Finite Group Databasebeta

← #14517694

Subgroups — #14517694

Order 1944.

832 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 832 conjugacy classes of subgroups — too many to draw. Only its 66 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
1C22×C3×AΣL(1,𝔽27)C211yesmaximal, characteristic, complemented
2C2×C3⋊#4175C211yesmaximal, characteristic, complemented
3C3⋊#66322C211yesmaximal, characteristic, complemented
4#10620179C311yesmaximal, characteristic, complemented
5#10619977—31nomaximal
6#10619642—33nomaximal
7C2×C3×AΣL(1,𝔽27)—21no—
8C2×C3×AΣL(1,𝔽27)C2211yescharacteristic
9C3⋊#4175—21no—
10C22×C34C2×C311yescharacteristic, complemented
11C2×C3⋊(C3⋊(C3⋊D3))C2×C311yescharacteristic, complemented
12C3⋊(C3⋊(C3⋊Dic3))C2×C311yescharacteristic, complemented
13C22×C3×3−1+2—21no—
14C22×C3×H(3,𝔽3)D311yescharacteristic, complemented
15C2×C3⋊((C3⋊D3)⋊C3)—31no—
16C3⋊((C3⋊Dic3)⋊C3)—31no—
17C22×AΣL(1,𝔽27)—23no—
18C22×AΣL(1,𝔽27)D313yescomplemented
19C2×#4175—33no—
20#66322—33no—
21C3×AΣL(1,𝔽27)D411yescharacteristic, complemented, Sylow
22#6312—31no—
23#6312—33no—
24#6312—91no—
25#6312—92no—
26#6312—99no—
27C3×#638—91no—
28#638⋊C3—92no—
29#638⋊C3—91no—
30C2×C34—21no—
31C2×C34C22×C311yescharacteristic
32C3⋊(C3⋊(C3⋊D3))—21no—
33C2×C3×3−1+2—22no—
34C2×C3×3−1+2—21no—
35C2×C3×H(3,𝔽3)—21no—
36C2×C3×H(3,𝔽3)C2×D311yescharacteristic
37C3⋊((C3⋊D3)⋊C3)—61no—
38C2×AΣL(1,𝔽27)—26no—
39C2×AΣL(1,𝔽27)—23no—
40C2×AΣL(1,𝔽27)—23no—
41C2×AΣL(1,𝔽27)C2×D313yes—
42#4175—63no—
43C22×C33—39no—
44C22×C33—32no—
45C22×C33—31no—
46C22×C33—31no—
47C22×C33C3×D311yescharacteristic
48C22×C33C3×D313yescomplemented
49C2×C3⋊(C3⋊D3)—31no—
50C2×C3⋊(C3⋊D3)—33no—
51C2×C3⋊(C3⋊D3)—91no—
52C2×C3⋊(C3⋊D3)—92no—
53C2×C3⋊(C3⋊D3)—99no—
54C2×C3×C3⋊D3—91no—
55C3⋊(C3⋊Dic3)—31no—
56C3⋊(C3⋊Dic3)—33no—
57C3⋊(C3⋊Dic3)—91no—
58C3⋊(C3⋊Dic3)—92no—
59C3⋊(C3⋊Dic3)—99no—
60C3×C3⋊Dic3—91no—
61C22×3−1+2—23no—
62C22×3−1+2—62no—
63C22×H(3,𝔽3)—21no—
64C22×H(3,𝔽3)—32no—
65C22×H(3,𝔽3)C3⋊D311yescharacteristic, complemented
66C22×C3×C9—61no—
67C2×(C3⋊D3)⋊C3—92no—
68C2×(C3⋊D3)⋊C3—91no—
69(C3⋊Dic3)⋊C3—92no—
70(C3⋊Dic3)⋊C3—91no—
71C34C3×D411yescharacteristic, complemented
72C3×3−1+2—21no—
73C3×H(3,𝔽3)#6711yescharacteristic, complemented
74AΣL(1,𝔽27)—23no—
75AΣL(1,𝔽27)#6713yescomplemented
