Finite Group Databasebeta

← #13139922

Subgroups — #13139922

Order 1728.

605 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 605 conjugacy classes of subgroups — too many to draw. Only its 159 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#11720303C212yesmaximal, complemented
4#11720302C212yesmaximal, complemented
5Dic3×#208—32nomaximal
6C22×C4×H(3,𝔽3)C2211yescharacteristic
7C22×#1084C412yes—
8C4×#1084C412yes—
9C4×#1084C2211yescharacteristic
10C4×#1084C2211yescharacteristic
11#3748⋊C3C418yescomplemented
12C2×(C3⋊(C3⋊C8))⋊C3C2214yes—
13C2×#3748D312yes—
14C2×C3×C4×Dic3—32no—
15C2×C3×C4×Dic3—61no—
16C2×C3×#208—32no—
17#63554—34no—
18Dic3×C3⋊C8—34no—
19C23×H(3,𝔽3)C2×C411yescharacteristic
20C2×C4×H(3,𝔽3)C2×C411yescharacteristic
21C2×C4×H(3,𝔽3)C2×C414yes—
22C2×C4×H(3,𝔽3)C2311yescharacteristic
23C2×#1084—24no—
24C2×#1084C2×C412yes—
25C2×#1084C2×C412yes—
26(C3⋊(C3⋊C8))⋊C3C2×C418yescomplemented
27Dic3×M16—92no—
28C4×#208—91no—
29C22×C32×C4—21no—
30C22×C32×C4C2×D312yes—
31C22×C3×Dic3—34no—
32C22×C3×Dic3—62no—
33#3748Dic318yescomplemented
34C2×C3⋊(C3⋊C8)C2×D314yes—
35C3×C4×Dic3—34no—
36C3×C4×Dic3—32no—
37C3×C4×Dic3—32no—
38C3×C4×Dic3—62no—
39C3×C4×Dic3—61no—
40C3×C4×Dic3—61no—
41C3×#208—38no—
42C2×C3×C3⋊C8—34no—
43C22×H(3,𝔽3)—22no—
44C22×H(3,𝔽3)C4212yes—
45C22×H(3,𝔽3)C22×C411yescharacteristic
46C4×H(3,𝔽3)C4212yes—
47C4×H(3,𝔽3)C22×C412yes—
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—
56#815—94no—
57C8×Dic3—94no—
58#803—92no—
59C4×C3⋊C8—92no—
60C23×C32—21no—
61C23×C32C4×D312yes—
62C2×C32×C4—24no—
63C2×C32×C4—21no—
64C2×C32×C4—21no—
65C2×C32×C4C4×D314yes—
66C2×C32×C4C4×D312yes—
67C2×C32×C4C2×Dic314yes—
68C2×C32×C4C22×D312yes—
69C2×C3×Dic3—34no—
70C2×C3×Dic3—34no—
71C2×C3×Dic3—62no—
72C2×C3×Dic3—62no—
73C2×C3×Dic3—68no—
74C2×C3×Dic3—68no—
75C3⋊(C3⋊C8)C2×Dic318yescomplemented
76C3×C3⋊C8—38no—
77C4×M16—271noSylow
78C2×H(3,𝔽3)—22no—
79C2×H(3,𝔽3)C2×C4211yescharacteristic
80C2×H(3,𝔽3)C2×M1612yes—
81C22×C3×C4—32no—
82C22×C3×C4—61no—
83C22×C3×C4—92no—
84C22×C3×C4D3211yescharacteristic
85C22×Dic3—34no—
86C22×Dic3—62no—
87C3×M16—98no—
88C2×C3×C8—94no—
89C3×C42—92no—
90C3×C42—91no—
91C3×C42—91no—
92C4×Dic3—32no—
93C4×Dic3—32no—
94C4×Dic3—34no—
95C4×Dic3—62no—
96C4×Dic3—61no—
97C4×Dic3—61no—
98#208—38no—
99#208—98no—
100C2×C3⋊C8—34no—
101C2×C3⋊C8—94no—
102C22×C32—24no—
103C22×C32—24no—
104C22×C32—22no—
105C22×C32—21no—
106C22×C32C4×Dic314yes—
107C22×C32C2×C4×D312yes—
108C32×C4—22no—
109C32×C4—22no—
110C32×C4C4×Dic314yes—
111C32×C4C2×C4×D312yes—
112C32×C4C22×Dic312yes—
