Subgroups — C₃×#69849 #11773692

Order 1152.

499 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×#208	C₂	1	2	yes	maximal
3	C₃×#4455	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4454	C₂	1	2	yes	maximal, complemented
5	C₃×C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1455	—	3	1	no	maximal
7	#69849	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃²×M₁₆	C₄	1	2	yes	—
9	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₃×#208	C₄	1	4	yes	—
11	C₃²×#97	C₄	1	2	yes	—
12	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₃×#803	C₄	1	4	yes	—
15	C₃×C₄×C₃⋊C₈	C₂²	1	2	yes	—
16	C₃×C₄×C₃⋊C₈	C₂²	1	2	yes	—
17	C₃×C₄×M₁₆	—	2	1	no	—
18	C₃×C₄×M₁₆	—	3	2	no	—
19	C₃×C₄×M₁₆	D₃	1	1	yes	characteristic
20	C₃×C₄×M₁₆	C₂×C₃	1	1	yes	characteristic
21	C₄×#208	C₂×C₃	1	2	yes	—
22	C₃×#322	—	3	2	no	—
23	C₃×#322	—	3	1	no	—
24	C₃×C₈²	—	3	1	no	—
25	#4455	C₂×C₃	1	1	yes	characteristic, complemented
26	#4454	C₂×C₃	1	2	yes	complemented
27	C₈×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
29	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
30	C₃²×M₁₆	—	2	4	no	—
31	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
32	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
33	C₃²×C₄²	C₂×C₄	1	2	yes	—
34	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
35	C₃²×C₄²	C₂³	1	1	yes	characteristic
36	C₃×#208	—	2	8	no	—
37	C₂×C₃×C₃⋊C₈	C₂×C₄	1	4	yes	—
38	C₂×C₃×C₃⋊C₈	C₂×C₄	1	4	yes	—
39	#1455	—	3	1	no	Sylow
40	C₂×C₃×M₁₆	—	2	2	no	—
41	C₂×C₃×M₁₆	—	3	4	no	—
42	C₂×C₃×M₁₆	Dic₃	1	2	yes	—
43	C₂×C₃×M₁₆	C₃×C₄	1	2	yes	—
44	C₂×C₃×C₄²	—	2	1	no	—
45	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
46	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
47	C₂×#208	C₃×C₄	1	4	yes	—
48	C₃×#97	—	2	2	no	—
49	C₃×#97	—	3	4	no	—
50	C₃×#97	Dic₃	1	2	yes	—
51	C₃×#97	C₃×C₄	1	2	yes	—
52	C₃×C₄×C₈	—	2	1	no	—
53	C₃×C₄×C₈	—	2	1	no	—
54	C₃×C₄×C₈	—	3	2	no	—
55	C₃×C₄×C₈	—	3	2	no	—
56	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
57	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
58	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
59	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
60	#803	C₃×C₄	1	4	yes	—
61	C₄×C₃⋊C₈	C₂²×C₃	1	2	yes	—
62	C₄×C₃⋊C₈	C₂²×C₃	1	2	yes	—
63	C₂³×C₃²	C₄²	1	1	yes	characteristic
64	C₂×C₃²×C₄	—	2	2	no	—
65	C₂×C₃²×C₄	—	2	4	no	—
66	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₄²	1	2	yes	—
68	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
69	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
70	C₃²×C₈	—	2	2	no	—
71	C₃²×C₈	M₁₆	1	4	yes	complemented
72	C₃×C₃⋊C₈	—	2	4	no	—
73	C₃×C₃⋊C₈	M₁₆	1	8	yes	complemented
74	C₄×M₁₆	—	3	2	no	—
75	C₄×M₁₆	C₃×D₃	1	1	yes	characteristic
76	#322	—	3	2	no	—
77	#322	—	3	1	no	—
78	C₈²	—	3	1	no	—
79	C₂²×C₃×C₄	—	2	2	no	—
80	C₂²×C₃×C₄	—	2	1	no	—
81	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
82	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
83	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
84	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
85	C₃×M₁₆	—	2	8	no	—
86	C₃×M₁₆	—	2	4	no	—
87	C₃×M₁₆	—	2	4	no	—
88	C₃×M₁₆	—	6	8	no	—
89	C₂×C₃×C₈	—	2	2	no	—
90	C₂×C₃×C₈	—	2	2	no	—
91	C₂×C₃×C₈	—	3	4	no	—
92	C₂×C₃×C₈	—	3	4	no	—
93	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
94	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
95	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
96	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
97	C₃×C₄²	—	2	2	no	—
98	C₃×C₄²	—	2	1	no	—
99	C₃×C₄²	—	2	1	no	—
100	C₃×C₄²	C₄×D₃	1	2	yes	—
101	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
102	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
103	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
104	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
