Finite Group Databasebeta

← #13135075

Subgroups — #13135075

Order 1728.

247 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 247 conjugacy classes of subgroups — too many to draw. Only its 55 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×#3745⋊C3C211yesmaximal, characteristic
2#11720552C211yesmaximal, characteristic, complemented
3#63639⋊C3C211yesmaximal, characteristic, complemented
4#10589128—31nomaximal
5#10589126—31nomaximal
6C2×C4⋊((C3⋊D3)⋊C3)—21no—
7C2×C8×H(3,𝔽3)C2211yescharacteristic
8#88538—21no—
9#3745⋊C3—21no—
10#3745⋊C3C412yes—
11(C3⋊(C3⋊C16))⋊C3—21no—
12C2×#3745D311yescharacteristic
13C2×C3×D24—31no—
14#63639D311yescharacteristic, complemented
15C3×#819—31no—
16C3×#819—31no—
17C3×#819—61no—
18C3×#812—31no—
19C22×(C3⋊D3)⋊C3—41no—
20C2×C4×H(3,𝔽3)D411yescharacteristic
21C4⋊((C3⋊D3)⋊C3)—22no—
22C4⋊((C3⋊D3)⋊C3)—41no—
23C8×H(3,𝔽3)C2×C411yescharacteristic
24C8×H(3,𝔽3)D411yescharacteristic
25#6180—41no—
26#4563—91no—
27#4516—91no—
28#4495—91no—
29C2×C4⋊(C3⋊D3)—21no—
30C2×C3×D12—61no—
31C2×C32×C8—21no—
32C2×C32×C8C2×D311yescharacteristic
33C2×C32×C8C2×D311yescharacteristic
34#3745—21no—
35#3745Dic312yes—
36C3×#208—62no—
37C3×#208—61no—
38C3×#208—61no—
39C3×D24—32no—
40C3×D24—61no—
41C3⋊(C3⋊C16)—21no—
42C3×C3⋊C16—61no—
43C22×H(3,𝔽3)SD1611yescharacteristic
44C2×(C3⋊D3)⋊C3—42no—
45C2×(C3⋊D3)⋊C3—82no—
46C4×H(3,𝔽3)C22⋊C411yescharacteristic
47C4×H(3,𝔽3)D811yescharacteristic
48C2×C3×D8—91no—
49C2×D24—31no—
50C2×D24—91no—
51C3×M32—91no—
52C3×#108—91no—
53#819—31no—
54#819—31no—
55#819—61no—
56#812—31no—
57#812—91no—
58C22×C3⋊D3—41no—
59C22×C3×D3—121no—
60C2×C32×C4—21no—
61C2×C32×C4D1211yescharacteristic
62C2×C32×C4#6711yescharacteristic
63C4⋊(C3⋊D3)—22no—
64C4⋊(C3⋊D3)—41no—
65C3×D12—62no—
66C3×D12—121no—
67C32×C8—21no—
68C32×C8—21no—
69C32×C8C4×D311yescharacteristic
70C32×C8C2×Dic311yescharacteristic
71C32×C8#6711yescharacteristic
72C32×C8#6711yescharacteristic
73C3×C3⋊C8—122no—
74C3×C3⋊C8—121no—
75C3×C3⋊C8—121no—
76#361—271noSylow
77C2×H(3,𝔽3)—21no—
78C2×H(3,𝔽3)#10211yescharacteristic
79(C3⋊D3)⋊C3—82no—
80C2×C3×D4—181no—
81C2×D12—61no—
82C2×D12—181no—
83C3×D8—92no—
84C3×D8—181no—
85C3×M16—181no—
86C2×C3×C8—31no—
87C2×C3×C8—31no—
88C2×C3×C8—61no—
89C2×C3×C8D3211yescharacteristic
90#208—61no—
91#208—61no—
92#208—62no—
93D24—32no—
94D24—61no—
95D24—92no—
96D24—181no—
97C3×C16—181no—
98C3⋊C16—61no—
99C3⋊C16—181no—
100C22×C32—21no—
101C22×C32#20411yescharacteristic
102C22×C32#21411yescharacteristic
103C2×C3⋊D3—42no—
104C2×C3⋊D3—82no—
105C2×C3×D3—122no—
106C2×C3×D3—242no—
107C32×C4—21no—
108C32×C4—21no—