113C3×Dic3—38no—
114C3×Dic3—64no—
115C3×Dic3—64no—
116C3×Dic3—64no—
117C2×M16—92no—
118C2×C42—91no—
119#97—272no—
120C4×C8—272no—
121H(3,𝔽3)C4×M1611yescharacteristic, complemented, Sylow
122C23×C3—32no—
123C23×C3—61no—
124C23×C3#62411yescharacteristic
125C2×C3×C4—34no—
126C2×C3×C4—34no—
127C2×C3×C4—32no—
128C2×C3×C4—32no—
129C2×C3×C4—64no—
130C2×C3×C4—61no—
131C2×C3×C4—61no—
132C2×C3×C4—92no—
133C2×C3×C4—92no—
134C2×C3×C4—184no—
135C2×C3×C4D3×Dic314yes—
136C2×C3×C4#62411yescharacteristic
137C2×C3×C4C2×D3211yescharacteristic
138C2×Dic3—34no—
139C2×Dic3—34no—
140C2×Dic3—62no—
141C2×Dic3—62no—
142C2×Dic3—68no—
143C2×Dic3—68no—
144C3×C8—98no—
145C3⋊C8—38no—
146C3⋊C8—98no—
147C2×C32—24no—
148C2×C32—24no—
149C2×C32—22no—
150C2×C32—21no—
151C2×C32D3×M1614yes—
152C2×C32C2×C4×Dic312yes—
153C22×C4—92no—
154C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
155M16—98no—
156C2×C8—94no—
157C42—92no—
158C42—91no—
159C42—91no—
160C22×C3—22no—
161C22×C3—32no—
162C22×C3—34no—
163C22×C3—64no—
164C22×C3—64no—
165C22×C3—61no—
166C22×C3—62no—
167C22×C3Dic3212yes—
168C22×C3C2×#62411yescharacteristic
169C3×C4—32no—
170C3×C4—32no—
171C3×C4—34no—
172C3×C4—62no—
173C3×C4—62no—
174C3×C4—94no—
175C3×C4—182no—
176C3×C4Dic3212yes—
177C3×C4C2×D3×Dic312yes—
178Dic3—38no—
179Dic3—64no—
180Dic3—64no—
181Dic3—64no—
182C32—21no—
183C32Dic3×M1612yescomplemented
184C23#619911yescharacteristic
185C2×C4—92no—
186C2×C4—92no—
187C2×C4—184no—
188C2×C4((C3⋊D3)⋊C3)⋊C414yes—
189C2×C4#619911yescharacteristic
190C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
191C8—98no—
192C2×C3—22no—
193C2×C3—32no—
194C2×C3—34no—
195C2×C3—64no—
196C2×C3—64no—
197C2×C3—61no—
198C2×C3—62no—
199C2×C3#6381812yes—
200C2×C3C2×Dic3211yescharacteristic
201C22—22no—
202C22#8844712yes—
203C22C2×#619911yescharacteristic
204C4—94no—
205C4—182no—
206C4#8844712yes—
207C4C2×((C3⋊D3)⋊C3)⋊C412yes—
208C3—32no—
209C3—61no—
210C3#1059106011yescharacteristic
211C2—22no—
212C2#1172106512yes—
213C2C2×#8844711yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C2×C42
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)C4×M16
LCSG ▷ C2×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. #11720303C2semidirect
4. #11720302C2semidirect
11. #3748⋊C3C4semidirect
26. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
33. #3748Dic3semidirect
49. #1084M16semidirect
75. C3⋊(C3⋊C8)C2×Dic3semidirect
121. H(3,𝔽3)C4×M16semidirect
183. C32Dic3×M16semidirect

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
#117203032
#117203022
Dic3×#2083