Finite Group Databasebeta

← #38741785

Subgroups — #38741785

Order 25216.

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

normal subgroups only

This group has 236 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#27161790C211yesmaximal, characteristic, complemented
2#27161453C211yesmaximal, characteristic, complemented
3#27161280C211yesmaximal, characteristic, complemented
4#27161275C211yesmaximal, characteristic
5#27160809C211yesmaximal, characteristic, complemented
6#27160806C211yesmaximal, characteristic, complemented
7#333×C197C211yesmaximal, characteristic, complemented
8C2×#15691347—21no—
9C2×#15691345—22no—
10C2×D1576—21no—
11#20263323C2211yescharacteristic
12#20263317—21no—
13#20263306—21no—
14#20263302—22no—
15C4×Dic394C2211yescharacteristic
16C4×D788C2211yescharacteristic
17#20263288—21no—
18C4⋊C4×D197—21no—
19#20263248—21no—
20#20263248—21no—
21C4×#13096802C2211yescharacteristic
22C4⋊Q8×C197C2211yescharacteristic
23C4⋊C8×C197C2211yescharacteristic
24C4⋊C8×C197C2211yescharacteristic
25C2×Dic394D411yescharacteristic
26C2×D788—21no—
27C2×D788—21no—
28C2×D788—21no—
29C2×D788D411yescharacteristic
30C2×C4×D197—22no—
31C2×C4×D197—21no—
32#15691347—42no—
33#15691345—44no—
34D1576—42no—
35C2×#13096802—21no—
36C4⋊Dic197—22no—
37C4⋊Dic197D411yescharacteristic
38C4⋊Dic197D411yescharacteristic
39#15691337—22no—
40#15691337—21no—
41#15691335—22no—
42#15691335—21no—
43C4×Dic197—21no—
44C2×Q8×C197D412yes—
45C2×C8×C197—21no—
46C2×C8×C197—21no—
47C4⋊C4×C197—21no—
48C4⋊C4×C197—21no—
49C42×C197C2311yescharacteristic
50C22×D197—21no—
51C22×D197—41no—
52Dic394—21no—
53Dic394—21no—
54D788—21no—
55D788—42no—
56D788—41no—
57D788—41no—
58D788—41no—
59D788—41no—
60D788SD1612yescomplemented
61C2×Dic197—21no—
62C2×Dic197—21no—
63C2×Dic197—21no—
64C4×D197—41no—
65C4×D197—41no—
66C4×D197—82no—
67#13096802—22no—
68Q8×C197—42no—
69C2×C4×C197—22no—
70C2×C4×C197C2×D411yescharacteristic
71C2×C4×C197C2×D411yescharacteristic
72C2×C4×C197C2×D411yescharacteristic
73C8×C197—41no—
74C8×C197—41no—
75C22×C197#12011yescharacteristic
76C2×D197—22no—
77C2×D197—41no—
78C2×D197—41no—
79C2×D197—81no—
80C2×D197—81no—
81C2×D197—81no—
82C4×C197—21no—
83C4×C197—42no—
84C4×C197C2×SD1611yescharacteristic
85C4×C197#13511yescharacteristic
86C4×C197AGL(1,ℤ/8)11yescharacteristic
87C4×C197#13711yescharacteristic
88Dic197—41no—
89Dic197—41no—
90Dic197—41no—
91C2×C197#44811yescharacteristic
92C2×C197#44911yescharacteristic
93C2×C197#45311yescharacteristic
94D197—42no—
95D197—161no—
96C197#162211yescharacteristic, complemented, Sylow
97#1622—1971nomaximal, Sylow
98#554—1971no—
99#492—1971no—
100#460—1971no—
101#459—1971no—
102#333D19711yescharacteristic, complemented
103Q8⋊C8—1971no—
104D4⋊C8—1971no—
105C2×SD16—3941no—
106C2×SD16—3942no—
107C2×D8—3941no—
108C4⋊Q8C2×D19711yescharacteristic
109#127—1971no—
110#125—3941no—
111C22⋊Q8—3941no—
112C22⋊Q8—3942no—
113C4×Q8—1971no—
114C4×D4—3941no—
115C4×D4—1971no—
116C2×C4⋊C4—3941no—
117C4⋊C8C2×D19711yescharacteristic
118C4⋊C8C2×D19711yescharacteristic
119#102—3941no—
120#102—3941no—
121C4×C8—1971no—
122C2×Q8—1971no—
123C2×Q8D78812yes—
124C2×D4—3941no—
125C2×D4—3941no—
126C2×D4—3941no—
127C2×D4—1971no—
128C22×C4—3942no—
129C22×C4—3941no—
130SD16—7882no—
131SD16—7884no—
132D8—7882no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—3941no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—3942no—
139C4⋊C4—3942no—
140C4⋊C4—3941no—
141C4⋊C4—1971no—
142C4⋊C4—1971no—
143C22⋊C4—3942no—
144C22⋊C4—3941no—
145C42—3941no—
146C42C22×D19711yescharacteristic
147C23—7881no—
148C23—3941no—
149Q8—42no—
150Q8—3941no—
151Q8—3941no—
152D4—7882no—
153D4—7881no—
154D4—7881no—
155D4—7881no—
156D4—7881no—
157D4—3941no—
158D4—1972no—
159C2×C4—22no—
160C2×C4—7881no—
161C2×C4—7881no—
162C2×C4—15762no—
163C2×C4—3941no—
164C2×C4—3941no—
165C2×C4—3941no—
166C2×C4D4×D19711yescharacteristic
167C2×C4D4×D19711yescharacteristic
168C2×C4C2×D78811yescharacteristic
169C8—41no—
170C8—41no—
171C8—3942no—
172C22—7881no—
173C22—7881no—
174C22—15761no—
175C22—15761no—
176C22—15761no—
177C22—3942no—
178C22#2026329611yescharacteristic
179C4—21no—
180C4—42no—
181C4—7881no—
182C4—7881no—
183C4—7881no—
184C4SD16×D19711yescharacteristic
185C4#2026334911yescharacteristic
186C4#2026335311yescharacteristic
187C4#2026335811yescharacteristic
188C2—7882no—
189C2—31521no—
190C2#2716121911yescharacteristic
191C2#2716122711yescharacteristic
192C2#2716124911yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#333×C197C2
socleC22×C197#120
prime radical#27161275C2
hypercenter#333D197
hypocenterC197#1622
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
1. #27161790C2semidirect
2. #27161453C2semidirect
3. #27161280C2semidirect
5. #27160809C2semidirect
6. #27160806C2semidirect
7. #333×C197C2semidirect
60. D788SD16semidirect
96. C197#1622semidirect
102. #333D197semidirect

Maximal subgroups

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

subgroupindex
#271617902
#271614532
#271612802
#271612752
#271608092
#271608062
#333×C1972
#1622197