Subgroups — C₂×#11772672 #72255481

Order 2304.

666 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10589365	C₂	1	1	yes	maximal, characteristic
2	C₂×#10589368	C₂	1	1	yes	maximal, characteristic
3	#11772672	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃²×M₆₄	C₂	1	1	yes	maximal, characteristic
5	C₂×#67600	—	3	4	no	maximal
6	C₂²×C₃²×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₈×C₃⋊Dic₃	C₄	1	1	yes	characteristic
8	C₂²×C₃⋊(C₃⋊C₁₆)	C₄	1	1	yes	characteristic
9	C₃²×M₆₄	C₄	1	2	yes	complemented
10	C₃²×M₆₄	C₂²	1	2	yes	—
11	C₂×C₃²×C₃₂	C₄	1	2	yes	complemented
12	#10589368	C₄	1	2	yes	—
13	#10589368	C₂²	1	2	yes	—
14	#10589365	C₄	1	2	yes	complemented
15	#10589365	C₂²	1	2	yes	—
16	C₂×C₃⋊(C₃⋊C₃₂)	C₄	1	2	yes	complemented
17	C₂×C₃×M₆₄	D₃	1	4	yes	—
18	C₂×#4502	—	3	4	no	—
19	C₂×#4499	—	3	4	no	—
20	#67600	—	3	16	no	—
21	C₂²×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₃⋊Dic₃	C₈	1	1	yes	characteristic
23	C₂²×C₃⋊(C₃⋊C₈)	C₈	1	1	yes	characteristic
24	C₂×C₃²×C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃²×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃²×C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃²×C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃²×C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃²×C₁₆	C₂³	1	1	yes	characteristic
30	C₈×C₃⋊Dic₃	C₈	1	2	yes	—
31	C₈×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
32	C₂×C₃⋊(C₃⋊C₁₆)	C₈	1	2	yes	—
33	C₂×C₃⋊(C₃⋊C₁₆)	C₈	1	2	yes	—
34	C₂×C₃⋊(C₃⋊C₁₆)	C₂×C₄	1	2	yes	—
35	C₃²×C₃₂	C₂×C₄	1	4	yes	complemented
36	C₃⋊(C₃⋊C₃₂)	C₂×C₄	1	4	yes	complemented
37	C₂×#1398	—	9	1	no	Sylow
38	C₂²×C₃×C₁₆	C₂×D₃	1	4	yes	—
39	C₂×C₈×Dic₃	—	3	4	no	—
40	C₂²×C₃⋊C₁₆	—	3	4	no	—
41	C₃×M₆₄	Dic₃	1	8	yes	complemented
42	C₃×M₆₄	C₂×D₃	1	8	yes	—
43	C₂×C₃×C₃₂	Dic₃	1	8	yes	complemented
44	#4502	—	3	8	no	—
45	#4502	—	3	8	no	—
46	#4499	—	3	8	no	—
47	#4499	—	3	8	no	—
48	C₂×C₃⋊C₃₂	—	3	8	no	—
49	C₂²×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₃⋊Dic₃	—	2	1	no	—
51	C₂×C₃²×C₈	C₄²	1	1	yes	characteristic
52	C₂×C₃²×C₈	C₄²	1	1	yes	characteristic
53	C₂×C₃²×C₈	C₂×C₈	1	1	yes	characteristic
54	C₂×C₃²×C₈	C₂×C₈	1	1	yes	characteristic
55	C₂×C₃²×C₈	C₂×C₈	1	1	yes	characteristic
56	C₂×C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
57	C₄×C₃⋊Dic₃	—	2	1	no	—
58	C₄×C₃⋊Dic₃	C₂×C₈	1	2	yes	—
