Subgroups — C₂×C₃×#2230 #10652242

Order 768.

558 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#443	C₂	1	1	yes	maximal, characteristic
3	C₃×#2230	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×#365	C₂	1	2	yes	maximal, complemented
5	C₂×C₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×#360	C₂	1	2	yes	maximal
7	C₂×#2230	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂×C₃×#135	C₂²	1	1	yes	characteristic
9	C₂×C₃×#131	—	2	1	no	—
10	C₂×C₃×C₄⋊Q₈	—	2	1	no	—
11	C₂×C₃×M₃₂	C₂²	1	2	yes	—
12	C₃×#498	—	2	2	no	—
13	C₃×#498	C₂²	1	4	yes	complemented
14	C₃×#443	—	2	2	no	—
15	C₃×#443	C₂²	1	4	yes	—
16	C₂×C₃×#108	C₂²	1	1	yes	characteristic
17	C₂×C₃×#106	—	2	1	no	—
18	C₂×C₃×#106	C₂²	1	2	yes	—
19	C₂×C₃×#104	—	2	1	no	—
20	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
21	C₃×#365	C₂²	1	8	yes	complemented
22	C₃×#364	C₂²	1	4	yes	complemented
23	C₃×#360	C₂²	1	8	yes	—
24	C₂×#498	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂×#443	C₂×C₃	1	1	yes	characteristic
26	#2230	C₂×C₃	1	8	yes	direct product factor
27	C₂×#365	C₂×C₃	1	2	yes	complemented
28	C₂×#364	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂×#360	C₂×C₃	1	2	yes	—
30	C₂×C₃×#41	—	2	1	no	—
31	C₂²×C₃×Q₈	—	4	1	no	—
32	C₃×#135	—	2	1	no	—
33	C₃×#135	—	2	1	no	—
34	C₃×#135	Q₈	1	2	yes	complemented
35	C₃×#135	C₂³	1	2	yes	—
36	C₂×C₃×Q₁₆	Q₈	1	1	yes	characteristic, complemented
37	C₂×C₃×SD₁₆	—	2	1	no	—
38	C₂×C₃×D₈	Q₈	1	1	yes	characteristic, complemented
39	C₃×#131	—	2	4	no	—
40	C₃×#131	—	4	2	no	—
41	C₂×C₃×M₁₆	—	2	1	no	—
42	C₂×C₃×M₁₆	—	4	1	no	—
43	C₂²×C₃×C₈	—	4	1	no	—
44	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
45	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
46	C₃×C₄⋊Q₈	—	2	4	no	—
47	C₃×C₄⋊Q₈	—	4	2	no	—
48	C₂×C₃×C₄⋊C₄	—	2	2	no	—
49	C₂×C₃×C₄⋊C₄	—	4	1	no	—
50	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
51	C₃×M₃₂	—	2	2	no	—
52	C₃×M₃₂	C₂³	1	4	yes	—
53	C₂×C₃×C₁₆	—	2	2	no	—
54	C₃×#108	—	2	1	no	—
55	C₃×#108	C₂³	1	2	yes	—
56	C₃×#106	—	2	4	no	—
57	C₃×#106	—	2	2	no	—
58	C₃×#106	C₂³	1	4	yes	—
59	C₃×#104	—	2	4	no	—
60	C₃×C₄×C₈	D₄	1	2	yes	—
61	C₃×C₄×C₈	C₂³	1	2	yes	—
62	C₂×#135	C₂²×C₃	1	1	yes	characteristic
63	C₂×#131	—	2	1	no	—
64	C₂×C₄⋊Q₈	—	2	1	no	—
65	C₂×M₃₂	C₂²×C₃	1	2	yes	—
66	#498	—	2	2	no	—
67	#498	C₂²×C₃	1	4	yes	complemented
68	#443	—	2	2	no	—
69	#443	C₂²×C₃	1	4	yes	—
70	C₂×#108	C₂²×C₃	1	1	yes	characteristic
