Subgroups — C₂×C₃×#5230 #11808179

Order 1152.

2241 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃²×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×Dic₃⋊C₄	C₂	1	3	yes	maximal
3	C₃×#5230	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×C₃⋊#95	C₂	1	3	yes	maximal
5	C₂×C₃×#395	—	3	1	no	maximal
6	C₂×#5230	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃²×Q₈	—	2	12	no	—
8	C₂²×C₃²×Q₈	C₂²	1	4	yes	—
9	C₂³×C₃²×C₄	C₂²	1	3	yes	—
10	C₂³×C₃×Dic₃	C₂²	1	1	yes	characteristic
11	C₂³×C₃×Dic₃	C₂²	1	3	yes	—
12	C₂×C₃×Dic₃⋊C₄	—	2	12	no	—
13	C₂×C₃×Dic₃⋊C₄	C₂²	1	12	yes	—
14	C₃×C₃⋊#95	C₂²	1	12	yes	—
15	C₂³×C₃×Q₈	—	2	1	no	—
16	C₂³×C₃×Q₈	D₃	1	1	yes	characteristic
17	C₂³×C₃×Q₈	C₂×C₃	1	1	yes	characteristic
18	C₂²×C₃×C₄⋊C₄	—	3	3	no	—
19	C₂²×Dic₃⋊C₄	C₂×C₃	1	3	yes	—
20	C₃×#395	—	3	8	no	—
21	C₂×C₃×#95	—	3	3	no	—
22	#5230	C₂×C₃	1	8	yes	direct product factor
23	C₂×C₃⋊#95	C₂×C₃	1	3	yes	—
24	C₂⁴×C₃²	C₂³	1	1	yes	characteristic
25	C₂×C₃²×Q₈	—	2	32	no	—
26	C₂×C₃²×Q₈	—	2	24	no	—
27	C₂²×C₃²×C₄	—	2	12	no	—
28	C₂²×C₃²×C₄	D₄	1	12	yes	—
29	C₂²×C₃²×C₄	C₂³	1	6	yes	—
30	C₂²×C₃×Dic₃	—	2	6	no	—
31	C₂²×C₃×Dic₃	—	2	6	no	—
32	C₂²×C₃×Dic₃	—	2	6	no	—
33	C₂²×C₃×Dic₃	Q₈	1	12	yes	complemented
34	C₂²×C₃×Dic₃	C₂³	1	2	yes	—
35	C₂²×C₃×Dic₃	C₂³	1	6	yes	—
36	C₃×Dic₃⋊C₄	—	2	24	no	—
37	C₂×#395	—	3	1	no	Sylow
38	C₂²×C₃×Q₈	—	2	24	no	—
39	C₂²×C₃×Q₈	—	2	12	no	—
40	C₂²×C₃×Q₈	—	2	12	no	—
41	C₂²×C₃×Q₈	—	2	4	no	—
42	C₂²×C₃×Q₈	C₂×D₃	1	4	yes	—
43	C₂²×C₃×Q₈	C₂²×C₃	1	4	yes	—
44	C₂³×C₃×C₄	—	2	3	no	—
45	C₂³×C₃×C₄	—	3	3	no	—
46	C₂³×C₃×C₄	—	3	1	no	—
47	C₂³×C₃×C₄	C₂×D₃	1	3	yes	—
48	C₂³×C₃×C₄	C₂²×C₃	1	3	yes	—
49	C₂³×Dic₃	C₂²×C₃	1	1	yes	characteristic
50	C₂³×Dic₃	C₂²×C₃	1	3	yes	—
51	C₂×C₃×C₄⋊C₄	—	3	12	no	—
52	C₂×C₃×C₄⋊C₄	—	6	12	no	—
53	C₂×Dic₃⋊C₄	—	2	12	no	—
54	C₂×Dic₃⋊C₄	C₂²×C₃	1	12	yes	—
55	C₃×#95	—	3	12	no	—
56	C₃⋊#95	C₂²×C₃	1	12	yes	—
57	C₂³×C₃²	C₂×D₄	1	6	yes	—
58	C₂³×C₃²	C₂×Q₈	1	6	yes	—
59	C₂³×C₃²	#41	1	2	yes	—
60	C₂³×C₃²	C₂⁴	1	1	yes	characteristic
61	C₃²×Q₈	—	2	32	no	—
62	C₂×C₃²×C₄	—	2	24	no	—
63	C₂×C₃²×C₄	—	2	12	no	—
