Finite Group Databasebeta

← #13135077

Subgroups — #13135077

Order 1728.

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

normal subgroups only

This group has 226 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
1#64113⋊C3C211yesmaximal, characteristic
2#11720552C211yesmaximal, characteristic, complemented
3C2×(C3⋊(C3⋊C16))⋊C3C211yesmaximal, characteristic, complemented
4#10589122—31nomaximal
5#10589120—31nomaximal
6#3828⋊C3—21no—
7C2×C8×H(3,𝔽3)C2211yescharacteristic
8#88538—21no—
9#3745⋊C3C411yescharacteristic
10#3744⋊C3—21no—
11#3746⋊C3C411yescharacteristic
12(C3⋊(C3⋊C16))⋊C3—21no—
13#64113D311yescharacteristic
14C3×#904—31no—
15C2×C3⋊(C3⋊C16)D311yescharacteristic, complemented
16C3×#819—31no—
17C3×#819—31no—
18C3×#819—61no—
19C2×C3×C3⋊C16—31no—
20C2×C4×H(3,𝔽3)D411yescharacteristic
21#638⋊C3—41no—
22C4⋊((C3⋊D3)⋊C3)—21no—
23C4×(C3⋊D3)⋊C3—41no—
24#634⋊C3—21no—
25C8×H(3,𝔽3)C2×C411yescharacteristic
26C8×H(3,𝔽3)D411yescharacteristic
27#6180—41no—
28#4567—91no—
29#4510—91no—
30#4497—91no—
31#3828—21no—
32C3×#235—61no—
33C2×C32×C8—21no—
34C2×C32×C8C2×D311yescharacteristic
35C2×C32×C8C2×D311yescharacteristic
36#3746Dic311yescharacteristic
37#3745Dic311yescharacteristic
38#3744—21no—
39C3×#208—62no—
40C3×#208—61no—
41C3×#208—61no—
42C3×Dic12—31no—
43C3×D24—31no—
44C3×#204—61no—
45C3⋊(C3⋊C16)—21no—
46C3×C3⋊C16—61no—
47C22×H(3,𝔽3)SD1611yescharacteristic
48C2×(C3⋊D3)⋊C3—41no—
49C4×H(3,𝔽3)C22⋊C411yescharacteristic
50C4×H(3,𝔽3)D811yescharacteristic
51(C3⋊Dic3)⋊C3—41no—
52C3×#135—91no—
53#904—31no—
54#904—91no—
55C2×C3×C16—91no—
56C3×#108—91no—
57#819—31no—
58#819—31no—
59#819—61no—
60C2×C3⋊C16—31no—
61C2×C3⋊C16—91no—
62C2×C32×C4—21no—
63C2×C32×C4D1211yescharacteristic
64C2×C32×C4#6711yescharacteristic
65#638—41no—
66C4⋊(C3⋊D3)—21no—
67C4×C3⋊D3—41no—
68#634—21no—
69C3×#67—121no—
70C3×D12—61no—
71C3×C4×D3—121no—
72C3×Dic6—61no—
73C32×C8—21no—
74C32×C8—21no—
75C32×C8C4×D311yescharacteristic
76C32×C8C2×Dic311yescharacteristic
77C32×C8#6711yescharacteristic
78C32×C8#6711yescharacteristic
79C3×C3⋊C8—122no—
80C3×C3⋊C8—121no—
81C3×C3⋊C8—121no—
82#359—271noSylow
83C2×H(3,𝔽3)—21no—
84C2×H(3,𝔽3)#10211yescharacteristic
85(C3⋊D3)⋊C3—81no—
86C3×#41—181no—
87#235—61no—
88#235—181no—
89C3×Q16—91no—
90C3×SD16—181no—
91C3×D8—91no—
92C3×M16—181no—
93C2×C3×C8—31no—
94C2×C3×C8—31no—
95C2×C3×C8—61no—
96C2×C3×C8D3211yescharacteristic
97#208—61no—
98#208—61no—
99#208—62no—
100Dic12—31no—
101Dic12—91no—
102D24—31no—
103D24—91no—
104#204—61no—
105#204—181no—
106C3×C16—181no—
107C3⋊C16—61no—
108C3⋊C16—181no—
109C22×C32—21no—
110C22×C32#20411yescharacteristic
111C22×C32#21411yescharacteristic
112C2×C3⋊D3—41no—
113C2×C3×D3—121no—
114C32×C4—21no—
115C32×C4—21no—
