Finite Group Databasebeta

← #11800017

Subgroups — #11800017

Order 1152.

372 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 93 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#10592592C211yesmaximal, characteristic
2#10592566C211yesmaximal, characteristic
3C32×#365C211yesmaximal, characteristic, complemented
4#10589383C211yesmaximal, characteristic, complemented
5#10589380C211yesmaximal, characteristic
6#10589362C211yesmaximal, characteristic, complemented
7#10589353C211yesmaximal, characteristic, complemented
8#69314—34nomaximal
9M16×C3⋊D3—21no—
10C2×#3746C2211yescharacteristic
11C2×#3743C2211yescharacteristic
12#64106—21no—
13C32×M32C2211yescharacteristic
14C32×#108C2211yescharacteristic
15C32×#106C2211yescharacteristic
16#63649—21no—
17#63646C2211yescharacteristic
18#63639C2211yescharacteristic
19#63637—21no—
20#4893—34no—
21#4867—34no—
22C3×#365D314yescomplemented
23#4517—34no—
24#4514—34no—
25#4496—34no—
26#4487—34no—
27C2×C4×C3⋊D3D411yescharacteristic
28C2×#634—21no—
29C32×M16—21no—
30C2×C32×C8C2311yescharacteristic
31C32×C4⋊C4—21no—
32#3752—41no—
33C4⋊(C3⋊Dic3)—41no—
34#3748—21no—
35C2×C3⋊(C3⋊C8)D411yescharacteristic
36#3746—21no—
37#3746—21no—
38#3743—21no—
39#3743—21no—
40#3743—41no—
41C8×C3⋊D3—41no—
42C32×C16—21no—
43C3⋊(C3⋊C16)—21no—
44#2239—91noSylow
45D3×M16—64no—
46C2×Dic12—34no—
47C2×ΓL(1,𝔽25)—34no—
48#897—64no—
49C3×M32C2×D314yes—
50C3×#108C2×D314yes—
51C3×#106C2×D314yes—
52#822—64no—
53#819—34no—
54#812—34no—
55#810—64no—
56C22×C3⋊D3SD1611yescharacteristic
57C2×C32×C4—41no—
58C2×C32×C4C2×D411yescharacteristic
59C2×C3⋊Dic3—41no—
60C2×C3⋊Dic3SD1611yescharacteristic
61C4×C3⋊D3—21no—
62C4×C3⋊D3—21no—
63#634—41no—
64#634—41no—
65C32×C8—41no—
66C32×C8#4111yescharacteristic
67C32×C8#4111yescharacteristic
68C3⋊(C3⋊C8)—21no—
69C3⋊(C3⋊C8)—41no—
70#470—91no—
71#430—91no—
72#365—91no—
73#365C3⋊D311yescharacteristic, complemented
74#362—91no—
75#362—91no—
76#349—91no—
77C2×C4×D3—34no—
78C2×Dic6—64no—
79C3×M16—24no—
80C2×C3×C8C22×D314yes—
81C3×C4⋊C4—24no—
82(C2×D3)⋊C4—124no—
83C4⋊Dic3—124no—
84#208—64no—
85C2×C3⋊C8—34no—
86Dic12—64no—
87Dic12—64no—
88ΓL(1,𝔽25)—64no—
89ΓL(1,𝔽25)—64no—
90ΓL(1,𝔽25)—124no—
91D3×C8—124no—
92C3×C16—24no—
93C3⋊C16—64no—
94C22×C32C2×SD1611yescharacteristic
95C2×C3⋊D3—21no—
96C2×C3⋊D3—41no—
97C32×C4—81no—
98C32×C4#12311yescharacteristic
99C32×C4AGL(1,ℤ/8)11yescharacteristic
100C3⋊Dic3—21no—
101C3⋊Dic3—81no—
102C2×Q16—91no—
103C2×M16—91no—
104C2×M16—181no—
105C22⋊Q8—181no—
106M32—91no—
107M32C2×C3⋊D311yescharacteristic
108#108—91no—
109#108—181no—
110#108C2×C3⋊D311yescharacteristic
111#106C2×C3⋊D311yescharacteristic
112#103—181no—
113C22×D3—34no—
114C2×C3×C4—44no—
115C2×C3×C4D3×D414yes—
116C2×Dic3—34no—
117C2×Dic3—124no—
118C4×D3—64no—
119C4×D3—64no—
120Dic6—124no—
121Dic6—124no—
122C3×C8—44no—
123C3×C8#23514yes—
124C3×C8#23914yes—
125C3⋊C8—64no—
126C3⋊C8—124no—
127C2×C32—21no—
128C2×C32#48111yescharacteristic
129C3⋊D3—41no—
130C2×Q8—181no—
131C22×C4—91no—
132Q16—181no—
133Q16—181no—
134M16—21no—
135M16—181no—
136M16—181no—
137M16—181no—
138M16—361no—
139C2×C8—91no—
140C2×C8—361no—
141C2×C8C22×C3⋊D311yescharacteristic
142C4⋊C4—21no—
143C4⋊C4—361no—
144C22⋊C4—361no—
145C16—21no—
146C16—181no—
147C22×C3D3×SD1614yes—
148C2×D3—64no—
149C2×D3—124no—
150C3×C4—84no—
151C3×C4#89414yes—
152C3×C4#91414yes—
153Dic3—64no—
154Dic3—244no—
155C32#223911yescharacteristic, complemented, Sylow
156C23—91no—
157Q8—361no—
158Q8—361no—
159C2×C4—41no—
160C2×C4—91no—
161C2×C4—181no—
162C2×C4—181no—
163C2×C4—361no—
164C2×C4D4×C3⋊D311yescharacteristic
165C8—41no—
166C8—181no—
167C8—361no—
168C8#382811yescharacteristic
169C8#383211yescharacteristic
170C2×C3—24no—
171C2×C3#486314yes—
172D3—124no—
173C22—181no—
174C22—361no—
175C22SD16×C3⋊D311yescharacteristic
176C4—81no—
177C4—181no—
178C4—721no—
179C4#6410311yescharacteristic
180C4#6412311yescharacteristic
181C3#6931414yescomplemented
182C2—21no—
183C2—361no—
184C2#1059256211yescharacteristic
subgroupG/H
centerC2#10592562
commutator, cosocleC2×C32×C8C23
FrattiniC2×C8C22×C3⋊D3
FittingC32×#365C2
socleC2×C32#481
prime radicalC22×C3⋊D3SD16
hypercenter#365C3⋊D3
hypocenterC32#2239
LCSG ▷ C2×C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #365

Complements

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

NH ≅ G/Ntype
3. C32×#365C2semidirect
4. #10589383C2semidirect
6. #10589362C2semidirect
7. #10589353C2semidirect
22. C3×#365D3semidirect
73. #365C3⋊D3semidirect
155. C32#2239semidirect
181. C3#69314semidirect

Maximal subgroups

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

subgroupindex
#105925922
#105925662
C32×#3652
#105893832
#105893802
#105893622
#105893532
#693143