Finite Group Databasebeta

← #13138718

Subgroups — #13138718

Order 1728.

320 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 97 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#64110⋊C3C211yesmaximal, characteristic
2#11720545C211yesmaximal, characteristic, complemented
3C2×(C3⋊(C3⋊C16))⋊C3C211yesmaximal, characteristic, complemented
4#11720247C212yesmaximal, complemented
5((C3⋊D3)⋊C3)⋊C16C212yesmaximal, complemented
6#10590624—31nomaximal
7#10590622—31nomaximal
8#3828⋊C3C411yescharacteristic
9C2×C8×H(3,𝔽3)C2211yescharacteristic
10#3748⋊C3C411yescharacteristic
11#3743⋊C3C412yes—
12C8×(C3⋊D3)⋊C3C2212yes—
13#88386C2212yes—
14(C3⋊(C3⋊C16))⋊C3C2211yescharacteristic
15(C3⋊(C3⋊C16))⋊C3C2211yescharacteristic
16#64110D311yescharacteristic
17C3×#901—31no—
18C2×C3⋊(C3⋊C16)D311yescharacteristic, complemented
19C3×#812—31no—
20C3×#812—31no—
21C3×#812—61no—
22C2×C3×C3⋊C16—31no—
23#63544—32no—
24#63543—32no—
25#63542—32no—
26D3×C3⋊C16—32no—
27C2×C4×H(3,𝔽3)C2×C411yescharacteristic
28#638⋊C3C812yes—
29C4⋊((C3⋊D3)⋊C3)C811yescharacteristic
30C4×(C3⋊D3)⋊C3C2×C412yes—
31#634⋊C3C811yescharacteristic
32C8×H(3,𝔽3)C2×C411yescharacteristic
33C8×H(3,𝔽3)C2311yescharacteristic
34(C3⋊(C3⋊C8))⋊C3C2×C412yes—
35#5140—91no—
36#4907—91no—
37#4901—91no—
38#3828Dic311yescharacteristic
39C3×#235—31no—
40C2×C32×C8—21no—
41C2×C32×C8C2×D311yescharacteristic
42C2×C32×C8C2×D311yescharacteristic
43#3748Dic311yescharacteristic
44#3743Dic312yes—
45C8×C3⋊D3C2×D312yes—
46C3×#208—31no—
47C3×ΓL(1,𝔽25)—32no—
48C3×D3×C8—32no—
49C3⋊(C3⋊C16)C2×D311yescharacteristic
50C3⋊(C3⋊C16)C2×D311yescharacteristic
51C3×C3⋊C16—32no—
52C3×C3⋊C16—31no—
53C3×C3⋊C16—32no—
54C3×C3⋊C16—31no—
55C3×C3⋊C16—62no—
56C22×H(3,𝔽3)C2×C811yescharacteristic
57C2×(C3⋊D3)⋊C3C2×C812yes—
58C4×H(3,𝔽3)C2×C811yescharacteristic
59C4×H(3,𝔽3)C22×C411yescharacteristic
60(C3⋊Dic3)⋊C3C2×C812yes—
61C3×#131—91no—
62#901—31no—
63#901—91no—
64C3×M32—91no—
65C2×C3×C16—91no—
66#812—31no—
67#812—31no—
68#812—61no—
69#812—92no—
70C2×C3⋊C16—31no—
71C2×C3⋊C16—91no—
72C2×C3⋊C16—92no—
73#798—92no—
74#798—92no—
75D3×C16—92no—
76D3×C16—92no—
77C2×C32×C4—21no—
78C2×C32×C4C4×D311yescharacteristic
79C2×C32×C4C2×Dic311yescharacteristic
80#638C3⋊C812yes—
81C4⋊(C3⋊D3)C3⋊C811yescharacteristic
82C4×C3⋊D3C2×Dic312yes—
83#634C3⋊C811yescharacteristic
84C3×#67—32no—
85C3×D12—31no—
86C3×C4×D3—32no—
87C3×Dic6—31no—
88C32×C8—21no—
89C32×C8—21no—
90C32×C8C4×D311yescharacteristic
91C32×C8C2×Dic311yescharacteristic
92C32×C8C22×D311yescharacteristic
93C32×C8C22×D311yescharacteristic
94C3⋊(C3⋊C8)C2×Dic312yes—
95C3×C3⋊C8—32no—
96#504—271noSylow
97C2×H(3,𝔽3)—21no—
98C2×H(3,𝔽3)C22×C811yescharacteristic
99(C3⋊D3)⋊C3—22no—
100C3×#41—91no—
101#235—31no—
102#235—91no—
103C3×M16—92no—
104C3×M16—91no—
105C2×C3×C8—31no—
106C2×C3×C8—31no—
107C2×C3×C8—61no—
108C2×C3×C8—92no—
109C2×C3×C8D3211yescharacteristic
110#208—31no—
111#208—91no—
112ΓL(1,𝔽25)—32no—
113ΓL(1,𝔽25)—92no—
114D3×C8—32no—
115D3×C8—92no—
116C3×C16—91no—
117C3×C16—91no—
118C3×C16—92no—
119C3⋊C16—32no—
120C3⋊C16—32no—
121C3⋊C16—31no—
122C3⋊C16—31no—
123C3⋊C16—62no—
124C3⋊C16—91no—
125C3⋊C16—91no—
126C22×C32—21no—
127C22×C32D3×C811yescharacteristic
