Finite Group Databasebeta

← C23×C3×#4183 #72276188

Subgroups — C23×C3×#4183 #72276188

Order 3888.

4430 conjugacy classes of subgroups; only the 390 normal subgroups are shown.

normal subgroups only

This group has 4430 conjugacy classes of subgroups — too many to draw. Only its 390 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
1C23×C3×AΣL(1,𝔽27)C211yesmaximal, characteristic, complemented
2C22×C3×#4183C2114yesmaximal, direct product factor
3C23×C32×C3⋊D3—31nomaximal
4C23×C3×AGL(1,ℤ/9)—32nomaximal
5C23×C3×(C3⋊D3)⋊C3—31nomaximal
6C23×#4183C311yesmaximal, characteristic, direct product factor
7C22×C3×AΣL(1,𝔽27)C2217yescomplemented
8C2×C3×#4183C22128yesdirect product factor
9C23×C34D311yescharacteristic, complemented
10C22×C32×C3⋊D3—314no—
11C23×C3×3−1+2D312yescomplemented
12C23×C3×H(3,𝔽3)D311yescharacteristic, complemented
13C22×C3×AGL(1,ℤ/9)—328no—
14C22×C3×(C3⋊D3)⋊C3—314no—
15C23×AΣL(1,𝔽27)—21no—
16C23×AΣL(1,𝔽27)—22no—
17C23×AΣL(1,𝔽27)—21no—
18C23×AΣL(1,𝔽27)C2×C311yescharacteristic, complemented
19C22×#4183C2×C3114yesdirect product factor
20C2×C3×AΣL(1,𝔽27)C2317yescomplemented
21C3×#4183C2318yesdirect product factor
22C23×C3×C3⋊D3—31no—
23C23×C3×C3⋊D3—32no—
24C23×C3×C3⋊D3—31no—
25C23×C3×C3⋊D3—31no—
26C23×C32×D3—93no—
27C23×C32×D3—91no—
28C23×AGL(1,ℤ/9)—34no—
29C23×AGL(1,ℤ/9)—32no—
30C23×(C3⋊D3)⋊C3—32no—
31C23×(C3⋊D3)⋊C3—31no—
32C23×C3×D9—32no—
33C22×C34C2×D317yescomplemented
34C2×C32×C3⋊D3—328no—
35C22×C3×3−1+2C2×D3114yescomplemented
36C22×C3×H(3,𝔽3)C2×D317yescomplemented
37C2×C3×AGL(1,ℤ/9)—356no—
38C2×C3×(C3⋊D3)⋊C3—328no—
39C22×AΣL(1,𝔽27)—27no—
40C22×AΣL(1,𝔽27)—214no—
41C22×AΣL(1,𝔽27)—27no—
42C22×AΣL(1,𝔽27)C22×C317yescomplemented
43C2×#4183C22×C3128yesdirect product factor
44C3×AΣL(1,𝔽27)C2411yescharacteristic, complemented, Sylow
45C23×C33—21no—
46C23×C33—33no—
47C23×C33—32no—
48C23×C33—31no—
49C23×C33—31no—
50C23×C33—63no—
51C23×C33C3×D311yescharacteristic, complemented
52C23×C33C3⋊D311yescharacteristic
53C22×C3×C3⋊D3—314no—
54C22×C3×C3⋊D3—328no—
55C22×C3×C3⋊D3—314no—
56C22×C3×C3⋊D3—314no—
57C22×C32×D3—942no—
58C22×C32×D3—914no—
59C23×3−1+2—22no—
60C23×3−1+2—34no—
61C23×3−1+2C3×D312yescomplemented
62C23×H(3,𝔽3)—21no—
63C23×H(3,𝔽3)—32no—
64C23×H(3,𝔽3)C3×D311yescharacteristic, complemented
65C23×C3×C9—32no—
66C22×AGL(1,ℤ/9)—356no—
67C22×AGL(1,ℤ/9)—328no—
68C22×(C3⋊D3)⋊C3—328no—
69C22×(C3⋊D3)⋊C3—314no—
70C22×C3×D9—328no—
71C2×C34C22×D317yescomplemented
72C32×C3⋊D3—38no—
