Subgroups — C₂×C₃×#2207 #10652183

Order 768.

806 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×SD₃₂	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×#493	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#2207	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×#357	C₂	1	2	yes	maximal, complemented
7	C₂×#2207	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂²×C₃×Q₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₃×D₈	—	2	1	no	—
10	C₂²×C₃×D₈	C₂²	1	2	yes	—
11	C₂×C₃×#127	—	2	1	no	—
12	C₂×C₃×C₄×Q₈	—	2	1	no	—
13	C₂×C₃×SD₃₂	—	2	8	no	—
14	C₂×C₃×SD₃₂	C₂²	1	8	yes	complemented
15	C₂²×C₃×C₁₆	C₂²	1	2	yes	—
16	C₃×#493	—	2	2	no	—
17	C₃×#493	C₂²	1	4	yes	—
18	C₃×C₄×Q₁₆	—	2	2	no	—
19	C₃×C₄×Q₁₆	C₂²	1	4	yes	complemented
20	C₂×C₃×#107	C₂²	1	1	yes	characteristic
21	C₂×C₃×#103	—	2	1	no	—
22	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
23	C₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
24	C₃×#357	C₂²	1	8	yes	complemented
25	C₂²×SD₃₂	C₂×C₃	1	2	yes	complemented
26	C₂×#493	C₂×C₃	1	1	yes	characteristic
27	C₂×C₄×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
28	#2207	C₂×C₃	1	8	yes	direct product factor
29	C₂×C₄⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂×#357	C₂×C₃	1	2	yes	complemented
31	C₂²×C₃×Q₈	—	2	1	no	—
32	C₂²×C₃×D₄	—	2	2	no	—
33	C₂²×C₃×D₄	—	4	1	no	—
34	C₂²×C₃×D₄	—	4	1	no	—
35	C₂×C₃×Q₁₆	—	2	1	no	—
36	C₂×C₃×Q₁₆	—	2	2	no	—
37	C₂×C₃×Q₁₆	D₄	1	4	yes	complemented
38	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
39	C₂×C₃×D₈	—	2	4	no	—
40	C₂×C₃×D₈	—	2	4	no	—
41	C₂×C₃×D₈	—	2	2	no	—
42	C₂×C₃×D₈	—	2	4	no	—
43	C₂×C₃×D₈	—	4	4	no	—
44	C₂×C₃×D₈	C₂³	1	4	yes	—
45	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
46	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
47	C₃×#127	—	2	4	no	—
48	C₃×#127	—	4	2	no	—
49	C₃×C₄×Q₈	—	2	4	no	—
50	C₃×C₄×Q₈	—	4	2	no	—
51	C₂×C₃×C₄⋊C₄	—	2	1	no	—
52	C₂×C₃×C₄⋊C₄	—	4	1	no	—
53	C₂×C₃×C₄²	—	4	1	no	—
54	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
55	C₃×SD₃₂	—	2	16	no	—
56	C₂×C₃×C₁₆	—	2	4	no	—
57	C₂×C₃×C₁₆	C₂³	1	4	yes	—
58	C₃×#107	—	2	1	no	—
59	C₃×#107	C₂³	1	2	yes	—
60	C₃×#103	—	2	4	no	—
61	C₃×C₄×C₈	D₄	1	2	yes	—
62	C₃×C₄×C₈	C₂³	1	2	yes	—
63	C₂²×Q₁₆	C₂²×C₃	1	1	yes	characteristic
64	C₂²×D₈	—	2	1	no	—
65	C₂²×D₈	C₂²×C₃	1	2	yes	—
66	C₂×#127	—	2	1	no	—
67	C₂×C₄×Q₈	—	2	1	no	—
68	C₂×SD₃₂	—	2	8	no	—
69	C₂×SD₃₂	C₂²×C₃	1	8	yes	complemented
70	C₂²×C₁₆	C₂²×C₃	1	2	yes	—
71	#493	—	2	2	no	—
72	#493	C₂²×C₃	1	4	yes	—
73	C₄×Q₁₆	—	2	2	no	—
