Subgroups — C₃×#79910 #11898255

Order 1152.

971 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#5577	C₂	1	2	yes	maximal, complemented
2	C₃²×C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4843	C₂	1	6	yes	maximal, complemented
4	C₃×#4695	C₂	1	3	yes	maximal, complemented
5	C₃×C₄×D₂₄	C₂	1	3	yes	maximal, complemented
6	C₃×#3391	—	3	1	no	maximal
7	#79910	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×#239	—	2	2	no	—
9	C₃²×C₄×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×D₂₄	C₂²	1	3	yes	—
11	C₂×C₃×D₂₄	C₂²	1	6	yes	—
12	C₃×#895	—	2	6	no	—
13	C₃×#874	C₂²	1	6	yes	—
14	C₃×C₄×D₁₂	C₂²	1	6	yes	—
15	C₃²×C₄⋊C₈	C₂²	1	3	yes	—
16	C₃²×C₄×C₈	C₂²	1	3	yes	—
17	C₃×#821	C₂²	1	6	yes	—
18	C₃×#818	C₂²	1	1	yes	characteristic
19	C₃×#550	—	3	2	no	—
20	#5577	C₂×C₃	1	2	yes	complemented
21	C₃×#493	—	3	3	no	—
22	C₃×#459	—	3	6	no	—
23	C₃×C₈×Q₈	—	2	1	no	—
24	C₃×C₈×Q₈	D₃	1	1	yes	characteristic, complemented
25	C₃×C₈×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
26	C₃×C₄×D₈	—	3	3	no	—
27	#4843	C₂×C₃	1	6	yes	complemented
28	#4695	C₂×C₃	1	3	yes	complemented
29	C₄×D₂₄	C₂×C₃	1	3	yes	complemented
30	C₂×C₃²×Q₈	D₄	1	1	yes	characteristic
31	C₃×#239	—	4	8	no	—
32	C₂×C₃×D₁₂	—	2	6	no	—
33	C₂×C₃×D₁₂	—	2	6	no	—
34	C₂×C₃×D₁₂	C₂³	1	6	yes	—
35	C₂×C₃×C₄×D₃	—	2	6	no	—
36	C₂×C₃²×C₈	C₂³	1	3	yes	—
37	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
38	C₃²×C₄⋊C₄	D₄	1	3	yes	—
39	C₃²×C₄²	C₂³	1	3	yes	—
40	C₃×(C₂×D₃)⋊C₄	—	2	6	no	—
41	C₃×C₄⋊Dic₃	C₂³	1	2	yes	—
42	C₃×D₂₄	—	2	6	no	—
43	C₃×D₂₄	—	4	6	no	—
44	#3391	—	3	1	no	Sylow
45	C₂×C₃×#41	—	6	2	no	—
46	C₂×#239	—	2	2	no	—
47	C₂×C₃×D₈	—	3	3	no	—
48	C₂×C₃×D₈	—	3	6	no	—
49	C₃×#127	—	3	6	no	—
50	C₃×#121	—	6	6	no	—
51	C₃×C₄×Q₈	—	2	1	no	—
52	C₃×C₄×Q₈	C₂×D₃	1	1	yes	characteristic
53	C₃×C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
54	C₃×C₄×D₄	—	3	6	no	—
55	C₂×D₂₄	C₂²×C₃	1	3	yes	—
56	C₂×D₂₄	C₂²×C₃	1	6	yes	—
57	#895	—	2	6	no	—
58	#874	C₂²×C₃	1	6	yes	—
59	C₄×D₁₂	C₂²×C₃	1	6	yes	—
60	C₃×#107	—	3	1	no	—
61	C₃×C₄⋊C₈	—	2	3	no	—
62	C₃×C₄⋊C₈	C₂×D₃	1	3	yes	—
63	C₃×C₄⋊C₈	C₂²×C₃	1	3	yes	—
64	C₃×#102	—	3	6	no	—
65	C₃×C₄×C₈	—	2	3	no	—
66	C₃×C₄×C₈	C₂×D₃	1	3	yes	—
67	C₃×C₄×C₈	C₂²×C₃	1	3	yes	—
