Finite Group Databasebeta

← #13139929

Subgroups — #13139929

Order 1728.

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

normal subgroups only

This group has 549 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
1C2×C4×#1084C211yesmaximal, characteristic
2C2×#3748⋊C3C212yesmaximal
3#11720304C214yesmaximal, complemented
4#10591115—32nomaximal
5C22×C4×H(3,𝔽3)C2211yescharacteristic
6C22×#1084C412yes—
7C4×#1084C411yescharacteristic
8C4×#1084C411yescharacteristic
9C4×#1084C2212yes—
10#3748⋊C3—24no—
11C2×(C3⋊(C3⋊C8))⋊C3C2214yes—
12C2×#3748D312yes—
13C2×C3×C4×Dic3—32no—
14C2×C3×C4×Dic3—61no—
15C2×C3×#208—32no—
16Dic3⋊(C3⋊C8)—38no—
17C23×H(3,𝔽3)C2×C411yescharacteristic
18C2×C4×H(3,𝔽3)C2×C411yescharacteristic
19C2×C4×H(3,𝔽3)D412yes—
20C2×C4×H(3,𝔽3)Q812yes—
21C2×C4×H(3,𝔽3)C2311yescharacteristic
22C2×#1084—24no—
23C2×#1084C2×C412yes—
24C2×#1084C2×C412yes—
25(C3⋊(C3⋊C8))⋊C3—24no—
26#5118—92no—
27#4924—91no—
28C22×C32×C4—21no—
29C22×C32×C4C2×D312yes—
30C22×C3×Dic3—34no—
31C22×C3×Dic3—62no—
32#3748—24no—
33C2×C3⋊(C3⋊C8)C2×D314yes—
34C3×C4×Dic3—32no—
35C3×C4×Dic3—32no—
36C3×C4×Dic3—34no—
37C3×C4×Dic3—61no—
38C3×C4×Dic3—61no—
39C3×C4×Dic3—62no—
40C3×#208—64no—
41C2×C3×C3⋊C8—34no—
42C22×H(3,𝔽3)—22no—
43C22×H(3,𝔽3)C4⋊C412yes—
44C22×H(3,𝔽3)C22×C411yescharacteristic
45C4×H(3,𝔽3)C4⋊C412yes—
46C4×H(3,𝔽3)C2×D411yescharacteristic
47C4×H(3,𝔽3)C2×Q811yescharacteristic
48#1084—22no—
49#1084M1614yescomplemented
50C2×C3×M16—92no—
51C2×C3×C42—91no—
52C2×C4×Dic3—32no—
53C2×C4×Dic3—61no—
54C2×#208—32no—
55C2×#208—92no—
56Dic3⋊C8—98no—
57C4⋊(C3⋊C8)—94no—
58C23×C32—21no—
59C23×C32C4×D312yes—
60C2×C32×C4—21no—
61C2×C32×C4—22no—
62C2×C32×C4—22no—
63C2×C32×C4—21no—
64C2×C32×C4Dic614yes—
65C2×C32×C4C4×D312yes—
66C2×C32×C4#6714yes—
67C2×C32×C4C22×D312yes—
68C2×C3×Dic3—34no—
69C2×C3×Dic3—34no—
70C2×C3×Dic3—62no—
71C2×C3×Dic3—62no—
72C2×C3×Dic3—68no—
73C2×C3×Dic3—68no—
74C3⋊(C3⋊C8)—24no—
75C3×C3⋊C8—64no—
76#423—271noSylow
77C2×H(3,𝔽3)—22no—
78C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
79C2×H(3,𝔽3)C2×M1612yes—
80C22×C3×C4—32no—
81C22×C3×C4—61no—
82C22×C3×C4—92no—
83C22×C3×C4D3211yescharacteristic
84C22×Dic3—34no—
85C22×Dic3—62no—
86C3×M16—184no—
87C2×C3×C8—94no—
88C3×C42—91no—
89C3×C42—91no—
90C3×C42—92no—
91C4×Dic3—34no—
92C4×Dic3—32no—
93C4×Dic3—32no—
94C4×Dic3—61no—
95C4×Dic3—61no—
96C4×Dic3—62no—
97#208—64no—
98#208—184no—
99C2×C3⋊C8—34no—
100C2×C3⋊C8—94no—
101C22×C32—24no—
102C22×C32—24no—
103C22×C32—22no—
104C22×C32—21no—
105C22×C32Dic3⋊C414yes—
106C22×C32C2×C4×D312yes—
107C32×C4—22no—
108C32×C4—21no—
109C32×C4—21no—
