Finite Group Databasebeta

← #13138716

Subgroups — #13138716

Order 1728.

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

normal subgroups only

This group has 322 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
2C2×#88386C211yesmaximal, characteristic, complemented
3#63639⋊C3C211yesmaximal, characteristic, complemented
4#11720247C212yesmaximal, complemented
5((C3⋊D3)⋊C3)⋊C16C212yesmaximal, complemented
6#10590624—31nomaximal
7#10590621—31nomaximal
8#3828⋊C3C411yescharacteristic
9C2×C8×H(3,𝔽3)C2211yescharacteristic
10#3748⋊C3C411yescharacteristic
11#3743⋊C3C412yes—
12C8×(C3⋊D3)⋊C3C2212yes—
13#88386C2211yescharacteristic
14#88386C2211yescharacteristic
15(C3⋊(C3⋊C16))⋊C3C2212yes—
16#64110D311yescharacteristic
17C3×#901—31no—
18#63639D311yescharacteristic, complemented
19C3×#812—31no—
20C2×C3×C3⋊C16—31no—
21C2×C3×C3⋊C16—31no—
22C2×C3×C3⋊C16—61no—
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×D312yes—
50C3×C3⋊C16—31no—
51C3×C3⋊C16—31no—
52C3×C3⋊C16—32no—
53C3×C3⋊C16—31no—
54C3×C3⋊C16—31no—
55C3×C3⋊C16—61no—
56C3×C3⋊C16—61no—
57C22×H(3,𝔽3)C2×C811yescharacteristic
58C2×(C3⋊D3)⋊C3C2×C812yes—
59C4×H(3,𝔽3)C2×C811yescharacteristic
60C4×H(3,𝔽3)C22×C411yescharacteristic
61(C3⋊Dic3)⋊C3C2×C812yes—
62C3×#131—91no—
63#901—31no—
64#901—91no—
65C3×M32—91no—
66C2×C3×C16—91no—
67#812—31no—
68#812—91no—
69#812—92no—
70C2×C3⋊C16—31no—
71C2×C3⋊C16—31no—
72C2×C3⋊C16—61no—
73C2×C3⋊C16—92no—
74#798—92no—
75#798—92no—
76D3×C16—92no—
77D3×C16—92no—
78C2×C32×C4—21no—
79C2×C32×C4C4×D311yescharacteristic
80C2×C32×C4C2×Dic311yescharacteristic
81#638C3⋊C812yes—
82C4⋊(C3⋊D3)C3⋊C811yescharacteristic
83C4×C3⋊D3C2×Dic312yes—
84#634C3⋊C811yescharacteristic
85C3×#67—32no—
86C3×D12—31no—
87C3×C4×D3—32no—
88C3×Dic6—31no—
89C32×C8—21no—
90C32×C8—21no—
91C32×C8C4×D311yescharacteristic
92C32×C8C2×Dic311yescharacteristic
93C32×C8C22×D311yescharacteristic
94C32×C8C22×D311yescharacteristic
95C3⋊(C3⋊C8)C2×Dic312yes—
96C3×C3⋊C8—32no—
97#504—271noSylow
98C2×H(3,𝔽3)—21no—
99C2×H(3,𝔽3)C22×C811yescharacteristic
100(C3⋊D3)⋊C3—22no—
101C3×#41—91no—
102#235—31no—
103#235—91no—
104C3×M16—92no—
105C3×M16—91no—
106C2×C3×C8—31no—
107C2×C3×C8—31no—
108C2×C3×C8—61no—
109C2×C3×C8—92no—
110C2×C3×C8D3211yescharacteristic
111#208—31no—
112#208—91no—
113ΓL(1,𝔽25)—32no—
114ΓL(1,𝔽25)—92no—
115D3×C8—32no—
116D3×C8—92no—
117C3×C16—92no—
118C3×C16—91no—
119C3×C16—91no—
120C3⋊C16—31no—
121C3⋊C16—31no—
122C3⋊C16—31no—
123C3⋊C16—31no—
124C3⋊C16—32no—
125C3⋊C16—61no—
126C3⋊C16—61no—
127C3⋊C16—92no—
128C22×C32—21no—
129C22×C32D3×C811yescharacteristic
130C22×C32C2×C3⋊C811yescharacteristic
131C2×C3⋊D3C2×C3⋊C812yes—
132C2×C3×D3—32no—
133C32×C4—21no—
134C32×C4—21no—
