Subgroups — C₃×#68928 #11786087

Order 1152.

421 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#5118	C₂	1	2	yes	maximal
2	C₂×C₃×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₃²×#341	C₂	1	1	yes	maximal, characteristic
4	C₃×#4558	C₂	1	2	yes	maximal
5	C₃×#4552	C₂	1	1	yes	maximal, characteristic
6	C₃×#1837	—	3	1	no	maximal
7	#68928	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
11	C₂×C₃×#208	C₂²	1	2	yes	—
12	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₃×Dic₃⋊C₈	—	2	4	no	—
14	C₃×C₈×Dic₃	C₄	1	2	yes	—
15	C₃×C₈×Dic₃	C₄	1	2	yes	—
16	C₃×#423	—	3	2	no	—
17	C₂×C₃×C₄×C₈	—	3	1	no	—
18	#5118	C₂×C₃	1	2	yes	—
19	C₂×C₈×Dic₃	C₂×C₃	1	1	yes	characteristic
20	C₃×#341	—	2	1	no	—
21	C₃×#341	—	3	2	no	—
22	C₃×#341	—	3	1	no	—
23	C₃×#341	D₃	1	1	yes	characteristic
24	C₃×#341	C₂×C₃	1	1	yes	characteristic
25	#4558	C₂×C₃	1	2	yes	—
26	#4552	C₂×C₃	1	1	yes	characteristic
27	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
28	C₂²×C₃×Dic₃	Q₈	1	2	yes	—
29	C₃²×M₁₆	—	2	2	no	—
30	C₃²×M₁₆	—	2	2	no	—
31	C₂×C₃²×C₈	—	2	2	no	—
32	C₂×C₃²×C₈	—	2	2	no	—
33	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
34	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄×Dic₃	D₄	1	2	yes	—
37	C₃×#208	—	2	2	no	—
38	C₃×#208	—	2	2	no	—
39	C₂×C₃×C₃⋊C₈	—	2	2	no	—
40	C₂×C₃×C₃⋊C₈	—	2	2	no	—
41	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
42	#1837	—	3	1	no	Sylow
43	C₂×C₃×M₁₆	—	2	2	no	—
44	C₂×C₃×M₁₆	—	3	2	no	—
45	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
46	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
47	C₂²×C₃×C₈	—	2	1	no	—
48	C₂²×C₃×C₈	—	3	1	no	—
49	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
50	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₃×C₄²	—	3	1	no	—
52	C₂×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
53	C₂×#208	C₂²×C₃	1	2	yes	—
54	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
55	C₃×C₄⋊C₈	—	6	4	no	—
56	C₃×C₄×C₈	—	3	2	no	—
57	C₃×C₄×C₈	—	3	2	no	—
58	Dic₃⋊C₈	—	2	4	no	—
59	C₈×Dic₃	C₃×C₄	1	2	yes	—
60	C₈×Dic₃	C₃×C₄	1	2	yes	—
61	C₂³×C₃²	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃²×C₄	#41	1	2	yes	—
65	C₂×C₃²×C₄	#41	1	2	yes	—
66	C₂×C₃×Dic₃	—	2	4	no	—
67	C₂×C₃×Dic₃	C₄⋊C₄	1	2	yes	—
68	C₂×C₃×Dic₃	C₄⋊C₄	1	2	yes	—
69	C₃²×C₈	—	2	2	no	—
70	C₃²×C₈	—	4	2	no	—
71	C₃×C₃⋊C₈	—	2	2	no	—
72	C₃×C₃⋊C₈	—	4	2	no	—
73	#423	—	3	2	no	—
74	C₂×C₄×C₈	—	3	1	no	—
75	#341	—	3	2	no	—
76	#341	—	3	1	no	—
77	#341	C₃×D₃	1	1	yes	characteristic
78	C₂²×C₃×C₄	—	2	1	no	—
79	C₂²×C₃×C₄	—	3	2	no	—
80	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
82	C₂²×Dic₃	C₃×Q₈	1	2	yes	—
83	C₃×M₁₆	—	2	4	no	—
