Finite Group Databasebeta

← #13135071

Subgroups — #13135071

Order 1728.

267 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 65 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#11720551C211yesmaximal, characteristic, complemented
3C2×(C3⋊(C3⋊C16))⋊C3C211yesmaximal, characteristic, complemented
4#10589116—31nomaximal
5#10589114—31nomaximal
6C2×C4⋊((C3⋊D3)⋊C3)—21no—
7C2×C8×H(3,𝔽3)C2211yescharacteristic
8C4⋊#1084—21no—
9#3745⋊C3—21no—
10#3745⋊C3C412yescomplemented
11(C3⋊(C3⋊C16))⋊C3—21no—
12C2×#3745D311yescharacteristic
13C2×C3×D24—31no—
14C2×C3⋊(C3⋊C16)D311yescharacteristic, complemented
15C3×#818—31no—
16C3×#818—31no—
17C3×#818—61no—
18C2×C3×C3⋊C16—31no—
19C22×(C3⋊D3)⋊C3—41no—
20C2×C4×H(3,𝔽3)D411yescharacteristic
21C2×#1084—41no—
22C4⋊((C3⋊D3)⋊C3)—22no—
23C4⋊((C3⋊D3)⋊C3)—41no—
24C8×H(3,𝔽3)C2×C411yescharacteristic
25C8×H(3,𝔽3)D411yescharacteristic
26#4562—91no—
27#4509—91no—
28#4491—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#3745Dic312yescomplemented
36C3×C4⋊Dic3—62no—
37C3×C4⋊Dic3—61no—
38C3×C4⋊Dic3—61no—
39C3×D24—32no—
40C3×D24—61no—
41C3⋊(C3⋊C16)—21no—
42C3×C3⋊C16—61no—
43C22×H(3,𝔽3)D811yescharacteristic
44C2×(C3⋊D3)⋊C3—42no—
45C2×(C3⋊D3)⋊C3—81no—
46C2×(C3⋊D3)⋊C3—81no—
47C4×H(3,𝔽3)C22⋊C411yescharacteristic
48C4×H(3,𝔽3)SD1611yescharacteristic
49#1084—81no—
50C2×C3×D8—91no—
51C2×D24—31no—
52C2×D24—91no—
53C2×C3×C16—91no—
54C3×#107—91no—
55#818—31no—
56#818—31no—
57#818—61no—
58C2×C3⋊C16—31no—
59C2×C3⋊C16—91no—
60C22×C3⋊D3—41no—
61C22×C3×D3—121no—
62C2×C32×C4—21no—
63C2×C32×C4D1211yescharacteristic
64C2×C32×C4#6711yescharacteristic
65C4⋊(C3⋊D3)—22no—
66C4⋊(C3⋊D3)—41no—
67C2×C3×Dic3—122no—
68C2×C3×Dic3—121no—
69C2×C3×Dic3—121no—
70C3×D12—62no—
71C3×D12—121no—
72C32×C8—21no—
73C32×C8—21no—
74C32×C8C4×D311yescharacteristic
75C32×C8C2×Dic311yescharacteristic
76C32×C8#6711yescharacteristic
77C32×C8#6711yescharacteristic
78#357—271noSylow
79C2×H(3,𝔽3)#10211yescharacteristic
80C2×H(3,𝔽3)D1611yescharacteristic
81C2×H(3,𝔽3)SD3211yescharacteristic
82(C3⋊D3)⋊C3—82no—
83C2×C3×D4—181no—
84C2×D12—61no—
85C2×D12—181no—
86C3×D8—92no—
87C3×D8—181no—
88C2×C3×C8—31no—
89C2×C3×C8—31no—
90C2×C3×C8—61no—
91C2×C3×C8D3211yescharacteristic
92C3×C4⋊C4—181no—
93C4⋊Dic3—61no—
94C4⋊Dic3—61no—
95C4⋊Dic3—62no—
96D24—32no—
97D24—61no—
98D24—92no—
99D24—181no—
100C3×C16—181no—
101C3⋊C16—61no—
102C3⋊C16—181no—
103C22×C32—21no—
104C22×C32D2411yescharacteristic
105C22×C32#21311yescharacteristic
106C2×C3⋊D3—42no—
107C2×C3⋊D3—81no—
108C2×C3⋊D3—81no—
109C2×C3×D3—122no—
110C2×C3×D3—241no—
111C2×C3×D3—241no—
112C32×C4—21no—
113C32×C4—21no—
114C32×C4#20411yescharacteristic
115C32×C4(C2×D3)⋊C411yescharacteristic
116C32×C4#21411yescharacteristic
117C32×C4#21711yescharacteristic
118C3×Dic3—242no—
119C3×Dic3—241no—
120C3×Dic3—241no—
121C2×D8—91no—
122C2×C16—91no—
123#107—91no—
124H(3,𝔽3)#35711yescharacteristic, complemented, Sylow
