Finite Group Databasebeta

← #13143892

Subgroups — #13143892

Order 1728.

781 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 781 conjugacy classes of subgroups — too many to draw. Only its 99 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
1#11722670C211yesmaximal, characteristic
2#11721861C211yesmaximal, characteristic, complemented
3#11721788C212yesmaximal, complemented
4H(3,𝔽3)×#101C211yesmaximal, characteristic, complemented
5#11720562C212yesmaximal, complemented
6C3×#4752—34nomaximal
7C2×Q8×H(3,𝔽3)C2211yescharacteristic
8#88748—22no—
9Q8×#272—22no—
10#88740—22no—
11#88740C2212yes—
12C2×#6229C2211yescharacteristic
13M16×H(3,𝔽3)C2212yes—
14#88540—22no—
15C4×#1084C2211yescharacteristic
16#88530—22no—
17#88528—22no—
18C3×#1007—34no—
19C3×#944—34no—
20C3×#909—38no—
21C32×#101D314yescomplemented
22C3×#825—38no—
23Q8×H(3,𝔽3)—22no—
24C2×C4×H(3,𝔽3)D412yes—
25C2×C4×H(3,𝔽3)C2311yescharacteristic
26#6235—22no—
27C2×#1084—21no—
28C2×#1084—21no—
29C4⋊#272—22no—
30C4⋊#272D412yes—
31C4×#272—24no—
32C4×#272—22no—
33#6229—22no—
34#6229—22no—
35#6229D412yes—
36C8×H(3,𝔽3)—22no—
37C3×#456—91no—
38#4752—34no—
39C2×C32×Q8C2×D314yes—
40C3×#239—68no—
41C3×D3×Q8—68no—
42C3×#235—38no—
43C3×#235—68no—
44C2×C3×Dic6—34no—
45C32×M16C2×D318yes—
46C3×Dic3⋊C4—68no—
47C3×C4×Dic3—34no—
48C3×Dic12—68no—
49C3×#204—68no—
50C22×H(3,𝔽3)C2×D411yescharacteristic
51C2×#272—22no—
52C4×H(3,𝔽3)—22no—
53C4×H(3,𝔽3)C2×D412yes—
54#1084—22no—
55#1084—22no—
56#1084—41no—
57C3×2−1+4—91no—
58#1007—34no—
59C3×#137—92no—
60C3×C4⋊Q8—91no—
61#944—34no—
62#909—38no—
63C3×#104—92no—
64C3×#101—34no—
65C3×#101C3⋊D311yescharacteristic
66#825—38no—
67C32×Q8—28no—
68C2×C32×C4D1218yes—
69C2×C32×C4C22×D314yes—
70C3×#67—68no—
71C2×C3×Dic3—64no—
72C2×C3×Dic3—64no—
73C3×D12—38no—
74C3×D12—68no—
75C3×C4×D3—616no—
76C3×C4×D3—68no—
77C3×Dic6—38no—
78C3×Dic6—68no—
79C3×Dic6—68no—
80C32×C8—28no—
81#456—91noSylow
82C2×H(3,𝔽3)—21no—
83C2×H(3,𝔽3)#12011yescharacteristic
84#272—42no—
85C3×#41—92no—
86C3×#41—182no—
87C3×#41—182no—
88C2×C3×Q8—34no—
89C2×C3×Q8—91no—
90C2×C3×Q8—182no—
91C2×C3×Q8C2×C3⋊D311yescharacteristic
92#239—68no—
93D3×Q8—68no—
94#235—38no—
95#235—68no—
96C2×Dic6—34no—
97C3×Q16—182no—
98C3×SD16—182no—
99C3×M16—38no—
100C3×M16C2×C3⋊D312yes—
101C3×C4⋊C4—182no—
102C3×C42—91no—
103Dic3⋊C4—68no—
104C4×Dic3—34no—
105Dic12—68no—
106#204—68no—
107C22×C32C2×D1214yes—
108C2×C3×D3—68no—
109C32×C4—28no—
110C32×C4D3×D418yes—
111C3×Dic3—68no—
112C3×Dic3—68no—
113C3×Dic3—124no—
1142−1+4—91no—
115#137—92no—
116C4⋊Q8—91no—
117#104—92no—
118#101#27211yescharacteristic, complemented
119H(3,𝔽3)#45611yescharacteristic, complemented, Sylow
120C3×Q8—22no—
121C3×Q8—68no—
122C3×Q8—92no—
123C3×Q8—182no—
124C3×Q8—182no—
125C3×D4—92no—
126C3×D4—182no—
127C3×D4—182no—
128C2×C3×C4—38no—
129C2×C3×C4—34no—
130C2×C3×C4—184no—
131C2×C3×C4—181no—
132C2×C3×C4—182no—
133C2×C3×C4—181no—
134C2×C3×C4C4⋊(C3⋊D3)12yes—
135C2×C3×C4C22×C3⋊D311yescharacteristic
136#67—68no—
137C2×Dic3—64no—
138C2×Dic3—64no—
139D12—38no—
140D12—68no—
141C4×D3—68no—
142C4×D3—616no—
143Dic6—38no—
144Dic6—68no—
145Dic6—68no—
146C3×C8—22no—
147C3×C8—68no—
148C2×C32—24no—
149C2×C32#88214yes—
150C3×D3—128no—
151#41—92no—
152#41—182no—
153#41—182no—
154C2×Q8—91no—
155C2×Q8—182no—
156C2×Q8C2×#27211yescharacteristic
157Q16—182no—
158SD16—182no—
159M16C2×#27212yes—
160C4⋊C4—182no—
161C42—91no—
162C22×C3—34no—
163C22×C3—182no—
164C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
165C2×D3—68no—
166C3×C4—22no—
167C3×C4—38no—
168C3×C4—68no—
169C3×C4—182no—
170C3×C4—182no—
171C3×C4—361no—
172C3×C4D4×C3⋊D312yes—
173Dic3—68no—
174Dic3—68no—
175Dic3—124no—
176C32#475214yescomplemented
177Q8—22no—
178Q8—92no—
179Q8—182no—
180Q8—182no—
181D4—92no—
182D4—182no—
183D4—182no—
184C2×C4—184no—
185C2×C4—181no—
186C2×C4—182no—
187C2×C4—181no—
188C2×C4C4⋊#27212yes—
189C2×C4C22×#27211yescharacteristic
190C8—22no—
191C2×C3—21no—
192C2×C3—34no—
193C2×C3—64no—
194C2×C3—362no—
195C2×C3#6409111yescharacteristic
196D3—128no—
197C22—182no—
198C22C2×C4⋊#27211yescharacteristic
199C4—22no—
200C4—182no—
201C4—182no—
202C4—361no—
203C4D4×#27212yes—
204C3—34no—
205C3#1059245111yescharacteristic
206C2—21no—
207C2—362no—
208C2#1172173811yescharacteristic
subgroupG/H
center, socleC2×C3#64091
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#101C2
prime radical#11722670C2
hypercenterC3×#101C3⋊D3
hypocenterH(3,𝔽3)#456
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#101

Complements

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

NH ≅ G/Ntype
2. #11721861C2semidirect
3. #11721788C2semidirect
4. H(3,𝔽3)×#101C2semidirect
5. #11720562C2semidirect
21. C32×#101D3semidirect
118. #101#272semidirect
119. H(3,𝔽3)#456semidirect
176. C32#4752semidirect

Maximal subgroups

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

subgroupindex
#117226702
#117218612
#117217882
H(3,𝔽3)×#1012
#117205622
C3×#47523