Finite Group Databasebeta

← #13143863

Subgroups — #13143863

Order 1728.

814 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C22×#6229C211yesmaximal, characteristic
2#11721824C211yesmaximal, characteristic
3C2×#88530C212yesmaximal, complemented
4H(3,𝔽3)×C22⋊C8C211yesmaximal, characteristic
5#11720549C212yesmaximal, complemented
6C3×#4739—34nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C22×#1084—21no—
9C2×#6229—21no—
10C2×#6229—24no—
11C2×#6229C2211yescharacteristic
12C2×#6229C2212yes—
13C2×C8×H(3,𝔽3)C2212yes—
14#88552—21no—
15C4⋊#1084C2211yescharacteristic
16#88540—21no—
17#88530—24no—
18C22×C3×Dic6—34no—
19C3×#924—34no—
20C2×C3×Dic12—38no—
21C32×C22⋊C8D314yes—
22C3×#816—38no—
23C23×H(3,𝔽3)D411yescharacteristic
24C2×C4×H(3,𝔽3)—22no—
25C2×C4×H(3,𝔽3)D411yescharacteristic
26C2×C4×H(3,𝔽3)C2311yescharacteristic
27C2×#1084—24no—
28C2×#1084—22no—
29C2×#1084—21no—
30C2×#1084—21no—
31#6229—24no—
32#6229—22no—
33#6229—22no—
34#6229D414yes—
35C8×H(3,𝔽3)—22no—
36C3×#451—91no—
37#4739—34no—
38C22×C32×C4C2×D314yes—
39C22×C3×Dic3—64no—
40C2×C3×Dic6—38no—
41C2×C3×Dic6—34no—
42C2×C3×Dic6—616no—
43C2×C3×Dic6—64no—
44C2×C32×C8C2×D318yes—
45C3×#217—64no—
46C3×C4⋊Dic3—34no—
47C3×Dic3⋊C4—64no—
48C3×Dic12—616no—
49C22×H(3,𝔽3)—21no—
50C22×H(3,𝔽3)—21no—
51C22×H(3,𝔽3)Q1612yescomplemented
52C22×H(3,𝔽3)C2×D411yescharacteristic
53C4×H(3,𝔽3)—21no—
54C4×H(3,𝔽3)C2×D412yes—
55#1084—24no—
56#1084—41no—
57#1084—41no—
58C22×C3×Q8—91no—
59C22×Dic6—34no—
60C2×C3×Q16—92no—
61C3×C22⋊Q8—91no—
62#924—34no—
63C2×Dic12—38no—
64C3×#103—92no—
65C3×C22⋊C8—34no—
66C3×C22⋊C8C3⋊D311yescharacteristic
67#816—38no—
68C23×C32D1214yes—
69C2×C32×C4—28no—
70C2×C32×C4D1214yes—
71C2×C32×C4C22×D314yes—
72C2×C3×Dic3—616no—
73C2×C3×Dic3—68no—
74C2×C3×Dic3—64no—
75C2×C3×Dic3—64no—
76C3×Dic6—316no—
77C3×Dic6—616no—
78C3×Dic6—68no—
79C3×Dic6—68no—
80C32×C8—28no—
81#451—91noSylow
82C2×H(3,𝔽3)—22no—
83C2×H(3,𝔽3)#12011yescharacteristic
84C2×H(3,𝔽3)C2×Q1611yescharacteristic
85C2×H(3,𝔽3)#13711yescharacteristic
86C2×C3×Q8—92no—
87C2×C3×Q8—91no—
88C2×C3×Q8—184no—
89C2×C3×Q8—181no—
90C22×C3×C4—34no—
91C22×C3×C4—181no—
92C22×C3×C4C2×C3⋊D311yescharacteristic
93C22×Dic3—64no—
94C2×Dic6—38no—
95C2×Dic6—34no—
96C2×Dic6—616no—
97C2×Dic6—64no—
98C3×Q16—184no—
99C2×C3×C8—38no—
100C2×C3×C8C2×C3⋊D312yes—
101C3×C4⋊C4—91no—
102C3×C4⋊C4—181no—
103C3×C22⋊C4—181no—
104#217—64no—
105C4⋊Dic3—34no—
106Dic3⋊C4—64no—
107Dic12—616no—
108C22×C32—24no—
109C22×C32—24no—
110C22×C32Dic1218yescomplemented
111C22×C32C2×D1214yes—
112C32×C4—24no—
113C32×C4D3×D418yes—
