Finite Group Databasebeta

← #13135175

Subgroups — #13135175

Order 1728.

397 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 101 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×#88538C211yesmaximal, characteristic
2#64135⋊C3C212yesmaximal
3#10589158—32nomaximal
4C22×C4×H(3,𝔽3)C2211yescharacteristic
5#88538—22no—
6C2×#6180C412yescomplemented
7#3757⋊C3—22no—
8C4⋊((C3⋊Dic3)⋊C3)—22no—
9C4×(C3⋊Dic3)⋊C3C414yescomplemented
10#64135D312yes—
11C3×#926—32no—
12C2×C3×#208—32no—
13C2×C3×#208—61no—
14C23×H(3,𝔽3)D411yescharacteristic
15C2×C4×H(3,𝔽3)C2×C411yescharacteristic
16C2×C4×H(3,𝔽3)C2×C412yes—
17C2×C4×H(3,𝔽3)D412yes—
18C2×C4×H(3,𝔽3)Q811yescharacteristic
19C2×(C3⋊Dic3)⋊C3—24no—
20#6180—22no—
21#4557—91no—
22#4531—92no—
23C22×C32×C4—21no—
24C22×C32×C4C2×D312yes—
25#3757—22no—
26C4⋊(C3⋊Dic3)—22no—
27C4×C3⋊Dic3Dic314yescomplemented
28C3×#217—62no—
29C3×C4⋊Dic3—62no—
30C3×C4×Dic3—34no—
31C3×#208—64no—
32C3×#208—62no—
33C3×#208—62no—
34C2×C3×C3⋊C8—34no—
35C2×C3×C3⋊C8—62no—
36C22×H(3,𝔽3)—41no—
37C22×H(3,𝔽3)C22⋊C412yes—
38C22×H(3,𝔽3)C4⋊C411yescharacteristic
39C4×H(3,𝔽3)C4211yescharacteristic
40C4×H(3,𝔽3)C22⋊C411yescharacteristic
41C4×H(3,𝔽3)C4⋊C412yes—
42(C3⋊Dic3)⋊C3—44no—
43C2×C3×M16—91no—
44C3×#117—92no—
45#926—32no—
46#926—92no—
47C2×#208—32no—
48C2×#208—61no—
49C23×C32—21no—
50C23×C32#6712yes—
51C2×C32×C4—22no—
52C2×C32×C4—21no—
53C2×C32×C4—22no—
54C2×C32×C4—21no—
55C2×C32×C4Dic612yes—
56C2×C32×C4C4×D312yes—
57C2×C32×C4C4×D312yes—
58C2×C32×C4D1212yes—
59C2×C32×C4C2×Dic312yes—
60C2×C32×C4#6712yes—
61C2×C3⋊Dic3—24no—
62C2×C3×Dic3—64no—
63C3×C3⋊C8—64no—
64C3×C3⋊C8—64no—
65H(3,ℤ/4)—271noSylow
66C2×H(3,𝔽3)—21no—
67C2×H(3,𝔽3)—22no—
68C2×H(3,𝔽3)#9511yescharacteristic
69C22×C3×C4—32no—
70C22×C3×C4—61no—
71C22×C3×C4D3211yescharacteristic
72C3×M16—182no—
73C2×C3×C8—92no—
74C3×C4⋊C4—182no—
75C3×C22⋊C4—182no—
76C3×C42—94no—
77#217—62no—
78#217—182no—
79C4⋊Dic3—62no—
80C4⋊Dic3—182no—
81C4×Dic3—34no—
82C4×Dic3—94no—
83#208—62no—
84#208—62no—
85#208—64no—
86C2×C3⋊C8—34no—
87C2×C3⋊C8—62no—
88C22×C32—22no—
89C22×C32—21no—
90C22×C32—42no—
91C22×C32—42no—
92C22×C32Dic3⋊C412yes—
93C22×C32(C2×D3)⋊C412yes—
94C22×C32#21712yes—
95C32×C4—21no—
96C32×C4—21no—
97C32×C4—22no—
98C32×C4C4×Dic312yes—
99C32×C4Dic3⋊C412yes—
100C32×C4C4⋊Dic312yes—
101C32×C4(C2×D3)⋊C412yes—
102C3⋊Dic3—44no—
103C3×Dic3—124no—
104C2×M16—91no—
105#117—92no—