109C32×C4D2411yescharacteristic
110C32×C4(C2×D3)⋊C411yescharacteristic
111C32×C4#21311yescharacteristic
112C32×C4#21711yescharacteristic
113C2×D8—91no—
114M32—91no—
115#108—91no—
116H(3,𝔽3)#36111yescharacteristic, complemented, Sylow
117C23×C3—361no—
118C22×D3—121no—
119C22×D3—361no—
120C3×D4—182no—
121C3×D4—361no—
122C2×C3×C4—31no—
123C2×C3×C4—31no—
124C2×C3×C4—61no—
125C2×C3×C4#62611yescharacteristic
126D12—62no—
127D12—121no—
128D12—182no—
129D12—361no—
130C3×C8—31no—
131C3×C8—31no—
132C3×C8—31no—
133C3×C8—31no—
134C3×C8—61no—
135C3×C8—61no—
136C3×C8—361no—
137C3×C8D3×Dic311yescharacteristic
138C3×C8#62511yescharacteristic
139C3⋊C8—121no—
140C3⋊C8—121no—
141C3⋊C8—122no—
142C2×C32—21no—
143C2×C32—21no—
144C2×C32—21no—
145C2×C32—41no—
146C2×C32#82111yescharacteristic
147C2×C32#83211yescharacteristic
148C3⋊D3—82no—
149C3×D3—242no—
150C2×D4—181no—
151D8—92no—
152D8—181no—
153M16—181no—
154C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
155C16—181no—
156C22×C3—31no—
157C22×C3—31no—
158C22×C3—61no—
159C22×C3—362no—
160C22×C3—722no—
161C22×C3#371611yescharacteristic
162C2×D3—122no—
163C2×D3—242no—
164C2×D3—362no—
165C2×D3—722no—
166C3×C4—31no—
167C3×C4—31no—
168C3×C4—31no—
169C3×C4—31no—
170C3×C4—61no—
171C3×C4—61no—
172C3×C4#371411yescharacteristic
173C3×C4(C2×D3)⋊Dic311yescharacteristic
174C32—21no—
175C32#451611yescharacteristic, complemented
176C32#456311yescharacteristic, complemented
177C23—361no—
178D4—182no—
179D4—361no—
180C2×C4#620011yescharacteristic
181C8—361no—
182C8((C3⋊D3)⋊C3)⋊C411yescharacteristic
183C8#619811yescharacteristic
184C2×C3—21no—
185C2×C3—31no—
186C2×C3—31no—
187C2×C3—61no—
188C2×C3—61no—
189C2×C3—61no—
190C2×C3—121no—
191C2×C3—722no—
192C2×C3#6356311yescharacteristic
193D3—242no—
194D3—722no—
195C22—362no—
196C22—722no—
197C22#8843811yescharacteristic
198C4#8843611yescharacteristic
199C4#8845011yescharacteristic
200C3—31no—
201C3—31no—
202C3—61no—
203C3#1058912711yescharacteristic
204C2—21no—
205C2—722no—
206C2#1172030811yescharacteristic
subgroupG/H
centerC2#11720308
commutatorC8×H(3,𝔽3)C2×C4
FrattiniC2×C3×C8D32
Fitting, cosocleC2×C8×H(3,𝔽3)C22
socleC2×C3#63563
prime radicalC2×#3745⋊C3C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#361
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11720552C2semidirect
3. #63639⋊C3C2semidirect
14. #63639D3semidirect
116. H(3,𝔽3)#361semidirect
175. C32#4516semidirect
176. C32#4563semidirect

Maximal subgroups

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

subgroupindex
C2×#3745⋊C32
#117205522
#63639⋊C32
#105891283
#105891263