76#638—91no—
77#638—92no—
78#638—91no—
79#638—273no—
80#638—273no—
81#638—279no—
82#638—2718no—
83#638—279no—
84C3×#67—271no—
85C3×#67—273no—
86C2×C33—21no—
87C2×C33—23no—
88C2×C33—32no—
89C2×C33—31no—
90C2×C33—39no—
91C2×C33—31no—
92C2×C33—61no—
93C2×C33—62no—
94C2×C33—61no—
95C2×C33—69no—
96C2×C33C2×C3×D313yes—
97C2×C33C2×C3×D311yescharacteristic
98C3⋊(C3⋊D3)—61no—
99C3⋊(C3⋊D3)—63no—
100C3⋊(C3⋊D3)—181no—
101C3⋊(C3⋊D3)—182no—
102C3⋊(C3⋊D3)—189no—
103C3×C3⋊D3—181no—
104C2×3−1+2—26no—
105C2×3−1+2—23no—
106C2×3−1+2—64no—
107C2×3−1+2—62no—
108C2×H(3,𝔽3)—22no—
109C2×H(3,𝔽3)—21no—
110C2×H(3,𝔽3)—21no—
111C2×H(3,𝔽3)—32no—
112C2×H(3,𝔽3)—62no—
113C2×H(3,𝔽3)C2×C3⋊D311yescharacteristic
114C2×C3×C9—62no—
115C2×C3×C9—61no—
116(C3⋊D3)⋊C3—182no—
117(C3⋊D3)⋊C3—181no—
118C22×C32—318no—
119C22×C32—33no—
120C22×C32—39no—
121C22×C32—33no—
122C22×C32—39no—
123C22×C32—31no—
124C22×C32—61no—
125C22×C32—93no—
126C22×C32(C3⋊D3)⋊C312yes—
127C22×C32(C3⋊D3)⋊C311yescharacteristic
128C22×C32C3×C3⋊D311yescharacteristic
129C2×C3⋊D3—91no—
130C2×C3⋊D3—92no—
131C2×C3⋊D3—91no—
132C2×C3⋊D3—273no—
133C2×C3⋊D3—273no—
134C2×C3⋊D3—279no—
135C2×C3⋊D3—2718no—
136C2×C3⋊D3—279no—
137C2×C3×D3—271no—
138C2×C3×D3—273no—
139C3⋊Dic3—91no—
140C3⋊Dic3—92no—
141C3⋊Dic3—91no—
142C3⋊Dic3—273no—
143C3⋊Dic3—273no—
144C3⋊Dic3—279no—
145C3⋊Dic3—2718no—
146C3⋊Dic3—279no—
147C3×Dic3—271no—
148C3×Dic3—273no—
149C22×C9—63no—
150C33—39no—
151C33—32no—
152C33—31no—
153C33—31no—
154C33C3×#6711yescharacteristic
155C33C3×#6713yescomplemented
1563−1+2—23no—
1573−1+2—62no—
158H(3,𝔽3)—21no—
159H(3,𝔽3)—32no—
160H(3,𝔽3)#63811yescharacteristic, complemented
161C3×C9—61no—
162C3×D4—811no—
163#67—273no—
164#67—271no—
165#67—812no—
166#67—811no—
167#67—819no—
168C2×C32—21no—
169C2×C32—22no—
170C2×C32—21no—
171C2×C32—318no—
172C2×C32—33no—
173C2×C32—39no—
174C2×C32—33no—
175C2×C32—39no—
176C2×C32—31no—
177C2×C32—63no—
178C2×C32—63no—
179C2×C32—69no—
180C2×C32—61no—
181C2×C32—62no—
182C2×C32—618no—
183C2×C32—69no—
184C2×C32—61no—
185C2×C32—93no—
186C2×C32—183no—
187C2×C32C2×(C3⋊D3)⋊C312yes—
188C2×C32C2×(C3⋊D3)⋊C311yescharacteristic
189C2×C32C2×C3×C3⋊D311yescharacteristic
190C3⋊D3—181no—
191C3⋊D3—182no—
192C3⋊D3—181no—
193C3⋊D3—543no—
194C3⋊D3—543no—
195C3⋊D3—549no—