59	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
60	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
61	C₂×C₃⋊(C₃⋊C₈)	C₂×C₈	1	2	yes	—
62	C₃²×C₁₆	C₄²	1	1	yes	characteristic
63	C₃²×C₁₆	C₄²	1	1	yes	characteristic
64	C₃²×C₁₆	C₂²×C₄	1	1	yes	characteristic
65	C₃²×C₁₆	C₂²×C₄	1	1	yes	characteristic
66	C₃⋊(C₃⋊C₁₆)	C₂×C₈	1	4	yes	—
67	C₂×M₆₄	—	9	1	no	—
68	C₂×M₆₄	C₃⋊D₃	1	1	yes	characteristic
69	C₂×#346	—	9	1	no	—
70	#1398	—	9	4	no	—
71	C₂²×C₃×C₈	C₄×D₃	1	4	yes	—
72	C₂×C₄×Dic₃	—	3	4	no	—
73	C₂²×C₃⋊C₈	—	3	4	no	—
74	C₂×C₃×C₁₆	C₄×D₃	1	4	yes	—
75	C₂×C₃×C₁₆	C₄×D₃	1	4	yes	—
76	C₂×C₃×C₁₆	C₄×D₃	1	4	yes	—
77	C₂×C₃×C₁₆	C₂×Dic₃	1	4	yes	—
78	C₂×C₃×C₁₆	C₂×Dic₃	1	4	yes	—
79	C₂×C₃×C₁₆	C₂²×D₃	1	4	yes	—
80	C₈×Dic₃	—	3	8	no	—
81	C₈×Dic₃	—	3	8	no	—
82	C₂×C₃⋊C₁₆	—	3	8	no	—
83	C₂×C₃⋊C₁₆	—	3	8	no	—
84	C₂×C₃⋊C₁₆	—	3	8	no	—
85	C₃×C₃₂	C₂×Dic₃	1	16	yes	complemented
86	C₃⋊C₃₂	—	3	16	no	—
87	C₂³×C₃²	M₃₂	1	1	yes	characteristic
88	C₂×C₃²×C₄	C₄×C₈	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₄×C₈	1	1	yes	characteristic
90	C₂×C₃²×C₄	M₃₂	1	1	yes	characteristic
91	C₂×C₃²×C₄	M₃₂	1	1	yes	characteristic
92	C₂×C₃²×C₄	M₃₂	1	1	yes	characteristic
93	C₂×C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₃⋊Dic₃	—	2	2	no	—
95	C₂×C₃⋊Dic₃	—	4	2	no	—
96	C₃²×C₈	C₄×C₈	1	1	yes	characteristic
97	C₃²×C₈	C₄×C₈	1	1	yes	characteristic
98	C₃²×C₈	C₂×C₄²	1	1	yes	characteristic
99	C₃²×C₈	C₂²×C₈	1	1	yes	characteristic
100	C₃⋊(C₃⋊C₈)	—	2	2	no	—
101	C₂²×C₁₆	—	9	1	no	—
102	C₂²×C₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
103	C₂×C₄×C₈	—	9	1	no	—
104	M₆₄	—	9	2	no	—
105	M₆₄	—	9	2	no	—
106	M₆₄	C₃⋊Dic₃	1	2	yes	complemented
107	M₆₄	C₂×C₃⋊D₃	1	2	yes	—
108	C₂×C₃₂	—	9	2	no	—
109	C₂×C₃₂	C₃⋊Dic₃	1	2	yes	complemented
110	#346	—	9	2	no	—
111	#346	—	9	2	no	—
112	C₂²×C₃×C₄	D₃×C₈	1	4	yes	—
113	C₂²×Dic₃	—	6	4	no	—
114	C₂×C₃×C₈	D₃×C₈	1	4	yes	—
115	C₂×C₃×C₈	D₃×C₈	1	4	yes	—
116	C₂×C₃×C₈	D₃×C₈	1	4	yes	—
117	C₂×C₃×C₈	C₄×Dic₃	1	4	yes	—
118	C₂×C₃×C₈	C₄×Dic₃	1	4	yes	—
119	C₂×C₃×C₈	C₂×C₄×D₃	1	4	yes	—
120	C₄×Dic₃	—	3	8	no	—
121	C₄×Dic₃	—	6	4	no	—
122	C₂×C₃⋊C₈	—	3	8	no	—
123	C₂×C₃⋊C₈	—	6	4	no	—
124	C₂×C₃⋊C₈	—	6	4	no	—
125	C₃×C₁₆	C₄×Dic₃	1	4	yes	—
126	C₃×C₁₆	C₄×Dic₃	1	4	yes	—
127	C₃×C₁₆	C₂×C₄×D₃	1	4	yes	—
128	C₃×C₁₆	C₂²×Dic₃	1	4	yes	—
129	C₃⋊C₁₆	—	3	16	no	—
130	C₂²×C₃²	—	2	2	no	—
