145Q8—41no—
146Q8—7882no—
147D4—21no—
148D4—21no—
149D4—15762no—
150D4—3941no—
151D4—3941no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—7881no—
157C2×C4—7881no—
158C2×C4—3941no—
159C2×C4—3942no—
160C2×C4D4×D19711yescharacteristic
161C2×C4D4×D19711yescharacteristic
162C2×C4C2×C22⋊D19711yescharacteristic
163C8—41no—
164C8—7881no—
165C8—7881no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—7881no—
170C22—7881no—
171C22—7881no—
172C22#2026329411yescharacteristic
173C4—21no—
174C4—41no—
175C4—81no—
176C4—7881no—
177C4—7882no—
178C4#2026334511yescharacteristic
179C4#2026334611yescharacteristic
180C4#2026335011yescharacteristic
181C4#2026335711yescharacteristic
182C2—81no—
183C2—15761no—
184C2#2716122111yescharacteristic
185C2#2716123911yescharacteristic
186C2#2716125511yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#474×C197C2
socleC22×C197#120
prime radical#27161794C2
hypercenter#474D197
hypocenterC197#1646
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #27161431C2semidirect
3. #27161281C2semidirect
4. #27160827C2semidirect
5. D4⋊#13096802C2semidirect
6. #27160806C2semidirect
7. #474×C197C2semidirect
93. C197#1646semidirect
97. #474D197semidirect

Maximal subgroups

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

subgroupindex
#271617942
#271614312
#271612812
#271608272
D4⋊#130968022
#271608062
#474×C1972
#1646197