Subgroups — C₂×C₃×#2153 #10652007

Order 768.

698 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×M₃₂	C₂	1	2	yes	maximal
2	C₂×C₃×#405	C₂	1	1	yes	maximal, characteristic
3	C₃×#2153	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×#364	C₂	1	4	yes	maximal, complemented
5	C₂×#2153	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂²×C₃×M₁₆	C₄	1	1	yes	characteristic
7	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×#117	C₄	1	1	yes	characteristic
9	C₂×C₃×M₃₂	—	2	4	no	—
10	C₂×C₃×M₃₂	—	2	2	no	—
11	C₂×C₃×M₃₂	C₂²	1	4	yes	—
12	C₂×C₃×M₃₂	C₂²	1	8	yes	—
13	C₂²×C₃×C₁₆	—	2	2	no	—
14	C₃×#405	C₄	1	4	yes	—
15	C₃×#405	C₂²	1	4	yes	—
16	C₂×C₃×#97	C₄	1	2	yes	—
17	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
18	C₃×#364	C₂²	1	16	yes	complemented
19	C₂²×M₃₂	C₂×C₃	1	2	yes	—
20	C₂×#405	C₂×C₃	1	1	yes	characteristic
21	#2153	C₂×C₃	1	8	yes	direct product factor
22	C₂×#364	C₂×C₃	1	4	yes	complemented
23	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×M₁₆	C₈	1	4	yes	—
25	C₂×C₃×M₁₆	C₈	1	4	yes	—
26	C₂×C₃×M₁₆	C₂×C₄	1	4	yes	—
27	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
29	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₃×C₈	D₄	1	2	yes	—
31	C₂²×C₃×C₈	D₄	1	2	yes	—
32	C₂²×C₃×C₈	Q₈	1	2	yes	—
33	C₂²×C₃×C₈	Q₈	1	2	yes	—
34	C₂²×C₃×C₈	C₂³	1	2	yes	—
35	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
36	C₃×#117	—	2	2	no	—
37	C₃×#117	C₂×C₄	1	4	yes	—
38	C₂×C₃×C₄⋊C₄	—	2	1	no	—
39	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
40	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
41	C₃×M₃₂	—	2	8	no	—
42	C₃×M₃₂	—	2	4	no	—
43	C₃×M₃₂	C₂³	1	8	yes	—
44	C₂×C₃×C₁₆	—	2	8	no	—
45	C₂×C₃×C₁₆	—	2	4	no	—
46	C₃×#97	C₂×C₄	1	8	yes	—
47	C₃×C₄×C₈	C₂×C₄	1	4	yes	—
48	C₃×C₄×C₈	C₂³	1	4	yes	—
49	C₂²×M₁₆	C₃×C₄	1	1	yes	characteristic
50	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
51	C₂×#117	C₃×C₄	1	1	yes	characteristic
52	C₂×M₃₂	—	2	2	no	—
53	C₂×M₃₂	—	2	4	no	—
54	C₂×M₃₂	C₂²×C₃	1	4	yes	—
55	C₂×M₃₂	C₂²×C₃	1	8	yes	—
56	C₂²×C₁₆	—	2	2	no	—
57	#405	C₃×C₄	1	4	yes	—
58	#405	C₂²×C₃	1	4	yes	—
59	C₂×#97	C₃×C₄	1	2	yes	—
60	C₂×C₄×C₈	C₂²×C₃	1	2	yes	—
61	#364	C₂²×C₃	1	16	yes	complemented
62	C₂⁴×C₃	M₁₆	1	1	yes	characteristic
63	C₂²×C₃×C₄	—	2	2	no	—
64	C₂²×C₃×C₄	C₄⋊C₄	1	2	yes	—
65	C₂²×C₃×C₄	C₄⋊C₄	1	2	yes	—
66	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
67	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
68	C₂²×C₃×C₄	M₁₆	1	1	yes	characteristic
69	C₂²×C₃×C₄	M₁₆	1	2	yes	—
70	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
72	C₃×M₁₆	C₂×C₈	1	16	yes	—
73	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
74	C₂×C₃×C₈	C₄⋊C₄	1	4	yes	—
75	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
76	C₂×C₃×C₈	C₄⋊C₄	1	4	yes	—
77	C₂×C₃×C₈	C₂×C₈	1	4	yes	—
78	C₂×C₃×C₈	C₂×C₈	1	4	yes	—
79	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
81	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
82	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
83	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
86	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