74	C₄×Q₁₆	C₂²×C₃	1	4	yes	complemented
75	C₂×#107	C₂²×C₃	1	1	yes	characteristic
76	C₂×#103	—	2	1	no	—
77	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
78	C₄⋊C₁₆	C₂²×C₃	1	4	yes	complemented
79	#357	C₂²×C₃	1	8	yes	complemented
80	C₂⁴×C₃	—	4	2	no	—
81	C₂×C₃×Q₈	—	2	4	no	—
82	C₂×C₃×Q₈	—	2	2	no	—
83	C₂×C₃×Q₈	—	4	1	no	—
84	C₂×C₃×Q₈	—	4	2	no	—
85	C₂×C₃×D₄	—	2	4	no	—
86	C₂×C₃×D₄	—	4	4	no	—
87	C₂×C₃×D₄	—	4	4	no	—
88	C₂×C₃×D₄	—	4	4	no	—
89	C₂×C₃×D₄	—	4	2	no	—
90	C₂×C₃×D₄	—	4	4	no	—
91	C₂×C₃×D₄	—	8	4	no	—
92	C₂×C₃×D₄	—	8	4	no	—
93	C₂²×C₃×C₄	—	4	1	no	—
94	C₂²×C₃×C₄	—	4	1	no	—
95	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
96	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
97	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
98	C₃×Q₁₆	—	2	4	no	—
99	C₃×Q₁₆	—	2	2	no	—
100	C₃×Q₁₆	C₂×D₄	1	4	yes	complemented
101	C₃×D₈	—	2	4	no	—
102	C₃×D₈	—	2	4	no	—
103	C₃×D₈	—	4	8	no	—
104	C₃×D₈	—	4	4	no	—
105	C₂×C₃×C₈	—	2	2	no	—
106	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
107	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
109	C₂×C₃×C₈	#41	1	2	yes	—
110	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
111	C₃×C₄⋊C₄	—	2	2	no	—
112	C₃×C₄⋊C₄	—	4	1	no	—
113	C₃×C₄⋊C₄	—	4	4	no	—
114	C₃×C₄²	—	4	4	no	—
115	C₃×C₄²	D₈	1	2	yes	—
116	C₃×C₄²	C₂×D₄	1	2	yes	—
117	C₃×C₁₆	—	2	4	no	—
118	C₂²×Q₈	—	2	1	no	—
119	C₂²×D₄	—	2	2	no	—
120	C₂²×D₄	—	4	1	no	—
121	C₂²×D₄	—	4	1	no	—
122	C₂×Q₁₆	—	2	1	no	—
123	C₂×Q₁₆	—	2	2	no	—
124	C₂×Q₁₆	C₃×D₄	1	4	yes	complemented
125	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
126	C₂×D₈	—	2	4	no	—
127	C₂×D₈	—	2	4	no	—
128	C₂×D₈	—	2	2	no	—
129	C₂×D₈	—	2	4	no	—
130	C₂×D₈	—	4	4	no	—
131	C₂×D₈	C₂³×C₃	1	4	yes	—
132	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
133	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
134	#127	—	2	4	no	—
135	#127	—	4	2	no	—
136	C₄×Q₈	—	2	4	no	—
137	C₄×Q₈	—	4	2	no	—
138	C₂×C₄⋊C₄	—	2	1	no	—
139	C₂×C₄⋊C₄	—	4	1	no	—
140	C₂×C₄²	—	4	1	no	—
141	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
142	SD₃₂	—	2	16	no	—
143	C₂×C₁₆	—	2	4	no	—
144	C₂×C₁₆	C₂³×C₃	1	4	yes	—
145	#107	—	2	1	no	—
146	#107	C₂³×C₃	1	2	yes	—
147	#103	—	2	4	no	—
148	C₄×C₈	C₃×D₄	1	2	yes	—
149	C₄×C₈	C₂³×C₃	1	2	yes	—
150	C₂³×C₃	—	4	4	no	—
151	C₂³×C₃	—	8	4	no	—
152	C₂³×C₃	—	8	4	no	—
153	C₂³×C₃	—	8	4	no	—
154	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
155	C₃×Q₈	—	2	4	no	—
156	C₃×Q₈	—	4	2	no	—
157	C₃×Q₈	—	4	4	no	—
158	C₃×D₄	—	4	4	no	—
159	C₃×D₄	—	4	4	no	—
160	C₃×D₄	—	8	8	no	—
