Finite Group Databasebeta

← #13135092

Subgroups — #13135092

Order 1728.

229 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 229 conjugacy classes of subgroups — too many to draw. Only its 59 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×#3743⋊C3C211yesmaximal, characteristic
2#11720545C211yesmaximal, characteristic, complemented
3#63639⋊C3C211yesmaximal, characteristic, complemented
4#10589110—31nomaximal
5#10589109—31nomaximal
6C2×C4×(C3⋊D3)⋊C3C411yescharacteristic
7C2×C8×H(3,𝔽3)C2211yescharacteristic
8C2×(C3⋊(C3⋊C8))⋊C3C411yescharacteristic
9#3743⋊C3—21no—
10#3743⋊C3—21no—
11#88386—21no—
12(C3⋊(C3⋊C16))⋊C3—21no—
13C2×#3743D311yescharacteristic
14C2×C3×ΓL(1,𝔽25)—31no—
15#63639D311yescharacteristic, complemented
16C3×#812—31no—
17C3×#812—31no—
18C3×#812—31no—
19C3×#812—61no—
20C22×(C3⋊D3)⋊C3C811yescharacteristic
21C2×C4×H(3,𝔽3)C2×C411yescharacteristic
22C2×(C3⋊Dic3)⋊C3C811yescharacteristic
23C4×(C3⋊D3)⋊C3—21no—
24C4×(C3⋊D3)⋊C3—21no—
25C8×H(3,𝔽3)D411yescharacteristic
26C8×H(3,𝔽3)D411yescharacteristic
27(C3⋊(C3⋊C8))⋊C3—21no—
28#4553—91no—
29#4514—91no—
30#4514—91no—
31C2×C4×C3⋊D3Dic311yescharacteristic
32C2×C3×C4×D3—31no—
33C2×C32×C8—21no—
34C2×C32×C8C2×D311yescharacteristic
35C2×C32×C8C2×D311yescharacteristic
36C2×C3⋊(C3⋊C8)Dic311yescharacteristic
37#3743—21no—
38#3743—21no—
39C2×C3×C3⋊C8—31no—
40C3×ΓL(1,𝔽25)—61no—
41C3×ΓL(1,𝔽25)—61no—
42C3⋊(C3⋊C16)—21no—
43C3×C3⋊C16—62no—
44C3×C3⋊C16—61no—
45C3×C3⋊C16—61no—
46C3×C3⋊C16—61no—
47C22×H(3,𝔽3)C2×C811yescharacteristic
48C2×(C3⋊D3)⋊C3—21no—
49C2×(C3⋊D3)⋊C3—41no—
50C4×H(3,𝔽3)C22⋊C411yescharacteristic
51C4×H(3,𝔽3)M1611yescharacteristic
52(C3⋊Dic3)⋊C3—21no—
53C2×C3×M16—91no—
54C2×ΓL(1,𝔽25)—31no—
55C2×ΓL(1,𝔽25)—91no—
56C3×M32—91no—
57C3×M32—91no—
58#812—31no—
59#812—31no—
60#812—31no—
61#812—61no—
62#812—91no—
63C22×C3⋊D3C3⋊C811yescharacteristic
64C22×C3×D3—31no—
65C2×C32×C4—21no—
66C2×C32×C4C4×D311yescharacteristic
67C2×C32×C4C2×Dic311yescharacteristic
68C2×C3⋊Dic3C3⋊C811yescharacteristic
69C4×C3⋊D3—21no—
70C4×C3⋊D3—21no—
71C2×C3×Dic3—31no—
72C3×C4×D3—61no—
73C3×C4×D3—61no—
74C32×C8—21no—
75C32×C8—21no—
76C32×C8D1211yescharacteristic
77C32×C8#6711yescharacteristic
78C32×C8#6711yescharacteristic
79C32×C8#6711yescharacteristic
80C3⋊(C3⋊C8)—21no—
81C3×C3⋊C8—61no—
82#349—271noSylow
83C2×H(3,𝔽3)—21no—
84C2×H(3,𝔽3)C22⋊C811yescharacteristic
85(C3⋊D3)⋊C3—41no—
86C22×C3×C4—91no—
87C2×C4×D3—31no—
88C2×C4×D3—91no—
89C3×M16—181no—
90C3×M16—181no—
91C2×C3×C8—31no—
92C2×C3×C8—31no—
93C2×C3×C8—61no—
94C2×C3×C8—91no—
95C2×C3×C8D3211yescharacteristic
96C2×C3⋊C8—31no—
97C2×C3⋊C8—91no—
98ΓL(1,𝔽25)—61no—
99ΓL(1,𝔽25)—61no—
100ΓL(1,𝔽25)—181no—
101ΓL(1,𝔽25)—181no—
102C3×C16—181no—
103C3×C16—181no—
104C3⋊C16—61no—
105C3⋊C16—61no—
106C3⋊C16—62no—
107C3⋊C16—61no—
108C3⋊C16—181no—
109C22×C32—21no—
110C22×C32D3×C811yescharacteristic
111C22×C32C2×C3⋊C811yescharacteristic
112C2×C3⋊D3—21no—