196C3⋊D3—5418no—
197C3⋊D3—549no—
198C3×D3—541no—
199C3×D3—543no—
200C2×C9—66no—
201C2×C9—63no—
202C22×C3—39no—
203C22×C3—32no—
204C22×C3—31no—
205C22×C3—91no—
206C22×C3—181no—
207C22×C3—811no—
208C22×C3#417513yescomplemented
209C22×C3C3⋊((C3⋊D3)⋊C3)11yescharacteristic
210C2×D3—273no—
211C2×D3—271no—
212C2×D3—812no—
213C2×D3—811no—
214C2×D3—819no—
215C3×C4—811no—
216Dic3—273no—
217Dic3—271no—
218Dic3—812no—
219Dic3—811no—
220Dic3—819no—
221C32—39no—
222C32—318no—
223C32—33no—
224C32—39no—
225C32—33no—
226C32—31no—
227C32—61no—
228C32—93no—
229C32#638⋊C312yes—
230C32#638⋊C311yescharacteristic
231C32C3×#63811yescharacteristic
232C9—63no—
233D4—811noSylow
234C2×C3—21no—
235C2×C3—23no—
236C2×C3—39no—
237C2×C3—32no—
238C2×C3—31no—
239C2×C3—62no—
240C2×C3—61no—
241C2×C3—69no—
242C2×C3—91no—
243C2×C3—181no—
244C2×C3—182no—
245C2×C3—181no—
246C2×C3—1621no—
247C2×C3C2×#417513yes—
248C2×C3C2×C3⋊((C3⋊D3)⋊C3)11yescharacteristic
249D3—541no—
250D3—543no—
251D3—1622no—
252D3—1621no—
253D3—1629no—
254C22—811no—
255C22C3⋊#417511yescharacteristic, complemented
256C4—811no—
257C3—39no—
258C3—32no—
259C3—31no—
260C3—91no—
261C3—181no—
262C3#1061964213yescomplemented
263C3#1061997711yescharacteristic
264C2—21no—
265C2—1621no—
266C2C2×C3⋊#417511yescharacteristic
subgroupG/H
centerC2C2×C3⋊#4175
commutator, cosocleC2×C34C22×C3
FrattiniC2×C32C2×C3×C3⋊D3
FittingC22×C3×AΣL(1,𝔽27)C2
socleC2×C32C2×(C3⋊D3)⋊C3
hypercenterC22C3⋊#4175
hypocenterC34C3×D4
LCSG ▷ C2×C34 ▷ C34
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3×AΣL(1,𝔽27)C2semidirect
2. C2×C3⋊#4175C2semidirect
3. C3⋊#66322C2semidirect
4. #10620179C3semidirect
10. C22×C34C2×C3semidirect
11. C2×C3⋊(C3⋊(C3⋊D3))C2×C3semidirect
12. C3⋊(C3⋊(C3⋊Dic3))C2×C3semidirect
14. C22×C3×H(3,𝔽3)D3semidirect
18. C22×AΣL(1,𝔽27)D3semidirect
21. C3×AΣL(1,𝔽27)D4semidirect
48. C22×C33C3×D3semidirect
65. C22×H(3,𝔽3)C3⋊D3semidirect
71. C34C3×D4semidirect
73. C3×H(3,𝔽3)#67semidirect
75. AΣL(1,𝔽27)#67semidirect
155. C33C3×#67semidirect
160. H(3,𝔽3)#638semidirect
208. C22×C3#4175semidirect
255. C22C3⋊#4175semidirect
262. C3#10619642semidirect

Maximal subgroups

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

subgroupindex
C22×C3×AΣL(1,𝔽27)2
C2×C3⋊#41752
C3⋊#663222
#106201793
#106199773
#106196423