105	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
106	#208	—	2	8	no	—
107	C₂×C₃⋊C₈	C₂×C₃×C₄	1	4	yes	—
108	C₂×C₃⋊C₈	C₂×C₃×C₄	1	4	yes	—
109	C₂²×C₃²	—	2	1	no	—
110	C₂²×C₃²	—	2	2	no	—
111	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
112	C₃²×C₄	—	2	1	no	—
113	C₃²×C₄	—	2	2	no	—
114	C₃²×C₄	C₂×M₁₆	1	4	yes	—
115	C₃²×C₄	C₂×M₁₆	1	2	yes	—
116	C₂×M₁₆	—	3	4	no	—
117	C₂×M₁₆	C₃×Dic₃	1	2	yes	—
118	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
119	#97	—	3	4	no	—
120	#97	C₃×Dic₃	1	2	yes	—
121	C₄×C₈	—	3	2	no	—
122	C₄×C₈	—	3	2	no	—
123	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
124	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
125	C₂³×C₃	—	2	1	no	—
126	C₂³×C₃	C₄×Dic₃	1	1	yes	characteristic
127	C₂³×C₃	C₃×C₄²	1	1	yes	characteristic
128	C₂×C₃×C₄	—	2	4	no	—
129	C₂×C₃×C₄	—	2	4	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	2	4	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	2	2	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	4	4	no	—
138	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
139	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
140	C₂×C₃×C₄	C₃×C₄²	1	1	yes	characteristic
141	C₂×C₃×C₄	C₃×C₄²	1	2	yes	—
142	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
143	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
144	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
145	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
146	C₃×C₈	—	2	4	no	—
147	C₃×C₈	—	2	2	no	—
148	C₃×C₈	—	2	2	no	—
149	C₃×C₈	—	2	4	no	—
150	C₃×C₈	—	3	8	no	—
151	C₃×C₈	—	6	4	no	—
152	C₃×C₈	#208	1	4	yes	complemented
153	C₃×C₈	C₃×M₁₆	1	4	yes	complemented
154	C₃⋊C₈	—	2	4	no	—
155	C₃⋊C₈	C₃×M₁₆	1	8	yes	complemented
156	C₂×C₃²	—	4	1	no	—
157	C₂×C₃²	C₄×M₁₆	1	1	yes	characteristic
158	C₂×C₃²	C₄×M₁₆	1	2	yes	—
159	C₂²×C₄	C₃×C₄×D₃	1	2	yes	—
160	C₂²×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
161	M₁₆	—	2	4	no	—
162	M₁₆	—	6	8	no	—
163	C₂×C₈	—	3	4	no	—
164	C₂×C₈	—	3	4	no	—
165	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
166	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
167	C₄²	C₃×C₄×D₃	1	2	yes	—
168	C₄²	C₂×C₃×Dic₃	1	1	yes	characteristic
169	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
170	C₂²×C₃	—	2	2	no	—
171	C₂²×C₃	—	2	1	no	—
172	C₂²×C₃	—	2	1	no	—
173	C₂²×C₃	—	2	2	no	—
174	C₂²×C₃	—	2	2	no	—
175	C₂²×C₃	—	2	1	no	—
176	C₂²×C₃	—	4	2	no	—
177	C₂²×C₃	C₂×C₄×Dic₃	1	1	yes	characteristic
178	C₂²×C₃	C₂×C₃×C₄²	1	1	yes	characteristic
179	C₃×C₄	—	2	2	no	—
180	C₃×C₄	—	2	1	no	—
181	C₃×C₄	—	2	1	no	—
182	C₃×C₄	—	2	2	no	—
183	C₃×C₄	—	2	2	no	—
184	C₃×C₄	—	2	4	no	—
185	C₃×C₄	—	2	2	no	—
186	C₃×C₄	—	4	2	no	—
187	C₃×C₄	D₃×M₁₆	1	4	yes	—
188	C₃×C₄	C₂×#208	1	2	yes	—
189	C₃×C₄	C₂×C₃×M₁₆	1	4	yes	—
190	C₃×C₄	C₂×C₃×M₁₆	1	2	yes	—
191	C₃²	#1455	1	1	yes	characteristic, complemented, Sylow
192	C₂³	C₃×C₄×Dic₃	1	1	yes	characteristic
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	C₃×C₄×Dic₃	1	1	yes	characteristic
196	C₂×C₄	C₃×C₄×Dic₃	1	2	yes	—
197	C₂×C₄	C₂×C₃×C₄×D₃	1	2	yes	—
198	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
199	C₈	—	2	2	no	—
200	C₈	—	3	8	no	—
201	C₈	—	6	4	no	—
202	C₈	C₃×#208	1	4	yes	complemented
203	C₂×C₃	—	2	1	no	—
204	C₂×C₃	—	2	2	no	—
205	C₂×C₃	—	4	2	no	—
206	C₂×C₃	—	4	1	no	—
207	C₂×C₃	—	4	1	no	—
208	C₂×C₃	C₄×#208	1	1	yes	characteristic
209	C₂×C₃	Dic₃×M₁₆	1	2	yes	—
210	C₂×C₃	C₃×C₄×M₁₆	1	1	yes	characteristic
211	C₂×C₃	C₃×C₄×M₁₆	1	2	yes	—
212	C₂²	—	2	1	no	—
213	C₂²	—	2	2	no	—
214	C₂²	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	2	no	—
217	C₄	C₃×D₃×M₁₆	1	4	yes	—
218	C₄	C₂×C₃×#208	1	2	yes	—
219	C₃	—	2	1	no	—
220	C₃	#69849	1	1	yes	characteristic, direct product factor
221	C₃	C₃×#1455	1	1	yes	characteristic, complemented
222	C₂	—	4	1	no	—
223	C₂	C₃×C₄×#208	1	1	yes	characteristic
224	C₂	C₃×Dic₃×M₁₆	1	2	yes	—