131	C₂²×C₃²	#346	1	1	yes	characteristic
132	C₂²×C₃²	#346	1	1	yes	characteristic
133	C₂²×C₃²	C₂×M₃₂	1	1	yes	characteristic
134	C₃²×C₄	#346	1	1	yes	characteristic
135	C₃²×C₄	#346	1	1	yes	characteristic
136	C₃²×C₄	C₂×C₄×C₈	1	1	yes	characteristic
137	C₃²×C₄	C₂×M₃₂	1	1	yes	characteristic
138	C₃⋊Dic₃	—	4	2	no	—
139	C₂²×C₈	—	9	1	no	—
140	C₂²×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
141	C₂×C₄²	—	9	1	no	—
142	C₂×C₁₆	—	9	2	no	—
143	C₂×C₁₆	—	9	2	no	—
144	C₂×C₁₆	—	9	2	no	—
145	C₂×C₁₆	C₄×C₃⋊D₃	1	1	yes	characteristic
146	C₂×C₁₆	C₄×C₃⋊D₃	1	1	yes	characteristic
147	C₂×C₁₆	C₄×C₃⋊D₃	1	1	yes	characteristic
148	C₂×C₁₆	C₂×C₃⋊Dic₃	1	1	yes	characteristic
149	C₂×C₁₆	C₂×C₃⋊Dic₃	1	1	yes	characteristic
150	C₂×C₁₆	C₂²×C₃⋊D₃	1	1	yes	characteristic
151	C₄×C₈	—	9	2	no	—
152	C₄×C₈	—	9	2	no	—
153	C₃₂	—	9	4	no	—
154	C₃₂	C₂×C₃⋊Dic₃	1	4	yes	complemented
155	C₂³×C₃	#798	1	4	yes	—
156	C₂×C₃×C₄	#798	1	4	yes	—
157	C₂×C₃×C₄	#798	1	4	yes	—
158	C₂×C₃×C₄	#798	1	4	yes	—
159	C₂×C₃×C₄	C₈×Dic₃	1	4	yes	—
160	C₂×C₃×C₄	C₈×Dic₃	1	4	yes	—
161	C₂×C₃×C₄	C₂×D₃×C₈	1	4	yes	—
162	C₂×Dic₃	—	6	8	no	—
163	C₂×Dic₃	—	12	8	no	—
164	C₃×C₈	C₈×Dic₃	1	4	yes	—
165	C₃×C₈	C₈×Dic₃	1	4	yes	—
166	C₃×C₈	C₂×D₃×C₈	1	4	yes	—
167	C₃×C₈	C₂×C₄×Dic₃	1	4	yes	—
168	C₃⋊C₈	—	6	8	no	—
169	C₂×C₃²	—	2	1	no	—
170	C₂×C₃²	—	2	1	no	—
171	C₂×C₃²	#1398	1	2	yes	complemented
172	C₂×C₃²	C₂×#346	1	1	yes	characteristic
173	C₂²×C₄	—	18	1	no	—
174	C₂²×C₄	C₈×C₃⋊D₃	1	1	yes	characteristic
175	C₂×C₈	—	9	2	no	—
176	C₂×C₈	—	18	1	no	—
177	C₂×C₈	—	18	1	no	—
178	C₂×C₈	C₈×C₃⋊D₃	1	1	yes	characteristic
179	C₂×C₈	C₈×C₃⋊D₃	1	1	yes	characteristic
180	C₂×C₈	C₈×C₃⋊D₃	1	1	yes	characteristic
181	C₂×C₈	C₄×C₃⋊Dic₃	1	1	yes	characteristic
182	C₂×C₈	C₄×C₃⋊Dic₃	1	1	yes	characteristic
183	C₂×C₈	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
184	C₄²	—	9	2	no	—
185	C₄²	—	18	1	no	—
186	C₁₆	—	9	4	no	—
187	C₁₆	C₄×C₃⋊Dic₃	1	1	yes	characteristic
188	C₁₆	C₄×C₃⋊Dic₃	1	1	yes	characteristic
189	C₁₆	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
190	C₁₆	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
191	C₂²×C₃	—	2	8	no	—
192	C₂²×C₃	#4502	1	4	yes	—
193	C₂²×C₃	#4502	1	4	yes	—
194	C₂²×C₃	C₂×#798	1	4	yes	—
195	C₃×C₄	#4502	1	4	yes	—
196	C₃×C₄	#4502	1	4	yes	—
197	C₃×C₄	C₂×#798	1	4	yes	—
198	C₃×C₄	C₂×C₈×Dic₃	1	4	yes	—
199	Dic₃	—	12	8	no	—
