Subgroups — C₂×C₃²×#1369 #72255494

Order 2304.

666 conjugacy classes of subgroups.

The subgroup lattice has 231 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#364	C₂	1	2	yes	maximal
2	C₂×C₃²×#346	C₂	1	1	yes	maximal, characteristic
3	C₃²×#1369	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×#1369	C₃	1	4	yes	maximal, direct product factor
5	C₂×C₃²×M₃₂	—	2	2	no	—
6	C₂²×C₃²×C₁₆	C₄	1	1	yes	characteristic
7	C₂²×C₃²×C₁₆	C₄	1	1	yes	characteristic
8	C₂×C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₃²×#364	—	2	2	no	—
10	C₃²×#364	C₂²	1	4	yes	—
11	C₃²×#346	C₄	1	2	yes	—
12	C₃²×#346	C₂²	1	2	yes	—
13	C₂×C₃×#364	C₂×C₃	1	8	yes	—
14	C₂×C₃×#346	C₂×C₃	1	4	yes	—
15	C₃×#1369	C₂×C₃	1	16	yes	direct product factor
16	C₂²×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₃²×C₈	D₄	1	1	yes	characteristic
18	C₂×C₃²×C₄²	Q₈	1	1	yes	characteristic
19	C₃²×M₃₂	—	2	4	no	—
20	C₃²×M₃₂	—	4	2	no	—
21	C₂×C₃²×C₁₆	—	2	1	no	—
22	C₂×C₃²×C₁₆	—	2	1	no	—
23	C₂×C₃²×C₁₆	—	4	2	no	—
24	C₂×C₃²×C₁₆	C₈	1	2	yes	—
25	C₂×C₃²×C₁₆	C₈	1	2	yes	—
26	C₂×C₃²×C₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₃²×C₁₆	C₂×C₄	1	2	yes	—
28	C₃²×C₄×C₈	C₂×C₄	1	1	yes	characteristic
29	C₃²×C₄×C₈	D₄	1	1	yes	characteristic
30	C₃²×C₄×C₈	Q₈	1	1	yes	characteristic
31	C₃²×C₄×C₈	C₂³	1	1	yes	characteristic
32	C₂×#1369	C₃²	1	1	yes	characteristic, direct product factor, Sylow
33	C₂×C₃×M₃₂	—	2	8	no	—
34	C₂²×C₃×C₁₆	C₃×C₄	1	4	yes	—
35	C₂²×C₃×C₁₆	C₃×C₄	1	4	yes	—
36	C₂×C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
37	C₃×#364	—	2	8	no	—
38	C₃×#364	C₂²×C₃	1	16	yes	—
39	C₃×#346	C₃×C₄	1	8	yes	—
40	C₃×#346	C₂²×C₃	1	8	yes	—
41	C₂²×C₃²×C₄	—	2	1	no	—
42	C₂²×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₃²×C₈	C₂×C₈	1	1	yes	characteristic
46	C₂×C₃²×C₈	C₂×C₈	1	1	yes	characteristic
47	C₂×C₃²×C₈	M₁₆	1	1	yes	characteristic
48	C₂×C₃²×C₈	M₁₆	1	1	yes	characteristic
49	C₂×C₃²×C₈	SD₁₆	1	2	yes	—
50	C₂×C₃²×C₈	SD₁₆	1	2	yes	—
51	C₂×C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₃²×C₈	C₂×D₄	1	1	yes	characteristic
53	C₃²×C₄²	—	2	1	no	—
54	C₃²×C₄²	C₄⋊C₄	1	1	yes	characteristic
55	C₃²×C₄²	C₂×Q₈	1	1	yes	characteristic
56	C₃²×C₁₆	—	2	2	no	—
57	C₃²×C₁₆	—	4	4	no	—
58	C₃²×C₁₆	C₂×C₈	1	4	yes	—