68	#821	C₂²×C₃	1	6	yes	—
69	#818	C₂²×C₃	1	1	yes	characteristic
70	C₂²×C₃×D₃	—	2	6	no	—
71	C₃²×Q₈	D₈	1	4	yes	complemented
72	C₂×C₃²×C₄	C₂×D₄	1	3	yes	—
73	C₂×C₃²×C₄	C₂×D₄	1	3	yes	—
74	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
75	C₂×C₃×Dic₃	—	2	2	no	—
76	C₃×D₁₂	—	2	6	no	—
77	C₃×D₁₂	—	2	6	no	—
78	C₃×D₁₂	—	4	12	no	—
79	C₃×D₁₂	—	8	6	no	—
80	C₃×C₄×D₃	—	4	12	no	—
81	C₃²×C₈	—	2	3	no	—
82	C₃²×C₈	#41	1	2	yes	complemented
83	#550	—	3	2	no	—
84	#493	—	3	3	no	—
85	#459	—	3	6	no	—
86	C₈×Q₈	C₃×D₃	1	1	yes	characteristic, complemented
87	C₄×D₈	—	3	3	no	—
88	C₃×#41	—	12	8	no	—
89	C₂×C₃×Q₈	—	2	1	no	—
90	C₂×C₃×Q₈	D₁₂	1	1	yes	characteristic
91	C₂×C₃×Q₈	C₃×D₄	1	1	yes	characteristic
92	C₂×C₃×D₄	—	3	6	no	—
93	C₂×C₃×D₄	—	6	6	no	—
94	C₂×C₃×D₄	—	6	6	no	—
95	C₂²×C₃×C₄	—	6	6	no	—
96	#239	—	4	8	no	—
97	C₂×D₁₂	—	2	6	no	—
98	C₂×D₁₂	—	2	6	no	—
99	C₂×D₁₂	C₂³×C₃	1	6	yes	—
100	C₂×C₄×D₃	—	2	6	no	—
101	C₃×D₈	—	6	6	no	—
102	C₃×D₈	—	12	6	no	—
103	C₂×C₃×C₈	—	2	1	no	—
104	C₂×C₃×C₈	—	2	3	no	—
105	C₂×C₃×C₈	C₂²×D₃	1	3	yes	—
106	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
107	C₂×C₃×C₈	C₂³×C₃	1	3	yes	—
108	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
109	C₃×C₄⋊C₄	—	2	3	no	—
110	C₃×C₄⋊C₄	—	3	2	no	—
111	C₃×C₄⋊C₄	D₁₂	1	3	yes	—
112	C₃×C₄⋊C₄	C₃×D₄	1	3	yes	—
113	C₃×C₂²⋊C₄	—	6	6	no	—
114	C₃×C₄²	—	2	3	no	—
115	C₃×C₄²	C₂²×D₃	1	3	yes	—
116	C₃×C₄²	C₂³×C₃	1	3	yes	—
117	(C₂×D₃)⋊C₄	—	2	6	no	—
118	C₄⋊Dic₃	C₂³×C₃	1	2	yes	—
119	D₂₄	—	2	6	no	—
120	D₂₄	—	4	6	no	—
121	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₃×D₃	—	4	6	no	—
123	C₂×C₃×D₃	—	4	6	no	—
124	C₂×C₃×D₃	—	4	6	no	—
125	C₃²×C₄	—	2	3	no	—
126	C₃²×C₄	C₂×D₈	1	6	yes	—
127	C₃²×C₄	C₂×#41	1	1	yes	characteristic
128	C₃²×C₄	H(5,𝔽₂)	1	1	yes	characteristic
129	C₃×Dic₃	—	4	2	no	—
130	C₂×#41	—	6	2	no	—
131	C₂×D₈	—	3	3	no	—
132	C₂×D₈	—	3	6	no	—
133	#127	—	3	6	no	—
134	#121	—	6	6	no	—
135	C₄×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
136	C₄×D₄	—	3	6	no	—
137	#107	—	3	1	no	—
138	C₄⋊C₈	C₂×C₃×D₃	1	3	yes	—
139	#102	—	3	6	no	—
140	C₄×C₈	C₂×C₃×D₃	1	3	yes	—
141	C₂³×C₃	—	6	6	no	—
142	C₂²×D₃	—	2	6	no	—
143	C₃×Q₈	—	2	4	no	—
144	C₃×Q₈	D₂₄	1	4	yes	complemented
145	C₃×Q₈	C₃×D₈	1	4	yes	complemented
