Finite Group Databasebeta

← #14517693

Subgroups — #14517693

Order 1944.

612 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 60 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⋊#4174C211yesmaximal, characteristic, complemented
3C3⋊#66321C211yesmaximal, characteristic, complemented
4C32×#638—31nomaximal
5#10619994C311yesmaximal, characteristic, complemented
6#10619641—33nomaximal
7C2×C3×AΣL(1,𝔽27)—21no—
8C2×C3×AΣL(1,𝔽27)C2211yescharacteristic
9C3⋊#4174—21no—
10C22×C34D311yescharacteristic, complemented
11C2×C32×C3⋊D3—31no—
12C32×C3⋊Dic3—31no—
13C22×C3×3−1+2—21no—
14C22×C3×H(3,𝔽3)C2×C311yescharacteristic, complemented
15C2×C3⋊#272C2×C311yescharacteristic, complemented
16C3⋊#1084C2×C311yescharacteristic, complemented
17C22×AΣL(1,𝔽27)—23no—
18C22×AΣL(1,𝔽27)D313yescomplemented
19C2×#4174—33no—
20#66321—33no—
21C3×AΣL(1,𝔽27)D411yescharacteristic, complemented, Sylow
22C3×#638—33no—
23C3×#638—31no—
24C3×#638—91no—
25C32×#67—92no—
26C32×#67—91no—
27C32×#67—91no—
28#6235—33no—
29#6235—92no—
30C2×C34—21no—
31C2×C34C2×D311yescharacteristic
32C32×C3⋊D3—61no—
33C2×C3×3−1+2—22no—
34C2×C3×3−1+2—21no—
35C2×C3×H(3,𝔽3)—21no—
36C2×C3×H(3,𝔽3)C22×C311yescharacteristic
37C3⋊#272—21no—
38C2×AΣL(1,𝔽27)—26no—
39C2×AΣL(1,𝔽27)—23no—
40C2×AΣL(1,𝔽27)—23no—
41C2×AΣL(1,𝔽27)C2×D313yes—
42#4174—63no—
43C22×C33—21no—
44C22×C33—33no—
45C22×C33—32no—
46C22×C33—31no—
47C22×C33—31no—
48C22×C33—63no—
49C22×C33C3×D311yescharacteristic
50C22×C33C3⋊D311yescharacteristic, complemented
51C2×C3×C3⋊D3—33no—
52C2×C3×C3⋊D3—31no—
53C2×C3×C3⋊D3—91no—
54C2×C32×D3—92no—
55C2×C32×D3—91no—
56C2×C32×D3—91no—
57C3×C3⋊Dic3—33no—
58C3×C3⋊Dic3—31no—
59C3×C3⋊Dic3—91no—
60C32×Dic3—92no—
61C32×Dic3—91no—
62C32×Dic3—91no—
63C22×3−1+2—23no—
64C22×3−1+2—62no—
65C22×H(3,𝔽3)—32no—
66C22×H(3,𝔽3)C3×D313yescomplemented
67C22×C3×C9—61no—
68C2×#272—33no—
69C2×#272—92no—
70#1084—33no—
71#1084—92no—
72C34#6711yescharacteristic, complemented
73C3×3−1+2—21no—
74C3×H(3,𝔽3)C3×D411yescharacteristic, complemented
75AΣL(1,𝔽27)—23no—
76AΣL(1,𝔽27)#6713yescomplemented
77C32×D4—271no—
78#638—31no—
79#638—91no—
80C3×#67—93no—
81C3×#67—96no—
82C3×#67—93no—
83C3×#67—92no—
84C3×#67—91no—
85C3×#67—91no—
86C3×#67—273no—
87C2×C33—21no—
88C2×C33—22no—
89C2×C33—21no—
90C2×C33—21no—
91C2×C33—33no—
92C2×C33—32no—
93C2×C33—31no—
