Finite Group Databasebeta

← #13138894

Subgroups — #13138894

Order 1728.

425 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 425 conjugacy classes of subgroups — too many to draw. Only its 81 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×#88527C211yesmaximal, characteristic
2#11721065C212yesmaximal
3#63646⋊C3C212yesmaximal, complemented
4#11720316C212yesmaximal, complemented
5#10590912—32nomaximal
6C2×C4×#272C2211yescharacteristic
7C2×C8×H(3,𝔽3)C2211yescharacteristic
8C2×#6180C2211yescharacteristic
9#88527C414yes—
10#3748⋊C3C2214yes—
11#88435—22no—
12#88434—22no—
13C2×C3×ΓL(1,𝔽25)—32no—
14C2×C3×ΓL(1,𝔽25)—61no—
15D3×#208—34no—
16#63646D312yescomplemented
17C3×#819—32no—
18#63577—34no—
19C22×#272Q811yescharacteristic
20C2×C4×H(3,𝔽3)C2311yescharacteristic
21C2×#1084Q811yescharacteristic
22C4×#272C2×C412yes—
23C4×#272D412yes—
24C8×H(3,𝔽3)C2×C412yes—
25#6180C2×C412yes—
26(C3⋊(C3⋊C8))⋊C3—24no—
27#5125—91no—
28#4893—92no—
29C2×C3×C4×D3—32no—
30C2×C3×C4×D3—61no—
31C2×C32×C8—21no—
32C2×C32×C8C2×D312yes—
33#3748C2×D314yes—
34C3×#208—34no—
35C2×C3×C3⋊C8—32no—
36C2×C3×C3⋊C8—61no—
37C3×ΓL(1,𝔽25)—38no—
38C3×ΓL(1,𝔽25)—64no—
39#3711—64no—
40D3×C3⋊C8—64no—
41C22×H(3,𝔽3)C2×Q811yescharacteristic
42C2×#272—41no—
43C2×#272C4⋊C412yes—
44C4×H(3,𝔽3)C22×C411yescharacteristic
45C4×H(3,𝔽3)C2×D411yescharacteristic
46#1084C4⋊C412yes—
47C2×C3×M16—91no—
48C2×#208—92no—
49D3×M16—94no—
50C2×ΓL(1,𝔽25)—32no—
51C2×ΓL(1,𝔽25)—61no—
52C3×#108—92no—
53#822—94no—
54#819—32no—
55#819—92no—
56#819—92no—
57C22×C3×D3—32no—
58C22×C3×D3—61no—
59C2×C32×C4—21no—
60C2×C32×C4C22×D312yes—
61C2×C3×Dic3—32no—
62C2×C3×Dic3—61no—
63C3×C4×D3—34no—
64C3×C4×D3—34no—
65C3×C4×D3—62no—
66C3×C4×D3—62no—
67C32×C8—22no—
68C32×C8C4×D314yes—
69C3⋊(C3⋊C8)—24no—
70C3×C3⋊C8—34no—
71C3×C3⋊C8—62no—
72C3×C3⋊C8—64no—
73#430—271noSylow
74C2×H(3,𝔽3)—21no—
75C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
76#272—22no—
77C22×C3×C4—91no—
78C2×C4×D3—32no—
79C2×C4×D3—61no—
80C3×M16—94no—
81C3×M16—94no—
82C2×C3×C8—32no—
83C2×C3×C8—61no—
84C2×C3×C8—91no—
85C2×C3×C8D3211yescharacteristic
86#208—34no—
87#208—94no—
88#208—182no—
89C2×C3⋊C8—32no—
90C2×C3⋊C8—61no—
91C2×C3⋊C8—182no—
92ΓL(1,𝔽25)—38no—
93ΓL(1,𝔽25)—64no—
94ΓL(1,𝔽25)—184no—
95D3×C8—184no—
96C22×C32—21no—
97C22×C32D3×Q812yes—
98C2×C3×D3—34no—
99C2×C3×D3—62no—
100C2×C3×D3—122no—
101C2×C3×D3—122no—
102C32×C4—21no—
103C32×C4—21no—
104C32×C4C2×C4×D312yes—
105C32×C4D3×D412yes—
106C3×Dic3—34no—
107C3×Dic3—62no—
108C2×M16—91no—
109C2×M16—272no—
110#108—92no—
111#108—272no—
112H(3,𝔽3)#43011yescharacteristic, complemented, Sylow
113C23×C3—91no—
114C22×D3—32no—
115C22×D3—61no—
116C2×C3×C4—32no—
117C2×C3×C4—61no—
118C2×C3×C4—91no—
119C2×C3×C4—92no—
120C2×C3×C4—92no—
121C2×C3×C4C2×D3211yescharacteristic
122C2×Dic3—32no—
123C2×Dic3—61no—
124C4×D3—34no—
125C4×D3—34no—
126C4×D3—62no—
127C4×D3—62no—
128C3×C8—34no—
129C3×C8—62no—
130C3×C8—92no—
131C3×C8—184no—
132C3×C8#62412yes—
133C3⋊C8—34no—
134C3⋊C8—62no—
135C3⋊C8—64no—
136C3⋊C8—184no—
137C2×C32—22no—
138C2×C32—22no—
139C2×C32—21no—
140C2×C32D3×C4⋊C412yes—
141C3×D3—64no—
142C3×D3—122no—
143C22×C4—91no—
144M16—94no—
145M16—94no—
146M16—542no—
147C2×C8—91no—
148C2×C8—542no—
149C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
150C22×C3—32no—
151C22×C3—61no—
152C22×C3—92no—
153C22×C3—361no—
154C22×C3#379711yescharacteristic
155C2×D3—34no—
156C2×D3—62no—
157C2×D3—122no—
158C2×D3—122no—
159C3×C4—32no—
160C3×C4—32no—
161C3×C4—61no—
162C3×C4—61no—
163C3×C4—92no—
164C3×C4#380211yescharacteristic
165C3×C4C2×#62411yescharacteristic
166Dic3—34no—
167Dic3—62no—
168C32—21no—
169C32#489312yescomplemented
170C23—91no—
171C2×C4—91no—
172C2×C4—92no—
173C2×C4—92no—
174C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
175C8—92no—
176C8—184no—
177C8#619912yes—
178C2×C3—21no—
179C2×C3—32no—
180C2×C3—62no—
181C2×C3—62no—
182C2×C3—61no—
183C2×C3—182no—
184C2×C3#6390011yescharacteristic
185D3—64no—
186D3—122no—
187C22—92no—
188C22—361no—
189C22#8865111yescharacteristic
190C4—92no—
191C4#8865511yescharacteristic
192C4C2×#619911yescharacteristic
193C3—32no—
194C3—61no—
195C3#1059091411yescharacteristic
196C2—21no—
197C2—182no—
198C2#1172115211yescharacteristic
subgroupG/H
centerC4#88655
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC2×C3#63900
prime radicalC22×#272Q8
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#430
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. #63646⋊C3C2semidirect
4. #11720316C2semidirect
16. #63646D3semidirect
112. H(3,𝔽3)#430semidirect
169. C32#4893semidirect

Maximal subgroups

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

subgroupindex
C2×#885272
#117210652
#63646⋊C32
#117203162
#105909123