Finite Group Databasebeta

← #38741782

Subgroups — #38741782

Order 25216.

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

normal subgroups only

This group has 236 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, complemented
2#27161451C211yesmaximal, characteristic, complemented
3#27161276C211yesmaximal, characteristic, complemented
4#27160827C211yesmaximal, characteristic, complemented
5#27160809C211yesmaximal, characteristic, complemented
6D4⋊#13096802C211yesmaximal, characteristic, complemented
7#459×C197C211yesmaximal, characteristic
8C2×#15691346—22no—
9C2×#15691346—21no—
10#20263326C2211yescharacteristic
11#20263317—21no—
12#20263305—21no—
13#20263303—22no—
14C4×Dic394C2211yescharacteristic
15D4×Dic197—21no—
16C2×C4⋊Dic197—21no—
17C4⋊#13096802C2211yescharacteristic
18#20263247—21no—
19C4×#13096802C2211yescharacteristic
20C2×D8×C197—21no—
21#127×C197C2211yescharacteristic
22C4×D4×C197C2211yescharacteristic
23C4⋊C8×C197C2211yescharacteristic
24#102×C197—21no—
25C2×Dic394D411yescharacteristic
26C2×Dic394D412yes—
27C22×Dic197—22no—
28#15691346—44no—
29#15691346—42no—
30C2×#13096802—21no—
31C2×#13096802—21no—
32C4⋊Dic197—22no—
33C4⋊Dic197—21no—
34C4⋊Dic197—21no—
35C4⋊Dic197D411yescharacteristic
36#15691337—22no—
37#15691337—21no—
38C22⋊Dic197—22no—
39C4×Dic197—21no—
40C2×D4×C197—21no—
41C2×D4×C197—21no—
42C2×D4×C197—21no—
43C2×D4×C197D411yescharacteristic
44C22×C4×C197—21no—
45D8×C197—42no—
46C2×C8×C197—21no—
47C4⋊C4×C197D411yescharacteristic
48C22⋊C4×C197—21no—
49C42×C197C2311yescharacteristic
50Dic394—21no—
51Dic394—21no—
52Dic394—42no—
53C2×Dic197—21no—
54C2×Dic197—21no—
55C2×Dic197—22no—
56C2×Dic197—82no—
57#13096802—22no—
58#13096802—41no—
59C23×C197—21no—
60C23×C197—41no—
61D4×C197—21no—
62D4×C197—42no—
63D4×C197—41no—
64D4×C197—41no—
65D4×C197—41no—
66D4×C197—41no—
67D4×C197SD1612yescomplemented
68C2×C4×C197—21no—
69C2×C4×C197—41no—
70C2×C4×C197—41no—
71C2×C4×C197C2×D411yescharacteristic
72C2×C4×C197C2×D411yescharacteristic
73C2×C4×C197C2×D411yescharacteristic
74C8×C197—41no—
75C22×C197—22no—
76C22×C197—41no—
77C22×C197—41no—
78C22×C197—81no—
79C22×C197—81no—
80C22×C197—81no—
81C22×C197#12011yescharacteristic
82C4×C197—21no—
83C4×C197—41no—
84C4×C197C2×SD1611yescharacteristic
85C4×C197#13511yescharacteristic
86C4×C197AGL(1,ℤ/8)11yescharacteristic
87C4×C197#13711yescharacteristic
88Dic197—41no—
89Dic197—41no—
90Dic197—42no—
91C2×C197—42no—
92C2×C197—161no—
93C2×C197#44811yescharacteristic
94C2×C197#44911yescharacteristic
95C2×C197#45311yescharacteristic
96C197#162211yescharacteristic, complemented, Sylow
97#1622—1971nomaximal, Sylow
98#554—1971no—
99#492—1971no—
100#460—1971no—
101#459D19711yescharacteristic
102#333—1971no—
103Q8⋊C8—1971no—
104D4⋊C8—1971no—
105C2×SD16—3941no—
106C2×SD16—3942no—
107C2×D8—21no—
108C4⋊Q8—1971no—
109#127C2×D19711yescharacteristic
110#125—3941no—
111C22⋊Q8—3942no—
112C22⋊Q8—3941no—
113C4×Q8—1971no—
114C4×D4—3941no—
115C4×D4C2×D19711yescharacteristic
116C2×C4⋊C4—3941no—
117C4⋊C8—1971no—
118C4⋊C8C2×D19711yescharacteristic
119#102—21no—
120#102—3941no—
121C4×C8—1971no—
122C2×Q8—1972no—
123C2×Q8—1971no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4C22⋊D19711yescharacteristic
128C22×C4—21no—
129C22×C4—3942no—
130SD16—7882no—
131SD16—7884no—
132D8—42no—
133C2×C8—21no—
134C2×C8—3941no—
135C2×C8—3941no—
136C4⋊C4—3942no—
137C4⋊C4—3942no—
138C4⋊C4—3941no—
139C4⋊C4—3941no—
140C4⋊C4—3941no—
141C4⋊C4—1971no—
142C4⋊C4C22⋊D19711yescharacteristic
143C22⋊C4—21no—
144C22⋊C4—3942no—
145C42—3941no—
146C42C22×D19711yescharacteristic
147C23—21no—
148C23—41no—
149Q8—7882no—
150Q8—3941no—
151Q8—3941no—
152D4—21no—
153D4—42no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4#1569134612yescomplemented
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—15762no—
163C2×C4—3941no—
164C2×C4—3941no—
165C2×C4—3942no—
166C2×C4D4×D19711yescharacteristic
167C2×C4D4×D19711yescharacteristic
168C2×C4C2×C22⋊D19711yescharacteristic
169C8—41no—
170C8—7881no—
171C8—3942no—
172C22—22no—
173C22—41no—
174C22—41no—
175C22—81no—
176C22—81no—
177C22—81no—
178C22#2026329411yescharacteristic
179C4—21no—
180C4—41no—
181C4—7881no—
182C4—7881no—
183C4—7882no—
184C4C2×#1569134611yescharacteristic
185C4#2026334511yescharacteristic
186C4#2026335011yescharacteristic
187C4#2026335611yescharacteristic
188C2—42no—
189C2—161no—
190C2#2716121711yescharacteristic
191C2#2716122111yescharacteristic
192C2#2716124511yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting, prime radical#459×C197C2
socleC22×C197#120
hypercenter#459D197
hypocenterC197#1622
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #27161786C2semidirect
2. #27161451C2semidirect
3. #27161276C2semidirect
4. #27160827C2semidirect
5. #27160809C2semidirect
6. D4⋊#13096802C2semidirect
67. D4×C197SD16semidirect
96. C197#1622semidirect
158. D4#15691346semidirect

Maximal subgroups

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

subgroupindex
#271617862
#271614512
#271612762
#271608272
#271608092
D4⋊#130968022
#459×C1972
#1622197