Finite Group Databasebeta

← #38741878

Subgroups — #38741878

Order 25216.

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

normal subgroups only

This group has 228 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#27161744C211yesmaximal, characteristic
2#27161431C211yesmaximal, characteristic, complemented
3#27161375C211yesmaximal, characteristic, complemented
4#27161370C211yesmaximal, characteristic
5#27160809C211yesmaximal, characteristic, complemented
6#27160806C211yesmaximal, characteristic
7#332×C197C211yesmaximal, characteristic, complemented
8C2×Q8×D197—21no—
9#20263326C2211yescharacteristic
10#20263325—21no—
11#20263304—22no—
12#20263302—21no—
13C4×Dic394C2211yescharacteristic
14C4×D788C2211yescharacteristic
15#20263288—21no—
16#20263260—21no—
17#20263258—21no—
18#20263251—21no—
19#20263249—22no—
20#20263248—21no—
21C4×#13096802C2211yescharacteristic
22C4⋊Q8×C197C2211yescharacteristic
23C4⋊C8×C197C2211yescharacteristic
24C4⋊C8×C197C2211yescharacteristic
25C2×Dic394—22no—
26C2×Dic394—21no—
27C2×Dic394D411yescharacteristic
28Q8×D197—42no—
29Q8×D197—42no—
30C2×D788D411yescharacteristic
31C2×C4×D197—22no—
32C2×C4×D197—21no—
33C2×#13096802—21no—
34C4⋊Dic197—21no—
35C4⋊Dic197—21no—
36C4⋊Dic197D411yescharacteristic
37C4⋊Dic197D411yescharacteristic
38#15691337—22no—
39#15691337—21no—
40#15691335—22no—
41#15691335—21no—
42C4×Dic197—21no—
43C2×Q8×C197—21no—
44C2×C8×C197—21no—
45C2×C8×C197—21no—
46C4⋊C4×C197—21no—
47C4⋊C4×C197D412yes—
48C42×C197C2311yescharacteristic
49C22×D197—21no—
50Dic394—21no—
51Dic394—44no—
52Dic394—42no—
53Dic394SD1612yes—
54D788—21no—
55D788Q1612yes—
56C2×Dic197—21no—
57C2×Dic197—21no—
58C2×Dic197—21no—
59C2×Dic197—41no—
60C4×D197—44no—
61C4×D197—41no—
62C4×D197—41no—
63#13096802—41no—
64Q8×C197—41no—
65Q8×C197—41no—
66C2×C4×C197—22no—
67C2×C4×C197C2×D411yescharacteristic
68C2×C4×C197C2×D411yescharacteristic
69C2×C4×C197C2×D411yescharacteristic
70C8×C197—41no—
71C8×C197—41no—
72C22×C197#12011yescharacteristic
73C2×D197—22no—
74C2×D197—41no—
75C2×D197—41no—
76C4×C197—21no—
77C4×C197—42no—
78C4×C197C2×SD1611yescharacteristic
79C4×C197C2×Q1611yescharacteristic
80C4×C197AGL(1,ℤ/8)11yescharacteristic
81C4×C197#13711yescharacteristic
82Dic197—22no—
83Dic197—41no—
84Dic197—41no—
85Dic197—81no—
86C2×C197#45011yescharacteristic
87C2×C197#45111yescharacteristic
88C2×C197#45411yescharacteristic
89D197—42no—
90C197#163711yescharacteristic, complemented, Sylow
91#1637—1971nomaximal, Sylow
92D4×Q8—1971no—
93#487—1971no—
94#477—1971no—
95#476—1971no—
96#332D19711yescharacteristic, complemented
97Q8⋊C8—1971no—
98D4⋊C8—1971no—
99C22×Q8—3941no—
100C4⋊Q8—3941no—
101C4⋊Q8—1971no—
102C4⋊Q8C2×D19711yescharacteristic
103C22⋊Q8—3941no—
104C22⋊Q8—3942no—
105C4×Q8—1971no—
106C4×D4—3941no—
107C4×D4—1971no—
108#107—3941no—
109#106—3941no—
110C4⋊C8C2×D19711yescharacteristic
111C4⋊C8C2×D19711yescharacteristic
112#103—3941no—
113#103—3942no—
114#102—3941no—
115C4×C8—1971no—
116C2×Q8—21no—
117C2×Q8—7882no—
118C2×Q8—7882no—
119C2×Q8—3942no—
120C2×Q8—3941no—
121C2×Q8—1971no—
122C2×D4—1971no—
123C22×C4—3942no—
124C22×C4—3941no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—3941no—
128C4⋊C4—21no—
129C4⋊C4—3942no—
130C4⋊C4—3941no—
131C4⋊C4—3941no—
132C4⋊C4—3941no—
133C4⋊C4—1971no—
134C4⋊C4—1971no—
135C4⋊C4D78812yes—
136C22⋊C4—3942no—
137C22⋊C4—3941no—
138C42—3941no—
139C42C22×D19711yescharacteristic
140C23—3941no—
141Q8—41no—
142Q8—41no—
143Q8—7882no—
144Q8—7884no—
145Q8—3941no—
146Q8—1972no—
147D4—3941no—
148D4—1972no—
149C2×C4—22no—
150C2×C4—7884no—
151C2×C4—7881no—
152C2×C4—7881no—
153C2×C4—7881no—
154C2×C4—3941no—
155C2×C4—3941no—
156C2×C4—3941no—
157C2×C4D4×D19711yescharacteristic
158C2×C4D4×D19711yescharacteristic
159C2×C4C2×D78811yescharacteristic
160C8—41no—
161C8—41no—
162C8—7881no—
163C22—7881no—
164C22—7881no—
165C22—3942no—
166C22#2026329611yescharacteristic
167C4—21no—
168C4—42no—
169C4—7881no—
170C4—7881no—
171C4—15761no—
172C4—3942no—
173C4SD16×D19711yescharacteristic
174C4Q16×D19711yescharacteristic
175C4#2026335011yescharacteristic
176C4#2026335511yescharacteristic
177C2—7882no—
178C2#2716123111yescharacteristic
179C2#2716123611yescharacteristic
180C2#2716125411yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#332×C197C2
socleC22×C197#120
prime radicalC2×D788D4
hypercenter#332D197
hypocenterC197#1637
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #27161431C2semidirect
3. #27161375C2semidirect
5. #27160809C2semidirect
7. #332×C197C2semidirect
90. C197#1637semidirect
96. #332D197semidirect

Maximal subgroups

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

subgroupindex
#271617442
#271614312
#271613752
#271613702
#271608092
#271608062
#332×C1972
#1637197