135C32×C4D3×C811yescharacteristic
136C32×C4C2×C3⋊C811yescharacteristic
137C32×C4C2×C4×D311yescharacteristic
138C32×C4C22×Dic311yescharacteristic
139C3⋊Dic3C2×C3⋊C812yes—
140C3×Dic3—32no—
141#131—91no—
142M32—91no—
143M32—272no—
144C2×C16—91no—
145C2×C16—272no—
146H(3,𝔽3)#50411yescharacteristic, complemented, Sylow
147C3×Q8—91no—
148C3×D4—91no—
149C3×D4—92no—
150C2×C3×C4—31no—
151C2×C3×C4—31no—
152C2×C3×C4—61no—
153C2×C3×C4—92no—
154C2×C3×C4D3×Dic311yescharacteristic
155#67—32no—
156#67—92no—
157D12—31no—
158D12—91no—
159C4×D3—32no—
160C4×D3—92no—
161Dic6—31no—
162Dic6—91no—
163C3×C8—31no—
164C3×C8—31no—
165C3×C8—31no—
166C3×C8—31no—
167C3×C8—61no—
168C3×C8—61no—
169C3×C8—92no—
170C3×C8D3×Dic311yescharacteristic
171C3×C8C2×D3211yescharacteristic
172C3⋊C8—32no—
173C3⋊C8—92no—
174C2×C32—22no—
175C2×C32—21no—
176C2×C32—21no—
177C2×C32—21no—
178C2×C32C2×D3×C811yescharacteristic
179C2×C32C22×C3⋊C811yescharacteristic
180C3⋊D3—22no—
181C3×D3—62no—
182#41—91no—
183M16—91no—
184M16—92no—
185C2×C8—92no—
186C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
187C16—91no—
188C16—91no—
189C16—92no—
190C22×C3—31no—
191C22×C3—31no—
192C22×C3—61no—
193C22×C3—92no—
194C22×C3D3×C3⋊C811yescharacteristic
195C2×D3—32no—
196C2×D3—92no—
197C3×C4—31no—
198C3×C4—31no—
199C3×C4—31no—
200C3×C4—31no—
201C3×C4—61no—
202C3×C4—61no—
203C3×C4—92no—
204C3×C4D3×C3⋊C811yescharacteristic
205C3×C4C2×D3×Dic311yescharacteristic
206Dic3—32no—
207Dic3—92no—
208C32—21no—
209C32#490711yescharacteristic, complemented
210C32#514011yescharacteristic, complemented
211Q8—91no—
212D4—92no—
213D4—91no—
214C2×C4—92no—
215C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
216C8—92no—
217C8((C3⋊D3)⋊C3)⋊C411yescharacteristic
218C8C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
219C2×C3—21no—
220C2×C3—31no—
221C2×C3—31no—
222C2×C3—61no—
223C2×C3—61no—
224C2×C3—62no—
225C2×C3—61no—
226C2×C3—182no—
227C2×C3C2×D3×C3⋊C811yescharacteristic
228D3—62no—
229D3—182no—
230C22—92no—
231C22((C3⋊D3)⋊C3)⋊C811yescharacteristic
232C4—92no—
233C4((C3⋊D3)⋊C3)⋊C811yescharacteristic
234C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
235C3—31no—
236C3—31no—
237C3—61no—
238C3#1059062211yescharacteristic
239C2—21no—
240C2—182no—
241C2C2×((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. C2×#88386C2semidirect
3. #63639⋊C3C2semidirect
4. #11720247C2semidirect
5. ((C3⋊D3)⋊C3)⋊C16C2semidirect
18. #63639D3semidirect
146. H(3,𝔽3)#504semidirect
209. C32#4907semidirect
210. C32#5140semidirect

Maximal subgroups

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

subgroupindex
#64110⋊C32
C2×#883862
#63639⋊C32
#117202472
((C3⋊D3)⋊C3)⋊C162
#105906243
#105906213