Finite Group Databasebeta

← #11920603

Subgroups — #11920603

Order 1152.

199 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 199 conjugacy classes of subgroups — too many to draw. Only its 27 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#5753⋊C3C211yesmaximal, characteristic, complemented
2C3×#5407C211yesmaximal, characteristic, complemented
3C3⋊#4628C211yesmaximal, characteristic, complemented
4#73139—31nomaximal
5#71095—31nomaximal
6#1010⋊C3—21no—
7C3×#986—21no—
8C3×#131⋊C3C2211yescharacteristic
9#63545—41nomaximal
10#5753D311yescharacteristic, complemented
11#578⋊C3—31no—
12#5407D311yescharacteristic, complemented
13C3×#472—31no—
14C2×#827—31no—
15#5147—31no—
16#4628—31no—
17#4628—61no—
18#4561—31no—
19#4497—31no—
20#4496—31no—
21C3×#41⋊C3D411yescharacteristic
22D3×SL(2,𝔽3)—41no—
23C3×GL(2,𝔽3)—41no—
24C3×D24—41no—
25C3×D24—41no—
26C3⋊(C3⋊C16)—41no—
27#2193—91noSylow
28#1010—21no—
292−1+4⋊C3—61no—
30#986—21no—
31C3×AGL(1,ℤ/8)—61no—
32C2×C3×D8—31no—
33C3×#131C2×D311yescharacteristic
34#909—61no—
35C2×Dic12—31no—
36#904—31no—
37#131⋊C3—21no—
38#131⋊C3C2×D311yescharacteristic
39C3×#108—31no—
40C3×#100—61no—
41#827—61no—
42#827—62no—
43#826—61no—
44#826—121no—
45#812—31no—
46C2×C3⋊C16—31no—
47C3×D12—81no—
48C3×D12—81no—
49C3×SL(2,𝔽3)D811yescharacteristic
50C32×C8—41no—
51#578—31no—
52#509—91no—
53C2×SD32—91no—
54#472—31no—
55#362—91no—
56#359—91no—
57#350—91no—
58C3×#41#6711yescharacteristic
59C2×C3×D4—61no—
60D3×Q8—41no—
61#237—121no—
62#235—61no—
63C2×Dic6—61no—
64#41⋊C3—21no—
65#41⋊C3#6711yescharacteristic
66C2×SL(2,𝔽3)—121no—
67GL(2,𝔽3)—41no—
68C3×SD16—121no—
69C3×D8—61no—
70C3×D8—61no—
71C3×D8—121no—
72C3×D8—121no—
73C3×M16—31no—
74C3×M16—61no—
75C2×C3×C8—31no—
76Dic12—32no—
77Dic12—31no—
78D24—41no—
79D24S411yescharacteristic
80#204—61no—
81C3⋊C16—61no—
82C3⋊C16—61no—
83C3⋊C16—121no—
84C3⋊C16—241no—
85C2×C3×D3—161no—
86C2×C3×D3—161no—
87C32×C4—41no—
882−1+4—61no—
89#137—181no—
90AGL(1,ℤ/8)—61no—
91#135—91no—
92C2×Q16—91no—
93C2×D8—31no—
94#131D3211yescharacteristic
95SD32—182no—
96SD32—181no—
97D16—181no—
98M32—91no—
99C2×C16—91no—
100#108—31no—
101#100—61no—
102C23×C3—121no—
103C3×Q8#21311yescharacteristic
104C3×D4—31no—
105C3×D4—121no—
106C3×D4—121no—
107C3×D4—241no—
108C2×C3×C4—31no—
109#67—121no—
110C2×Dic3—121no—
111D12—21no—
112D12—81no—
113C4×D3—121no—
114Dic6—61no—
115Dic6—121no—
116SL(2,𝔽3)—21no—
117SL(2,𝔽3)#21311yescharacteristic
118C3×C8—31no—
119C3×C8—41no—
120C3×C8—81no—
121C3×C8—121no—
122C3×C8C2×S411yescharacteristic
123C2×C32—41no—
124C3×D3—321no—
125C3×D3—321no—
126#41—181no—
127#41—361no—
128#41#62511yescharacteristic
129C2×Q8—121no—
130C2×Q8—181no—
131C2×D4—61no—
132Q16—92no—
133Q16—91no—
134SD16—121no—
135SD16—181no—
136D8—31no—
137D8—61no—
138D8—61no—
139D8—121no—
140M16—31no—
141M16—61no—
142C2×C8—31no—
143C16—181no—
144C16—181no—
145C22×C3—31no—
146C22×C3—241no—
147C22×C3—481no—
148C22×C3—481no—
149C2×D3—41no—
150C2×D3—161no—
151C3×C4—31no—
152C3×C4—41no—
153C3×C4—81no—
154C3×C4GL(2,ℤ/4)11yescharacteristic
155Dic3—121no—
156C32—41noSylow
157C23—121no—
158Q8—181no—
159Q8—361no—
160Q8#371311yescharacteristic
161D4—31no—
162D4—61no—
163D4—121no—
164D4—121no—
165D4—361no—
166C2×C4—31no—
167C2×C4—361no—
168C2×C4—361no—
169C8—31no—
170C8—121no—
171C8D3×S411yescharacteristic
172C2×C3—41no—
173C2×C3—61no—
174C2×C3—81no—
175C2×C3—481no—
176C2×C3—961no—
177C2×C3#541511yescharacteristic
178D3—81no—
179D3—321no—
180C22—31no—
181C22—121no—
182C22—241no—
183C22—481no—
184C4—31no—
185C4—361no—
186C4#6420911yescharacteristic
187C3—41no—
188C3—81no—
189C3#7313911yescharacteristic, complemented
190C2—61no—
191C2—241no—
192C2—481no—
193C2#1059318311yescharacteristic
subgroupG/H
centerC2#10593183
commutator, cosocleC3×#131⋊C3C22
Frattini, hypercenterC8D3×S4
FittingC3×#131C2×D3
socleC2×C3#5415
hypocenterC3×SL(2,𝔽3)D8
LCSG ▷ C3×#131⋊C3 ▷ C3×#41⋊C3 ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. #5753⋊C3C2semidirect
2. C3×#5407C2semidirect
3. C3⋊#4628C2semidirect
10. #5753D3semidirect
12. #5407D3semidirect
189. C3#73139semidirect

Maximal subgroups

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

subgroupindex
#5753⋊C32
C3×#54072
C3⋊#46282
#731393
#710953
#635454