Subgroups — C₂³×C₃×#213 #11918533

Order 1152.

4633 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃²×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₃×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×#213	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₃×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₃×D₈	—	3	1	no	maximal
6	C₂³×#213	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂⁵×C₃²	—	2	1	no	—
8	C₂⁴×C₃×D₃	—	2	1	no	—
9	C₂²×C₃²×D₄	—	2	7	no	—
10	C₂²×C₃²×D₄	C₂²	1	14	yes	complemented
11	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
12	C₂²×C₃×D₁₂	—	2	7	no	—
13	C₂²×C₃×D₁₂	C₂²	1	14	yes	complemented
14	C₂×C₃×#213	C₂²	1	112	yes	direct product factor
15	C₂²×C₃×C₃⋊C₈	C₂²	1	14	yes	complemented
16	C₂³×C₃×D₄	—	2	1	no	—
17	C₂³×C₃×D₄	—	3	1	no	—
18	C₂³×C₃×D₄	D₃	1	1	yes	characteristic, complemented
19	C₂³×C₃×D₄	C₂×C₃	1	1	yes	characteristic, complemented
20	C₂³×D₁₂	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂²×C₃×D₈	—	3	28	no	—
22	C₂³×C₃×C₈	—	3	1	no	—
23	C₂²×#213	C₂×C₃	1	28	yes	direct product factor
24	C₂³×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂⁴×C₃²	—	2	14	no	—
26	C₂⁴×C₃²	—	4	8	no	—
27	C₂⁴×C₃²	D₄	1	1	yes	characteristic
28	C₂³×C₃×D₃	—	2	14	no	—
29	C₂³×C₃×D₃	—	4	8	no	—
30	C₂×C₃²×D₄	—	2	42	no	—
31	C₂×C₃²×D₄	C₂³	1	28	yes	complemented
32	C₂²×C₃²×C₄	D₄	1	7	yes	—
33	C₂²×C₃²×C₄	C₂³	1	7	yes	—
34	C₂×C₃×D₁₂	—	2	42	no	—
35	C₂×C₃×D₁₂	C₂³	1	28	yes	complemented
36	C₃×#213	C₂³	1	64	yes	direct product factor
37	C₂×C₃×C₃⋊C₈	C₂³	1	28	yes	complemented
38	C₂³×D₈	—	3	1	no	Sylow
39	C₂⁵×C₃	—	2	2	no	—
40	C₂⁵×C₃	—	2	1	no	—
41	C₂⁵×C₃	—	2	1	no	—
42	C₂⁵×C₃	—	6	1	no	—
43	C₂⁴×D₃	—	2	1	no	—
44	C₂²×C₃×D₄	—	2	14	no	—
45	C₂²×C₃×D₄	—	2	7	no	—
46	C₂²×C₃×D₄	—	2	7	no	—
47	C₂²×C₃×D₄	—	2	14	no	—
48	C₂²×C₃×D₄	—	3	14	no	—
49	C₂²×C₃×D₄	—	6	7	no	—
50	C₂²×C₃×D₄	C₂×D₃	1	14	yes	complemented
51	C₂²×C₃×D₄	C₂²×C₃	1	14	yes	complemented
52	C₂³×C₃×C₄	—	2	1	no	—
53	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
54	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
55	C₂²×D₁₂	—	2	7	no	—
56	C₂²×D₁₂	C₂²×C₃	1	14	yes	complemented
57	C₂×C₃×D₈	—	3	112	no	—
58	C₂²×C₃×C₈	—	3	14	no	—
59	C₂×#213	C₂²×C₃	1	112	yes	direct product factor
60	C₂²×C₃⋊C₈	C₂²×C₃	1	14	yes	complemented
61	C₂³×C₃²	—	2	28	no	—
62	C₂³×C₃²	—	4	56	no	—
63	C₂³×C₃²	D₈	1	8	yes	complemented
64	C₂³×C₃²	C₂×D₄	1	7	yes	—
65	C₂²×C₃×D₃	—	2	28	no	—