64	C₂×C₃²×C₄	C₂×D₄	1	12	yes	—
65	C₂×C₃×Dic₃	—	2	24	no	—
66	C₂×C₃×Dic₃	—	2	6	no	—
67	C₂×C₃×Dic₃	—	2	6	no	—
68	C₂×C₃×Dic₃	—	2	6	no	—
69	C₂×C₃×Dic₃	—	4	4	no	—
70	C₂×C₃×Dic₃	C₂×Q₈	1	12	yes	complemented
71	C₂³×Q₈	C₃×D₃	1	1	yes	characteristic
72	C₂²×C₄⋊C₄	—	3	3	no	—
73	#395	—	3	8	no	—
74	C₂×#95	—	3	3	no	—
75	C₂⁴×C₃	—	2	1	no	—
76	C₂⁴×C₃	C₂²×D₃	1	1	yes	characteristic
77	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
78	C₂×C₃×Q₈	—	2	64	no	—
79	C₂×C₃×Q₈	—	2	32	no	—
80	C₂×C₃×Q₈	—	2	32	no	—
81	C₂×C₃×Q₈	—	2	48	no	—
82	C₂×C₃×Q₈	—	2	24	no	—
83	C₂×C₃×Q₈	—	2	24	no	—
84	C₂²×C₃×C₄	—	2	24	no	—
85	C₂²×C₃×C₄	—	2	12	no	—
86	C₂²×C₃×C₄	—	2	12	no	—
87	C₂²×C₃×C₄	—	2	12	no	—
88	C₂²×C₃×C₄	—	2	6	no	—
89	C₂²×C₃×C₄	—	3	12	no	—
90	C₂²×C₃×C₄	—	3	6	no	—
91	C₂²×C₃×C₄	—	3	2	no	—
92	C₂²×C₃×C₄	—	6	6	no	—
93	C₂²×C₃×C₄	—	6	6	no	—
94	C₂²×C₃×C₄	—	6	6	no	—
95	C₂²×C₃×C₄	#67	1	12	yes	—
96	C₂²×C₃×C₄	C₃×D₄	1	12	yes	—
97	C₂²×C₃×C₄	C₂²×D₃	1	6	yes	—
98	C₂²×C₃×C₄	C₂³×C₃	1	6	yes	—
99	C₂²×Dic₃	—	2	6	no	—
100	C₂²×Dic₃	—	2	6	no	—
101	C₂²×Dic₃	—	2	6	no	—
102	C₂²×Dic₃	C₃×Q₈	1	12	yes	complemented
103	C₂²×Dic₃	C₂³×C₃	1	2	yes	—
104	C₂²×Dic₃	C₂³×C₃	1	6	yes	—
105	C₃×C₄⋊C₄	—	6	24	no	—
106	Dic₃⋊C₄	—	2	24	no	—
107	C₂²×C₃²	#120	1	4	yes	—
108	C₂²×C₃²	C₂²⋊Q₈	1	12	yes	—
109	C₂²×C₃²	C₄⋊Q₈	1	12	yes	—
110	C₂²×C₃²	C₂²×D₄	1	3	yes	—
111	C₂²×C₃²	C₂²×Q₈	1	3	yes	—
112	C₂²×C₃²	C₂×#41	1	1	yes	characteristic
113	C₃²×C₄	—	2	12	no	—
114	C₃×Dic₃	—	2	12	no	—
115	C₃×Dic₃	—	4	2	no	—
116	C₂²×Q₈	—	2	12	no	—
117	C₂²×Q₈	C₂×C₃×D₃	1	4	yes	—
118	C₂³×C₄	—	3	3	no	—
119	C₂³×C₄	—	3	1	no	—
120	C₂³×C₄	C₂×C₃×D₃	1	3	yes	—
121	C₂×C₄⋊C₄	—	3	12	no	—
122	C₂×C₄⋊C₄	—	6	12	no	—
123	#95	—	3	12	no	—
124	C₂³×C₃	—	2	1	no	—
125	C₂³×C₃	—	2	6	no	—
126	C₂³×C₃	—	2	2	no	—
127	C₂³×C₃	—	2	6	no	—
128	C₂³×C₃	D₃×Q₈	1	6	yes	—
129	C₂³×C₃	#239	1	2	yes	—
130	C₂³×C₃	C₂×#67	1	6	yes	—
131	C₂³×C₃	C₂×C₃×D₄	1	6	yes	—
132	C₂³×C₃	C₂×C₃×Q₈	1	6	yes	—
133	C₂³×C₃	C₃×#41	1	2	yes	—
134	C₂³×C₃	C₂³×D₃	1	1	yes	characteristic
135	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
136	C₃×Q₈	—	2	64	no	—