73C2×C3×3−1+2C22×D3114yescomplemented
74C2×C3×H(3,𝔽3)C22×D317yescomplemented
75C3×AGL(1,ℤ/9)—316no—
76C3×(C3⋊D3)⋊C3—38no—
77C2×AΣL(1,𝔽27)—27no—
78C2×AΣL(1,𝔽27)—214no—
79C2×AΣL(1,𝔽27)—27no—
80C2×AΣL(1,𝔽27)C23×C317yescomplemented
81#4183C23×C318yesdirect product factor
82C24×C32—271no—
83C23×C3⋊D3—31no—
84C23×C3⋊D3—31no—
85C23×C3×D3—93no—
86C23×C3×D3—96no—
87C23×C3×D3—93no—
88C23×C3×D3—91no—
89C23×C3×D3—92no—
90C23×C3×D3—91no—
91C23×C3×D3—271no—
92C23×D9—32no—
93C22×C33—27no—
94C22×C33—321no—
95C22×C33—314no—
96C22×C33—37no—
97C22×C33—37no—
98C22×C33—621no—
99C22×C33C2×C3×D317yescomplemented
100C22×C33C2×C3⋊D317yes—
101C2×C3×C3⋊D3—328no—
102C2×C3×C3⋊D3—356no—
103C2×C3×C3⋊D3—328no—
104C2×C3×C3⋊D3—328no—
105C2×C32×D3—984no—
106C2×C32×D3—928no—
107C22×3−1+2—214no—
108C22×3−1+2—328no—
109C22×3−1+2C2×C3×D3114yescomplemented
110C22×H(3,𝔽3)—27no—
111C22×H(3,𝔽3)—314no—
112C22×H(3,𝔽3)C2×C3×D317yescomplemented
113C22×C3×C9—314no—
114C2×AGL(1,ℤ/9)—3112no—
115C2×AGL(1,ℤ/9)—356no—
116C2×(C3⋊D3)⋊C3—356no—
117C2×(C3⋊D3)⋊C3—328no—
118C2×C3×D9—356no—
119C34C23×D311yescharacteristic, complemented
120C3×3−1+2C23×D312yescomplemented
121C3×H(3,𝔽3)C23×D311yescharacteristic, complemented
122AΣL(1,𝔽27)—21no—
123AΣL(1,𝔽27)—22no—
124AΣL(1,𝔽27)—21no—
125AΣL(1,𝔽27)C24×C311yescharacteristic, complemented
126C23×C32—31no—
127C23×C32—31no—
128C23×C32—33no—
129C23×C32—36no—
130C23×C32—33no—
131C23×C32—31no—
132C23×C32—61no—
133C23×C32—63no—
134C23×C32—66no—
135C23×C32—63no—
136C23×C32—61no—
137C23×C32—61no—
138C23×C32—91no—
139C23×C32—181no—
140C23×C32—2714no—
141C23×C32#27211yescharacteristic
142C23×C32#27212yes—
143C23×C32C3×C3⋊D311yescharacteristic
144C22×C3⋊D3—314no—
145C22×C3⋊D3—314no—
146C22×C3×D3—942no—
147C22×C3×D3—984no—
148C22×C3×D3—942no—
149C22×C3×D3—914no—
150C22×C3×D3—928no—
151C22×C3×D3—914no—
152C22×C3×D3—2714no—
153C23×C9—32no—
154C23×C9—62no—
155C22×D9—328no—
156C2×C33—27no—
157C2×C33—321no—
158C2×C33—314no—
159C2×C33—37no—
160C2×C33—37no—
161C2×C33—621no—
162C2×C33C22×C3×D317yescomplemented
163C2×C33C22×C3⋊D317yes—
164C3×C3⋊D3—38no—
165C3×C3⋊D3—316no—
166C3×C3⋊D3—38no—
167C3×C3⋊D3—38no—
168C32×D3—924no—
169C32×D3—98no—
170C2×3−1+2—214no—
171C2×3−1+2—328no—
172C2×3−1+2C22×C3×D3114yescomplemented
173C2×H(3,𝔽3)—27no—
174C2×H(3,𝔽3)—314no—
175C2×H(3,𝔽3)C22×C3×D317yescomplemented
176C2×C3×C9—314no—
177AGL(1,ℤ/9)—332no—
178AGL(1,ℤ/9)—316no—
179(C3⋊D3)⋊C3—316no—
180(C3⋊D3)⋊C3—38no—
181C3×D9—316no—