66	C₂²×C₃×D₃	—	4	56	no	—
67	C₃²×D₄	—	2	28	no	—
68	C₃²×D₄	C₂⁴	1	8	yes	complemented
69	C₂×C₃²×C₄	C₂×D₄	1	21	yes	—
70	C₂×C₃²×C₄	C₂⁴	1	7	yes	—
71	C₃×D₁₂	—	2	28	no	—
72	C₃×D₁₂	C₂⁴	1	8	yes	complemented
73	C₃×C₃⋊C₈	C₂⁴	1	8	yes	complemented
74	C₂³×D₄	—	3	1	no	—
75	C₂³×D₄	C₃×D₃	1	1	yes	characteristic, complemented
76	C₂²×D₈	—	3	28	no	—
77	C₂³×C₈	—	3	1	no	—
78	C₂⁴×C₃	—	2	28	no	—
79	C₂⁴×C₃	—	2	14	no	—
80	C₂⁴×C₃	—	2	14	no	—
81	C₂⁴×C₃	—	2	1	no	—
82	C₂⁴×C₃	—	4	16	no	—
83	C₂⁴×C₃	—	4	8	no	—
84	C₂⁴×C₃	—	4	8	no	—
85	C₂⁴×C₃	—	6	14	no	—
86	C₂⁴×C₃	—	12	8	no	—
87	C₂⁴×C₃	#67	1	1	yes	characteristic
88	C₂⁴×C₃	C₃×D₄	1	1	yes	characteristic
89	C₂³×D₃	—	2	14	no	—
90	C₂³×D₃	—	4	8	no	—
91	C₂×C₃×D₄	—	2	84	no	—
92	C₂×C₃×D₄	—	2	42	no	—
93	C₂×C₃×D₄	—	2	42	no	—
94	C₂×C₃×D₄	—	2	28	no	—
95	C₂×C₃×D₄	—	3	28	no	—
96	C₂×C₃×D₄	—	6	42	no	—
97	C₂×C₃×D₄	C₂²×D₃	1	28	yes	complemented
98	C₂×C₃×D₄	C₂³×C₃	1	28	yes	complemented
99	C₂²×C₃×C₄	—	2	7	no	—
100	C₂²×C₃×C₄	—	2	7	no	—
101	C₂²×C₃×C₄	#67	1	7	yes	—
102	C₂²×C₃×C₄	C₃×D₄	1	7	yes	—
103	C₂²×C₃×C₄	C₂²×D₃	1	7	yes	—
104	C₂²×C₃×C₄	C₂³×C₃	1	7	yes	—
105	C₂×D₁₂	—	2	42	no	—
106	C₂×D₁₂	C₂³×C₃	1	28	yes	complemented
107	C₃×D₈	—	3	64	no	—
108	C₂×C₃×C₈	—	3	28	no	—
109	#213	C₂³×C₃	1	64	yes	direct product factor
110	C₂×C₃⋊C₈	C₂³×C₃	1	28	yes	complemented
111	C₂²×C₃²	—	2	8	no	—
112	C₂²×C₃²	—	4	56	no	—
113	C₂²×C₃²	C₂×D₈	1	28	yes	complemented
114	C₂²×C₃²	C₂²×D₄	1	7	yes	—
115	C₂×C₃×D₃	—	2	8	no	—
116	C₂×C₃×D₃	—	4	56	no	—
117	C₃²×C₄	C₂²×D₄	1	7	yes	—
118	C₃²×C₄	C₂⁵	1	1	yes	characteristic
119	C₂⁵	—	2	1	no	—
120	C₂⁵	—	6	1	no	—
121	C₂²×D₄	—	2	7	no	—
122	C₂²×D₄	—	3	14	no	—
123	C₂²×D₄	—	6	7	no	—
124	C₂²×D₄	C₂×C₃×D₃	1	14	yes	complemented
125	C₂³×C₄	C₂×C₃×D₃	1	1	yes	characteristic
126	C₂×D₈	—	3	112	no	—
127	C₂²×C₈	—	3	14	no	—
128	C₂³×C₃	—	2	56	no	—
129	C₂³×C₃	—	2	28	no	—
130	C₂³×C₃	—	2	28	no	—
131	C₂³×C₃	—	2	8	no	—
132	C₂³×C₃	—	2	7	no	—
133	C₂³×C₃	—	4	112	no	—
134	C₂³×C₃	—	4	56	no	—
135	C₂³×C₃	—	4	56	no	—
136	C₂³×C₃	—	6	28	no	—
137	C₂³×C₃	—	12	56	no	—
138	C₂³×C₃	#213	1	8	yes	direct product factor
139	C₂³×C₃	C₃×D₈	1	8	yes	complemented
140	C₂³×C₃	C₂×#67	1	7	yes	—
141	C₂³×C₃	C₂×C₃×D₄	1	7	yes	—
142	C₂²×D₃	—	2	28	no	—
143	C₂²×D₃	—	4	56	no	—
144	C₃×D₄	—	2	56	no	—
145	C₃×D₄	—	2	28	no	—
146	C₃×D₄	—	2	28	no	—
147	C₃×D₄	—	2	8	no	—
148	C₃×D₄	—	3	8	no	—
149	C₃×D₄	—	6	28	no	—
150	C₃×D₄	C₂³×D₃	1	8	yes	complemented
151	C₃×D₄	C₂⁴×C₃	1	8	yes	complemented
