Subgroups — C₃²×#4842 #13162282

Order 1728.

800 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×D₁₂	C₂	1	1	yes	maximal, characteristic
2	C₃²×#869	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃³×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#821	C₂	1	2	yes	maximal
5	C₃²×#817	C₂	1	2	yes	maximal, complemented
6	C₃²×#476	—	3	1	no	maximal
7	C₃×#4842	C₃	1	4	yes	maximal, direct product factor
8	C₂×C₃²×D₁₂	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄×D₃	—	2	1	no	—
10	C₂×C₃²×Dic₆	—	2	1	no	—
11	C₂×C₃³×C₈	C₂²	1	2	yes	—
12	C₃³×C₄²	C₂²	1	1	yes	characteristic
13	C₃²×(C₂×D₃)⋊C₄	—	2	1	no	—
14	C₃²×C₄⋊Dic₃	—	2	1	no	—
15	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
16	C₃²×C₄⋊Dic₃	C₂²	1	2	yes	—
17	C₃²×C₄⋊Q₈	—	3	1	no	—
18	C₃²×C₄×D₄	—	3	1	no	—
19	C₃×C₄×D₁₂	C₂×C₃	1	4	yes	—
20	C₃×#869	C₂×C₃	1	4	yes	complemented
21	C₃²×#106	—	3	2	no	—
22	C₃²×C₄⋊C₈	—	2	4	no	—
23	C₃²×C₄⋊C₈	D₃	1	1	yes	characteristic, complemented
24	C₃²×C₄⋊C₈	C₂×C₃	1	4	yes	complemented
25	C₃²×#102	—	3	2	no	—
26	C₃×#821	C₂×C₃	1	8	yes	—
27	C₃×#817	C₂×C₃	1	8	yes	complemented
28	C₂²×C₃²×D₃	—	2	1	no	—
29	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
30	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
31	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
32	C₂×C₃²×Dic₃	—	2	2	no	—
33	C₂×C₃²×Dic₃	—	2	1	no	—
34	C₃²×D₁₂	—	2	1	no	—
35	C₃²×D₁₂	Q₈	1	2	yes	complemented
36	C₃²×C₄×D₃	—	4	2	no	—
37	C₃²×Dic₆	—	4	2	no	—
38	C₃³×C₈	—	2	2	no	—
39	C₃×#476	—	3	4	no	—
40	#4842	C₃²	1	1	yes	characteristic, direct product factor
41	C₂×C₃²×Q₈	—	6	1	no	—
42	C₂×C₃²×D₄	—	3	1	no	—
43	C₂²×C₃²×C₄	—	6	1	no	—
44	C₂×C₃×D₁₂	C₂²×C₃	1	4	yes	—
45	C₂×C₃×C₄×D₃	—	2	4	no	—
46	C₂×C₃×Dic₆	—	2	4	no	—
47	C₂×C₃²×C₈	—	2	8	no	—
48	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
49	C₂×C₃²×C₈	C₂²×C₃	1	8	yes	—
50	C₃²×C₄⋊C₄	—	3	1	no	—
51	C₃²×C₄⋊C₄	—	3	2	no	—
52	C₃²×C₄⋊C₄	—	6	1	no	—
53	C₃²×C₂²⋊C₄	—	6	1	no	—
54	C₃²×C₄²	—	2	4	no	—
55	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
56	C₃²×C₄²	C₂²×C₃	1	4	yes	—
57	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
58	C₃×C₄⋊Dic₃	—	2	4	no	—
59	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
60	C₃×C₄⋊Dic₃	C₂²×C₃	1	8	yes	—
61	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃²×D₃	—	2	2	no	—
63	C₂×C₃²×D₃	—	4	1	no	—
64	C₂×C₃²×D₃	—	4	1	no	—
65	C₃³×C₄	—	2	1	no	—
66	C₃³×C₄	SD₁₆	1	2	yes	—
67	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
68	C₃³×C₄	#41	1	1	yes	characteristic
69	C₃²×Dic₃	—	4	1	no	—