128C22×C32C2×C3⋊C811yescharacteristic
129C2×C3⋊D3C2×C3⋊C812yes—
130C2×C3×D3—32no—
131C32×C4—21no—
132C32×C4—21no—
133C32×C4D3×C811yescharacteristic
134C32×C4C2×C3⋊C811yescharacteristic
135C32×C4C2×C4×D311yescharacteristic
136C32×C4C22×Dic311yescharacteristic
137C3⋊Dic3C2×C3⋊C812yes—
138C3×Dic3—32no—
139#131—91no—
140M32—91no—
141M32—272no—
142C2×C16—91no—
143C2×C16—272no—
144H(3,𝔽3)#50411yescharacteristic, complemented, Sylow
145C3×Q8—91no—
146C3×D4—91no—
147C3×D4—92no—
148C2×C3×C4—31no—
149C2×C3×C4—31no—
150C2×C3×C4—61no—
151C2×C3×C4—92no—
152C2×C3×C4D3×Dic311yescharacteristic
153#67—32no—
154#67—92no—
155D12—31no—
156D12—91no—
157C4×D3—32no—
158C4×D3—92no—
159Dic6—31no—
160Dic6—91no—
161C3×C8—31no—
162C3×C8—31no—
163C3×C8—31no—
164C3×C8—31no—
165C3×C8—61no—
166C3×C8—61no—
167C3×C8—92no—
168C3×C8D3×Dic311yescharacteristic
169C3×C8C2×D3211yescharacteristic
170C3⋊C8—32no—
171C3⋊C8—92no—
172C2×C32—21no—
173C2×C32—21no—
174C2×C32—21no—
175C2×C32—41no—
176C2×C32C2×D3×C811yescharacteristic
177C2×C32C22×C3⋊C811yescharacteristic
178C3⋊D3—22no—
179C3×D3—62no—
180#41—91no—
181M16—91no—
182M16—92no—
183C2×C8—92no—
184C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
185C16—91no—
186C16—91no—
187C16—92no—
188C22×C3—31no—
189C22×C3—31no—
190C22×C3—61no—
191C22×C3—92no—
192C22×C3D3×C3⋊C811yescharacteristic
193C2×D3—32no—
194C2×D3—92no—
195C3×C4—31no—
196C3×C4—31no—
197C3×C4—31no—
198C3×C4—31no—
199C3×C4—61no—
200C3×C4—61no—
201C3×C4—92no—
202C3×C4D3×C3⋊C811yescharacteristic
203C3×C4C2×D3×Dic311yescharacteristic
204Dic3—32no—
205Dic3—92no—
206C32—21no—
207C32#490111yescharacteristic, complemented
208C32#514011yescharacteristic, complemented
209Q8—91no—
210D4—92no—
211D4—91no—
212C2×C4—92no—
213C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
214C8—92no—
215C8((C3⋊D3)⋊C3)⋊C411yescharacteristic
216C8C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
217C2×C3—21no—
218C2×C3—31no—
219C2×C3—31no—
220C2×C3—61no—
221C2×C3—61no—
222C2×C3—61no—
223C2×C3—121no—
224C2×C3—182no—
225C2×C3C2×D3×C3⋊C811yescharacteristic
226D3—62no—
227D3—182no—
228C22—92no—
229C22((C3⋊D3)⋊C3)⋊C811yescharacteristic
230C4—92no—
231C4((C3⋊D3)⋊C3)⋊C811yescharacteristic
232C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
233C3—31no—
234C3—31no—
235C3—61no—
236C3#1059062111yescharacteristic
237C2—21no—
238C2—182no—
239C2C2×((C3⋊D3)⋊C3)⋊C811yescharacteristic
subgroupG/H
centerC8C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C22×C8
FrattiniC3×C8C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC2×C3C2×D3×C3⋊C8
prime radical#3828⋊C3C4
cosocleC8×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#504
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C8 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11720545C2semidirect
3. C2×(C3⋊(C3⋊C16))⋊C3C2semidirect
4. #11720247C2semidirect
5. ((C3⋊D3)⋊C3)⋊C16C2semidirect
18. C2×C3⋊(C3⋊C16)D3semidirect
144. H(3,𝔽3)#504semidirect
207. C32#4901semidirect
208. C32#5140semidirect

Maximal subgroups

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

subgroupindex
#64110⋊C32
#117205452
C2×(C3⋊(C3⋊C16))⋊C32
#117202472
((C3⋊D3)⋊C3)⋊C162
#105906243
#105906223