200	C₃²	C₂×#1398	1	1	yes	characteristic, complemented, Sylow
201	C₂³	#63625	1	1	yes	characteristic
202	C₂×C₄	—	18	2	no	—
203	C₂×C₄	—	36	2	no	—
204	C₂×C₄	#63625	1	1	yes	characteristic
205	C₂×C₄	#63625	1	1	yes	characteristic
206	C₂×C₄	#63625	1	1	yes	characteristic
207	C₂×C₄	C₈×C₃⋊Dic₃	1	1	yes	characteristic
208	C₂×C₄	C₈×C₃⋊Dic₃	1	1	yes	characteristic
209	C₂×C₄	C₂×C₈×C₃⋊D₃	1	1	yes	characteristic
210	C₈	—	18	2	no	—
211	C₈	C₈×C₃⋊Dic₃	1	1	yes	characteristic
212	C₈	C₈×C₃⋊Dic₃	1	1	yes	characteristic
213	C₈	C₂×C₈×C₃⋊D₃	1	1	yes	characteristic
214	C₈	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
215	C₂×C₃	—	2	4	no	—
216	C₂×C₃	—	2	4	no	—
217	C₂×C₃	#67600	1	8	yes	complemented
218	C₂×C₃	C₂×#4502	1	4	yes	—
219	C₂²	—	2	2	no	—
220	C₂²	#10589368	1	1	yes	characteristic
221	C₂²	#10589368	1	1	yes	characteristic
222	C₂²	C₂×#63625	1	1	yes	characteristic
223	C₄	—	36	2	no	—
224	C₄	#10589368	1	1	yes	characteristic
225	C₄	#10589368	1	1	yes	characteristic
226	C₄	C₂×#63625	1	1	yes	characteristic
227	C₄	C₂×C₈×C₃⋊Dic₃	1	1	yes	characteristic
228	C₃	C₂×#67600	1	4	yes	complemented
229	C₂	—	2	1	no	—
230	C₂	—	2	1	no	—
231	C₂	#11772672	1	2	yes	direct product factor
232	C₂	C₂×#10589368	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×C₃⋊Dic₃
commutator	C₃²×C₄	C₂×C₄×C₈
Frattini	C₂×C₁₆	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×M₆₄	C₂
socle	C₂²×C₃²	#346
prime radical	C₂³×C₃²	M₃₂
cosocle	C₂×C₃²×C₁₆	C₂³
hypercenter	C₂×M₆₄	C₃⋊D₃
hypocenter	C₃²	C₂×#1398
LCS	G ▷ C₃²×C₄ ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
3. #11772672	C₂	direct (N × H)
9. C₃²×M₆₄	C₄	semidirect
11. C₂×C₃²×C₃₂	C₄	semidirect
14. #10589365	C₄	semidirect
16. C₂×C₃⋊(C₃⋊C₃₂)	C₄	semidirect
35. C₃²×C₃₂	C₂×C₄	semidirect
36. C₃⋊(C₃⋊C₃₂)	C₂×C₄	semidirect
41. C₃×M₆₄	Dic₃	semidirect
43. C₂×C₃×C₃₂	Dic₃	semidirect
85. C₃×C₃₂	C₂×Dic₃	semidirect
106. M₆₄	C₃⋊Dic₃	semidirect
109. C₂×C₃₂	C₃⋊Dic₃	semidirect
154. C₃₂	C₂×C₃⋊Dic₃	semidirect
171. C₂×C₃²	#1398	semidirect
200. C₃²	C₂×#1398	semidirect
217. C₂×C₃	#67600	semidirect
228. C₃	C₂×#67600	semidirect
231. C₂	#11772672	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×#10589365	2
C₂×#10589368	2
#11772672	2
C₂×C₃²×M₆₄	2
C₂×#67600	3

Subgroups — C2×#11772672 #72255481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11772672 #72255481