Finite Group Databasebeta

← #13144087

Subgroups — #13144087

Order 1728.

594 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 125 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×#88530C212yesmaximal
2#11721715C211yesmaximal, characteristic
3C4×#6229C211yesmaximal, characteristic
4H(3,𝔽3)×C4⋊C8C211yesmaximal, characteristic
5#11720549C212yesmaximal
6C3×#4849—34nomaximal
7C2×#6229C2212yes—
8C2×#6229C2211yescharacteristic
9C2×C8×H(3,𝔽3)C2212yes—
10C42×H(3,𝔽3)C2211yescharacteristic
11C4⋊#1084—21no—
12C4⋊#1084—21no—
13C4⋊#1084C2211yescharacteristic
14#88540—21no—
15C4×#1084—21no—
16#88530—24no—
17C2×C3×Dic12—38no—
18C3×#869—34no—
19C3×C4×Dic6—34no—
20C32×C4⋊C8D314yes—
21C3×#816—38no—
22C2×C4×H(3,𝔽3)D411yescharacteristic
23C2×C4×H(3,𝔽3)D411yescharacteristic
24C2×C4×H(3,𝔽3)C2311yescharacteristic
25C2×#1084—21no—
26C2×#1084—22no—
27C2×#1084—21no—
28#6229—21no—
29#6229—22no—
30#6229—22no—
31#6229D412yes—
32C8×H(3,𝔽3)—22no—
33C3×#462—91no—
34#4849—34no—
35C2×C3×Dic6—34no—
36C2×C3×Dic6—38no—
37C2×C32×C8C2×D318yes—
38C32×C42C2×D314yes—
39C3×C4⋊Dic3—34no—
40C3×C4⋊Dic3—64no—
41C3×C4⋊Dic3—64no—
42C3×Dic3⋊C4—64no—
43C3×C4×Dic3—64no—
44C3×Dic12—616no—
45C22×H(3,𝔽3)C2×D411yescharacteristic
46C4×H(3,𝔽3)—21no—
47C4×H(3,𝔽3)Q1612yescomplemented
48C4×H(3,𝔽3)C2×D411yescharacteristic
49C4×H(3,𝔽3)#4111yescharacteristic
50#1084—22no—
51#1084—42no—
52#1084—41no—
53C2×C3×Q16—92no—
54C3×C4⋊Q8—91no—
55C3×C4×Q8—91no—
56C2×Dic12—38no—
57#869—34no—
58C4×Dic6—34no—
59C3×C4⋊C8—34no—
60C3×C4⋊C8C3⋊D311yescharacteristic
61C3×#103—92no—
62#816—38no—
63C2×C32×C4D1214yes—
64C2×C32×C4D1214yes—
65C2×C32×C4C22×D314yes—
66C2×C3×Dic3—64no—
67C2×C3×Dic3—64no—
68C2×C3×Dic3—68no—
69C3×Dic6—38no—
70C3×Dic6—64no—
71C3×Dic6—68no—
72C3×Dic6—68no—
73C32×C8—28no—
74#462—91noSylow
75C2×H(3,𝔽3)#12111yescharacteristic
76C2×H(3,𝔽3)C2×Q1611yescharacteristic
77C2×H(3,𝔽3)#13711yescharacteristic
78C2×C3×Q8—91no—
79C2×C3×Q8—92no—
80C2×Dic6—34no—
81C2×Dic6—38no—
82C3×Q16—184no—
83C2×C3×C8—38no—
84C2×C3×C8C2×C3⋊D312yes—
85C3×C4⋊C4—91no—
86C3×C4⋊C4—181no—
87C3×C4⋊C4—181no—
88C3×C4⋊C4—181no—
89C3×C42—34no—
90C3×C42—181no—
91C3×C42C2×C3⋊D311yescharacteristic
92C4⋊Dic3—34no—
93C4⋊Dic3—64no—
94C4⋊Dic3—64no—
95Dic3⋊C4—64no—
96C4×Dic3—64no—
97Dic12—616no—
98C22×C32C2×D1214yes—
99C32×C4—24no—
100C32×C4Dic1218yescomplemented
101C32×C4D3×D414yes—
102C32×C4#23914yes—
103C3×Dic3—68no—
104C3×Dic3—128no—
105C3×Dic3—124no—
106C2×Q16—92no—
107C4⋊Q8—91no—
108C4×Q8—91no—
109C4⋊C8#27211yescharacteristic
