Subgroups — C₃×#69864 #11797386

Order 1152.

327 conjugacy classes of subgroups.

The subgroup lattice has 239 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×#4553	C₂	1	2	yes	maximal
3	C₃×#4466	C₂	1	2	yes	maximal, complemented
4	C₃×#4463	C₂	1	2	yes	maximal, complemented
5	C₃×#2165	—	3	1	no	maximal
6	#69864	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂×C₃²×M₁₆	C₄	1	1	yes	characteristic
8	C₂×C₃²×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄²	C₄	1	1	yes	characteristic
10	C₃²×#97	C₄	1	1	yes	characteristic
11	C₃²×#97	C₂²	1	1	yes	characteristic
12	C₃²×C₄×C₈	C₄	1	1	yes	characteristic
13	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₃×#812	C₂²	1	4	yes	—
15	C₃×C₄×M₁₆	—	2	1	no	—
16	C₃×C₄×M₁₆	D₃	1	1	yes	characteristic
17	C₃×C₄×M₁₆	C₂×C₃	1	1	yes	characteristic
18	C₃×#364	—	3	2	no	—
19	C₃×#349	—	3	2	no	—
20	C₃×#347	—	3	2	no	—
21	#4553	C₂×C₃	1	2	yes	—
22	#4466	C₂×C₃	1	2	yes	complemented
23	#4463	C₂×C₃	1	2	yes	complemented
24	C₂²×C₃²×C₄	—	2	1	no	—
25	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
26	C₃²×M₁₆	—	2	2	no	—
27	C₃²×M₁₆	—	2	2	no	—
28	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
32	C₃²×C₄²	—	2	1	no	—
33	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
34	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
35	C₃×C₃⋊C₁₆	—	2	4	no	—
36	#2165	—	3	1	no	Sylow
37	C₂×C₃×M₁₆	—	2	1	no	—
38	C₂×C₃×M₁₆	—	2	1	no	—
39	C₂×C₃×M₁₆	Dic₃	1	1	yes	characteristic
40	C₂×C₃×M₁₆	C₃×C₄	1	1	yes	characteristic
41	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
42	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
43	C₂×C₃×C₄²	—	2	1	no	—
44	C₂×C₃×C₄²	Dic₃	1	1	yes	characteristic
45	C₂×C₃×C₄²	C₃×C₄	1	1	yes	characteristic
46	C₃×M₃₂	—	3	4	no	—
47	C₃×#97	—	2	1	no	—
48	C₃×#97	—	2	1	no	—
49	C₃×#97	Dic₃	1	1	yes	characteristic
50	C₃×#97	C₃×C₄	1	1	yes	characteristic
51	C₃×#97	C₂×D₃	1	1	yes	characteristic
52	C₃×#97	C₂²×C₃	1	1	yes	characteristic
53	C₃×C₄×C₈	—	2	1	no	—
54	C₃×C₄×C₈	—	2	1	no	—
55	C₃×C₄×C₈	Dic₃	1	1	yes	characteristic
56	C₃×C₄×C₈	C₃×C₄	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	#812	C₂²×C₃	1	4	yes	—
60	C₂³×C₃²	M₁₆	1	1	yes	characteristic
61	C₂×C₃²×C₄	—	2	1	no	—
62	C₂×C₃²×C₄	—	2	1	no	—
63	C₂×C₃²×C₄	—	2	1	no	—
64	C₂×C₃²×C₄	—	2	1	no	—
65	C₂×C₃²×C₄	—	4	2	no	—
66	C₂×C₃²×C₄	M₁₆	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
68	C₃²×C₈	—	2	1	no	—
69	C₃²×C₈	—	2	1	no	—
70	C₃²×C₈	M₁₆	1	2	yes	—
71	C₃²×C₈	#41	1	2	yes	—
72	C₄×M₁₆	C₃×D₃	1	1	yes	characteristic
73	#364	—	3	2	no	—
74	#349	—	3	2	no	—
75	#347	—	3	2	no	—
76	C₂²×C₃×C₄	—	2	2	no	—
77	C₂²×C₃×C₄	—	2	1	no	—
78	C₂²×C₃×C₄	—	2	1	no	—
79	C₂²×C₃×C₄	—	2	1	no	—
80	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
82	C₃×M₁₆	—	2	4	no	—
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	C₂×C₃×C₈	—	2	1	no	—
89	C₂×C₃×C₈	—	2	1	no	—
90	C₂×C₃×C₈	—	2	1	no	—
91	C₂×C₃×C₈	—	2	1	no	—
92	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
93	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
94	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
95	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
96	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
97	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
98	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
99	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
100	C₃×C₄²	—	2	2	no	—
101	C₃×C₄²	—	2	1	no	—
102	C₃×C₄²	—	2	1	no	—
103	C₃×C₄²	—	2	1	no	—
104	C₃×C₄²	—	2	1	no	—
105	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
106	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
107	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
108	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
109	C₃×C₁₆	—	6	4	no	—
110	C₃⋊C₁₆	—	2	4	no	—
111	C₂²×C₃²	—	2	1	no	—
112	C₂²×C₃²	—	4	1	no	—
113	C₂²×C₃²	C₂×M₁₆	1	1	yes	characteristic
114	C₃²×C₄	—	2	2	no	—
115	C₃²×C₄	—	2	1	no	—
116	C₃²×C₄	—	4	1	no	—
117	C₃²×C₄	#117	1	1	yes	characteristic
118	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