70	C₃²×Dic₃	—	4	2	no	—
71	C₃×C₄⋊Q₈	—	3	4	no	—
72	C₃×C₄×D₄	—	3	4	no	—
73	C₄×D₁₂	C₂×C₃²	1	1	yes	characteristic
74	#869	C₂×C₃²	1	1	yes	characteristic, complemented
75	C₃×#106	—	3	8	no	—
76	C₃×C₄⋊C₈	—	2	4	no	—
77	C₃×C₄⋊C₈	C₃×D₃	1	4	yes	complemented
78	C₃×C₄⋊C₈	C₂×C₃²	1	1	yes	characteristic, complemented
79	C₃×#102	—	3	8	no	—
80	#821	C₂×C₃²	1	2	yes	—
81	#817	C₂×C₃²	1	2	yes	complemented
82	C₂³×C₃²	—	6	1	no	—
83	C₂²×C₃×D₃	—	2	4	no	—
84	C₃²×Q₈	—	12	2	no	—
85	C₃²×D₄	—	3	2	no	—
86	C₃²×D₄	—	6	1	no	—
87	C₂×C₃²×C₄	—	2	4	no	—
88	C₂×C₃²×C₄	—	2	4	no	—
89	C₂×C₃²×C₄	—	2	4	no	—
90	C₂×C₃²×C₄	—	6	1	no	—
91	C₂×C₃²×C₄	—	6	2	no	—
92	C₂×C₃²×C₄	—	12	2	no	—
93	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
94	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
96	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
97	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
98	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
99	C₂×C₃×Dic₃	—	2	8	no	—
100	C₂×C₃×Dic₃	—	2	4	no	—
101	C₃×D₁₂	—	2	4	no	—
102	C₃×D₁₂	C₃×Q₈	1	8	yes	complemented
103	C₃×C₄×D₃	—	4	8	no	—
104	C₃×Dic₆	—	4	8	no	—
105	C₃²×C₈	—	2	2	no	—
106	C₃²×C₈	—	2	8	no	—
107	C₃²×C₈	—	4	8	no	—
108	#476	—	3	1	no	Sylow
109	C₂×C₃³	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₃³	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₃³	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₃²×D₃	—	4	2	no	—
113	C₂×C₃×Q₈	—	6	4	no	—
114	C₂×C₃×D₄	—	3	4	no	—
115	C₂²×C₃×C₄	—	6	4	no	—
116	C₂×D₁₂	C₂²×C₃²	1	1	yes	characteristic
117	C₂×C₄×D₃	—	2	1	no	—
118	C₂×Dic₆	—	2	1	no	—
119	C₂×C₃×C₈	—	2	8	no	—
120	C₂×C₃×C₈	C₂×C₃×D₃	1	8	yes	—
121	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
122	C₃×C₄⋊C₄	—	3	8	no	—
123	C₃×C₄⋊C₄	—	3	4	no	—
124	C₃×C₄⋊C₄	—	6	4	no	—
125	C₃×C₂²⋊C₄	—	6	4	no	—
126	C₃×C₄²	—	2	4	no	—
127	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
128	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
129	(C₂×D₃)⋊C₄	—	2	1	no	—
130	C₄⋊Dic₃	—	2	1	no	—
131	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
132	C₄⋊Dic₃	C₂²×C₃²	1	2	yes	—
133	C₂²×C₃²	—	2	4	no	—
134	C₂²×C₃²	—	6	2	no	—
135	C₂²×C₃²	—	12	1	no	—
136	C₂²×C₃²	—	12	1	no	—
137	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
138	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
139	C₂×C₃×D₃	—	2	8	no	—
140	C₂×C₃×D₃	—	4	4	no	—
141	C₂×C₃×D₃	—	4	4	no	—
142	C₃²×C₄	—	2	1	no	—
143	C₃²×C₄	—	2	8	no	—
144	C₃²×C₄	—	2	4	no	—
145	C₃²×C₄	—	2	4	no	—
146	C₃²×C₄	—	2	4	no	—
147	C₃²×C₄	—	4	4	no	—
148	C₃²×C₄	—	12	1	no	—
