Finite Group Databasebeta

← #13143804

Subgroups — #13143804

Order 1728.

517 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 517 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C4×#6229C211yesmaximal, characteristic
2H(3,𝔽3)×#97C211yesmaximal, characteristic
3#11720548C211yesmaximal, characteristic
4#11720547C211yesmaximal, characteristic
5#11720547C211yesmaximal, characteristic
6C8×#1084C211yesmaximal, characteristic
7C4⋊#6180C211yesmaximal, characteristic
8C3×#4701—34nomaximal
9C2×#6229C411yescharacteristic
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C42×H(3,𝔽3)C2211yescharacteristic
13C4⋊#1084C411yescharacteristic
14#88540C411yescharacteristic
15#88540C411yescharacteristic
16C4×#1084C2211yescharacteristic
17C4×#1084C2211yescharacteristic
18C2×#6180C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20C3×C4×Dic6—34no—
21C32×#97D314yes—
22C3×#815—34no—
23C3×Dic3⋊C8—34no—
24C3×Dic3⋊C8—34no—
25C3×C8×Dic3—34no—
26C3×C4⋊(C3⋊C8)—34no—
27C2×C4×H(3,𝔽3)C2×C411yescharacteristic
28C2×C4×H(3,𝔽3)C2×C411yescharacteristic
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×#1084C2×C411yescharacteristic
31C2×#1084C2×C411yescharacteristic
32C2×#1084C2×C411yescharacteristic
33C2×#1084C2×C411yescharacteristic
34#6229—22no—
35C8×H(3,𝔽3)—21no—
36C8×H(3,𝔽3)Q812yescomplemented
37#6180—21no—
38#6180—21no—
39C3×#446—91no—
40#4701—34no—
41C2×C3×Dic6—34no—
42C2×C32×C8C2×D314yes—
43C2×C32×C8C2×D314yes—
44C32×C42C2×D314yes—
45C3×C4⋊Dic3—34no—
46C3×Dic3⋊C4—34no—
47C3×Dic3⋊C4—34no—
48C3×C4×Dic3—34no—
49C3×C4×Dic3—34no—
50C2×C3×C3⋊C8—34no—
51C2×C3×C3⋊C8—34no—
52C22×H(3,𝔽3)C22×C411yescharacteristic
53C4×H(3,𝔽3)—21no—
54C4×H(3,𝔽3)—21no—
55C4×H(3,𝔽3)C2×Q811yescharacteristic
56C4×H(3,𝔽3)#4111yescharacteristic
57#1084—21no—
58#1084—21no—
59#1084—21no—
60#1084M1612yes—
61C3×C4×Q8—91no—
62C4×Dic6—34no—
63C3×C4⋊C8—91no—
64C3×C4⋊C8—91no—
65C3×C4⋊C8—91no—
66C3×#97—34no—
67C3×#97—91no—
68C3×#97C3⋊D311yescharacteristic
69C3×C4×C8—91no—
70#815—34no—
71Dic3⋊C8—34no—
72Dic3⋊C8—34no—
73C8×Dic3—34no—
74C4⋊(C3⋊C8)—34no—
75C2×C32×C4C4×D314yes—
76C2×C32×C4C4×D314yes—
77C2×C32×C4C22×D314yes—
78C2×C3×Dic3—34no—
79C2×C3×Dic3—34no—
80C2×C3×Dic3—34no—
81C2×C3×Dic3—34no—
82C3×Dic6—68no—
83C32×C8—24no—
84C32×C8Dic618yescomplemented
85C3×C3⋊C8—64no—
86C3×C3⋊C8—64no—
87#446—91noSylow
88C2×H(3,𝔽3)C4×Q811yescharacteristic
89C2×H(3,𝔽3)C2×M1611yescharacteristic
90C2×H(3,𝔽3)#13111yescharacteristic
91C2×C3×Q8—91no—
92C2×Dic6—34no—
93C2×C3×C8—34no—
94C2×C3×C8—34no—
95C2×C3×C8—91no—
96C2×C3×C8—91no—
97C2×C3×C8C2×C3⋊D311yescharacteristic
98C2×C3×C8C2×C3⋊D311yescharacteristic
99C3×C4⋊C4—91no—
100C3×C4⋊C4—91no—
101C3×C4⋊C4—91no—
102C3×C42—34no—
103C3×C42—91no—
104C3×C42—91no—
105C3×C42C2×C3⋊D311yescharacteristic
106C4⋊Dic3—34no—
107Dic3⋊C4—34no—
108Dic3⋊C4—34no—
109C4×Dic3—34no—
110C4×Dic3—34no—
111C2×C3⋊C8—34no—
112C2×C3⋊C8—34no—