59	C₂×#364	C₂×C₃²	1	2	yes	—
60	C₂×#346	C₂×C₃²	1	1	yes	characteristic
61	#1369	C₂×C₃²	1	4	yes	direct product factor
62	C₂²×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
63	C₂²×C₃×C₈	C₃×D₄	1	4	yes	—
64	C₂×C₃×C₄²	C₃×Q₈	1	4	yes	—
65	C₃×M₃₂	—	2	16	no	—
66	C₃×M₃₂	—	4	8	no	—
67	C₂×C₃×C₁₆	—	2	4	no	—
68	C₂×C₃×C₁₆	—	2	4	no	—
69	C₂×C₃×C₁₆	—	4	8	no	—
70	C₂×C₃×C₁₆	C₃×C₈	1	8	yes	—
71	C₂×C₃×C₁₆	C₃×C₈	1	8	yes	—
72	C₂×C₃×C₁₆	C₂×C₃×C₄	1	8	yes	—
73	C₂×C₃×C₁₆	C₂×C₃×C₄	1	8	yes	—
74	C₃×C₄×C₈	C₂×C₃×C₄	1	4	yes	—
75	C₃×C₄×C₈	C₃×D₄	1	4	yes	—
76	C₃×C₄×C₈	C₃×Q₈	1	4	yes	—
77	C₃×C₄×C₈	C₂³×C₃	1	4	yes	—
78	C₂³×C₃²	#108	1	1	yes	characteristic
79	C₂×C₃²×C₄	—	2	1	no	—
80	C₂×C₃²×C₄	—	2	1	no	—
81	C₂×C₃²×C₄	—	4	2	no	—
82	C₂×C₃²×C₄	C₄⋊C₈	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₄⋊C₈	1	1	yes	characteristic
84	C₂×C₃²×C₄	#106	1	1	yes	characteristic
85	C₂×C₃²×C₄	#106	1	1	yes	characteristic
86	C₂×C₃²×C₄	#108	1	1	yes	characteristic
87	C₂×C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₃²×C₈	C₄⋊C₈	1	1	yes	characteristic
89	C₃²×C₈	C₄⋊C₈	1	1	yes	characteristic
90	C₃²×C₈	#106	1	2	yes	—
91	C₃²×C₈	C₂²×C₈	1	1	yes	characteristic
92	C₃²×C₈	C₂×M₁₆	1	1	yes	characteristic
93	C₃²×C₈	C₂×SD₁₆	1	2	yes	—
94	C₂×M₃₂	—	2	2	no	—
95	C₂²×C₁₆	C₃²×C₄	1	1	yes	characteristic
96	C₂²×C₁₆	C₃²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
98	#364	—	2	2	no	—
99	#364	C₂²×C₃²	1	4	yes	—
100	#346	C₃²×C₄	1	2	yes	—
101	#346	C₂²×C₃²	1	2	yes	—
102	C₂²×C₃×C₄	—	2	4	no	—
103	C₂²×C₃×C₄	C₃×C₄⋊C₄	1	4	yes	—
104	C₂×C₃×C₈	C₃×C₄⋊C₄	1	4	yes	—
105	C₂×C₃×C₈	C₃×C₄⋊C₄	1	4	yes	—
106	C₂×C₃×C₈	C₂×C₃×C₈	1	4	yes	—
107	C₂×C₃×C₈	C₂×C₃×C₈	1	4	yes	—
108	C₂×C₃×C₈	C₃×M₁₆	1	4	yes	—
109	C₂×C₃×C₈	C₃×M₁₆	1	4	yes	—
110	C₂×C₃×C₈	C₃×SD₁₆	1	8	yes	—
111	C₂×C₃×C₈	C₃×SD₁₆	1	8	yes	—
112	C₂×C₃×C₈	C₂²×C₃×C₄	1	4	yes	—
113	C₂×C₃×C₈	C₂×C₃×D₄	1	4	yes	—
114	C₃×C₄²	—	2	4	no	—
115	C₃×C₄²	C₃×C₄⋊C₄	1	4	yes	—
116	C₃×C₄²	C₂×C₃×Q₈	1	4	yes	—
117	C₃×C₁₆	—	2	8	no	—
118	C₃×C₁₆	—	4	16	no	—
119	C₃×C₁₆	C₂×C₃×C₈	1	16	yes	—
120	C₂²×C₃²	—	2	2	no	—
121	C₂²×C₃²	#334	1	1	yes	characteristic
122	C₂²×C₃²	#334	1	1	yes	characteristic
123	C₂²×C₃²	C₂×#108	1	1	yes	characteristic
124	C₃²×C₄	—	4	1	no	—
125	C₃²×C₄	—	4	1	no	—
126	C₃²×C₄	#334	1	1	yes	characteristic