94C2×C33—31no—
95C2×C33—61no—
96C2×C33—62no—
97C2×C33—61no—
98C2×C33—63no—
99C2×C33—63no—
100C2×C33—66no—
101C2×C33C2×C3×D311yescharacteristic
102C2×C33C2×C3⋊D311yescharacteristic
103C3×C3⋊D3—63no—
104C3×C3⋊D3—61no—
105C3×C3⋊D3—181no—
106C32×D3—182no—
107C32×D3—181no—
108C32×D3—181no—
109C2×3−1+2—26no—
110C2×3−1+2—23no—
111C2×3−1+2—64no—
112C2×3−1+2—62no—
113C2×H(3,𝔽3)—23no—
114C2×H(3,𝔽3)—32no—
115C2×H(3,𝔽3)—62no—
116C2×H(3,𝔽3)C2×C3×D313yes—
117C2×C3×C9—62no—
118C2×C3×C9—61no—
119#272—63no—
120#272—182no—
121C22×C32—31no—
122C22×C32—31no—
123C22×C32—33no—
124C22×C32—36no—
125C22×C32—33no—
126C22×C32—33no—
127C22×C32—61no—
128C22×C32—63no—
129C22×C32—66no—
130C22×C32—63no—
131C22×C32—61no—
132C22×C32—91no—
133C22×C32—181no—
134C22×C32—271no—
135C22×C32(C3⋊D3)⋊C311yescharacteristic
136C22×C32(C3⋊D3)⋊C312yes—
137C22×C32C3×C3⋊D311yescharacteristic
138C2×C3⋊D3—31no—
139C2×C3⋊D3—91no—
140C2×C3×D3—93no—
141C2×C3×D3—96no—
142C2×C3×D3—93no—
143C2×C3×D3—92no—
144C2×C3×D3—91no—
145C2×C3×D3—91no—
146C2×C3×D3—273no—
147C32×C4—271no—
148C3⋊Dic3—31no—
149C3⋊Dic3—91no—
150C3×Dic3—93no—
151C3×Dic3—96no—
152C3×Dic3—93no—
153C3×Dic3—92no—
154C3×Dic3—91no—
155C3×Dic3—91no—
156C3×Dic3—273no—
157C22×C9—63no—
158C33—21no—
159C33—33no—
160C33—32no—
161C33—31no—
162C33—31no—
163C33—63no—
164C33C3×#6711yescharacteristic
165C33#63811yescharacteristic, complemented
1663−1+2—23no—
1673−1+2—62no—
168H(3,𝔽3)—32no—
169H(3,𝔽3)C3×#6713yescomplemented
170C3×C9—61no—
171C3×D4—273no—
172C3×D4—271no—
173#67—92no—
174#67—91no—
175#67—91no—
176#67—273no—
177C2×C32—21no—
178C2×C32—22no—
179C2×C32—21no—
180C2×C32—31no—
181C2×C32—31no—
182C2×C32—33no—
183C2×C32—36no—
184C2×C32—33no—
185C2×C32—33no—
186C2×C32—61no—
187C2×C32—63no—
188C2×C32—61no—
189C2×C32—61no—
190C2×C32—63no—
191C2×C32—62no—
192C2×C32—62no—
193C2×C32—66no—
194C2×C32—63no—
195C2×C32—66no—
196C2×C32—63no—
197C2×C32—66no—
198C2×C32—612no—
199C2×C32—61no—
200C2×C32—63no—
201C2×C32—66no—
202C2×C32—91no—
203C2×C32—181no—
204C2×C32—181no—
205C2×C32—182no—
206C2×C32—541no—
207C2×C32C2×(C3⋊D3)⋊C311yescharacteristic
208C2×C32C2×(C3⋊D3)⋊C312yes—
209C2×C32C2×C3×C3⋊D311yescharacteristic
210C3⋊D3—61no—
211C3⋊D3—181no—
212C3×D3—183no—
213C3×D3—186no—
214C3×D3—183no—
215C3×D3—182no—
216C3×D3—181no—
217C3×D3—181no—
218C3×D3—543no—
219C2×C9—66no—