149	C₃²×C₄	—	12	2	no	—
150	C₃²×C₄	#204	1	2	yes	—
151	C₃²×C₄	C₃×SD₁₆	1	8	yes	—
152	C₃²×C₄	#237	1	1	yes	characteristic
153	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
154	C₃²×C₄	C₂×C₃×Q₈	1	4	yes	—
155	C₃²×C₄	C₃×#41	1	4	yes	—
156	C₃×Dic₃	—	4	4	no	—
157	C₃×Dic₃	—	4	8	no	—
158	C₄⋊Q₈	—	3	1	no	—
159	C₄×D₄	—	3	1	no	—
160	#106	—	3	2	no	—
161	C₄⋊C₈	C₃²×D₃	1	1	yes	characteristic, complemented
162	#102	—	3	2	no	—
163	C₃³	#476	1	1	yes	characteristic, complemented, Sylow
164	C₂³×C₃	—	6	4	no	—
165	C₂²×D₃	—	2	1	no	—
166	C₃×Q₈	—	12	8	no	—
167	C₃×D₄	—	3	8	no	—
168	C₃×D₄	—	6	4	no	—
169	C₂×C₃×C₄	—	2	4	no	—
170	C₂×C₃×C₄	—	2	4	no	—
171	C₂×C₃×C₄	—	2	4	no	—
172	C₂×C₃×C₄	—	6	4	no	—
173	C₂×C₃×C₄	—	6	8	no	—
174	C₂×C₃×C₄	—	12	8	no	—
175	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
176	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
177	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
178	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
180	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
181	C₂×Dic₃	—	2	2	no	—
182	C₂×Dic₃	—	2	1	no	—
183	D₁₂	—	2	1	no	—
184	D₁₂	C₃²×Q₈	1	2	yes	complemented
185	C₄×D₃	—	4	2	no	—
186	Dic₆	—	4	2	no	—
187	C₃×C₈	—	2	8	no	—
188	C₃×C₈	—	2	2	no	—
189	C₃×C₈	—	4	8	no	—
190	C₂×C₃²	—	2	4	no	—
191	C₂×C₃²	—	2	4	no	—
192	C₂×C₃²	—	2	4	no	—
193	C₂×C₃²	—	12	2	no	—
194	C₂×C₃²	#897	1	1	yes	characteristic
195	C₂×C₃²	C₂×#204	1	1	yes	characteristic
196	C₂×C₃²	#908	1	1	yes	characteristic
197	C₂×C₃²	C₃×C₂²⋊Q₈	1	4	yes	—
198	C₂×C₃²	C₂×C₃×SD₁₆	1	4	yes	—
199	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	4	yes	—
200	C₃×D₃	—	4	8	no	—
201	C₂×Q₈	—	6	1	no	—
202	C₂×D₄	—	3	1	no	—
203	C₂²×C₄	—	6	1	no	—
204	C₂×C₈	C₂×C₃²×D₃	1	2	yes	—
205	C₄⋊C₄	—	3	2	no	—
206	C₄⋊C₄	—	3	1	no	—
207	C₄⋊C₄	—	6	1	no	—
208	C₂²⋊C₄	—	6	1	no	—
209	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
210	C₂²×C₃	—	2	4	no	—
211	C₂²×C₃	—	6	8	no	—
212	C₂²×C₃	—	12	4	no	—
213	C₂²×C₃	—	12	4	no	—
214	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
215	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
216	C₂×D₃	—	2	2	no	—
217	C₂×D₃	—	4	1	no	—
218	C₂×D₃	—	4	1	no	—
219	C₃×C₄	—	2	4	no	—
220	C₃×C₄	—	2	1	no	—
221	C₃×C₄	—	2	4	no	—
222	C₃×C₄	—	2	4	no	—
223	C₃×C₄	—	2	8	no	—
224	C₃×C₄	—	4	4	no	—
225	C₃×C₄	—	12	4	no	—
226	C₃×C₄	—	12	8	no	—
227	C₃×C₄	C₃×#204	1	8	yes	—
228	C₃×C₄	C₃²×SD₁₆	1	2	yes	—
229	C₃×C₄	C₃×#237	1	4	yes	—
230	C₃×C₄	C₃×D₃×Q₈	1	4	yes	—
231	C₃×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
232	C₃×C₄	C₃²×#41	1	1	yes	characteristic