137	C₃×Q₈	—	2	32	no	—
138	C₃×Q₈	—	2	32	no	—
139	C₂×C₃×C₄	—	2	48	no	—
140	C₂×C₃×C₄	—	2	24	no	—
141	C₂×C₃×C₄	—	2	24	no	—
142	C₂×C₃×C₄	—	2	24	no	—
143	C₂×C₃×C₄	—	2	12	no	—
144	C₂×C₃×C₄	—	2	12	no	—
145	C₂×C₃×C₄	—	2	12	no	—
146	C₂×C₃×C₄	—	3	12	no	—
147	C₂×C₃×C₄	—	6	24	no	—
148	C₂×C₃×C₄	—	6	6	no	—
149	C₂×C₃×C₄	—	6	6	no	—
150	C₂×C₃×C₄	—	6	6	no	—
151	C₂×C₃×C₄	—	12	4	no	—
152	C₂×C₃×C₄	C₂×#67	1	12	yes	—
153	C₂×C₃×C₄	C₂×C₃×D₄	1	12	yes	—
154	C₂×Dic₃	—	2	24	no	—
155	C₂×Dic₃	—	2	6	no	—
156	C₂×Dic₃	—	2	6	no	—
157	C₂×Dic₃	—	2	6	no	—
158	C₂×Dic₃	—	4	4	no	—
159	C₂×Dic₃	C₂×C₃×Q₈	1	12	yes	complemented
160	C₂×C₃²	#395	1	8	yes	complemented
161	C₂×C₃²	C₂×#120	1	1	yes	characteristic
162	C₂×C₃²	C₂×C₂²⋊Q₈	1	3	yes	—
163	C₂×C₃²	C₂×C₄⋊Q₈	1	3	yes	—
164	C₂⁴	C₂²×C₃×D₃	1	1	yes	characteristic
165	C₂×Q₈	—	2	32	no	—
166	C₂×Q₈	—	2	24	no	—
167	C₂²×C₄	—	2	12	no	—
168	C₂²×C₄	—	3	6	no	—
169	C₂²×C₄	—	3	12	no	—
170	C₂²×C₄	—	3	2	no	—
171	C₂²×C₄	—	6	6	no	—
172	C₂²×C₄	—	6	6	no	—
173	C₂²×C₄	—	6	6	no	—
174	C₂²×C₄	C₃×#67	1	12	yes	—
175	C₂²×C₄	C₂²×C₃×D₃	1	6	yes	—
176	C₄⋊C₄	—	6	24	no	—
177	C₂²×C₃	—	2	12	no	—
178	C₂²×C₃	—	2	12	no	—
179	C₂²×C₃	—	2	4	no	—
180	C₂²×C₃	—	2	3	no	—
181	C₂²×C₃	—	2	1	no	—
182	C₂²×C₃	—	2	3	no	—
183	C₂²×C₃	#944	1	12	yes	—
184	C₂²×C₃	#946	1	12	yes	—
185	C₂²×C₃	#953	1	4	yes	—
186	C₂²×C₃	C₃×#120	1	4	yes	—
187	C₂²×C₃	C₃×C₂²⋊Q₈	1	12	yes	—
188	C₂²×C₃	C₃×C₄⋊Q₈	1	12	yes	—
189	C₂²×C₃	C₂×D₃×Q₈	1	3	yes	—
190	C₂²×C₃	C₂×#239	1	1	yes	characteristic
191	C₂²×C₃	C₂²×#67	1	3	yes	—
192	C₂²×C₃	C₂²×C₃×D₄	1	3	yes	—
193	C₂²×C₃	C₂²×C₃×Q₈	1	3	yes	—
194	C₂²×C₃	C₂×C₃×#41	1	1	yes	characteristic
195	C₃×C₄	—	2	24	no	—
196	C₃×C₄	—	2	12	no	—
197	C₃×C₄	—	2	12	no	—
198	C₃×C₄	—	6	12	no	—
199	C₃×C₄	—	12	2	no	—
200	Dic₃	—	2	12	no	—
201	Dic₃	—	4	2	no	—
202	C₃²	C₂×#395	1	1	yes	characteristic, complemented, Sylow
203	C₂³	C₃×D₃×Q₈	1	6	yes	—
204	C₂³	C₃×#239	1	2	yes	—
205	C₂³	C₂×C₃×#67	1	6	yes	—
206	C₂³	C₂³×C₃×D₃	1	1	yes	characteristic
207	Q₈	—	2	32	no	—
208	C₂×C₄	—	2	24	no	—
209	C₂×C₄	—	2	12	no	—
210	C₂×C₄	—	3	12	no	—
211	C₂×C₄	—	6	24	no	—