119	C₂×M₁₆	C₃×Dic₃	1	1	yes	characteristic
120	C₂×M₁₆	C₂×C₃×D₃	1	1	yes	characteristic
121	C₂×C₄²	C₃×Dic₃	1	1	yes	characteristic
122	M₃₂	—	3	4	no	—
123	#97	C₃×Dic₃	1	1	yes	characteristic
124	#97	C₂×C₃×D₃	1	1	yes	characteristic
125	C₄×C₈	C₃×Dic₃	1	1	yes	characteristic
126	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
127	C₂³×C₃	—	2	1	no	—
128	C₂³×C₃	#208	1	1	yes	characteristic
129	C₂³×C₃	C₃×M₁₆	1	1	yes	characteristic
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	4	2	no	—
145	C₂×C₃×C₄	—	4	2	no	—
146	C₂×C₃×C₄	—	4	4	no	—
147	C₂×C₃×C₄	#208	1	1	yes	characteristic
148	C₂×C₃×C₄	C₃×M₁₆	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
151	C₃×C₈	—	2	2	no	—
152	C₃×C₈	—	2	2	no	—
153	C₃×C₈	—	2	1	no	—
154	C₃×C₈	—	2	1	no	—
155	C₃×C₈	—	2	1	no	—
156	C₃×C₈	—	2	1	no	—
157	C₃×C₈	—	2	2	no	—
158	C₃×C₈	—	2	2	no	—
159	C₃×C₈	#208	1	2	yes	—
160	C₃×C₈	C₃×M₁₆	1	2	yes	—
161	C₃×C₈	#235	1	2	yes	—
162	C₃×C₈	C₃×#41	1	2	yes	—
163	C₂×C₃²	—	2	1	no	—
164	C₂×C₃²	—	4	1	no	—
165	C₂×C₃²	#432	1	1	yes	characteristic
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
168	M₁₆	—	2	2	no	—
169	M₁₆	—	2	2	no	—
170	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
171	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
172	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
173	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
174	C₄²	—	2	1	no	—
175	C₄²	C₂×C₃×Dic₃	1	1	yes	characteristic
176	C₄²	C₂×C₃×Dic₃	1	1	yes	characteristic
177	C₁₆	—	6	4	no	—
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	—	2	1	no	—
180	C₂²×C₃	—	2	1	no	—
181	C₂²×C₃	—	2	1	no	—
182	C₂²×C₃	—	4	1	no	—
183	C₂²×C₃	—	4	1	no	—
184	C₂²×C₃	—	4	2	no	—
185	C₂²×C₃	C₂×#208	1	1	yes	characteristic
186	C₂²×C₃	C₂×C₃×M₁₆	1	1	yes	characteristic
187	C₃×C₄	—	2	2	no	—
188	C₃×C₄	—	2	4	no	—
189	C₃×C₄	—	2	2	no	—
190	C₃×C₄	—	2	2	no	—
191	C₃×C₄	—	2	1	no	—
192	C₃×C₄	—	2	1	no	—
193	C₃×C₄	—	2	1	no	—
194	C₃×C₄	—	2	1	no	—
195	C₃×C₄	—	4	2	no	—
196	C₃×C₄	—	4	1	no	—
197	C₃×C₄	—	4	1	no	—
198	C₃×C₄	C₂×#208	1	1	yes	characteristic
199	C₃×C₄	#926	1	1	yes	characteristic
200	C₃×C₄	C₃×#117	1	1	yes	characteristic
201	C₃×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
202	C₃²	#2165	1	1	yes	characteristic, complemented, Sylow
203	C₂³	C₃×#208	1	1	yes	characteristic
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	C₃×#208	1	1	yes	characteristic
210	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
211	C₈	—	2	1	no	—
212	C₈	—	2	1	no	—
213	C₈	C₃×#208	1	2	yes	—
214	C₈	C₃×#235	1	2	yes	—
215	C₂×C₃	—	2	2	no	—
216	C₂×C₃	—	2	1	no	—
217	C₂×C₃	—	2	1	no	—
218	C₂×C₃	—	2	1	no	—
219	C₂×C₃	—	4	2	no	—
220	C₂×C₃	—	4	1	no	—
221	C₂×C₃	—	4	1	no	—
222	C₂×C₃	#4926	1	1	yes	characteristic
223	C₂×C₃	C₃×#432	1	1	yes	characteristic
224	C₂²	—	2	1	no	—
225	C₂²	—	4	1	no	—
226	C₂²	C₂×C₃×#208	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	1	no	—
229	C₄	—	4	1	no	—
230	C₄	C₂×C₃×#208	1	1	yes	characteristic
231	C₄	C₃×#926	1	1	yes	characteristic
232	C₃	—	2	1	no	—
233	C₃	#69864	1	1	yes	characteristic, direct product factor
234	C₃	C₃×#2165	1	1	yes	characteristic, complemented
235	C₂	—	2	1	no	—
236	C₂	—	4	1	no	—
237	C₂	C₃×#4926	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	#926
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂×C₈	C₂²×C₃×D₃
Fitting	C₃²×C₄×M₁₆	C₂
socle	C₂×C₃²	#432
prime radical	C₂³×C₃²	M₁₆
cosocle	C₂×C₃×C₈	C₂³×C₃
hypercenter	C₃×C₄×M₁₆	D₃
hypocenter	C₃	C₃×#2165
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₄ ◁ C₃×#97 ◁ C₃×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₃×#4466	C₂	semidirect
4. C₃×#4463	C₂	semidirect
6. #69864	C₃	direct (N × H)
22. #4466	C₂×C₃	semidirect
23. #4463	C₂×C₃	semidirect
202. C₃²	#2165	semidirect
233. C₃	#69864	direct (N × H)
234. C₃	C₃×#2165	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×#4553	2
C₃×#4466	2
C₃×#4463	2
C₃×#2165	3
#69864	3

Subgroups — C3×#69864 #11797386

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69864 #11797386