110C32×C4Dic3⋊C414yes—
111C32×C4C2×Dic612yes—
112C32×C4C2×#6712yes—
113C3×Dic3—38no—
114C3×Dic3—64no—
115C3×Dic3—64no—
116C3×Dic3—64no—
117C2×M16—92no—
118C2×C42—91no—
119C4⋊C8—274no—
120H(3,𝔽3)#42311yescharacteristic, complemented, Sylow
121C23×C3—32no—
122C23×C3—61no—
123C23×C3#62411yescharacteristic
124C2×C3×C4—34no—
125C2×C3×C4—34no—
126C2×C3×C4—32no—
127C2×C3×C4—32no—
128C2×C3×C4—62no—
129C2×C3×C4—62no—
130C2×C3×C4—61no—
131C2×C3×C4—61no—
132C2×C3×C4—92no—
133C2×C3×C4—92no—
134C2×C3×C4—184no—
135C2×C3×C4#62411yescharacteristic
136C2×C3×C4#62512yes—
137C2×C3×C4#62712yes—
138C2×C3×C4C2×D3211yescharacteristic
139C2×Dic3—34no—
140C2×Dic3—34no—
141C2×Dic3—62no—
142C2×Dic3—62no—
143C2×Dic3—68no—
144C2×Dic3—68no—
145C3×C8—184no—
146C3⋊C8—64no—
147C3⋊C8—184no—
148C2×C32—24no—
149C2×C32—24no—
150C2×C32—22no—
151C2×C32—21no—
152C2×C32D3×M1614yes—
153C2×C32C2×Dic3⋊C412yes—
154C22×C4—92no—
155C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
156M16—184no—
157C2×C8—94no—
158C42—91no—
159C42—91no—
160C42—92no—
161C22×C3—22no—
162C22×C3—32no—
163C22×C3—34no—
164C22×C3—64no—
165C22×C3—64no—
166C22×C3—61no—
167C22×C3—62no—
168C22×C3#372412yes—
169C22×C3C2×#62411yescharacteristic
170C3×C4—32no—
171C3×C4—32no—
172C3×C4—34no—
173C3×C4—61no—
174C3×C4—61no—
175C3×C4—62no—
176C3×C4—94no—
177C3×C4—182no—
178C3×C4#372412yes—
179C3×C4C2×#62511yescharacteristic
180C3×C4C2×#62711yescharacteristic
181Dic3—38no—
182Dic3—64no—
183Dic3—64no—
184Dic3—64no—
185C32—21no—
186C32#511812yescomplemented
187C23#619911yescharacteristic
188C2×C4—92no—
189C2×C4—92no—
190C2×C4—184no—
191C2×C4#619612yes—
192C2×C4#619812yes—
193C2×C4#619911yescharacteristic
194C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
195C8—184no—
196C2×C3—22no—
197C2×C3—32no—
198C2×C3—34no—
199C2×C3—64no—
200C2×C3—64no—
201C2×C3—61no—
202C2×C3—62no—
203C2×C3#6381812yes—
204C2×C3C2×#372411yescharacteristic
205C22—22no—
206C22#8844912yes—
207C22C2×#619911yescharacteristic
208C4—94no—
209C4—182no—
210C4#8844912yes—
211C4C2×#619611yescharacteristic
212C4C2×#619811yescharacteristic
213C3—32no—
214C3—61no—
215C3#1059112011yescharacteristic
216C2—22no—
217C2#1172106512yes—
218C2C2×#8844911yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#624
prime radicalC23×H(3,𝔽3)C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#423
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #11720304C2semidirect
49. #1084M16semidirect
120. H(3,𝔽3)#423semidirect
186. C32#5118semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#10842
C2×#3748⋊C32
#117203042
#105911153