152	C₂×C₃×C₄	—	2	7	no	—
153	C₂×C₃×C₄	—	2	21	no	—
154	C₂×C₃×C₄	C₂×#67	1	21	yes	—
155	C₂×C₃×C₄	C₂×C₃×D₄	1	21	yes	—
156	C₂×C₃×C₄	C₂³×D₃	1	7	yes	—
157	C₂×C₃×C₄	C₂⁴×C₃	1	7	yes	—
158	D₁₂	—	2	28	no	—
159	D₁₂	C₂⁴×C₃	1	8	yes	complemented
160	C₃×C₈	—	3	8	no	—
161	C₃⋊C₈	C₂⁴×C₃	1	8	yes	complemented
162	C₂×C₃²	—	4	8	no	—
163	C₂×C₃²	C₂²×D₈	1	14	yes	complemented
164	C₂×C₃²	C₂³×D₄	1	1	yes	characteristic
165	C₃×D₃	—	4	8	no	—
166	C₂⁴	—	2	14	no	—
167	C₂⁴	—	4	8	no	—
168	C₂⁴	—	6	14	no	—
169	C₂⁴	—	12	8	no	—
170	C₂⁴	C₃×#67	1	1	yes	characteristic
171	C₂×D₄	—	2	42	no	—
172	C₂×D₄	—	3	28	no	—
173	C₂×D₄	—	6	42	no	—
174	C₂×D₄	C₂²×C₃×D₃	1	28	yes	complemented
175	C₂²×C₄	C₃×#67	1	7	yes	—
176	C₂²×C₄	C₂²×C₃×D₃	1	7	yes	—
177	D₈	—	3	64	no	—
178	C₂×C₈	—	3	28	no	—
179	C₂²×C₃	—	2	16	no	—
180	C₂²×C₃	—	2	8	no	—
181	C₂²×C₃	—	2	8	no	—
182	C₂²×C₃	—	2	28	no	—
183	C₂²×C₃	—	2	7	no	—
184	C₂²×C₃	—	4	112	no	—
185	C₂²×C₃	—	4	56	no	—
186	C₂²×C₃	—	4	56	no	—
187	C₂²×C₃	—	6	8	no	—
188	C₂²×C₃	—	12	56	no	—
189	C₂²×C₃	C₂×#213	1	28	yes	direct product factor
190	C₂²×C₃	C₂×C₃×D₈	1	28	yes	complemented
191	C₂²×C₃	C₂²×#67	1	7	yes	—
192	C₂²×C₃	C₂²×C₃×D₄	1	7	yes	—
193	C₂×D₃	—	2	8	no	—
194	C₂×D₃	—	4	56	no	—
195	C₃×C₄	—	2	7	no	—
196	C₃×C₄	—	2	1	no	—
197	C₃×C₄	C₂²×#67	1	7	yes	—
198	C₃×C₄	C₂²×C₃×D₄	1	7	yes	—
199	C₃×C₄	C₂⁴×D₃	1	1	yes	characteristic
200	C₃×C₄	C₂⁵×C₃	1	1	yes	characteristic
201	C₃²	C₂³×D₈	1	1	yes	characteristic, complemented, Sylow
202	C₂³	—	2	28	no	—
203	C₂³	—	4	56	no	—
204	C₂³	—	6	28	no	—
205	C₂³	—	12	56	no	—
206	C₂³	C₃×#213	1	8	yes	direct product factor
207	C₂³	C₂×C₃×#67	1	7	yes	—
208	D₄	—	2	28	no	—
209	D₄	—	3	8	no	—
210	D₄	—	6	28	no	—
211	D₄	C₂³×C₃×D₃	1	8	yes	complemented
212	C₂×C₄	C₂×C₃×#67	1	21	yes	—
213	C₂×C₄	C₂³×C₃×D₃	1	7	yes	—
214	C₈	—	3	8	no	—
215	C₂×C₃	—	2	14	no	—
216	C₂×C₃	—	2	1	no	—
217	C₂×C₃	—	4	16	no	—
218	C₂×C₃	—	4	8	no	—
219	C₂×C₃	—	4	8	no	—
220	C₂×C₃	—	12	8	no	—
221	C₂×C₃	C₂²×#213	1	14	yes	direct product factor
222	C₂×C₃	C₂²×C₃×D₈	1	14	yes	complemented
223	C₂×C₃	C₂³×#67	1	1	yes	characteristic
224	C₂×C₃	C₂³×C₃×D₄	1	1	yes	characteristic
225	D₃	—	4	8	no	—
226	C₂²	—	2	8	no	—
227	C₂²	—	4	56	no	—
228	C₂²	—	6	8	no	—
229	C₂²	—	12	56	no	—
230	C₂²	C₂×C₃×#213	1	28	yes	direct product factor
231	C₂²	C₂²×C₃×#67	1	7	yes	—
232	C₄	C₂²×C₃×#67	1	7	yes	—
233	C₄	C₂⁴×C₃×D₃	1	1	yes	characteristic
234	C₃	—	2	1	no	—
235	C₃	C₂³×#213	1	1	yes	characteristic, direct product factor