161	C₃×D₄	—	8	4	no	—
162	C₃×D₄	—	8	8	no	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	4	2	no	—
165	C₂×C₃×C₄	—	4	4	no	—
166	C₂×C₃×C₄	—	4	2	no	—
167	C₂×C₃×C₄	—	8	2	no	—
168	C₂×C₃×C₄	SD₃₂	1	4	yes	complemented
169	C₂×C₃×C₄	#121	1	2	yes	—
170	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
171	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
173	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
174	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
175	C₃×C₈	—	2	2	no	—
176	C₃×C₈	#121	1	2	yes	—
177	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
178	C₃×C₈	C₂×#41	1	1	yes	characteristic
179	C₂⁴	—	4	2	no	—
180	C₂×Q₈	—	2	4	no	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	4	1	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×D₄	—	2	4	no	—
185	C₂×D₄	—	4	4	no	—
186	C₂×D₄	—	4	4	no	—
187	C₂×D₄	—	4	4	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂×D₄	—	4	4	no	—
190	C₂×D₄	—	8	4	no	—
191	C₂×D₄	—	8	4	no	—
192	C₂²×C₄	—	4	1	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
195	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
197	Q₁₆	—	2	4	no	—
198	Q₁₆	—	2	2	no	—
199	Q₁₆	C₂×C₃×D₄	1	4	yes	complemented
200	D₈	—	2	4	no	—
201	D₈	—	2	4	no	—
202	D₈	—	4	8	no	—
203	D₈	—	4	4	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
206	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
207	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
208	C₂×C₈	C₃×#41	1	2	yes	—
209	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	4	1	no	—
212	C₄⋊C₄	—	4	4	no	—
213	C₄²	—	4	4	no	—
214	C₄²	C₃×D₈	1	2	yes	—
215	C₄²	C₂×C₃×D₄	1	2	yes	—
216	C₁₆	—	2	4	no	—
217	C₂²×C₃	—	8	4	no	—
218	C₂²×C₃	—	8	4	no	—
219	C₂²×C₃	—	8	4	no	—
220	C₂²×C₃	—	16	8	no	—
221	C₂²×C₃	#459	1	2	yes	—
222	C₂²×C₃	C₂×SD₃₂	1	2	yes	—
223	C₂²×C₃	#509	1	2	yes	—
224	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	4	4	no	—
227	C₃×C₄	—	8	2	no	—
228	C₃×C₄	#459	1	2	yes	—
229	C₃×C₄	C₂×SD₃₂	1	4	yes	complemented
230	C₃×C₄	C₂×#121	1	1	yes	characteristic
231	C₃×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
232	C₂³	—	4	4	no	—
233	C₂³	—	8	4	no	—
234	C₂³	—	8	4	no	—
235	C₂³	—	8	4	no	—
236	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
237	Q₈	—	2	4	no	—
238	Q₈	—	4	2	no	—
239	Q₈	—	4	4	no	—
240	D₄	—	4	4	no	—
241	D₄	—	4	4	no	—
242	D₄	—	8	8	no	—
243	D₄	—	8	8	no	—
244	D₄	—	8	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	8	2	no	—
250	C₂×C₄	C₃×SD₃₂	1	4	yes	complemented
251	C₂×C₄	C₃×#121	1	2	yes	—