182C24×C3—272no—
183C24×C3—271no—
184C24×C3—271no—
185C23×D3—93no—
186C23×D3—91no—
187C23×D3—271no—
188C22×C32—37no—
189C22×C32—37no—
190C22×C32—321no—
191C22×C32—342no—
192C22×C32—321no—
193C22×C32—37no—
194C22×C32—67no—
195C22×C32—621no—
196C22×C32—642no—
197C22×C32—621no—
198C22×C32—67no—
199C22×C32—67no—
200C22×C32—97no—
201C22×C32—187no—
202C22×C32—2728no—
203C22×C32C2×#27217yes—
204C22×C32C2×#272114yes—
205C22×C32C2×C3×C3⋊D317yes—
206C2×C3⋊D3—328no—
207C2×C3⋊D3—328no—
208C2×C3×D3—984no—
209C2×C3×D3—9168no—
210C2×C3×D3—984no—
211C2×C3×D3—928no—
212C2×C3×D3—956no—
213C2×C3×D3—928no—
214C2×C3×D3—2728no—
215C22×C9—314no—
216C22×C9—614no—
217C2×D9—356no—
218C33—21no—
219C33—33no—
220C33—32no—
221C33—31no—
222C33—31no—
223C33—63no—
224C33C23×C3×D311yescharacteristic, complemented
225C33C23×C3⋊D311yescharacteristic
2263−1+2—22no—
2273−1+2—34no—
2283−1+2C23×C3×D312yescomplemented
229H(3,𝔽3)—21no—
230H(3,𝔽3)—32no—
231H(3,𝔽3)C23×C3×D311yescharacteristic, complemented
232C3×C9—32no—
233C23×C3—21no—
234C23×C3—33no—
235C23×C3—32no—
236C23×C3—31no—
237C23×C3—63no—
238C23×C3—91no—
239C23×C3—181no—
240C23×C3—2728no—
241C23×C3—2714no—
242C23×C3—2714no—
243C23×C3#418311yescharacteristic, direct product factor
244C23×C3C3×#27211yescharacteristic
245C22×D3—942no—
246C22×D3—914no—
247C22×D3—2714no—
248C2×C32—37no—
249C2×C32—37no—
250C2×C32—321no—
251C2×C32—342no—
252C2×C32—321no—
253C2×C32—37no—
254C2×C32—67no—
255C2×C32—621no—
256C2×C32—642no—
257C2×C32—621no—
258C2×C32—67no—
259C2×C32—67no—
260C2×C32—97no—
261C2×C32—187no—
262C2×C32—278no—
263C2×C32C22×#27217yes—
264C2×C32C22×#272114yes—
265C2×C32C22×C3×C3⋊D317yes—
266C3⋊D3—38no—
267C3⋊D3—38no—
268C3×D3—924no—
269C3×D3—948no—
270C3×D3—924no—
271C3×D3—98no—
272C3×D3—98no—
273C3×D3—916no—
274C3×D3—278no—
275C2×C9—314no—
276C2×C9—614no—
277D9—316no—
278C24—271noSylow
279C22×C3—27no—
280C22×C3—321no—
281C22×C3—314no—
282C22×C3—37no—
283C22×C3—621no—
284C22×C3—97no—
285C22×C3—187no—
286C22×C3—2756no—
287C22×C3—2728no—
288C22×C3—2728no—
289C22×C3C2×#418317yesdirect product factor
290C22×C3C2×C3×#27217yes—
291C2×D3—984no—
292C2×D3—928no—
293C2×D3—2728no—
294C32—33no—
295C32—31no—
296C32—31no—
297C32—36no—
298C32—33no—
299C32—31no—
300C32—61no—
301C32—63no—
302C32—66no—
303C32—63no—
304C32—61no—
305C32—61no—
306C32—91no—
307C32—181no—
308C32C23×#27211yescharacteristic
309C32C23×#27212yes—
310C32C23×C3×C3⋊D311yescharacteristic
311C9—32no—
312C9—62no—
313C23—2714no—