127	C₃²×C₄	#334	1	1	yes	characteristic
128	C₃²×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
129	C₃²×C₄	C₂×#106	1	1	yes	characteristic
130	C₂²×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
131	C₂²×C₈	C₃²×D₄	1	1	yes	characteristic
132	C₂×C₄²	C₃²×Q₈	1	1	yes	characteristic
133	M₃₂	—	2	4	no	—
134	M₃₂	—	4	2	no	—
135	C₂×C₁₆	—	2	1	no	—
136	C₂×C₁₆	—	2	1	no	—
137	C₂×C₁₆	—	4	2	no	—
138	C₂×C₁₆	C₃²×C₈	1	2	yes	—
139	C₂×C₁₆	C₃²×C₈	1	2	yes	—
140	C₂×C₁₆	C₂×C₃²×C₄	1	2	yes	—
141	C₂×C₁₆	C₂×C₃²×C₄	1	2	yes	—
142	C₄×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
143	C₄×C₈	C₃²×D₄	1	1	yes	characteristic
144	C₄×C₈	C₃²×Q₈	1	1	yes	characteristic
145	C₄×C₈	C₂³×C₃²	1	1	yes	characteristic
146	C₂³×C₃	C₃×#108	1	4	yes	—
147	C₂×C₃×C₄	—	2	4	no	—
148	C₂×C₃×C₄	—	2	4	no	—
149	C₂×C₃×C₄	—	4	8	no	—
150	C₂×C₃×C₄	C₃×C₄⋊C₈	1	4	yes	—
151	C₂×C₃×C₄	C₃×C₄⋊C₈	1	4	yes	—
152	C₂×C₃×C₄	C₃×#106	1	4	yes	—
153	C₂×C₃×C₄	C₃×#106	1	4	yes	—
154	C₂×C₃×C₄	C₃×#108	1	4	yes	—
155	C₂×C₃×C₄	C₂×C₃×C₄⋊C₄	1	4	yes	—
156	C₃×C₈	C₃×C₄⋊C₈	1	4	yes	—
157	C₃×C₈	C₃×C₄⋊C₈	1	4	yes	—
158	C₃×C₈	C₃×#106	1	8	yes	—
159	C₃×C₈	C₂²×C₃×C₈	1	4	yes	—
160	C₃×C₈	C₂×C₃×M₁₆	1	4	yes	—
161	C₃×C₈	C₂×C₃×SD₁₆	1	8	yes	—
162	C₂×C₃²	—	2	1	no	—
163	C₂×C₃²	—	2	1	no	—
164	C₂×C₃²	#1369	1	2	yes	direct product factor
165	C₂×C₃²	C₂×#334	1	1	yes	characteristic
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	C₃²×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₈	C₃²×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₈	C₃²×C₄⋊C₄	1	1	yes	characteristic
170	C₂×C₈	C₂×C₃²×C₈	1	1	yes	characteristic
171	C₂×C₈	C₂×C₃²×C₈	1	1	yes	characteristic
172	C₂×C₈	C₃²×M₁₆	1	1	yes	characteristic
173	C₂×C₈	C₃²×M₁₆	1	1	yes	characteristic
174	C₂×C₈	C₃²×SD₁₆	1	2	yes	—
175	C₂×C₈	C₃²×SD₁₆	1	2	yes	—
176	C₂×C₈	C₂²×C₃²×C₄	1	1	yes	characteristic
177	C₂×C₈	C₂×C₃²×D₄	1	1	yes	characteristic
178	C₄²	—	2	1	no	—
179	C₄²	C₃²×C₄⋊C₄	1	1	yes	characteristic
180	C₄²	C₂×C₃²×Q₈	1	1	yes	characteristic
181	C₁₆	—	2	2	no	—
182	C₁₆	—	4	4	no	—
183	C₁₆	C₂×C₃²×C₈	1	4	yes	—
184	C₂²×C₃	—	2	8	no	—
185	C₂²×C₃	C₃×#334	1	4	yes	—
186	C₂²×C₃	C₃×#334	1	4	yes	—
187	C₂²×C₃	C₂×C₃×#108	1	4	yes	—
188	C₃×C₄	—	4	4	no	—
189	C₃×C₄	—	4	4	no	—
190	C₃×C₄	C₃×#334	1	4	yes	—