89	C₃×C₄⋊C₄	—	2	4	no	—
90	C₃×C₂²⋊C₄	—	2	4	no	—
91	C₃×C₄²	—	2	2	no	—
92	C₃×C₄²	C₂²×C₄	1	4	yes	—
93	C₃×C₁₆	—	2	8	no	—
94	C₂³×C₄	C₂×C₃×C₄	1	1	yes	characteristic
95	C₂×M₁₆	C₃×C₈	1	4	yes	—
96	C₂×M₁₆	C₃×C₈	1	4	yes	—
97	C₂×M₁₆	C₂×C₃×C₄	1	4	yes	—
98	C₂²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
99	C₂²×C₈	C₂×C₃×C₄	1	2	yes	—
100	C₂²×C₈	C₂×C₃×C₄	1	2	yes	—
101	C₂²×C₈	C₃×D₄	1	2	yes	—
102	C₂²×C₈	C₃×D₄	1	2	yes	—
103	C₂²×C₈	C₃×Q₈	1	2	yes	—
104	C₂²×C₈	C₃×Q₈	1	2	yes	—
105	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
106	C₂²×C₈	C₂³×C₃	1	2	yes	—
107	#117	—	2	2	no	—
108	#117	C₂×C₃×C₄	1	4	yes	—
109	C₂×C₄⋊C₄	—	2	1	no	—
110	C₂×C₂²⋊C₄	—	2	1	no	—
111	C₂×C₄²	C₂×C₃×C₄	1	2	yes	—
112	M₃₂	—	2	8	no	—
113	M₃₂	—	2	4	no	—
114	M₃₂	C₂³×C₃	1	8	yes	—
115	C₂×C₁₆	—	2	4	no	—
116	C₂×C₁₆	—	2	8	no	—
117	#97	C₂×C₃×C₄	1	8	yes	—
118	C₄×C₈	C₂×C₃×C₄	1	4	yes	—
119	C₄×C₈	C₂³×C₃	1	4	yes	—
120	C₂³×C₃	—	4	2	no	—
121	C₂³×C₃	C₄⋊C₈	1	2	yes	—
122	C₂³×C₃	C₄⋊C₈	1	2	yes	—
123	C₂³×C₃	C₂×M₁₆	1	1	yes	characteristic
124	C₂³×C₃	C₂×M₁₆	1	2	yes	—
125	C₂×C₃×C₄	—	2	4	no	—
126	C₂×C₃×C₄	—	4	4	no	—
127	C₂×C₃×C₄	C₄⋊C₈	1	2	yes	—
128	C₂×C₃×C₄	C₄⋊C₈	1	4	yes	—
129	C₂×C₃×C₄	C₄⋊C₈	1	4	yes	—
130	C₂×C₃×C₄	C₄⋊C₈	1	2	yes	—
131	C₂×C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
132	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂²×C₈	1	4	yes	—
137	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
139	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
140	C₃×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₃×C₈	C₂×C₄⋊C₄	1	4	yes	—
142	C₃×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₃×C₈	C₂²×C₈	1	8	yes	—
144	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
145	C₃×C₈	C₂²×Q₈	1	1	yes	characteristic
146	C₂⁴	C₃×M₁₆	1	1	yes	characteristic
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	C₃×C₄⋊C₄	1	2	yes	—
149	C₂²×C₄	C₃×C₄⋊C₄	1	2	yes	—
150	C₂²×C₄	C₂×C₃×C₈	1	2	yes	—
151	C₂²×C₄	C₂×C₃×C₈	1	2	yes	—
152	C₂²×C₄	C₃×M₁₆	1	1	yes	characteristic
153	C₂²×C₄	C₃×M₁₆	1	2	yes	—
154	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
155	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
156	M₁₆	C₂×C₃×C₈	1	16	yes	—
157	C₂×C₈	C₃×C₄⋊C₄	1	2	yes	—
158	C₂×C₈	C₃×C₄⋊C₄	1	2	yes	—
159	C₂×C₈	C₃×C₄⋊C₄	1	4	yes	—
160	C₂×C₈	C₃×C₄⋊C₄	1	4	yes	—
161	C₂×C₈	C₂×C₃×C₈	1	4	yes	—
162	C₂×C₈	C₂×C₃×C₈	1	4	yes	—
163	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
164	C₂×C₈	C₂²×C₃×C₄	1	2	yes	—
165	C₂×C₈	C₂²×C₃×C₄	1	4	yes	—
166	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
167	C₂×C₈	C₂×C₃×D₄	1	4	yes	—
168	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
169	C₂×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
170	C₂×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
171	C₂×C₈	C₂×C₃×Q₈	1	4	yes	—
172	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
173	C₄⋊C₄	—	2	4	no	—
174	C₂²⋊C₄	—	2	4	no	—
175	C₄²	—	2	2	no	—
176	C₄²	C₂²×C₃×C₄	1	4	yes	—
177	C₁₆	—	2	8	no	—
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	—	2	4	no	—
180	C₂²×C₃	—	4	2	no	—
181	C₂²×C₃	—	4	2	no	—