146	C₃×D₄	—	6	6	no	—
147	C₃×D₄	—	6	6	no	—
148	C₃×D₄	—	12	12	no	—
149	C₃×D₄	—	24	6	no	—
150	C₂×C₃×C₄	—	2	1	no	—
151	C₂×C₃×C₄	—	2	3	no	—
152	C₂×C₃×C₄	—	2	3	no	—
153	C₂×C₃×C₄	—	6	2	no	—
154	C₂×C₃×C₄	—	12	12	no	—
155	C₂×C₃×C₄	C₂×D₁₂	1	3	yes	—
156	C₂×C₃×C₄	C₂×D₁₂	1	3	yes	—
157	C₂×C₃×C₄	C₂×C₃×D₄	1	3	yes	—
158	C₂×C₃×C₄	C₂×C₃×D₄	1	3	yes	—
159	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
161	C₂×Dic₃	—	2	2	no	—
162	D₁₂	—	2	6	no	—
163	D₁₂	—	2	6	no	—
164	D₁₂	—	4	12	no	—
165	D₁₂	—	8	6	no	—
166	C₄×D₃	—	4	12	no	—
167	C₃×C₈	—	2	3	no	—
168	C₃×C₈	—	2	3	no	—
169	C₃×C₈	—	2	2	no	—
170	C₃×C₈	—	4	3	no	—
171	C₃×C₈	#239	1	2	yes	complemented
172	C₃×C₈	C₃×#41	1	2	yes	complemented
173	C₂×C₃²	#550	1	1	yes	characteristic
174	C₂×C₃²	C₂²×D₈	1	1	yes	characteristic
175	C₂×C₃²	#576	1	1	yes	characteristic
176	C₃×D₃	—	8	6	no	—
177	#41	—	12	8	no	—
178	C₂×Q₈	C₃×D₁₂	1	1	yes	characteristic
179	C₂×D₄	—	3	6	no	—
180	C₂×D₄	—	6	6	no	—
181	C₂×D₄	—	6	6	no	—
182	C₂²×C₄	—	6	6	no	—
183	D₈	—	6	6	no	—
184	D₈	—	12	6	no	—
185	C₂×C₈	C₂²×C₃×D₃	1	3	yes	—
186	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
187	C₄⋊C₄	—	3	2	no	—
188	C₄⋊C₄	C₃×D₁₂	1	3	yes	—
189	C₂²⋊C₄	—	6	6	no	—
190	C₄²	C₂²×C₃×D₃	1	3	yes	—
191	C₂²×C₃	—	2	1	no	—
192	C₂²×C₃	—	12	6	no	—
193	C₂²×C₃	—	12	6	no	—
194	C₂²×C₃	—	12	6	no	—
195	C₂²×C₃	C₂²×D₁₂	1	1	yes	characteristic
196	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
197	C₂×D₃	—	4	6	no	—
198	C₂×D₃	—	4	6	no	—
199	C₂×D₃	—	4	6	no	—
200	C₃×C₄	—	2	3	no	—
201	C₃×C₄	—	2	3	no	—
202	C₃×C₄	—	2	6	no	—
203	C₃×C₄	—	2	1	no	—
204	C₃×C₄	—	2	1	no	—
205	C₃×C₄	—	4	3	no	—
206	C₃×C₄	—	12	2	no	—
207	C₃×C₄	C₂×D₂₄	1	6	yes	—
208	C₃×C₄	C₂×C₃×D₈	1	6	yes	—
209	C₃×C₄	C₂×#239	1	1	yes	characteristic
210	C₃×C₄	#1009	1	1	yes	characteristic
211	C₃×C₄	C₂×C₃×#41	1	1	yes	characteristic
212	C₃×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
213	Dic₃	—	4	2	no	—
214	C₃²	#3391	1	1	yes	characteristic, complemented, Sylow
215	C₂³	—	6	6	no	—
216	Q₈	C₃×D₂₄	1	4	yes	complemented
217	D₄	—	6	6	no	—
218	D₄	—	6	6	no	—
219	D₄	—	12	12	no	—
220	D₄	—	24	6	no	—
221	C₂×C₄	—	6	2	no	—
222	C₂×C₄	—	12	12	no	—
223	C₂×C₄	C₂×C₃×D₁₂	1	3	yes	—
224	C₂×C₄	C₂×C₃×D₁₂	1	3	yes	—
