Finite Group Databasebeta

← #13144518

Subgroups — #13144518

Order 1728.

495 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 111 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
1H(3,𝔽3)×#125C211yesmaximal, characteristic
2C4×#6229C211yesmaximal, characteristic
3#11720532C211yesmaximal, characteristic
4#11720532C211yesmaximal, characteristic
5#11720530C211yesmaximal, characteristic
6#11720529C211yesmaximal, characteristic
7C4⋊#6180C211yesmaximal, characteristic
8C3×#5063—34nomaximal
9C2×#6229C2211yescharacteristic
10C4⋊C4×H(3,𝔽3)—21no—
11C4⋊C4×H(3,𝔽3)—21no—
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C42×H(3,𝔽3)C2211yescharacteristic
15C4⋊#1084C2211yescharacteristic
16#88540—21no—
17C4×#1084—21no—
18C2×#6180C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20C32×#125D314yes—
21C3×C4×Dic6—34no—
22C3×#810—34no—
23C3×#810—34no—
24C3×#808—34no—
25C3×#807—34no—
26C3×C4⋊(C3⋊C8)—34no—
27C2×C4×H(3,𝔽3)—21no—
28C2×C4×H(3,𝔽3)—21no—
29C2×C4×H(3,𝔽3)D411yescharacteristic
30C2×C4×H(3,𝔽3)D411yescharacteristic
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×#1084—21no—
33C2×#1084—21no—
34#6229—21no—
35#6229Q812yes—
36#6180—21no—
37#6180—21no—
38C3×#479—91no—
39#5063—34no—
40C2×C3×Dic6—34no—
41C32×C4⋊C4—24no—
42C32×C4⋊C4—24no—
43C32×C4⋊C4C2×D314yes—
44C32×C4⋊C4C2×D314yes—
45C32×C42C2×D314yes—
46C3×C4⋊Dic3—34no—
47C3×Dic3⋊C4—64no—
48C3×C4×Dic3—64no—
49C2×C3×C3⋊C8—34no—
50C2×C3×C3⋊C8—34no—
51C22×H(3,𝔽3)C2×D411yescharacteristic
52C4×H(3,𝔽3)—21no—
53C4×H(3,𝔽3)—21no—
54C4×H(3,𝔽3)—41no—
55C4×H(3,𝔽3)—41no—
56C4×H(3,𝔽3)C2×Q811yescharacteristic
57C4×H(3,𝔽3)#4111yescharacteristic
58#1084—22no—
59#1084—41no—
60C3×#125—34no—
61C3×#125C3⋊D311yescharacteristic
62C3×C4×Q8—91no—
63C4×Dic6—34no—
64C3×#107—91no—
65C3×#106—91no—
66C3×C4⋊C8—91no—
67C3×#103—91no—
68C3×#103—91no—
69#810—34no—
70#810—34no—
71#808—34no—
72#807—34no—
73C4⋊(C3⋊C8)—34no—
74C2×C32×C4—24no—
75C2×C32×C4—24no—
76C2×C32×C4#6714yes—
77C2×C32×C4#6714yes—
78C2×C32×C4C22×D314yes—
79C2×C3×Dic3—64no—
80C2×C3×Dic3—64no—
81C3×Dic6—38no—
82C3×Dic6—64no—
83C3×C3⋊C8—64no—
84C3×C3⋊C8—64no—
85#479—91noSylow
86C2×H(3,𝔽3)C22⋊Q811yescharacteristic
87C2×H(3,𝔽3)#13511yescharacteristic
88C2×H(3,𝔽3)#13711yescharacteristic
89C2×C3×Q8—91no—
90C2×Dic6—34no—
91C2×C3×C8—91no—
92C2×C3×C8—91no—
93C3×C4⋊C4—21no—
94C3×C4⋊C4—21no—
95C3×C4⋊C4—34no—
96C3×C4⋊C4—34no—
97C3×C4⋊C4—64no—
98C3×C4⋊C4—64no—
99C3×C4⋊C4—91no—
100C3×C4⋊C4—181no—
101C3×C4⋊C4C2×C3⋊D311yescharacteristic
102C3×C4⋊C4C2×C3⋊D311yescharacteristic
103C3×C42—34no—
104C3×C42—181no—
105C3×C42C2×C3⋊D311yescharacteristic
106C4⋊Dic3—34no—
107Dic3⋊C4—64no—
