Finite Group Databasebeta

← #38741802

Subgroups — #38741802

Order 25216.

256 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 55 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#27161732C211yesmaximal, characteristic, complemented
2#27161427C211yesmaximal, characteristic, complemented
3#27161275C211yesmaximal, characteristic, complemented
4#27161275C211yesmaximal, characteristic, complemented
5#27160806C211yesmaximal, characteristic, complemented
6#27160806C211yesmaximal, characteristic, complemented
7#332×C197C211yesmaximal, characteristic, complemented
8C2×#15691359—21no—
9C2×D1576—21no—
10C2×D1576—21no—
11#20263323C2211yescharacteristic
12#20263311—22no—
13#20263302—21no—
14#20263301—21no—
15C4×D788C2211yescharacteristic
16C4×D788C2211yescharacteristic
17C4⋊C4×D197—21no—
18#20263248—21no—
19#20263248—21no—
20#20263246—22no—
21C4×#13096802C2211yescharacteristic
22C4⋊Q8×C197C2211yescharacteristic
23C4⋊C8×C197C2211yescharacteristic
24C4⋊C8×C197C2211yescharacteristic
25#15691359—42no—
26#15691359—42no—
27C2×D788—22no—
28C2×D788—21no—
29C2×D788—21no—
30C2×D788—21no—
31C2×D788D411yescharacteristic
32C2×D788D411yescharacteristic
33C2×C4×D197—22no—
34C2×C4×D197—21no—
35C2×C4×D197—21no—
36D1576—42no—
37D1576—42no—
38C2×#13096802—21no—
39C4⋊Dic197D411yescharacteristic
40C4⋊Dic197D411yescharacteristic
41#15691337—22no—
42#15691335—22no—
43#15691335—21no—
44#15691335—21no—
45C2×Q8×C197—21no—
46C2×C8×C197—21no—
47C2×C8×C197—21no—
48C4⋊C4×C197—21no—
49C4⋊C4×C197D412yes—
50C42×C197C2311yescharacteristic
51C22×D197—21no—
52C22×D197—21no—
53C22×D197—41no—
54D788—21no—
55D788—21no—
56D788—21no—
57D788—42no—
58D788—44no—
59D788—41no—
60D788—41no—
61D788—41no—
62D788—41no—
63D788D812yescomplemented
64C2×Dic197—21no—
65C2×Dic197—21no—
66C4×D197—44no—
67C4×D197—41no—
68C4×D197—41no—
69C4×D197—41no—
70C4×D197—41no—
71#13096802—41no—
72Q8×C197—41no—
73Q8×C197—41no—
74C2×C4×C197—22no—
75C2×C4×C197C2×D411yescharacteristic
76C2×C4×C197C2×D411yescharacteristic
77C2×C4×C197C2×D411yescharacteristic
78C8×C197—41no—
79C8×C197—41no—
80C22×C197#12011yescharacteristic
81C2×D197—22no—
82C2×D197—41no—
83C2×D197—41no—
84C2×D197—41no—
85C2×D197—41no—
86C2×D197—41no—
87C2×D197—81no—
88C2×D197—81no—
89C2×D197—81no—
90C4×C197—21no—
91C4×C197—42no—
92C4×C197C2×D811yescharacteristic
93C4×C197#13511yescharacteristic
94C4×C197AGL(1,ℤ/8)11yescharacteristic
95C4×C197AGL(1,ℤ/8)11yescharacteristic
96Dic197—41no—
97Dic197—41no—
98C2×C197#44711yescharacteristic
99C2×C197#44911yescharacteristic
100C2×C197#45411yescharacteristic
101D197—42no—
102D197—81no—
103D197—161no—
104C197#162511yescharacteristic, complemented, Sylow
105#1625—1971nomaximal, Sylow
106#547—1971no—
107#486—1971no—
108#459—1971no—
109#459—1971no—
110#332D19711yescharacteristic, complemented
111D4⋊C8—1971no—
112D4⋊C8—1971no—
113C2×#41—3941no—
114C2×D8—3941no—
115C2×D8—3941no—
116C4⋊Q8C2×D19711yescharacteristic
117#127—1971no—
118#123—3942no—
119C22⋊Q8—3941no—
120#121—3941no—
121C4×D4—1971no—
122C4×D4—1971no—
123C2×C4⋊C4—3941no—
124C4⋊C8C2×D19711yescharacteristic
125C4⋊C8C2×D19711yescharacteristic
126#102—3941no—
127#102—3941no—
128#102—3942no—
129C4×C8—1971no—
130#41—7882no—
131#41—7882no—
132C2×Q8—21no—
133C2×D4—3942no—
134C2×D4—3941no—
135C2×D4—3941no—
136C2×D4—3941no—
137C2×D4—1971no—
138C2×D4—1971no—
139C22×C4—3942no—
140C22×C4—3941no—
141C22×C4—3941no—
142D8—7882no—
143D8—7882no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—3941no—
147C4⋊C4—21no—
148C4⋊C4—3942no—
149C4⋊C4—1971no—
150C4⋊C4—1971no—
151C4⋊C4D78812yes—
152C22⋊C4—3942no—
153C22⋊C4—3941no—
154C22⋊C4—3941no—
155C42C22×D19711yescharacteristic
156C23—7881no—
157C23—3941no—
158C23—3941no—
159Q8—41no—
160Q8—41no—
161D4—7882no—
162D4—7884no—
163D4—7881no—
164D4—7881no—
165D4—7881no—
166D4—7881no—
167D4—3941no—
168D4—3941no—
169D4—3941no—
170D4—1972no—
171C2×C4—22no—
172C2×C4—7884no—
173C2×C4—7881no—
174C2×C4—7881no—
175C2×C4—7881no—
176C2×C4—7881no—
177C2×C4—3941no—
178C2×C4—3941no—
179C2×C4D4×D19711yescharacteristic
180C2×C4D4×D19711yescharacteristic
181C2×C4C2×D78811yescharacteristic
182C8—41no—
183C8—41no—
184C8—7881no—
185C22—7881no—
186C22—7881no—
187C22—7881no—
188C22—7881no—
189C22—7881no—
190C22—15761no—
191C22—15761no—
192C22—15761no—
193C22—3942no—
194C22#2026329611yescharacteristic
195C4—21no—
196C4—42no—
197C4—7881no—
198C4—7881no—
199C4D8×D19711yescharacteristic
200C4#2026334911yescharacteristic
201C4#2026335311yescharacteristic
202C4#2026335311yescharacteristic
203C2—7882no—
204C2—15761no—
205C2—31521no—
206C2#2716121411yescharacteristic
207C2#2716122711yescharacteristic
208C2#2716125411yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#332×C197C2
socleC22×C197#120
hypercenter#332D197
hypocenterC197#1625
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
1. #27161732C2semidirect
2. #27161427C2semidirect
3. #27161275C2semidirect
4. #27161275C2semidirect
5. #27160806C2semidirect
6. #27160806C2semidirect
7. #332×C197C2semidirect
63. D788D8semidirect
104. C197#1625semidirect
110. #332D197semidirect

Maximal subgroups

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

subgroupindex
#271617322
#271614272
#271612752
#271608062
#332×C1972
#1625197