225	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
226	C₈	—	2	3	no	—
227	C₈	C₃×#239	1	2	yes	complemented
228	C₂×C₃	—	2	1	no	—
229	C₂×C₃	—	2	1	no	—
230	C₂×C₃	—	2	1	no	—
231	C₂×C₃	—	24	6	no	—
232	C₂×C₃	#5577	1	1	yes	characteristic
233	C₂×C₃	C₂²×D₂₄	1	1	yes	characteristic
234	C₂×C₃	#5752	1	1	yes	characteristic
235	C₂×C₃	C₃×#550	1	1	yes	characteristic
236	C₂×C₃	C₂²×C₃×D₈	1	1	yes	characteristic
237	C₂×C₃	C₃×#576	1	1	yes	characteristic
238	D₃	—	8	6	no	—
239	C₂²	—	12	6	no	—
240	C₂²	—	12	6	no	—
241	C₂²	—	12	6	no	—
242	C₂²	C₂²×C₃×D₁₂	1	1	yes	characteristic
243	C₄	—	2	3	no	—
244	C₄	—	12	2	no	—
245	C₄	C₂×C₃×D₂₄	1	6	yes	—
246	C₄	C₂×C₃×#239	1	1	yes	characteristic
247	C₄	C₃×#1009	1	1	yes	characteristic
248	C₃	—	2	1	no	—
249	C₃	#79910	1	1	yes	characteristic, direct product factor
250	C₃	C₃×#3391	1	1	yes	characteristic, complemented
251	C₂	—	24	6	no	—
252	C₂	C₃×#5577	1	1	yes	characteristic
253	C₂	C₂²×C₃×D₂₄	1	1	yes	characteristic
254	C₂	C₃×#5752	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂²×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₃²×C₈×Q₈	C₂
socle	C₂²×C₃²	C₂²×D₄
hypercenter	C₃×C₈×Q₈	D₃
hypocenter	C₃	C₃×#3391
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×Q₈ ◁ C₃×C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×#5577	C₂	semidirect
2. C₃²×C₈×Q₈	C₂	semidirect
3. C₃×#4843	C₂	semidirect
4. C₃×#4695	C₂	semidirect
5. C₃×C₄×D₂₄	C₂	semidirect
7. #79910	C₃	direct (N × H)
20. #5577	C₂×C₃	semidirect
24. C₃×C₈×Q₈	D₃	semidirect
25. C₃×C₈×Q₈	C₂×C₃	semidirect
27. #4843	C₂×C₃	semidirect
28. #4695	C₂×C₃	semidirect
29. C₄×D₂₄	C₂×C₃	semidirect
71. C₃²×Q₈	D₈	semidirect
82. C₃²×C₈	#41	semidirect
86. C₈×Q₈	C₃×D₃	semidirect
144. C₃×Q₈	D₂₄	semidirect
145. C₃×Q₈	C₃×D₈	semidirect
171. C₃×C₈	#239	semidirect
172. C₃×C₈	C₃×#41	semidirect
214. C₃²	#3391	semidirect
216. Q₈	C₃×D₂₄	semidirect
227. C₈	C₃×#239	semidirect
249. C₃	#79910	direct (N × H)
250. C₃	C₃×#3391	semidirect

Maximal subgroups

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

subgroup	index
C₃×#5577	2
C₃²×C₈×Q₈	2
C₃×#4843	2
C₃×#4695	2
C₃×C₄×D₂₄	2
C₃×#3391	3
#79910	3

Subgroups — C3×#79910 #11898255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#79910 #11898255