212	C₂×C₄	—	6	6	no	—
213	C₂×C₄	—	6	6	no	—
214	C₂×C₄	—	6	6	no	—
215	C₂×C₄	—	12	4	no	—
216	C₂×C₄	C₂×C₃×#67	1	12	yes	—
217	C₂×C₃	—	2	3	no	—
218	C₂×C₃	—	2	3	no	—
219	C₂×C₃	—	2	8	no	—
220	C₂×C₃	—	2	1	no	—
221	C₂×C₃	#5230	1	8	yes	direct product factor
222	C₂×C₃	C₃×#395	1	8	yes	complemented
223	C₂×C₃	C₂×#944	1	3	yes	—
224	C₂×C₃	C₂×#946	1	3	yes	—
225	C₂×C₃	C₂×#953	1	1	yes	characteristic
226	C₂×C₃	C₂×C₃×#120	1	1	yes	characteristic
227	C₂×C₃	C₂×C₃×C₂²⋊Q₈	1	3	yes	—
228	C₂×C₃	C₂×C₃×C₄⋊Q₈	1	3	yes	—
229	C₂²	C₃×#944	1	12	yes	—
230	C₂²	C₃×#946	1	12	yes	—
231	C₂²	C₃×#953	1	4	yes	—
232	C₂²	C₂×C₃×D₃×Q₈	1	3	yes	—
233	C₂²	C₂×C₃×#239	1	1	yes	characteristic
234	C₂²	C₂²×C₃×#67	1	3	yes	—
235	C₄	—	2	12	no	—
236	C₄	—	6	12	no	—
237	C₄	—	12	2	no	—
238	C₃	—	2	1	no	—
239	C₃	C₂×#5230	1	1	yes	characteristic, direct product factor
240	C₃	C₂×C₃×#395	1	1	yes	characteristic, complemented
241	C₂	C₃×#5230	1	8	yes	direct product factor
242	C₂	C₂×C₃×#944	1	3	yes	—
243	C₂	C₂×C₃×#946	1	3	yes	—
244	C₂	C₂×C₃×#953	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₃	C₂²×D₃
commutator, cosocle	C₂³×C₃	C₂⁴×C₃
Frattini	C₂³	C₂³×C₃×D₃
Fitting	C₂³×C₃²×Q₈	C₂
socle, prime radical	C₂⁴×C₃²	C₂³
hypercenter	C₂³×C₃×Q₈	D₃
hypocenter	C₃	C₂×C₃×#395
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴×C₃ ◁ C₂³×C₃×Q₈

Complements

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

N	H ≅ G/N	type
3. C₃×#5230	C₂	direct (N × H)
6. C₂×#5230	C₃	direct (N × H)
22. #5230	C₂×C₃	direct (N × H)
33. C₂²×C₃×Dic₃	Q₈	semidirect
70. C₂×C₃×Dic₃	C₂×Q₈	semidirect
102. C₂²×Dic₃	C₃×Q₈	semidirect
159. C₂×Dic₃	C₂×C₃×Q₈	semidirect
160. C₂×C₃²	#395	semidirect
202. C₃²	C₂×#395	semidirect
221. C₂×C₃	#5230	direct (N × H)
222. C₂×C₃	C₃×#395	semidirect
239. C₃	C₂×#5230	direct (N × H)
240. C₃	C₂×C₃×#395	semidirect
241. C₂	C₃×#5230	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×Q₈	2
C₂²×C₃×Dic₃⋊C₄	2
C₃×#5230	2
C₂×C₃×C₃⋊#95	2
C₂×C₃×#395	3
C₂×#5230	3

Subgroups — C2×C3×#5230 #11808179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#5230 #11808179