110#103—92no—
111H(3,𝔽3)#46211yescharacteristic, complemented, Sylow
112C3×Q8—92no—
113C3×Q8—181no—
114C3×Q8—182no—
115C3×Q8—182no—
116C2×C3×C4—34no—
117C2×C3×C4—34no—
118C2×C3×C4—34no—
119C2×C3×C4—181no—
120C2×C3×C4—181no—
121C2×C3×C4—182no—
122C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
123C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
124C2×C3×C4C22×C3⋊D311yescharacteristic
125C2×Dic3—64no—
126C2×Dic3—64no—
127C2×Dic3—68no—
128Dic6—38no—
129Dic6—64no—
130Dic6—68no—
131Dic6—68no—
132C3×C8—22no—
133C3×C8—68no—
134C2×C32#89514yes—
135C2×C32C2×Dic1214yes—
136C2×C32#90914yes—
137C2×Q8—91no—
138C2×Q8—92no—
139Q16—184no—
140C2×C8C2×#27212yes—
141C4⋊C4—91no—
142C4⋊C4—181no—
143C4⋊C4—181no—
144C4⋊C4—181no—
145C42—181no—
146C42C2×#27211yescharacteristic
147C22×C3—34no—
148C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
149C3×C4—21no—
150C3×C4—34no—
151C3×C4—34no—
152C3×C4—38no—
153C3×C4—64no—
154C3×C4—182no—
155C3×C4—362no—
156C3×C4—361no—
157C3×C4#374612yes—
158C3×C4D4×C3⋊D311yescharacteristic
159C3×C4#383211yescharacteristic
160Dic3—68no—
161Dic3—128no—
162Dic3—124no—
163C32#484914yescomplemented
164Q8—92no—
165Q8—181no—
166Q8—182no—
167Q8—182no—
168C2×C4—181no—
169C2×C4—181no—
170C2×C4—182no—
171C2×C4C4⋊#27211yescharacteristic
172C2×C4C4⋊#27211yescharacteristic
173C2×C4C22×#27211yescharacteristic
174C8—22no—
175C2×C3—34no—
176C2×C3—34no—
177C2×C3—34no—
178C2×C3#6410411yescharacteristic
179C2×C3C2×#374611yescharacteristic
180C2×C3#6411811yescharacteristic
181C22C2×C4⋊#27211yescharacteristic
182C4—21no—
183C4—182no—
184C4—362no—
185C4—361no—
186C4#8853012yescomplemented
187C4D4×#27211yescharacteristic
188C4#8874811yescharacteristic
189C3—34no—
190C3#1059254811yescharacteristic
191C2#1172176311yescharacteristic
192C2C2×#8853011yescharacteristic
193C2#1172178811yescharacteristic
subgroupG/H
center, socleC22×C3C2×C4⋊(C3⋊D3)
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×C4⋊C8C2
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC3×C4⋊C8C3⋊D3
hypocenterH(3,𝔽3)#462
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4⋊C8

Complements

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

NH ≅ G/Ntype
47. C4×H(3,𝔽3)Q16semidirect
100. C32×C4Dic12semidirect
111. H(3,𝔽3)#462semidirect
163. C32#4849semidirect
186. C4#88530semidirect

Maximal subgroups

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

subgroupindex
C2×#885302
#117217152
C4×#62292
H(3,𝔽3)×C4⋊C82
#117205492
C3×#48493