116C32×C4D2411yescharacteristic
117C32×C4(C2×D3)⋊C411yescharacteristic
118C32×C4#21311yescharacteristic
119C32×C4#21711yescharacteristic
120C3⋊Dic3—41no—
121C3×Dic3—121no—
122#135—91no—
123C2×C16—91no—
124#108—91no—
125H(3,𝔽3)#35911yescharacteristic, complemented, Sylow
126C3×Q8—181no—
127C3×D4—181no—
128C3×D4—361no—
129C2×C3×C4—31no—
130C2×C3×C4—31no—
131C2×C3×C4—61no—
132C2×C3×C4—361no—
133C2×C3×C4#62611yescharacteristic
134#67—121no—
135#67—361no—
136D12—61no—
137D12—181no—
138C4×D3—121no—
139C4×D3—361no—
140Dic6—61no—
141Dic6—181no—
142C3×C8—31no—
143C3×C8—31no—
144C3×C8—31no—
145C3×C8—31no—
146C3×C8—61no—
147C3×C8—61no—
148C3×C8—361no—
149C3×C8D3×Dic311yescharacteristic
150C3×C8#62511yescharacteristic
151C3⋊C8—121no—
152C3⋊C8—121no—
153C3⋊C8—122no—
154C2×C32—21no—
155C2×C32—21no—
156C2×C32—21no—
157C2×C32—41no—
158C2×C32#82111yescharacteristic
159C2×C32#83211yescharacteristic
160C3⋊D3—81no—
161C3×D3—241no—
162#41—181no—
163Q16—91no—
164SD16—181no—
165D8—91no—
166M16—181no—
167C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
168C16—181no—
169C22×C3—31no—
170C22×C3—31no—
171C22×C3—61no—
172C22×C3—361no—
173C22×C3#371611yescharacteristic
174C2×D3—121no—
175C2×D3—361no—
176C3×C4—31no—
177C3×C4—31no—
178C3×C4—31no—
179C3×C4—31no—
180C3×C4—61no—
181C3×C4—61no—
182C3×C4—361no—
183C3×C4#371411yescharacteristic
184C3×C4(C2×D3)⋊Dic311yescharacteristic
185Dic3—121no—
186Dic3—361no—
187C32—21no—
188C32#451011yescharacteristic, complemented
189C32#456711yescharacteristic, complemented
190Q8—181no—
191D4—181no—
192D4—361no—
193C2×C4—361no—
194C2×C4#620011yescharacteristic
195C8—361no—
196C8((C3⋊D3)⋊C3)⋊C411yescharacteristic
197C8#619811yescharacteristic
198C2×C3—21no—
199C2×C3—31no—
200C2×C3—31no—
201C2×C3—61no—
202C2×C3—61no—
203C2×C3—61no—
204C2×C3—121no—
205C2×C3—721no—
206C2×C3#6356311yescharacteristic
207D3—241no—
208D3—721no—
209C22—361no—
210C22#8843811yescharacteristic
211C4—361no—
212C4#8843611yescharacteristic
213C4#8845011yescharacteristic
214C3—31no—
215C3—31no—
216C3—61no—
217C3#1058912111yescharacteristic
218C2—21no—
219C2—721no—
220C2#1172030811yescharacteristic
subgroupG/H
centerC4#88436
commutatorC8×H(3,𝔽3)C2×C4
FrattiniC2×C3×C8D32
Fitting, cosocleC2×C8×H(3,𝔽3)C22
socleC2×C3#63563
prime radical#64113⋊C3C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#359
LCSG ▷ C8×H(3,𝔽3) ▷ 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
2. #11720552C2semidirect
3. C2×(C3⋊(C3⋊C16))⋊C3C2semidirect
15. C2×C3⋊(C3⋊C16)D3semidirect
125. H(3,𝔽3)#359semidirect
188. C32#4510semidirect
189. C32#4567semidirect

Maximal subgroups

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

subgroupindex
#64113⋊C32
#117205522
C2×(C3⋊(C3⋊C16))⋊C32
#105891223
#105891203