220C2×C9—63no—
221C22×C3—21no—
222C22×C3—32no—
223C22×C3—31no—
224C22×C3—33no—
225C22×C3—63no—
226C22×C3—93no—
227C22×C3—273no—
228C22×C3—271no—
229C22×C3#417411yescharacteristic, complemented
230C22×C3C3⋊((C3⋊D3)⋊C3)11yescharacteristic
231C2×D3—92no—
232C2×D3—91no—
233C2×D3—91no—
234C2×D3—273no—
235C3×C4—273no—
236C3×C4—271no—
237Dic3—92no—
238Dic3—91no—
239Dic3—91no—
240Dic3—273no—
241C32—33no—
242C32—31no—
243C32—31no—
244C32—36no—
245C32—33no—
246C32—33no—
247C32—61no—
248C32—63no—
249C32—66no—
250C32—63no—
251C32—61no—
252C32—91no—
253C32—181no—
254C32#638⋊C311yescharacteristic
255C32#638⋊C312yes—
256C32C3×#63811yescharacteristic
257C9—63no—
258D4—271noSylow
259C2×C3—22no—
260C2×C3—21no—
261C2×C3—21no—
262C2×C3—21no—
263C2×C3—32no—
264C2×C3—31no—
265C2×C3—33no—
266C2×C3—62no—
267C2×C3—61no—
268C2×C3—63no—
269C2×C3—66no—
270C2×C3—63no—
271C2×C3—93no—
272C2×C3—183no—
273C2×C3—543no—
274C2×C3—541no—
275C2×C3C2×#417411yescharacteristic
276C2×C3C2×C3⋊((C3⋊D3)⋊C3)11yescharacteristic
277D3—182no—
278D3—181no—
279D3—181no—
280D3—543no—
281C22—271no—
282C22C3⋊#417411yescharacteristic, complemented
283C4—271no—
284C3—21no—
285C3—32no—
286C3—31no—
287C3—33no—
288C3—63no—
289C3—93no—
290C3#1061964111yescharacteristic, complemented
291C3#1061997711yescharacteristic
292C2—21no—
293C2—541no—
294C2C2×C3⋊#417411yescharacteristic
subgroupG/H
centerC2×C3C2×C3⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C3×H(3,𝔽3)C22×C3
FrattiniC2×C32C2×C3×C3⋊D3
FittingC22×C3×AΣL(1,𝔽27)C2
socleC2×C32C2×(C3⋊D3)⋊C3
hypercenterC22×C3C3⋊((C3⋊D3)⋊C3)
hypocenterC3×H(3,𝔽3)C3×D4
LCSG ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C22×C3×AΣL(1,𝔽27)C2semidirect
2. C2×C3⋊#4174C2semidirect
3. C3⋊#66321C2semidirect
5. #10619994C3semidirect
10. C22×C34D3semidirect
14. C22×C3×H(3,𝔽3)C2×C3semidirect
15. C2×C3⋊#272C2×C3semidirect
16. C3⋊#1084C2×C3semidirect
18. C22×AΣL(1,𝔽27)D3semidirect
21. C3×AΣL(1,𝔽27)D4semidirect
50. C22×C33C3⋊D3semidirect
66. C22×H(3,𝔽3)C3×D3semidirect
72. C34#67semidirect
74. C3×H(3,𝔽3)C3×D4semidirect
76. AΣL(1,𝔽27)#67semidirect
165. C33#638semidirect
169. H(3,𝔽3)C3×#67semidirect
229. C22×C3#4174semidirect
282. C22C3⋊#4174semidirect
290. C3#10619641semidirect

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⋊#41742
C3⋊#663212
C32×#6383
#106199943
#106196413