	subgroup	G/H
center	C₃×C₄²	C₂²×D₃
commutator	C₂²×C₃	C₂×C₃×C₄²
Frattini	C₄²	C₂²×C₃×D₃
Fitting	C₃²×C₄×M₁₆	C₂
socle	C₂²×C₃²	C₂×C₄²
prime radical	C₂³×C₃²	C₄²
cosocle	C₃×C₄²	C₂³×C₃
hypercenter	C₃×C₄×M₁₆	D₃
hypocenter	C₃	C₃×#1455
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₄² ◁ C₃×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₃×#4455	C₂	semidirect
4. C₃×#4454	C₂	semidirect
5. C₃×C₈×C₃⋊C₈	C₂	semidirect
7. #69849	C₃	direct (N × H)
25. #4455	C₂×C₃	semidirect
26. #4454	C₂×C₃	semidirect
27. C₈×C₃⋊C₈	C₂×C₃	semidirect
71. C₃²×C₈	M₁₆	semidirect
73. C₃×C₃⋊C₈	M₁₆	semidirect
152. C₃×C₈	#208	semidirect
153. C₃×C₈	C₃×M₁₆	semidirect
155. C₃⋊C₈	C₃×M₁₆	semidirect
191. C₃²	#1455	semidirect
202. C₈	C₃×#208	semidirect
220. C₃	#69849	direct (N × H)
221. C₃	C₃×#1455	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×M₁₆	2
C₃×C₄×#208	2
C₃×#4455	2
C₃×#4454	2
C₃×C₈×C₃⋊C₈	2
C₃×#1455	3
#69849	3

Subgroups — C3×#69849 #11773692

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69849 #11773692