114C3×Dic3—616no—
115C3×Dic3—124no—
116C3×Dic3—124no—
117C22×Q8—91no—
118C2×Q16—92no—
119C22⋊Q8—91no—
120#103—92no—
121C22⋊C8#27211yescharacteristic
122H(3,𝔽3)#45111yescharacteristic, complemented, Sylow
123C23×C3—34no—
124C23×C3C4⋊(C3⋊D3)11yescharacteristic
125C3×Q8—94no—
126C3×Q8—184no—
127C3×Q8—182no—
128C3×Q8—182no—
129C2×C3×C4—22no—
130C2×C3×C4—34no—
131C2×C3×C4—34no—
132C2×C3×C4—68no—
133C2×C3×C4—184no—
134C2×C3×C4—182no—
135C2×C3×C4—181no—
136C2×C3×C4—181no—
137C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
138C2×C3×C4C22×C3⋊D311yescharacteristic
139C2×Dic3—68no—
140C2×Dic3—616no—
141C2×Dic3—64no—
142C2×Dic3—64no—
143Dic6—316no—
144Dic6—68no—
145Dic6—616no—
146Dic6—68no—
147C3×C8—22no—
148C3×C8—68no—
149C2×C32—28no—
150C2×C32#88214yes—
151C2×C32C2×Dic1214yes—
152C2×C32#90914yes—
153C2×Q8—92no—
154C2×Q8—91no—
155C2×Q8—184no—
156C2×Q8—181no—
157C22×C4—181no—
158C22×C4C2×#27211yescharacteristic
159Q16—184no—
160C2×C8C2×#27212yes—
161C4⋊C4—91no—
162C4⋊C4—181no—
163C22⋊C4—181no—
164C22×C3—21no—
165C22×C3—21no—
166C22×C3—34no—
167C22×C3—38no—
168C22×C3—64no—
169C22×C3—64no—
170C22×C3#374612yes—
171C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
172C3×C4—21no—
173C3×C4—38no—
174C3×C4—64no—
175C3×C4—184no—
176C3×C4—361no—
177C3×C4—361no—
178C3×C4D4×C3⋊D312yes—
179Dic3—616no—
180Dic3—124no—
181Dic3—124no—
182C32#473914yescomplemented
183C23C4⋊#27211yescharacteristic
184Q8—94no—
185Q8—184no—
186Q8—182no—
187Q8—182no—
188C2×C4—22no—
189C2×C4—184no—
190C2×C4—182no—
191C2×C4—181no—
192C2×C4—181no—
193C2×C4C4⋊#27211yescharacteristic
194C2×C4C22×#27211yescharacteristic
195C8—22no—
196C2×C3—22no—
197C2×C3—34no—
198C2×C3—34no—
199C2×C3—34no—
200C2×C3—68no—
201C2×C3#6409111yescharacteristic
202C2×C3C2×#374611yescharacteristic
203C2×C3#6411811yescharacteristic
204C22—21no—
205C22—21no—
206C22#8853012yescomplemented
207C22C2×C4⋊#27211yescharacteristic
208C4—21no—
209C4—184no—
210C4—361no—
211C4—361no—
212C4D4×#27212yes—
213C3—34no—
214C3#1059243811yescharacteristic
215C2—22no—
216C2#1172173811yescharacteristic
217C2C2×#8853011yescharacteristic
218C2#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)#451
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
3. C2×#88530C2semidirect
5. #11720549C2semidirect
51. C22×H(3,𝔽3)Q16semidirect
110. C22×C32Dic12semidirect
122. H(3,𝔽3)#451semidirect
182. C32#4739semidirect
206. C22#88530semidirect

Maximal subgroups

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

subgroupindex
C22×#62292
#117218242
C2×#885302
H(3,𝔽3)×C22⋊C82
#117205492
C3×#47393