84	C₃×M₁₆	—	2	2	no	—
85	C₃×M₁₆	—	2	2	no	—
86	C₃×M₁₆	—	2	2	no	—
87	C₃×M₁₆	—	2	2	no	—
88	C₃×M₁₆	—	4	2	no	—
89	C₃×M₁₆	—	6	2	no	—
90	C₃×M₁₆	—	6	2	no	—
91	C₂×C₃×C₈	—	2	2	no	—
92	C₂×C₃×C₈	—	2	2	no	—
93	C₂×C₃×C₈	—	2	2	no	—
94	C₂×C₃×C₈	—	2	2	no	—
95	C₂×C₃×C₈	—	2	2	no	—
96	C₂×C₃×C₈	—	3	2	no	—
97	C₂×C₃×C₈	—	4	2	no	—
98	C₂×C₃×C₈	—	4	2	no	—
99	C₂×C₃×C₈	—	6	2	no	—
100	C₂×C₃×C₈	—	6	2	no	—
101	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
102	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
103	C₃×C₄²	—	3	1	no	—
104	C₃×C₄²	—	3	1	no	—
105	C₃×C₄²	—	3	2	no	—
106	C₄×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
107	C₄×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
108	C₄×Dic₃	C₃×D₄	1	2	yes	—
109	#208	—	2	2	no	—
110	#208	—	2	2	no	—
111	C₂×C₃⋊C₈	—	2	2	no	—
112	C₂×C₃⋊C₈	—	2	2	no	—
113	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
114	C₂²×C₃²	—	2	2	no	—
115	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₃²	#125	1	2	yes	—
117	C₃²×C₄	#117	1	1	yes	characteristic
118	C₃²×C₄	#117	1	1	yes	characteristic
119	C₃²×C₄	#124	1	2	yes	—
120	C₃×Dic₃	—	2	2	no	—
121	C₃×Dic₃	#108	1	4	yes	—
122	C₂×M₁₆	—	3	2	no	—
123	C₂×M₁₆	C₂×C₃×D₃	1	2	yes	—
124	C₂²×C₈	—	3	1	no	—
125	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
126	C₂×C₄²	—	3	1	no	—
127	C₄⋊C₈	—	6	4	no	—
128	C₄×C₈	—	3	2	no	—
129	C₄×C₈	—	3	2	no	—
130	C₂³×C₃	—	2	1	no	—
131	C₂³×C₃	D₃×Q₈	1	1	yes	characteristic
132	C₂³×C₃	C₂×C₃×Q₈	1	1	yes	characteristic
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	3	2	no	—
138	C₂×C₃×C₄	—	3	2	no	—
139	C₂×C₃×C₄	—	6	4	no	—
140	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
141	C₂×C₃×C₄	#235	1	2	yes	—
142	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
143	C₂×C₃×C₄	#237	1	2	yes	—
144	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
146	C₂×C₃×C₄	C₃×#41	1	2	yes	—
147	C₂×C₃×C₄	C₃×#41	1	2	yes	—
148	C₂×Dic₃	—	2	4	no	—
149	C₂×Dic₃	C₃×C₄⋊C₄	1	2	yes	—
150	C₂×Dic₃	C₃×C₄⋊C₄	1	2	yes	—
151	C₃×C₈	—	2	2	no	—
152	C₃×C₈	—	2	2	no	—
153	C₃×C₈	—	4	2	no	—
154	C₃×C₈	—	4	4	no	—
155	C₃×C₈	—	4	2	no	—
156	C₃×C₈	—	4	2	no	—
157	C₃×C₈	—	6	2	no	—
158	C₃×C₈	—	12	2	no	—
159	C₃⋊C₈	—	2	2	no	—
160	C₃⋊C₈	—	4	2	no	—
161	C₂×C₃²	—	2	2	no	—
162	C₂×C₃²	#382	1	1	yes	characteristic
163	C₂×C₃²	C₂×#108	1	2	yes	—
164	C₂²×C₄	—	3	2	no	—
165	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
166	M₁₆	—	2	2	no	—
167	M₁₆	—	2	2	no	—
168	M₁₆	—	6	2	no	—
169	M₁₆	—	6	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	3	2	no	—
173	C₂×C₈	—	6	2	no	—
174	C₂×C₈	—	6	2	no	—
175	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
