Finite Group Databasebeta

← #13143842

Subgroups — #13143842

Order 1728.

649 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 649 conjugacy classes of subgroups — too many to draw. Only its 141 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
2H(3,𝔽3)×C22⋊C8C211yesmaximal, characteristic
3#11720573C211yesmaximal, characteristic
4#11720547C212yesmaximal, complemented
5C8×#1084C212yesmaximal, complemented
6C3×#4718—34nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C22×#1084C411yescharacteristic
9C22×#1084C411yescharacteristic
10C2×C8×H(3,𝔽3)C2212yes—
11C4×#1084C411yescharacteristic
12C4×#1084C411yescharacteristic
13C4×#1084C2211yescharacteristic
14C4×#1084C2211yescharacteristic
15C2×#6180C2212yes—
16C2×C3×C4×Dic3—34no—
17C32×C22⋊C8D314yes—
18C3×#830—34no—
19C3×Dic3⋊C8—38no—
20C3×C8×Dic3—38no—
21C23×H(3,𝔽3)C2×C411yescharacteristic
22C2×C4×H(3,𝔽3)—22no—
23C2×C4×H(3,𝔽3)C2×C411yescharacteristic
24C2×C4×H(3,𝔽3)C2311yescharacteristic
25C2×#1084—22no—
26C2×#1084C814yescomplemented
27C2×#1084C2×C411yescharacteristic
28C2×#1084C2×C411yescharacteristic
29C2×#1084C2×C411yescharacteristic
30C2×#1084C2×C411yescharacteristic
31C8×H(3,𝔽3)—22no—
32#6180—22no—
33C3×#431—91no—
34#4718—34no—
35C22×C32×C4C2×D314yes—
36C22×C3×Dic3—34no—
37C22×C3×Dic3—34no—
38C2×C32×C8C2×D318yes—
39C3×C4×Dic3—34no—
40C3×C4×Dic3—34no—
41C3×C4×Dic3—34no—
42C3×C4×Dic3—34no—
43C2×C3×C3⋊C8—38no—
44C22×H(3,𝔽3)—21no—
45C22×H(3,𝔽3)—21no—
46C22×H(3,𝔽3)C2×C812yes—
47C22×H(3,𝔽3)C22×C411yescharacteristic
48C4×H(3,𝔽3)—21no—
49C4×H(3,𝔽3)#4112yes—
50#1084—21no—
51#1084C2×C812yes—
52#1084C2×C812yes—
53#1084M1612yes—
54C2×C3×C42—91no—
55C2×C4×Dic3—34no—
56C3×C4⋊C8—92no—
57C3×C22⋊C8—34no—
58C3×C22⋊C8—91no—
59C3×C22⋊C8C3⋊D311yescharacteristic
60C3×C4×C8—92no—
61#830—34no—
62Dic3⋊C8—38no—
63C8×Dic3—38no—
64C23×C32C4×D314yes—
65C2×C32×C4—28no—
66C2×C32×C4C4×D314yes—
67C2×C32×C4C22×D314yes—
68C2×C3×Dic3—316no—
69C2×C3×Dic3—34no—
70C2×C3×Dic3—34no—
71C2×C3×Dic3—34no—
72C2×C3×Dic3—34no—
73C2×C3×Dic3—68no—
74C32×C8—28no—
75C3×C3⋊C8—68no—
76#431—91noSylow
77C2×H(3,𝔽3)—22no—
78C2×H(3,𝔽3)#11711yescharacteristic
79C2×H(3,𝔽3)C22×C811yescharacteristic
80C2×H(3,𝔽3)C2×M1611yescharacteristic
81C22×C3×C4—34no—
82C22×C3×C4—91no—
83C22×C3×C4—91no—
84C22×C3×C4C2×C3⋊D311yescharacteristic
85C22×Dic3—34no—
86C22×Dic3—34no—
87C2×C3×C8—38no—
88C2×C3×C8—92no—
89C2×C3×C8C2×C3⋊D312yes—
90C3×C42—91no—
91C3×C42—91no—
92C3×C42—91no—
93C3×C42—91no—
94C4×Dic3—34no—
95C4×Dic3—34no—
96C4×Dic3—34no—
97C4×Dic3—34no—
98C2×C3⋊C8—38no—
99C22×C32—24no—
100C22×C32—24no—
101C22×C32D3×C818yes—
102C22×C32C2×C4×D314yes—
103C32×C4—24no—
104C32×C4#23718yes—