125C23×C3—361no—
126C22×D3—121no—
127C22×D3—361no—
128C3×D4—182no—
129C3×D4—361no—
130C2×C3×C4—31no—
131C2×C3×C4—31no—
132C2×C3×C4—61no—
133C2×C3×C4—361no—
134C2×C3×C4#62611yescharacteristic
135C2×Dic3—121no—
136C2×Dic3—121no—
137C2×Dic3—122no—
138D12—62no—
139D12—121no—
140D12—182no—
141D12—361no—
142C3×C8—31no—
143C3×C8—31no—
144C3×C8—31no—
145C3×C8—31no—
146C3×C8—61no—
147C3×C8—61no—
148C3×C8D3×Dic311yescharacteristic
149C3×C8#62511yescharacteristic
150C2×C32—21no—
151C2×C32—21no—
152C2×C32—21no—
153C2×C32D4811yescharacteristic
154C2×C32#80011yescharacteristic
155C2×C32#82111yescharacteristic
156C2×C32#82611yescharacteristic
157C2×C32#82711yescharacteristic
158C2×C32#83211yescharacteristic
159C3⋊D3—82no—
160C3×D3—242no—
161C2×D4—181no—
162D8—92no—
163D8—181no—
164C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
165C4⋊C4—181no—
166C16—181no—
167C22×C3—31no—
168C22×C3—31no—
169C22×C3—61no—
170C22×C3—362no—
171C22×C3—721no—
172C22×C3—721no—
173C22×C3#371411yescharacteristic
174C2×D3—122no—
175C2×D3—241no—
176C2×D3—241no—
177C2×D3—362no—
178C2×D3—721no—
179C2×D3—721no—
180C3×C4—31no—
181C3×C4—31no—
182C3×C4—31no—
183C3×C4—31no—
184C3×C4—61no—
185C3×C4—61no—
186C3×C4—721no—
187C3×C4#371611yescharacteristic
188C3×C4(C2×D3)⋊Dic311yescharacteristic
189Dic3—241no—
190Dic3—241no—
191Dic3—242no—
192C32—21no—
193C32#450911yescharacteristic, complemented
194C32#456211yescharacteristic, complemented
195C23—361no—
196D4—182no—
197D4—361no—
198C2×C4—361no—
199C2×C4#620011yescharacteristic
200C8((C3⋊D3)⋊C3)⋊C411yescharacteristic
201C8#619811yescharacteristic
202C2×C3—31no—
203C2×C3—31no—
204C2×C3—31no—
205C2×C3—31no—
206C2×C3—31no—
207C2×C3—31no—
208C2×C3—61no—
209C2×C3—61no—
210C2×C3—61no—
211C2×C3—722no—
212C2×C3#6354611yescharacteristic
213C2×C3#6354811yescharacteristic
214C2×C3#6356311yescharacteristic
215D3—242no—
216D3—722no—
217C22—362no—
218C22—721no—
219C22—721no—
220C22#8843611yescharacteristic
221C4—721no—
222C4#8843811yescharacteristic
223C4#8845011yescharacteristic
224C3—31no—
225C3—31no—
226C3—61no—
227C3#1058911511yescharacteristic
228C2—722no—
229C2#1172025311yescharacteristic
230C2#1172025511yescharacteristic
231C2#1172030811yescharacteristic
subgroupG/H
centerC22#88436
commutatorC8×H(3,𝔽3)C2×C4
FrattiniC2×C3×C8D32
Fitting, cosocleC2×C8×H(3,𝔽3)C22
socleC22×C3#3714
prime radicalC2×#3745⋊C3C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#357
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11720551C2semidirect
3. C2×(C3⋊(C3⋊C16))⋊C3C2semidirect
10. #3745⋊C3C4semidirect
14. C2×C3⋊(C3⋊C16)D3semidirect
35. #3745Dic3semidirect
124. H(3,𝔽3)#357semidirect
193. C32#4509semidirect
194. C32#4562semidirect

Maximal subgroups

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

subgroupindex
C2×#3745⋊C32
#117205512
C2×(C3⋊(C3⋊C16))⋊C32
#105891163
#105891143