191	C₃×C₄	C₃×#334	1	4	yes	—
192	C₃×C₄	C₂×C₃×C₄⋊C₈	1	4	yes	—
193	C₃×C₄	C₂×C₃×#106	1	4	yes	—
194	C₃²	C₂×#1369	1	1	yes	characteristic, direct product factor, Sylow
195	C₂³	C₃²×#108	1	1	yes	characteristic
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	C₃²×C₄⋊C₈	1	1	yes	characteristic
200	C₂×C₄	C₃²×C₄⋊C₈	1	1	yes	characteristic
201	C₂×C₄	C₃²×#106	1	1	yes	characteristic
202	C₂×C₄	C₃²×#106	1	1	yes	characteristic
203	C₂×C₄	C₃²×#108	1	1	yes	characteristic
204	C₂×C₄	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
205	C₈	C₃²×C₄⋊C₈	1	1	yes	characteristic
206	C₈	C₃²×C₄⋊C₈	1	1	yes	characteristic
207	C₈	C₃²×#106	1	2	yes	—
208	C₈	C₂²×C₃²×C₈	1	1	yes	characteristic
209	C₈	C₂×C₃²×M₁₆	1	1	yes	characteristic
210	C₈	C₂×C₃²×SD₁₆	1	2	yes	—
211	C₂×C₃	—	2	4	no	—
212	C₂×C₃	—	2	4	no	—
213	C₂×C₃	C₃×#1369	1	8	yes	direct product factor
214	C₂×C₃	C₂×C₃×#334	1	4	yes	—
215	C₂²	—	2	2	no	—
216	C₂²	C₃²×#334	1	1	yes	characteristic
217	C₂²	C₃²×#334	1	1	yes	characteristic
218	C₂²	C₂×C₃²×#108	1	1	yes	characteristic
219	C₄	—	4	1	no	—
220	C₄	—	4	1	no	—
221	C₄	C₃²×#334	1	1	yes	characteristic
222	C₄	C₃²×#334	1	1	yes	characteristic
223	C₄	C₂×C₃²×C₄⋊C₈	1	1	yes	characteristic
224	C₄	C₂×C₃²×#106	1	1	yes	characteristic
225	C₃	C₂×C₃×#1369	1	4	yes	direct product factor
226	C₂	—	2	1	no	—
227	C₂	—	2	1	no	—
228	C₂	C₃²×#1369	1	2	yes	direct product factor
229	C₂	C₂×C₃²×#334	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²×C₄	#106
commutator	C₈	C₂²×C₃²×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₃²
socle	C₂²×C₃²	#334
prime radical	C₂³×C₃²	#108
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ C₂²×C₃²×C₈ ◁ C₂×C₃²×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃²×#1369	C₂	direct (N × H)
4. C₂×C₃×#1369	C₃	direct (N × H)
15. C₃×#1369	C₂×C₃	direct (N × H)
32. C₂×#1369	C₃²	direct (N × H)
61. #1369	C₂×C₃²	direct (N × H)
164. C₂×C₃²	#1369	direct (N × H)
194. C₃²	C₂×#1369	direct (N × H)
213. C₂×C₃	C₃×#1369	direct (N × H)
225. C₃	C₂×C₃×#1369	direct (N × H)
228. C₂	C₃²×#1369	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×#364	2
C₂×C₃²×#346	2
C₃²×#1369	2
C₂×C₃×#1369	3

Subgroups — C2×C32×#1369 #72255494

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×#1369 #72255494