176	C₄²	—	3	1	no	—
177	C₄²	—	3	1	no	—
178	C₄²	—	3	2	no	—
179	C₂²×C₃	—	2	2	no	—
180	C₂²×C₃	—	2	2	no	—
181	C₂²×C₃	—	2	2	no	—
182	C₂²×C₃	—	2	1	no	—
183	C₂²×C₃	—	4	2	no	—
184	C₂²×C₃	#889	1	2	yes	—
185	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
186	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
187	C₂²×C₃	C₃×#125	1	2	yes	—
188	C₃×C₄	—	2	1	no	—
189	C₃×C₄	—	2	2	no	—
190	C₃×C₄	—	2	1	no	—
191	C₃×C₄	—	3	4	no	—
192	C₃×C₄	—	6	2	no	—
193	C₃×C₄	#872	1	1	yes	characteristic
194	C₃×C₄	#877	1	1	yes	characteristic
195	C₃×C₄	#885	1	2	yes	—
196	C₃×C₄	C₃×#117	1	1	yes	characteristic
197	C₃×C₄	C₃×#117	1	1	yes	characteristic
198	C₃×C₄	C₃×#124	1	2	yes	—
199	Dic₃	—	2	2	no	—
200	Dic₃	C₃×#108	1	4	yes	—
201	C₃²	#1837	1	1	yes	characteristic, complemented, Sylow
202	C₂³	C₃×D₃×Q₈	1	1	yes	characteristic
203	C₂×C₄	—	3	2	no	—
204	C₂×C₄	—	3	2	no	—
205	C₂×C₄	—	6	4	no	—
206	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
207	C₂×C₄	C₃×#235	1	2	yes	—
208	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
209	C₂×C₄	C₃×#237	1	2	yes	—
210	C₈	—	2	2	no	—
211	C₈	—	4	2	no	—
212	C₈	—	6	2	no	—
213	C₈	—	12	2	no	—
214	C₂×C₃	—	2	2	no	—
215	C₂×C₃	—	2	2	no	—
216	C₂×C₃	—	2	1	no	—
217	C₂×C₃	—	2	2	no	—
218	C₂×C₃	—	4	2	no	—
219	C₂×C₃	#4650	1	1	yes	characteristic
220	C₂×C₃	D₃×#108	1	2	yes	—
221	C₂×C₃	C₃×#382	1	1	yes	characteristic
222	C₂×C₃	C₂×C₃×#108	1	2	yes	—
223	C₂²	—	2	2	no	—
224	C₂²	C₃×#889	1	2	yes	—
225	C₂²	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
226	C₄	—	3	4	no	—
227	C₄	—	6	2	no	—
228	C₄	C₃×#872	1	1	yes	characteristic
229	C₄	C₃×#877	1	1	yes	characteristic
230	C₄	C₃×#885	1	2	yes	—
231	C₃	—	2	1	no	—
232	C₃	#68928	1	1	yes	characteristic, direct product factor
233	C₃	C₃×#1837	1	1	yes	characteristic, complemented
234	C₂	—	2	2	no	—
235	C₂	C₃×#4650	1	1	yes	characteristic
236	C₂	C₃×D₃×#108	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₃²×#341	C₂
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂³×C₃²	C₂×Q₈
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₃×#341	D₃
hypocenter	C₃	C₃×#1837
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₃×#341

Complements

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

N	H ≅ G/N	type
7. #68928	C₃	direct (N × H)
201. C₃²	#1837	semidirect
232. C₃	#68928	direct (N × H)
233. C₃	C₃×#1837	semidirect

Maximal subgroups

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

subgroup	index
C₃×#5118	2
C₂×C₃×C₈×Dic₃	2
C₃²×#341	2
C₃×#4558	2
C₃×#4552	2
C₃×#1837	3
#68928	3

Subgroups — C3×#68928 #11786087

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68928 #11786087