113C2×C3⋊D3—41no—
114C2×C3×D3—61no—
115C2×C3×D3—121no—
116C32×C4—21no—
117C32×C4—21no—
118C32×C4ΓL(1,𝔽25)11yescharacteristic
119C32×C4#20811yescharacteristic
120C32×C4(C2×D3)⋊C411yescharacteristic
121C32×C4#21711yescharacteristic
122C3⋊Dic3—21no—
123C3×Dic3—61no—
124C2×M16—91no—
125M32—91no—
126M32—91no—
127H(3,𝔽3)#34911yescharacteristic, complemented, Sylow
128C23×C3—91no—
129C22×D3—31no—
130C22×D3—91no—
131C2×C3×C4—31no—
132C2×C3×C4—31no—
133C2×C3×C4—61no—
134C2×C3×C4—91no—
135C2×C3×C4—181no—
136C2×C3×C4—181no—
137C2×C3×C4D3×Dic311yescharacteristic
138C2×Dic3—31no—
139C2×Dic3—91no—
140C4×D3—61no—
141C4×D3—61no—
142C4×D3—181no—
143C4×D3—181no—
144C3×C8—31no—
145C3×C8—31no—
146C3×C8—31no—
147C3×C8—31no—
148C3×C8—61no—
149C3×C8—61no—
150C3×C8—181no—
151C3×C8#62511yescharacteristic
152C3×C8#62611yescharacteristic
153C3⋊C8—61no—
154C3⋊C8—181no—
155C2×C32—21no—
156C2×C32—21no—
157C2×C32—21no—
158C2×C32—41no—
159C2×C32(C2×D3)⋊C811yescharacteristic
160C2×C32#83011yescharacteristic
161C3⋊D3—41no—
162C3×D3—121no—
163C22×C4—91no—
164M16—181no—
165M16—181no—
166C2×C8—91no—
167C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
168C16—181no—
169C16—181no—
170C22×C3—31no—
171C22×C3—31no—
172C22×C3—61no—
173C22×C3—181no—
174C22×C3—361no—
175C22×C3D3×C3⋊C811yescharacteristic
176C2×D3—61no—
177C2×D3—121no—
178C2×D3—181no—
179C2×D3—361no—
180C3×C4—31no—
181C3×C4—31no—
182C3×C4—31no—
183C3×C4—31no—
184C3×C4—61no—
185C3×C4—61no—
186C3×C4—181no—
187C3×C4#371111yescharacteristic
188C3×C4(C2×D3)⋊Dic311yescharacteristic
189Dic3—61no—
190Dic3—181no—
191C32—21no—
192C32#451411yescharacteristic, complemented
193C32#455311yescharacteristic, complemented
194C23—91no—
195C2×C4—91no—
196C2×C4—181no—
197C2×C4—181no—
198C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
199C8—181no—
200C8#619811yescharacteristic
201C8#620011yescharacteristic
202C2×C3—21no—
203C2×C3—31no—
204C2×C3—31no—
205C2×C3—61no—
206C2×C3—61no—
207C2×C3—61no—
208C2×C3—121no—
209C2×C3—361no—
210C2×C3(C2×D3)⋊(C3⋊C8)11yescharacteristic
211D3—121no—
212D3—361no—
213C22—181no—
214C22—361no—
215C22((C3⋊D3)⋊C3)⋊C811yescharacteristic
216C4—181no—
217C4#8843011yescharacteristic
218C4#8845011yescharacteristic
219C3—31no—
220C3—31no—
221C3—61no—
222C3#1058910911yescharacteristic
223C2—21no—
224C2—361no—
225C2#1172031711yescharacteristic
subgroupG/H
centerC4#88450
commutatorC22×H(3,𝔽3)C2×C8
FrattiniC2×C3×C8D32
Fitting, cosocleC2×C8×H(3,𝔽3)C22
socleC2×C3(C2×D3)⋊(C3⋊C8)
prime radicalC22×(C3⋊D3)⋊C3C8
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#349
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11720545C2semidirect
3. #63639⋊C3C2semidirect
15. #63639D3semidirect
127. H(3,𝔽3)#349semidirect
192. C32#4514semidirect
193. C32#4553semidirect

Maximal subgroups

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

subgroupindex
C2×#3743⋊C32
#117205452
#63639⋊C32
#105891103
#105891093