Finite Group Databasebeta

← #13144769

Subgroups — #13144769

Order 1728.

550 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×Q16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#11721859C212yesmaximal
3#11721776C211yesmaximal, characteristic
4#11720584C212yesmaximal, complemented
5#11720552C211yesmaximal, characteristic, complemented
6C3×#5192—34nomaximal
7C2×Q8×H(3,𝔽3)C2212yes—
8#88740C2211yescharacteristic
9Q16×H(3,𝔽3)—21no—
10Q16×H(3,𝔽3)—21no—
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12#88550—22no—
13#88549—22no—
14#88538C2211yescharacteristic
15#88538C2212yes—
16#88527—21no—
17C8×#272—21no—
18C2×C32×Q16D314yescomplemented
19C3×#942—38no—
20C3×#901—34no—
21C3×#836—38no—
22C3×#819—34no—
23Q8×H(3,𝔽3)—22no—
24Q8×H(3,𝔽3)—22no—
25C2×C4×H(3,𝔽3)—22no—
26C2×C4×H(3,𝔽3)C2311yescharacteristic
27#6235—21no—
28C4⋊#272D411yescharacteristic
29C4×#272—21no—
30#6229D411yescharacteristic
31C8×H(3,𝔽3)D411yescharacteristic
32C8×H(3,𝔽3)D411yescharacteristic
33#6180—21no—
34#6180—22no—
35C3×#473—91no—
36#5192—34no—
37C2×C32×Q8C2×D318yes—
38C3×#235—34no—
39C32×Q16—24no—
40C32×Q16—24no—
41C2×C32×C8C2×D314yes—
42C3×#216—68no—
43C3×#215—68no—
44C3×#208—34no—
45C3×#208—38no—
46C3×ΓL(1,𝔽25)—64no—
47C3×D3×C8—64no—
48C22×H(3,𝔽3)#4111yescharacteristic
49C2×#272—21no—
50C4×H(3,𝔽3)—42no—
51C4×H(3,𝔽3)C2×D411yescharacteristic
52C4×H(3,𝔽3)C2×D411yescharacteristic
53#1084—21no—
54C3×#137—92no—
55C2×C3×Q16—34no—
56C2×C3×Q16C3⋊D311yescharacteristic
57C3×#131—91no—
58#942—38no—
59#901—34no—
60C3×#108—91no—
61C3×#101—92no—
62#836—38no—
63#819—34no—
64C32×Q8—28no—
65C32×Q8—28no—
66C2×C32×C4—28no—
67C2×C32×C4C22×D314yes—
68C3×#67—64no—
69C3×D12—34no—
70C3×C4×D3—64no—
71C3×Dic6—34no—
72C32×C8#6714yes—
73C32×C8#6714yes—
74C3×C3⋊C8—68no—
75C3×C3⋊C8—64no—
76#473—91noSylow
77C2×H(3,𝔽3)—21no—
78C2×H(3,𝔽3)#12111yescharacteristic
79#272—41no—
80C3×#41—91no—
81C2×C3×Q8—38no—
82C2×C3×Q8C2×C3⋊D312yes—
83#235—34no—
84C3×Q16—21no—
85C3×Q16—21no—
86C3×Q16—64no—
87C3×Q16—64no—
88C3×Q16—182no—
89C3×SD16—182no—
90C3×M16—91no—
91C3×M16—92no—
92C3×M16—181no—
93C2×C3×C8—34no—
94C2×C3×C8—181no—
95C2×C3×C8C2×C3⋊D311yescharacteristic
96#216—68no—
97#215—68no—
98#208—38no—
99#208—34no—
100ΓL(1,𝔽25)—64no—
101D3×C8—64no—
102C22×C32#23714yes—
103C2×C3×D3—64no—
104C32×C4—48no—
105C32×C4D3×D414yes—
106C32×C4C2×#6714yes—
107C3×Dic3—64no—
108#137—92no—
109C2×Q16#27211yescharacteristic, complemented
110#131—91no—
111#108—91no—
112#101—92no—
113H(3,𝔽3)#47311yescharacteristic, complemented, Sylow
114C3×Q8—22no—
115C3×Q8—22no—
116C3×Q8—68no—
117C3×Q8—68no—
118C3×Q8—91no—
119C3×D4—91no—
120C3×D4—181no—
121C2×C3×C4—22no—
122C2×C3×C4—34no—
123C2×C3×C4—68no—
124C2×C3×C4—181no—
125C2×C3×C4C22×C3⋊D311yescharacteristic
126#67—64no—
127D12—34no—
128C4×D3—64no—
129Dic6—34no—
130C3×C8—34no—
131C3×C8—34no—
132C3×C8—182no—
133C3×C8—181no—
134C3×C8#63811yescharacteristic
135C3×C8#63811yescharacteristic
136C3⋊C8—64no—
137C3⋊C8—68no—
138C2×C32—24no—
139C2×C32#93814yes—
140C3×D3—124no—
141#41—91no—
142C2×Q8C2×#27212yes—
143Q16—21no—
144Q16—21no—
145Q16—182no—
146SD16—182no—
147M16—91no—
148M16—92no—
149M16—181no—
150C2×C8—181no—
151C2×C8C2×#27211yescharacteristic
152C22×C3—34no—
153C22×C3—181no—
154C22×C3#383011yescharacteristic
155C2×D3—64no—
156C3×C4—34no—
157C3×C4—34no—
158C3×C4—42no—
159C3×C4—128no—
160C3×C4—181no—
161C3×C4D4×C3⋊D311yescharacteristic
162C3×C4C2×#63811yescharacteristic
163Dic3—64no—
164C32#519214yescomplemented
165Q8—22no—
166Q8—22no—
167Q8—91no—
168D4—91no—
169D4—181no—
170C2×C4—22no—
171C2×C4—181no—
172C2×C4C22×#27211yescharacteristic
173C8—182no—
174C8—181no—
175C8#623511yescharacteristic
176C8#623511yescharacteristic
177C2×C3—21no—
178C2×C3—34no—
179C2×C3—64no—
180C2×C3—361no—
181C2×C3#6414711yescharacteristic
182D3—124no—
183C22—181no—
184C22#8874411yescharacteristic
185C4—42no—
186C4—181no—
187C4D4×#27211yescharacteristic
188C4C2×#623511yescharacteristic
189C3—34no—
190C3#1059289111yescharacteristic
191C2—21no—
192C2—361no—
193C2#1172184811yescharacteristic
subgroupG/H
center, socleC2×C3#64147
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC2×Q16×H(3,𝔽3)C2
prime radical#88740C22
hypercenterC2×C3×Q16C3⋊D3
hypocenterH(3,𝔽3)#473
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×Q16

Complements

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

NH ≅ G/Ntype
1. C2×Q16×H(3,𝔽3)C2semidirect
4. #11720584C2semidirect
5. #11720552C2semidirect
18. C2×C32×Q16D3semidirect
109. C2×Q16#272semidirect
113. H(3,𝔽3)#473semidirect
164. C32#5192semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×H(3,𝔽3)2
#117218592
#117217762
#117205842
#117205522
C3×#51923