314C23C3×#418311yescharacteristic, direct product factor
315C2×C3—27no—
316C2×C3—321no—
317C2×C3—314no—
318C2×C3—37no—
319C2×C3—621no—
320C2×C3—97no—
321C2×C3—187no—
322C2×C3—278no—
323C2×C3—2716no—
324C2×C3—278no—
325C2×C3C22×#418317yesdirect product factor
326C2×C3C22×C3×#27217yes—
327D3—924no—
328D3—98no—
329D3—278no—
330C22—2728no—
331C22C2×C3×#418317yesdirect product factor
332C3—21no—
333C3—33no—
334C3—31no—
335C3—32no—
336C3—63no—
337C3—91no—
338C3—181no—
339C3C23×#418311yescharacteristic, direct product factor
340C3C23×C3×#27211yescharacteristic
341C2—278no—
342C2C22×C3×#418317yesdirect product factor
subgroupG/H
center, hypercenterC23×C3#4183
commutator, cosocle, hypocenterAΣL(1,𝔽27)C24×C3
FrattiniC32C23×C3×C3⋊D3
FittingC23×C3×AΣL(1,𝔽27)C2
socleC23×C32#272
LCSG ▷ AΣL(1,𝔽27)
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C23×C3×AΣL(1,𝔽27)C2semidirect
2. C22×C3×#4183C2direct (N × H)
6. C23×#4183C3direct (N × H)
7. C22×C3×AΣL(1,𝔽27)C22semidirect
8. C2×C3×#4183C22direct (N × H)
9. C23×C34D3semidirect
11. C23×C3×3−1+2D3semidirect
12. C23×C3×H(3,𝔽3)D3semidirect
18. C23×AΣL(1,𝔽27)C2×C3semidirect
19. C22×#4183C2×C3direct (N × H)
20. C2×C3×AΣL(1,𝔽27)C23semidirect
21. C3×#4183C23direct (N × H)
33. C22×C34C2×D3semidirect
35. C22×C3×3−1+2C2×D3semidirect
36. C22×C3×H(3,𝔽3)C2×D3semidirect
42. C22×AΣL(1,𝔽27)C22×C3semidirect
43. C2×#4183C22×C3direct (N × H)
44. C3×AΣL(1,𝔽27)C24semidirect
51. C23×C33C3×D3semidirect
61. C23×3−1+2C3×D3semidirect
64. C23×H(3,𝔽3)C3×D3semidirect
71. C2×C34C22×D3semidirect
73. C2×C3×3−1+2C22×D3semidirect
74. C2×C3×H(3,𝔽3)C22×D3semidirect
80. C2×AΣL(1,𝔽27)C23×C3semidirect
81. #4183C23×C3direct (N × H)
99. C22×C33C2×C3×D3semidirect
109. C22×3−1+2C2×C3×D3semidirect
112. C22×H(3,𝔽3)C2×C3×D3semidirect
119. C34C23×D3semidirect
120. C3×3−1+2C23×D3semidirect
121. C3×H(3,𝔽3)C23×D3semidirect
125. AΣL(1,𝔽27)C24×C3semidirect
162. C2×C33C22×C3×D3semidirect
172. C2×3−1+2C22×C3×D3semidirect
175. C2×H(3,𝔽3)C22×C3×D3semidirect
224. C33C23×C3×D3semidirect
228. 3−1+2C23×C3×D3semidirect
231. H(3,𝔽3)C23×C3×D3semidirect
243. C23×C3#4183direct (N × H)
289. C22×C3C2×#4183direct (N × H)
314. C23C3×#4183direct (N × H)
325. C2×C3C22×#4183direct (N × H)
331. C22C2×C3×#4183direct (N × H)
339. C3C23×#4183direct (N × H)
342. C2C22×C3×#4183direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×AΣL(1,𝔽27)2
C22×C3×#41832
C23×C32×C3⋊D33
C23×C3×AGL(1,ℤ/9)3
C23×C3×(C3⋊D3)⋊C33
C23×#41833