108C4×Dic3—64no—
109C2×C3⋊C8—34no—
110C2×C3⋊C8—34no—
111C22×C32C2×#6714yes—
112C32×C4—24no—
113C32×C4—24no—
114C32×C4—44no—
115C32×C4—44no—
116C32×C4D3×Q814yes—
117C32×C4#23914yes—
118C3×Dic3—68no—
119C3×Dic3—124no—
120#125#27211yescharacteristic
121C4×Q8—91no—
122#107—91no—
123#106—91no—
124C4⋊C8—91no—
125#103—91no—
126#103—91no—
127H(3,𝔽3)#47911yescharacteristic, complemented, Sylow
128C3×Q8—92no—
129C3×Q8—181no—
130C2×C3×C4—21no—
131C2×C3×C4—21no—
132C2×C3×C4—34no—
133C2×C3×C4—34no—
134C2×C3×C4—34no—
135C2×C3×C4—64no—
136C2×C3×C4—64no—
137C2×C3×C4—181no—
138C2×C3×C4—181no—
139C2×C3×C4#63811yescharacteristic
140C2×C3×C4#63811yescharacteristic
141C2×C3×C4C22×C3⋊D311yescharacteristic
142C2×Dic3—64no—
143C2×Dic3—64no—
144Dic6—38no—
145Dic6—64no—
146C3×C8—181no—
147C3×C8—181no—
148C3⋊C8—64no—
149C3⋊C8—64no—
150C2×C32#94614yes—
151C2×C32#95014yes—
152C2×C32#95114yes—
153C2×Q8—91no—
154C2×C8—91no—
155C2×C8—91no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—91no—
159C4⋊C4—181no—
160C4⋊C4C2×#27211yescharacteristic
161C4⋊C4C2×#27211yescharacteristic
162C42—181no—
163C42C2×#27211yescharacteristic
164C22×C3—34no—
165C22×C3C2×#63811yescharacteristic
166C3×C4—21no—
167C3×C4—21no—
168C3×C4—34no—
169C3×C4—34no—
170C3×C4—41no—
171C3×C4—41no—
172C3×C4—64no—
173C3×C4—64no—
174C3×C4—124no—
175C3×C4—124no—
176C3×C4—182no—
177C3×C4—361no—
178C3×C4Q8×C3⋊D311yescharacteristic
179C3×C4#383211yescharacteristic
180Dic3—68no—
181Dic3—124no—
182C32#506314yescomplemented
183Q8—92no—
184Q8—181no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—181no—
188C2×C4—181no—
189C2×C4#623511yescharacteristic
190C2×C4#623511yescharacteristic
191C2×C4C22×#27211yescharacteristic
192C8—181no—
193C8—181no—
194C2×C3—34no—
195C2×C3—34no—
196C2×C3—34no—
197C2×C3#6415511yescharacteristic
198C2×C3#6415911yescharacteristic
199C2×C3#6416011yescharacteristic
200C22C2×#623511yescharacteristic
201C4—21no—
202C4—21no—
203C4—41no—
204C4—41no—
205C4—182no—
206C4—361no—
207C4Q8×#27211yescharacteristic
208C4#8874811yescharacteristic
209C3—34no—
210C3#1059276211yescharacteristic
211C2#1172186311yescharacteristic
212C2#1172187111yescharacteristic
213C2#1172187211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#125C2
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC3×#125C3⋊D3
hypocenterH(3,𝔽3)#479
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#125

Complements

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

NH ≅ G/Ntype
127. H(3,𝔽3)#479semidirect
182. C32#5063semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
C4×#62292
#117205322
#117205302
#117205292
C4⋊#61802
C3×#50633