106H(3,𝔽3)H(3,ℤ/4)11yescharacteristic, complemented, Sylow
107C23×C3—32no—
108C23×C3—61no—
109C23×C3#62511yescharacteristic
110C2×C3×C4—32no—
111C2×C3×C4—32no—
112C2×C3×C4—32no—
113C2×C3×C4—32no—
114C2×C3×C4—32no—
115C2×C3×C4—32no—
116C2×C3×C4—61no—
117C2×C3×C4—62no—
118C2×C3×C4—61no—
119C2×C3×C4—62no—
120C2×C3×C4—184no—
121C2×C3×C4D3×Dic312yes—
122C2×C3×C4#62411yescharacteristic
123C2×C3×C4#62612yes—
124C2×C3×C4#62711yescharacteristic
125C2×Dic3—64no—
126C2×Dic3—184no—
127C3×C8—182no—
128C3⋊C8—64no—
129C3⋊C8—64no—
130C2×C32—22no—
131C2×C32—22no—
132C2×C32—22no—
133C2×C32—22no—
134C2×C32—21no—
135C2×C32—42no—
136C2×C32C3⋊#9512yes—
137C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
138M16—182no—
139C2×C8—92no—
140C4⋊C4—182no—
141C22⋊C4—182no—
142C42—94no—
143C22×C3—32no—
144C22×C3—32no—
145C22×C3—32no—
146C22×C3—41no—
147C22×C3—61no—
148C22×C3—62no—
149C22×C3—122no—
150C22×C3—122no—
151C22×C3(C2×D3)⋊Dic312yes—
152C22×C3#372411yescharacteristic
153C3×C4—32no—
154C3×C4—32no—
155C3×C4—32no—
156C3×C4—32no—
157C3×C4—61no—
158C3×C4—62no—
159C3×C4—61no—
160C3×C4—364no—
161C3×C4Dic3211yescharacteristic
162C3×C4#372211yescharacteristic
163C3×C4Dic3⋊Dic312yes—
164Dic3—124no—
165Dic3—364no—
166C32—21no—
167C32#453112yescomplemented
168C23#619811yescharacteristic
169C2×C4—184no—
170C2×C4#619611yescharacteristic
171C2×C4((C3⋊D3)⋊C3)⋊C412yes—
172C2×C4#619911yescharacteristic
173C2×C4#620012yes—
174C8—182no—
175C2×C3—22no—
176C2×C3—21no—
177C2×C3—32no—
178C2×C3—62no—
179C2×C3—62no—
180C2×C3—62no—
181C2×C3—62no—
182C2×C3—61no—
183C2×C3—122no—
184C2×C3#6357911yescharacteristic
185C22—41no—
186C22#8844911yescharacteristic
187C22#8845012yes—
188C4—364no—
189C4#8844711yescharacteristic
190C4#8844812yes—
191C4#8845111yescharacteristic
192C3—32no—
193C3—61no—
194C3#1058915711yescharacteristic
195C2—22no—
196C2—21no—
197C2#1172033411yescharacteristic
subgroupG/H
centerC4#88447
commutatorC4×H(3,𝔽3)C42
FrattiniC22×C3×C4D32
Fitting, cosocleC22×C4×H(3,𝔽3)C22
socleC2×C3#63579
prime radicalC23×H(3,𝔽3)D4
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)H(3,ℤ/4)
LCSG ▷ C4×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
6. C2×#6180C4semidirect
9. C4×(C3⋊Dic3)⋊C3C4semidirect
27. C4×C3⋊Dic3Dic3semidirect
106. H(3,𝔽3)H(3,ℤ/4)semidirect
167. C32#4531semidirect

Maximal subgroups

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

subgroupindex
C2×#885382
#64135⋊C32
#105891583