113C22×C32C2×C4×D314yes—
114C32×C4—24no—
115C32×C4—24no—
116C32×C4C2×Dic614yes—
117C32×C4#23514yes—
118C3×Dic3—38no—
119C3×Dic3—64no—
120C3×Dic3—64no—
121C3×Dic3—64no—
122C4×Q8—91no—
123C4⋊C8—91no—
124C4⋊C8—91no—
125C4⋊C8—91no—
126#97—91no—
127#97#27211yescharacteristic
128C4×C8—91no—
129H(3,𝔽3)#44611yescharacteristic, complemented, Sylow
130C3×Q8—182no—
131C2×C3×C4—34no—
132C2×C3×C4—34no—
133C2×C3×C4—34no—
134C2×C3×C4—91no—
135C2×C3×C4—91no—
136C2×C3×C4—91no—
137C2×C3×C4—91no—
138C2×C3×C4C4×C3⋊D311yescharacteristic
139C2×C3×C4C4×C3⋊D311yescharacteristic
140C2×C3×C4C22×C3⋊D311yescharacteristic
141C2×Dic3—34no—
142C2×Dic3—34no—
143C2×Dic3—34no—
144C2×Dic3—34no—
145Dic6—68no—
146C3×C8—21no—
147C3×C8—38no—
148C3×C8—64no—
149C3×C8—181no—
150C3×C8—181no—
151C3×C8#63412yes—
152C3⋊C8—64no—
153C3⋊C8—64no—
154C2×C32C4×Dic614yes—
155C2×C32D3×M1614yes—
156C2×C32#90714yes—
157C2×Q8—91no—
158C2×C8—91no—
159C2×C8—91no—
160C2×C8C2×#27211yescharacteristic
161C2×C8C2×#27211yescharacteristic
162C4⋊C4—91no—
163C4⋊C4—91no—
164C4⋊C4—91no—
165C42—91no—
166C42—91no—
167C42C2×#27211yescharacteristic
168C22×C3—34no—
169C22×C3C2×C4×C3⋊D311yescharacteristic
170C3×C4—21no—
171C3×C4—21no—
172C3×C4—34no—
173C3×C4—34no—
174C3×C4—64no—
175C3×C4—64no—
176C3×C4—92no—
177C3×C4—181no—
178C3×C4—181no—
179C3×C4—181no—
180C3×C4C2×#63411yescharacteristic
181C3×C4#382811yescharacteristic
182Dic3—38no—
183Dic3—64no—
184Dic3—64no—
185Dic3—64no—
186C32#470114yescomplemented
187Q8—182no—
188C2×C4—91no—
189C2×C4—91no—
190C2×C4—91no—
191C2×C4—91no—
192C2×C4C4×#27211yescharacteristic
193C2×C4C4×#27211yescharacteristic
194C2×C4C22×#27211yescharacteristic
195C8—21no—
196C8—181no—
197C8—181no—
198C8#622912yescomplemented
199C2×C3—34no—
200C2×C3—34no—
201C2×C3—34no—
202C2×C3C4×#63411yescharacteristic
203C2×C3M16×C3⋊D311yescharacteristic
204C2×C3#6411611yescharacteristic
205C22C2×C4×#27211yescharacteristic
206C4—21no—
207C4—21no—
208C4—92no—
209C4—181no—
210C4—181no—
211C4—181no—
212C4C2×#622911yescharacteristic
213C4#8874011yescharacteristic
214C3—34no—
215C3#1059240011yescharacteristic
216C2C4×#622911yescharacteristic
217C2M16×#27211yescharacteristic
218C2#1172178611yescharacteristic
subgroupG/H
center, FrattiniC2×C3×C4C22×C3⋊D3
commutator, prime radicalC22×H(3,𝔽3)C22×C4
FittingH(3,𝔽3)×#97C2
socleC22×C3C2×C4×C3⋊D3
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×#97C3⋊D3
hypocenterH(3,𝔽3)#446
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3×C4 ◁ C3×#97

Complements

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

NH ≅ G/Ntype
36. C8×H(3,𝔽3)Q8semidirect
84. C32×C8Dic6semidirect
129. H(3,𝔽3)#446semidirect
186. C32#4701semidirect
198. C8#6229semidirect

Maximal subgroups

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

subgroupindex
C4×#62292
H(3,𝔽3)×#972
#117205482
#117205472
C8×#10842
C4⋊#61802
C3×#47013