Finite Group Databasebeta

← #13143846

Subgroups — #13143846

Order 1728.

572 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 123 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×C4⋊#1084C211yesmaximal, characteristic
2#11721824C211yesmaximal, characteristic
3H(3,𝔽3)×C22⋊C8C211yesmaximal, characteristic
4#11720550C212yesmaximal, complemented
5#11720549C212yesmaximal, complemented
6C3×#4721—34nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C22×#1084—21no—
9C2×#6229C2211yescharacteristic
10C2×C8×H(3,𝔽3)C2212yes—
11#88552—21no—
12C4⋊#1084—21no—
13C4⋊#1084C2211yescharacteristic
14C4⋊#1084C2212yes—
15#88540—21no—
16C2×C3×C4⋊Dic3—34no—
17C3×#924—34no—
18C32×C22⋊C8D314yes—
19C3×#817—38no—
20C3×#816—38no—
21C23×H(3,𝔽3)D411yescharacteristic
22C2×C4×H(3,𝔽3)—22no—
23C2×C4×H(3,𝔽3)D411yescharacteristic
24C2×C4×H(3,𝔽3)C2311yescharacteristic
25C2×#1084—21no—
26C2×#1084—22no—
27C2×#1084—21no—
28C2×#1084—42no—
29#6229—22no—
30C8×H(3,𝔽3)—22no—
31C3×#483—91no—
32#4721—34no—
33C22×C32×C4C2×D314yes—
34C22×C3×Dic3—64no—
35C2×C3×Dic6—34no—
36C2×C32×C8C2×D318yes—
37C3×#217—64no—
38C3×C4⋊Dic3—34no—
39C3×C4⋊Dic3—38no—
40C3×C4⋊Dic3—64no—
41C3×Dic3⋊C4—64no—
42C22×H(3,𝔽3)—21no—
43C22×H(3,𝔽3)—21no—
44C22×H(3,𝔽3)SD1612yes—
45C22×H(3,𝔽3)C2×D411yescharacteristic
46C4×H(3,𝔽3)—21no—
47C4×H(3,𝔽3)#4112yes—
48#1084—41no—
49#1084—42no—
50#1084—41no—
51C3×C22⋊Q8—91no—
52C2×C3×C4⋊C4—91no—
53C2×C4⋊Dic3—34no—
54#924—34no—
55C3×#106—92no—
56C3×#103—92no—
57C3×C22⋊C8—34no—
58C3×C22⋊C8C3⋊D311yescharacteristic
59#817—38no—
60#816—38no—
61C23×C32D1214yes—
62C2×C32×C4—28no—
63C2×C32×C4D1214yes—
64C2×C32×C4C22×D314yes—
65C2×C3×Dic3—64no—
66C2×C3×Dic3—64no—
67C2×C3×Dic3—68no—
68C2×C3×Dic3—128no—
69C3×Dic6—68no—
70C32×C8—28no—
71#483—91noSylow
72C2×H(3,𝔽3)—22no—
73C2×H(3,𝔽3)#12311yescharacteristic
74C2×H(3,𝔽3)C2×SD1611yescharacteristic
75C2×H(3,𝔽3)#13711yescharacteristic
76C2×C3×Q8—91no—
77C22×C3×C4—34no—
78C22×C3×C4—181no—
79C22×C3×C4C2×C3⋊D311yescharacteristic
80C22×Dic3—64no—
81C2×Dic6—34no—
82C2×C3×C8—38no—
83C2×C3×C8C2×C3⋊D312yes—
84C3×C4⋊C4—92no—
85C3×C4⋊C4—91no—
86C3×C4⋊C4—181no—
87C3×C4⋊C4—181no—
88C3×C22⋊C4—181no—
89#217—64no—
90C4⋊Dic3—38no—
91C4⋊Dic3—34no—
92C4⋊Dic3—64no—
93Dic3⋊C4—64no—
94C22×C32—24no—
95C22×C32—24no—
96C22×C32#20418yes—
97C22×C32C2×D1214yes—
98C32×C4—24no—
99C32×C4#23718yes—
100C3×Dic3—124no—
101C3×Dic3—128no—
102C3×Dic3—124no—
103C22⋊Q8—91no—
104C2×C4⋊C4—91no—
105#106—92no—
106#103—92no—
107C22⋊C8#27211yescharacteristic
108H(3,𝔽3)#48311yescharacteristic, complemented, Sylow
109C23×C3—34no—
110C23×C3C4⋊(C3⋊D3)11yescharacteristic
111C3×Q8—182no—
112C2×C3×C4—22no—
113C2×C3×C4—34no—
114C2×C3×C4—34no—
115C2×C3×C4—68no—
116C2×C3×C4—181no—
117C2×C3×C4—181no—
118C2×C3×C4—182no—
119C2×C3×C4—362no—
120C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
121C2×C3×C4C22×C3⋊D311yescharacteristic
122C2×Dic3—64no—
123C2×Dic3—64no—
124C2×Dic3—68no—
125C2×Dic3—128no—
126Dic6—68no—
127C3×C8—22no—
128C3×C8—68no—
129C2×C32—28no—
130C2×C32#88614yes—
131C2×C32C2×#20414yes—
132C2×C32#90914yes—
133C2×Q8—91no—
134C22×C4—181no—
135C22×C4C2×#27211yescharacteristic
136C2×C8C2×#27212yes—
137C4⋊C4—92no—
138C4⋊C4—91no—
139C4⋊C4—181no—
140C4⋊C4—181no—
141C22⋊C4—181no—
142C22×C3—21no—
143C22×C3—21no—
144C22×C3—34no—
145C22×C3—38no—
146C22×C3—64no—
147C22×C3—64no—
148C22×C3#374412yes—
149C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
150C3×C4—21no—
151C3×C4—38no—
152C3×C4—64no—
153C3×C4—361no—
154C3×C4—362no—
155C3×C4—361no—
156C3×C4#383012yes—
157Dic3—124no—
158Dic3—128no—
159Dic3—124no—
160C32#472114yescomplemented
161C23C4⋊#27211yescharacteristic
162Q8—182no—
163C2×C4—22no—
164C2×C4—181no—
165C2×C4—181no—
166C2×C4—182no—
167C2×C4—362no—
168C2×C4C4⋊#27211yescharacteristic
169C2×C4C22×#27211yescharacteristic
170C8—22no—
171C2×C3—22no—
172C2×C3—34no—
173C2×C3—34no—
174C2×C3—34no—
175C2×C3—68no—
176C2×C3#6409511yescharacteristic
177C2×C3C2×#374411yescharacteristic
178C2×C3#6411811yescharacteristic
179C22—21no—
180C22—21no—
181C22#8852812yes—
182C22C2×C4⋊#27211yescharacteristic
183C4—21no—
184C4—361no—
185C4—362no—
186C4—361no—
187C4#8874412yes—
188C3—34no—
189C3#1059242011yescharacteristic
190C2—22no—
191C2#1172174211yescharacteristic
192C2C2×#8852811yescharacteristic
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)×C22⋊C8C2
prime radicalC23×H(3,𝔽3)D4
hypercenterC3×C22⋊C8C3⋊D3
hypocenterH(3,𝔽3)#483
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
4. #11720550C2semidirect
5. #11720549C2semidirect
108. H(3,𝔽3)#483semidirect
160. C32#4721semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10842
#117218242
H(3,𝔽3)×C22⋊C82
#117205502
#117205492
C3×#47213