236	C₃	C₂³×C₃×D₈	1	1	yes	characteristic, complemented
237	C₂	—	4	8	no	—
238	C₂	—	12	8	no	—
239	C₂	C₂²×C₃×#213	1	14	yes	direct product factor
240	C₂	C₂³×C₃×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₃	#67
commutator, cosocle	C₃×C₄	C₂⁵×C₃
Frattini	C₄	C₂⁴×C₃×D₃
Fitting	C₂³×C₃²×D₄	C₂
socle	C₂⁴×C₃²	D₄
hypercenter	C₂³×C₃×D₄	D₃
hypocenter	C₃	C₂³×C₃×D₈
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴×C₃ ◁ C₂³×C₃×C₄ ◁ C₂³×C₃×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃²×D₄	C₂	semidirect
2. C₂³×C₃×D₁₂	C₂	semidirect
3. C₂²×C₃×#213	C₂	direct (N × H)
4. C₂³×C₃×C₃⋊C₈	C₂	semidirect
6. C₂³×#213	C₃	direct (N × H)
10. C₂²×C₃²×D₄	C₂²	semidirect
13. C₂²×C₃×D₁₂	C₂²	semidirect
14. C₂×C₃×#213	C₂²	direct (N × H)
15. C₂²×C₃×C₃⋊C₈	C₂²	semidirect
18. C₂³×C₃×D₄	D₃	semidirect
19. C₂³×C₃×D₄	C₂×C₃	semidirect
20. C₂³×D₁₂	C₂×C₃	semidirect
23. C₂²×#213	C₂×C₃	direct (N × H)
24. C₂³×C₃⋊C₈	C₂×C₃	semidirect
31. C₂×C₃²×D₄	C₂³	semidirect
35. C₂×C₃×D₁₂	C₂³	semidirect
36. C₃×#213	C₂³	direct (N × H)
37. C₂×C₃×C₃⋊C₈	C₂³	semidirect
50. C₂²×C₃×D₄	C₂×D₃	semidirect
51. C₂²×C₃×D₄	C₂²×C₃	semidirect
56. C₂²×D₁₂	C₂²×C₃	semidirect
59. C₂×#213	C₂²×C₃	direct (N × H)
60. C₂²×C₃⋊C₈	C₂²×C₃	semidirect
63. C₂³×C₃²	D₈	semidirect
68. C₃²×D₄	C₂⁴	semidirect
72. C₃×D₁₂	C₂⁴	semidirect
73. C₃×C₃⋊C₈	C₂⁴	semidirect
75. C₂³×D₄	C₃×D₃	semidirect
97. C₂×C₃×D₄	C₂²×D₃	semidirect
98. C₂×C₃×D₄	C₂³×C₃	semidirect
106. C₂×D₁₂	C₂³×C₃	semidirect
109. #213	C₂³×C₃	direct (N × H)
110. C₂×C₃⋊C₈	C₂³×C₃	semidirect
113. C₂²×C₃²	C₂×D₈	semidirect
124. C₂²×D₄	C₂×C₃×D₃	semidirect
138. C₂³×C₃	#213	direct (N × H)
139. C₂³×C₃	C₃×D₈	semidirect
150. C₃×D₄	C₂³×D₃	semidirect
151. C₃×D₄	C₂⁴×C₃	semidirect
159. D₁₂	C₂⁴×C₃	semidirect
161. C₃⋊C₈	C₂⁴×C₃	semidirect
163. C₂×C₃²	C₂²×D₈	semidirect
174. C₂×D₄	C₂²×C₃×D₃	semidirect
189. C₂²×C₃	C₂×#213	direct (N × H)
190. C₂²×C₃	C₂×C₃×D₈	semidirect
201. C₃²	C₂³×D₈	semidirect
206. C₂³	C₃×#213	direct (N × H)
211. D₄	C₂³×C₃×D₃	semidirect
221. C₂×C₃	C₂²×#213	direct (N × H)
222. C₂×C₃	C₂²×C₃×D₈	semidirect
230. C₂²	C₂×C₃×#213	direct (N × H)
235. C₃	C₂³×#213	direct (N × H)
236. C₃	C₂³×C₃×D₈	semidirect
239. C₂	C₂²×C₃×#213	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃²×D₄	2
C₂³×C₃×D₁₂	2
C₂²×C₃×#213	2
C₂³×C₃×C₃⋊C₈	2
C₂³×C₃×D₈	3
C₂³×#213	3

Subgroups — C23×C3×#213 #11918533

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃×#213 #11918533