233	Dic₃	—	4	1	no	—
234	Dic₃	—	4	2	no	—
235	C₃²	—	2	4	no	—
236	C₃²	#4842	1	1	yes	characteristic, direct product factor
237	C₃²	C₃×#476	1	4	yes	complemented
238	C₂³	—	6	1	no	—
239	Q₈	—	12	2	no	—
240	D₄	—	3	2	no	—
241	D₄	—	6	1	no	—
242	C₂×C₄	—	6	1	no	—
243	C₂×C₄	—	6	2	no	—
244	C₂×C₄	—	12	2	no	—
245	C₂×C₄	C₃²×D₁₂	1	1	yes	characteristic
246	C₂×C₄	C₃²×D₁₂	1	1	yes	characteristic
247	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₂×C₃	—	2	4	no	—
250	C₂×C₃	—	2	4	no	—
251	C₂×C₃	—	2	4	no	—
252	C₂×C₃	—	12	8	no	—
253	C₂×C₃	C₃×#897	1	4	yes	—
254	C₂×C₃	C₂×C₃×#204	1	4	yes	—
255	C₂×C₃	C₃×#908	1	4	yes	—
256	C₂×C₃	C₃²×C₂²⋊Q₈	1	1	yes	characteristic
257	C₂×C₃	C₂×C₃²×SD₁₆	1	1	yes	characteristic
258	C₂×C₃	C₃²×AGL(1,ℤ/8)	1	1	yes	characteristic
259	D₃	—	4	2	no	—
260	C₂²	—	6	2	no	—
261	C₂²	—	12	1	no	—
262	C₂²	—	12	1	no	—
263	C₂²	C₂×C₃²×D₁₂	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	12	2	no	—
266	C₄	—	12	1	no	—
267	C₄	C₃²×#204	1	2	yes	—
268	C₄	C₃²×#237	1	1	yes	characteristic
269	C₄	C₃²×D₃×Q₈	1	1	yes	characteristic
270	C₃	—	2	4	no	—
271	C₃	C₃×#4842	1	4	yes	direct product factor
272	C₃	C₃²×#476	1	1	yes	characteristic, complemented
273	C₂	—	12	2	no	—
274	C₂	C₃²×#897	1	1	yes	characteristic
275	C₂	C₂×C₃²×#204	1	1	yes	characteristic
276	C₂	C₃²×#908	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂³×C₃²
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×C₄⋊C₈	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₂×C₃²×D₁₂	C₂²
hypercenter	C₃²×C₄⋊C₈	D₃
hypocenter	C₃	C₃²×#476
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ C₃²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₃²×#869	C₂	semidirect
3. C₃³×C₄⋊C₈	C₂	semidirect
5. C₃²×#817	C₂	semidirect
7. C₃×#4842	C₃	direct (N × H)
20. C₃×#869	C₂×C₃	semidirect
23. C₃²×C₄⋊C₈	D₃	semidirect
24. C₃²×C₄⋊C₈	C₂×C₃	semidirect
27. C₃×#817	C₂×C₃	semidirect
35. C₃²×D₁₂	Q₈	semidirect
40. #4842	C₃²	direct (N × H)
74. #869	C₂×C₃²	semidirect
77. C₃×C₄⋊C₈	C₃×D₃	semidirect
78. C₃×C₄⋊C₈	C₂×C₃²	semidirect
81. #817	C₂×C₃²	semidirect
102. C₃×D₁₂	C₃×Q₈	semidirect
161. C₄⋊C₈	C₃²×D₃	semidirect
163. C₃³	#476	semidirect
184. D₁₂	C₃²×Q₈	semidirect
236. C₃²	#4842	direct (N × H)
237. C₃²	C₃×#476	semidirect
271. C₃	C₃×#4842	direct (N × H)
272. C₃	C₃²×#476	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×D₁₂	2
C₃²×#869	2
C₃³×C₄⋊C₈	2
C₃²×#821	2
C₃²×#817	2
C₃²×#476	3
C₃×#4842	3

Subgroups — C32×#4842 #13162282

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4842 #13162282