Finite Group Databasebeta

← #38741801

Subgroups — #38741801

Order 25216.

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

normal subgroups only

This group has 256 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#27161427C211yesmaximal, characteristic, complemented
2#27161274C211yesmaximal, characteristic, complemented
3#27161274C211yesmaximal, characteristic, complemented
4#27160825C211yesmaximal, characteristic, complemented
5D4⋊#13096802C211yesmaximal, characteristic, complemented
6D4⋊#13096802C211yesmaximal, characteristic, complemented
7#547×C197C211yesmaximal, characteristic, complemented
8C2×#15691343—21no—
9C2×#15691343—21no—
10#20263323C2211yescharacteristic
11C4⋊#13096802C2211yescharacteristic
12C4⋊#13096802C2211yescharacteristic
13#20263246—22no—
14#20263246—21no—
15#20263246—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×D788—21no—
26C2×D788—21no—
27C2×D788—21no—
28#15691343—42no—
29#15691343—42no—
30C2×#13096802—21no—
31C2×#13096802—21no—
32C2×#13096802—21no—
33#41×C197—42no—
34#41×C197—42no—
35C2×Q8×C197—21no—
36C2×D4×C197—22no—
37C2×D4×C197D411yescharacteristic
38C2×D4×C197D411yescharacteristic
39C22×C4×C197—22no—
40C22×C4×C197—21no—
41C22×C4×C197—21no—
42C4⋊C4×C197—22no—
43C4⋊C4×C197—21no—
44C4⋊C4×C197D411yescharacteristic
45C4⋊C4×C197D411yescharacteristic
46C4⋊C4×C197D412yes—
47C22⋊C4×C197—22no—
48C22⋊C4×C197—21no—
49C22⋊C4×C197—21no—
50C42×C197C2311yescharacteristic
51C22×D197—41no—
52D788—42no—
53D788—41no—
54D788—41no—
55D788—41no—
56D788—41no—
57#13096802—41no—
58#13096802—41no—
59#13096802—41no—
60C23×C197—21no—
61C23×C197—21no—
62Q8×C197—41no—
63Q8×C197—41no—
64D4×C197—21no—
65D4×C197—21no—
66D4×C197—21no—
67D4×C197—44no—
68D4×C197D812yescomplemented
69C2×C4×C197—21no—
70C2×C4×C197—21no—
71C2×C4×C197—22no—
72C2×C4×C197—44no—
73C2×C4×C197—41no—
74C2×C4×C197—41no—
75C2×C4×C197—41no—
76C2×C4×C197—41no—
77C2×C4×C197C2×D411yescharacteristic
78C2×C4×C197C2×D411yescharacteristic
79C2×C4×C197C2×D411yescharacteristic
80C22×C197—22no—
81C22×C197—41no—
82C22×C197—41no—
83C22×C197—41no—
84C22×C197—41no—
85C22×C197—41no—
86C22×C197#12011yescharacteristic
87C2×D197—81no—
88C2×D197—81no—
89C2×D197—81no—
90C4×C197—21no—
91C4×C197—41no—
92C4×C197—42no—
93C4×C197—41no—
94C4×C197C2×D811yescharacteristic
95C4×C197#13511yescharacteristic
96C4×C197AGL(1,ℤ/8)11yescharacteristic
97C4×C197AGL(1,ℤ/8)11yescharacteristic
98C2×C197—42no—
99C2×C197—81no—
100C2×C197#44711yescharacteristic
101C2×C197#44911yescharacteristic
102C2×C197#45411yescharacteristic
103D197—161no—
104C197#162511yescharacteristic, complemented, Sylow
105#1625—1971nomaximal, Sylow
106#547D19711yescharacteristic, complemented
107#486—1971no—
108#459—1971no—
109#459—1971no—
110#332—1971no—
111D4⋊C8—1971no—
112D4⋊C8—1971no—
113C2×#41—21no—
114C2×D8—3941no—
115C2×D8—3941no—
116C4⋊Q8C2×D19711yescharacteristic
117#127—1971no—
118#123—22no—
119C22⋊Q8—21no—
120#121—21no—
121C4×D4C2×D19711yescharacteristic
122C4×D4C2×D19711yescharacteristic
123C2×C4⋊C4—21no—
124C4⋊C8—1971no—
125C4⋊C8—1971no—
126#102—3941no—
127#102—3941no—
128#102—3942no—
129C4×C8—1971no—
130#41—42no—
131#41—42no—
132C2×Q8—21no—
133C2×D4—22no—
134C2×D4—3941no—
135C2×D4—3941no—
136C2×D4—3941no—
137C2×D4C22⋊D19711yescharacteristic
138C2×D4C22⋊D19711yescharacteristic
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142D8—7882no—
143D8—7882no—
144C2×C8—3941no—
145C2×C8—3941no—
146C2×C8—3941no—
147C4⋊C4—22no—
148C4⋊C4—21no—
149C4⋊C4C22⋊D19711yescharacteristic
150C4⋊C4C22⋊D19711yescharacteristic
151C4⋊C4C22⋊D19712yes—
152C22⋊C4—22no—
153C22⋊C4—21no—
154C22⋊C4—21no—
155C42C22×D19711yescharacteristic
156C23—21no—
157C23—21no—
158C23—7881no—
159Q8—41no—
160Q8—41no—
161D4—21no—
162D4—21no—
163D4—21no—
164D4—44no—
165D4—7882no—
166D4—7881no—
167D4—7881no—
168D4—7881no—
169D4—7881no—
170D4#1569134312yescomplemented
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—44no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4C2×C22⋊D19711yescharacteristic
180C2×C4C2×C22⋊D19711yescharacteristic
181C2×C4C2×C22⋊D19711yescharacteristic
182C8—7881no—
183C8—7881no—
184C8—7881no—
185C22—22no—
186C22—41no—
187C22—41no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22—15761no—
192C22—15761no—
193C22—15761no—
194C22#2026329511yescharacteristic
195C4—21no—
196C4—41no—
197C4—42no—
198C4—41no—
199C4C2×#1569134311yescharacteristic
200C4#2026334611yescharacteristic
201C4#2026335111yescharacteristic
202C4#2026335211yescharacteristic
203C2—42no—
204C2—81no—
205C2—31521no—
206C2#2716121311yescharacteristic
207C2#2716122411yescharacteristic
208C2#2716125211yescharacteristic
subgroupG/H
centerC22#20263295
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#547×C197C2
socleC22×C197#120
hypercenter#547D197
hypocenterC197#1625
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. #27161427C2semidirect
2. #27161274C2semidirect
3. #27161274C2semidirect
4. #27160825C2semidirect
5. D4⋊#13096802C2semidirect
6. D4⋊#13096802C2semidirect
7. #547×C197C2semidirect
68. D4×C197D8semidirect
104. C197#1625semidirect
106. #547D197semidirect
170. D4#15691343semidirect

Maximal subgroups

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

subgroupindex
#271614272
#271612742
#271608252
D4⋊#130968022
#547×C1972
#1625197