252	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
253	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
254	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
255	C₂×C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
256	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	C₃×#121	1	2	yes	—
259	C₈	C₂²×C₃×D₄	1	1	yes	characteristic
260	C₈	C₂×C₃×#41	1	1	yes	characteristic
261	C₂×C₃	—	16	4	no	—
262	C₂×C₃	#2207	1	4	yes	direct product factor
263	C₂×C₃	C₂×#459	1	1	yes	characteristic
264	C₂×C₃	C₂²×SD₃₂	1	1	yes	characteristic
265	C₂×C₃	C₂×#509	1	1	yes	characteristic
266	C₂²	—	8	4	no	—
267	C₂²	—	8	4	no	—
268	C₂²	—	8	4	no	—
269	C₂²	—	16	8	no	—
270	C₂²	C₃×#459	1	2	yes	—
271	C₂²	C₂×C₃×SD₃₂	1	2	yes	—
272	C₂²	C₃×#509	1	2	yes	—
273	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	4	4	no	—
276	C₄	—	8	2	no	—
277	C₄	C₃×#459	1	2	yes	—
278	C₄	C₂×C₃×SD₃₂	1	4	yes	complemented
279	C₄	C₂×C₃×#121	1	1	yes	characteristic
280	C₄	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
281	C₃	C₂×#2207	1	1	yes	characteristic, direct product factor, Sylow
282	C₂	—	16	4	no	—
283	C₂	C₃×#2207	1	4	yes	direct product factor
284	C₂	C₂×C₃×#459	1	1	yes	characteristic
285	C₂	C₂²×C₃×SD₃₂	1	1	yes	characteristic
286	C₂	C₂×C₃×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
prime radical	C₂×C₃×#493	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×SD₃₂	C₂	semidirect
3. C₂×C₃×C₄×Q₁₆	C₂	semidirect
4. C₃×#2207	C₂	direct (N × H)
5. C₂×C₃×C₄⋊C₁₆	C₂	semidirect
6. C₂×C₃×#357	C₂	semidirect
7. C₂×#2207	C₃	direct (N × H)
14. C₂×C₃×SD₃₂	C₂²	semidirect
19. C₃×C₄×Q₁₆	C₂²	semidirect
23. C₃×C₄⋊C₁₆	C₂²	semidirect
24. C₃×#357	C₂²	semidirect
25. C₂²×SD₃₂	C₂×C₃	semidirect
27. C₂×C₄×Q₁₆	C₂×C₃	semidirect
28. #2207	C₂×C₃	direct (N × H)
29. C₂×C₄⋊C₁₆	C₂×C₃	semidirect
30. C₂×#357	C₂×C₃	semidirect
37. C₂×C₃×Q₁₆	D₄	semidirect
69. C₂×SD₃₂	C₂²×C₃	semidirect
74. C₄×Q₁₆	C₂²×C₃	semidirect
78. C₄⋊C₁₆	C₂²×C₃	semidirect
79. #357	C₂²×C₃	semidirect
100. C₃×Q₁₆	C₂×D₄	semidirect
124. C₂×Q₁₆	C₃×D₄	semidirect
168. C₂×C₃×C₄	SD₃₂	semidirect
199. Q₁₆	C₂×C₃×D₄	semidirect
229. C₃×C₄	C₂×SD₃₂	semidirect
250. C₂×C₄	C₃×SD₃₂	semidirect
262. C₂×C₃	#2207	direct (N × H)
278. C₄	C₂×C₃×SD₃₂	semidirect
281. C₃	C₂×#2207	direct (N × H)
283. C₂	C₃×#2207	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×SD₃₂	2
C₂×C₃×#493	2
C₂×C₃×C₄×Q₁₆	2
C₃×#2207	2
C₂×C₃×C₄⋊C₁₆	2
C₂×C₃×#357	2
C₂×#2207	3

Subgroups — C2×C3×#2207 #10652183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#2207 #10652183