182	C₂²×C₃	C₂×C₄⋊C₈	1	1	yes	characteristic
183	C₂²×C₃	C₂×C₄⋊C₈	1	4	yes	—
184	C₂²×C₃	C₂×C₄⋊C₈	1	1	yes	characteristic
185	C₂²×C₃	C₂²×M₁₆	1	1	yes	characteristic
186	C₃×C₄	—	4	4	no	—
187	C₃×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
188	C₃×C₄	C₂×C₄⋊C₈	1	4	yes	—
189	C₃×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
190	C₃×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
191	C₃×C₄	C₂³×C₈	1	1	yes	characteristic
192	C₂³	—	4	2	no	—
193	C₂³	C₃×C₄⋊C₈	1	2	yes	—
194	C₂³	C₃×C₄⋊C₈	1	2	yes	—
195	C₂³	C₂×C₃×M₁₆	1	1	yes	characteristic
196	C₂³	C₂×C₃×M₁₆	1	2	yes	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	C₃×C₄⋊C₈	1	2	yes	—
200	C₂×C₄	C₃×C₄⋊C₈	1	4	yes	—
201	C₂×C₄	C₃×C₄⋊C₈	1	4	yes	—
202	C₂×C₄	C₃×C₄⋊C₈	1	2	yes	—
203	C₂×C₄	C₂×C₃×C₄⋊C₄	1	4	yes	—
204	C₂×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
205	C₂×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
206	C₂×C₄	C₂²×C₃×C₈	1	1	yes	characteristic
207	C₂×C₄	C₂²×C₃×C₈	1	1	yes	characteristic
208	C₂×C₄	C₂²×C₃×C₈	1	4	yes	—
209	C₂×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
210	C₂×C₄	C₂×C₃×M₁₆	1	2	yes	—
211	C₂×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
212	C₈	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
213	C₈	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
214	C₈	C₂×C₃×C₄⋊C₄	1	4	yes	—
215	C₈	C₂²×C₃×C₈	1	8	yes	—
216	C₈	C₂²×C₃×D₄	1	1	yes	characteristic
217	C₈	C₂²×C₃×Q₈	1	1	yes	characteristic
218	C₂×C₃	—	2	1	no	—
219	C₂×C₃	—	2	1	no	—
220	C₂×C₃	—	2	4	no	—
221	C₂×C₃	#2153	1	2	yes	direct product factor
222	C₂×C₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
223	C₂²	—	2	2	no	—
224	C₂²	—	2	4	no	—
225	C₂²	—	4	2	no	—
226	C₂²	—	4	2	no	—
227	C₂²	C₂×C₃×C₄⋊C₈	1	1	yes	characteristic
228	C₂²	C₂×C₃×C₄⋊C₈	1	4	yes	—
229	C₂²	C₂×C₃×C₄⋊C₈	1	1	yes	characteristic
230	C₂²	C₂²×C₃×M₁₆	1	1	yes	characteristic
231	C₄	—	4	4	no	—
232	C₄	C₂×C₃×C₄⋊C₈	1	1	yes	characteristic
233	C₄	C₂×C₃×C₄⋊C₈	1	1	yes	characteristic
234	C₄	C₂×C₃×C₄⋊C₈	1	4	yes	—
235	C₄	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
236	C₄	C₂³×C₃×C₈	1	1	yes	characteristic
237	C₃	C₂×#2153	1	1	yes	characteristic, direct product factor, Sylow
238	C₂	—	2	1	no	—
239	C₂	—	2	1	no	—
240	C₂	—	2	4	no	—
241	C₂	C₃×#2153	1	2	yes	direct product factor
242	C₂	C₂²×C₃×C₄⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂×D₄
commutator	C₄	C₂³×C₃×C₈
Frattini, cosocle	C₂×C₈	C₂⁴×C₃
socle	C₂²×C₃	C₂×C₄⋊C₈
prime radical	C₂⁴×C₃	M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₂³×C₃×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×#2153	C₂	direct (N × H)
4. C₂×C₃×#364	C₂	semidirect
5. C₂×#2153	C₃	direct (N × H)
18. C₃×#364	C₂²	semidirect
21. #2153	C₂×C₃	direct (N × H)
22. C₂×#364	C₂×C₃	semidirect
61. #364	C₂²×C₃	semidirect
221. C₂×C₃	#2153	direct (N × H)
237. C₃	C₂×#2153	direct (N × H)
241. C₂	C₃×#2153	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×M₃₂	2
C₂×C₃×#405	2
C₃×#2153	2
C₂×C₃×#364	2
C₂×#2153	3

Subgroups — C2×C3×#2153 #10652007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#2153 #10652007