Finite Group Databasebeta

← #11800019

Subgroups — #11800019

Order 1152.

303 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 303 conjugacy classes of subgroups — too many to draw. Only its 64 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#10592888C211yesmaximal, characteristic
2#10590914C211yesmaximal, characteristic
3C3×#4565C212yesmaximal, complemented
4#10589137C212yesmaximal, complemented
5#10589110C211yesmaximal, characteristic
6#69267—32nomaximal
7C2×C32×Q16C2211yescharacteristic
8#64155—21no—
9C2×#3743C2211yescharacteristic
10#63818—21no—
11#63662—21no—
12#63644C2211yescharacteristic
13C3×#819C2212yes—
14C3×#812C2212yes—
15#63578—21no—
16#5189—32no—
17#5189—61no—
18#4893—32no—
19C3×#362—32no—
20#4565D312yescomplemented
21#4514—32no—
22#4513—32no—
23#4494—32no—
24C2×C32×Q8—21no—
25C2×C4×C3⋊D3D411yescharacteristic
26C32×Q16—22no—
27C2×C32×C8C2311yescharacteristic
28#3752—41no—
29C4⋊(C3⋊Dic3)—21no—
30#3750—41no—
31C2×C3⋊(C3⋊C8)D411yescharacteristic
32#3743—22no—
33C3×#208—22no—
34#3712—41no—
35#3710—41no—
36C3×C3⋊C16—22no—
37#2239—91noSylow
38C2×C3×Q16—21no—
39C2×C3×Q16C2×D312yes—
40#946—62no—
41#946—62no—
42D3×M16—62no—
43C2×ΓL(1,𝔽25)—32no—
44C2×ΓL(1,𝔽25)—61no—
45C3×M32—32no—
46C3×#108—32no—
47#835—62no—
48#835—62no—
49#822—62no—
50#819C2×D312yes—
51#817—32no—
52#817—61no—
53#812C2×D312yes—
54C22×C3⋊D3SD1611yescharacteristic
55C32×Q8—41no—
56C32×Q8—41no—
57C2×C32×C4—41no—
58C2×C32×C4C2×D411yescharacteristic
59C2×C3⋊Dic3—41no—
60C2×C3⋊Dic3SD1611yescharacteristic
61C4×C3⋊D3—21no—
62C4×C3⋊D3—21no—
63C32×C8#4112yes—
64C3⋊(C3⋊C8)—21no—
65C3×C3⋊C8—42no—
66#470—91no—
67#430—91no—
68#365—92no—
69#362—32no—
70#349—91no—
71C2×C3×Q8—22no—
72C2×C3×Q8—22no—
73C2×C4×D3—32no—
74C2×C4×D3—61no—
75C3×Q16—22no—
76C3×Q16—22no—
77C3×Q16—42no—
78C3×M16—62no—
79C2×C3×C8—21no—
80C2×C3×C8C22×D312yes—
81(C2×D3)⋊C4—122no—
82(C2×D3)⋊C4—122no—
83C4⋊Dic3—62no—
84C4⋊Dic3—62no—
85Dic3⋊C4—122no—
86Dic3⋊C4—122no—
87#208—22no—
88C2×C3⋊C8—32no—
89C2×C3⋊C8—61no—
90ΓL(1,𝔽25)—62no—
91ΓL(1,𝔽25)—62no—
92ΓL(1,𝔽25)—122no—
93ΓL(1,𝔽25)—122no—
94D3×C8—122no—
95C3×C16—62no—
96C3⋊C16—22no—
97C22×C32C2×SD1611yescharacteristic
98C2×C3⋊D3—21no—
99C2×C3⋊D3—41no—
100C32×C4—81no—
101C32×C4#12311yescharacteristic
102C32×C4AGL(1,ℤ/8)11yescharacteristic
103C3⋊Dic3—21no—
104C3⋊Dic3—81no—
105C2×Q16D3211yescharacteristic
106C2×M16—91no—
107C2×M16—181no—
108C22⋊Q8—181no—
109M32—32no—
110#108—32no—
111#108—181no—
112#106—91no—
113#103—181no—
114C22×D3—32no—
115C22×D3—61no—
116C3×Q8—42no—
117C3×Q8—42no—
118C3×Q8—42no—
119C3×Q8—42no—
120C2×C3×C4—21no—
121C2×C3×C4—42no—
122C2×C3×C4—42no—
123C2×C3×C4D3×D412yes—
124C2×Dic3—32no—
125C2×Dic3—61no—
126C2×Dic3—122no—
127C2×Dic3—122no—
128C4×D3—62no—
129C4×D3—62no—
130C4×D3—121no—
131C4×D3—121no—
132C3×C8—22no—
133C3×C8—122no—
134C3×C8#23512yes—
135C3×C8#23712yes—
136C3⋊C8—42no—
137C3⋊C8—62no—
138C3⋊C8—121no—
139C2×C32—21no—
140C2×C32#48111yescharacteristic
141C3⋊D3—41no—
142C2×Q8—21no—
143C22×C4—91no—
144Q16—22no—
145M16—62no—
146M16—182no—
147M16—361no—
148C2×C8—91no—
149C2×C8—361no—
150C2×C8C2×D3211yescharacteristic
151C4⋊C4—181no—
152C4⋊C4—361no—
153C22⋊C4—361no—
154C16—62no—
155C22×C3—21no—
156C22×C3D3×SD1612yes—
157C2×D3—62no—
158C2×D3—121no—
159C2×D3—122no—
160C2×D3—122no—
161C3×C4—21no—
162C3×C4—21no—
163C3×C4—82no—
164C3×C4—82no—
165C3×C4#88312yes—
166C3×C4#91112yes—
167Dic3—62no—
168Dic3—121no—
169Dic3—242no—
170Dic3—242no—
171C32#223911yescharacteristic, complemented, Sylow
172C23—91no—
173Q8—41no—
174Q8—41no—
175C2×C4—41no—
176C2×C4—91no—
177C2×C4—181no—
178C2×C4—181no—
179C2×C4—361no—
180C2×C4#381111yescharacteristic
181C8—122no—
182C8—181no—
183C8#380412yes—
184C2×C3—22no—
185C2×C3—22no—
186C2×C3—21no—
187C2×C3#477712yes—
188D3—122no—
189D3—241no—
190C22—181no—
191C22—361no—
192C22#6393011yescharacteristic
193C4—81no—
194C4—181no—
195C4—721no—
196C4#6392711yescharacteristic
197C4#6397511yescharacteristic
198C3—21no—
199C3#6926712yescomplemented
200C2—21no—
201C2—361no—
202C2#1059141911yescharacteristic
subgroupG/H
centerC2#10591419
commutator, cosocleC2×C32×C8C23
FrattiniC2×C8C2×D32
FittingC2×C32×Q16C22
socleC2×C32#481
prime radicalC22×C3⋊D3SD16
hypercenterC2×Q16D32
hypocenterC32#2239
LCSG ▷ C2×C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×Q16

Complements

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

NH ≅ G/Ntype
3. C3×#4565C2semidirect
4. #10589137C2semidirect
20. #4565D3semidirect
171. C32#2239semidirect
199. C3#69267semidirect

Maximal subgroups

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

subgroupindex
#105928882
#105909142
C3×#45652
#105891372
#105891102
#692673