105C3×Dic3—38no—
106C3×Dic3—38no—
107C3×Dic3—38no—
108C3×Dic3—64no—
109C2×C42—91no—
110C4⋊C8—92no—
111C22⋊C8—91no—
112C22⋊C8#27211yescharacteristic
113C4×C8—92no—
114H(3,𝔽3)#43111yescharacteristic, complemented, Sylow
115C23×C3—34no—
116C23×C3C4×C3⋊D311yescharacteristic
117C2×C3×C4—22no—
118C2×C3×C4—34no—
119C2×C3×C4—34no—
120C2×C3×C4—68no—
121C2×C3×C4—94no—
122C2×C3×C4—91no—
123C2×C3×C4—91no—
124C2×C3×C4—91no—
125C2×C3×C4—91no—
126C2×C3×C4—182no—
127C2×C3×C4C4×C3⋊D311yescharacteristic
128C2×C3×C4C22×C3⋊D311yescharacteristic
129C2×Dic3—34no—
130C2×Dic3—34no—
131C2×Dic3—34no—
132C2×Dic3—316no—
133C2×Dic3—34no—
134C2×Dic3—68no—
135C3×C8—22no—
136C3×C8—68no—
137C3×C8—182no—
138C3⋊C8—68no—
139C2×C32—28no—
140C2×C32#87714yes—
141C2×C32C2×D3×C814yes—
142C2×C32D3×M1614yes—
143C22×C4—91no—
144C22×C4—91no—
145C22×C4C2×#27211yescharacteristic
146C2×C8—92no—
147C2×C8C2×#27212yes—
148C42—91no—
149C42—91no—
150C42—91no—
151C42—91no—
152C22×C3—21no—
153C22×C3—21no—
154C22×C3—34no—
155C22×C3—38no—
156C22×C3—64no—
157C22×C3—64no—
158C22×C3C8×C3⋊D312yes—
159C22×C3C2×C4×C3⋊D311yescharacteristic
160C3×C4—21no—
161C3×C4—38no—
162C3×C4—64no—
163C3×C4—92no—
164C3×C4—92no—
165C3×C4—92no—
166C3×C4—181no—
167C3×C4#383012yes—
168Dic3—38no—
169Dic3—38no—
170Dic3—38no—
171Dic3—64no—
172C32#471814yescomplemented
173C23C4×#27211yescharacteristic
174C2×C4—22no—
175C2×C4—91no—
176C2×C4—91no—
177C2×C4—91no—
178C2×C4—94no—
179C2×C4—91no—
180C2×C4—182no—
181C2×C4C4×#27211yescharacteristic
182C2×C4C22×#27211yescharacteristic
183C8—22no—
184C8—182no—
185C2×C3—22no—
186C2×C3—34no—
187C2×C3—34no—
188C2×C3—34no—
189C2×C3—68no—
190C2×C3#6408611yescharacteristic
191C2×C3C2×C8×C3⋊D311yescharacteristic
192C2×C3M16×C3⋊D311yescharacteristic
193C22—21no—
194C22—21no—
195C22C8×#27212yes—
196C22C2×C4×#27211yescharacteristic
197C4—21no—
198C4—92no—
199C4—92no—
200C4—92no—
201C4—181no—
202C4#8874412yes—
203C3—34no—
204C3#1059241711yescharacteristic
205C2—22no—
206C2#1172173311yescharacteristic
207C2C2×C8×#27211yescharacteristic
208C2M16×#27211yescharacteristic
subgroupG/H
center, FrattiniC2×C3×C4C22×C3⋊D3
commutatorC2×H(3,𝔽3)C22×C8
FittingH(3,𝔽3)×C22⋊C8C2
socleC22×C3C2×C4×C3⋊D3
prime radicalC23×H(3,𝔽3)C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×C22⋊C8C3⋊D3
hypocenterH(3,𝔽3)#431
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3×C4 ◁ C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
4. #11720547C2semidirect
5. C8×#1084C2semidirect
26. C2×#1084C8semidirect
114. H(3,𝔽3)#431semidirect
172. C32#4718semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#10842
H(3,𝔽3)×C22⋊C82
#117205732
#117205472
C8×#10842
C3×#47183