71	C₂×#106	—	2	1	no	—
72	C₂×#106	C₂²×C₃	1	2	yes	—
73	C₂×#104	—	2	1	no	—
74	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
75	#365	C₂²×C₃	1	8	yes	complemented
76	#364	C₂²×C₃	1	4	yes	complemented
77	#360	C₂²×C₃	1	8	yes	—
78	C₃×#41	—	2	2	no	—
79	C₃×#41	—	2	2	no	—
80	C₃×#41	—	4	1	no	—
81	C₃×#41	—	4	1	no	—
82	C₂×C₃×Q₈	—	2	1	no	—
83	C₂×C₃×Q₈	—	4	4	no	—
84	C₂×C₃×Q₈	—	8	4	no	—
85	C₂×C₃×D₄	—	2	1	no	—
86	C₂×C₃×D₄	—	4	1	no	—
87	C₂²×C₃×C₄	—	2	1	no	—
88	C₂²×C₃×C₄	—	4	1	no	—
89	C₂²×C₃×C₄	—	4	2	no	—
90	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
91	C₃×Q₁₆	—	2	1	no	—
92	C₃×Q₁₆	C₂×Q₈	1	2	yes	complemented
93	C₃×SD₁₆	—	2	2	no	—
94	C₃×SD₁₆	—	2	2	no	—
95	C₃×D₈	—	2	1	no	—
96	C₃×D₈	C₂×Q₈	1	2	yes	complemented
97	C₃×M₁₆	—	2	2	no	—
98	C₃×M₁₆	—	4	1	no	—
99	C₃×M₁₆	—	4	4	no	—
100	C₂×C₃×C₈	—	4	1	no	—
101	C₂×C₃×C₈	—	4	4	no	—
102	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
103	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
104	C₂×C₃×C₈	SD₁₆	1	2	yes	—
105	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
106	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
107	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
108	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
109	C₂×C₃×C₈	#41	1	1	yes	characteristic
110	C₂×C₃×C₈	#41	1	1	yes	characteristic
111	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
112	C₃×C₄⋊C₄	—	2	4	no	—
113	C₃×C₄⋊C₄	—	4	4	no	—
114	C₃×C₄⋊C₄	—	4	2	no	—
115	C₃×C₄²	—	2	1	no	—
116	C₃×C₄²	C₂×D₄	1	2	yes	—
117	C₃×C₁₆	—	2	4	no	—
118	C₂×#41	—	2	1	no	—
119	C₂²×Q₈	—	4	1	no	—
120	#135	—	2	1	no	—
121	#135	—	2	1	no	—
122	#135	C₃×Q₈	1	2	yes	complemented
123	#135	C₂³×C₃	1	2	yes	—
124	C₂×Q₁₆	C₃×Q₈	1	1	yes	characteristic, complemented
125	C₂×SD₁₆	—	2	1	no	—
126	C₂×D₈	C₃×Q₈	1	1	yes	characteristic, complemented
127	#131	—	2	4	no	—
128	#131	—	4	2	no	—
129	C₂×M₁₆	—	2	1	no	—
130	C₂×M₁₆	—	4	1	no	—
131	C₂²×C₈	—	4	1	no	—
132	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
133	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
134	C₄⋊Q₈	—	2	4	no	—
135	C₄⋊Q₈	—	4	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	4	1	no	—
138	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
139	M₃₂	—	2	2	no	—
140	M₃₂	C₂³×C₃	1	4	yes	—
141	C₂×C₁₆	—	2	2	no	—
142	#108	—	2	1	no	—
143	#108	C₂³×C₃	1	2	yes	—
144	#106	—	2	4	no	—
145	#106	—	2	2	no	—
146	#106	C₂³×C₃	1	4	yes	—
147	#104	—	2	4	no	—
148	C₄×C₈	C₃×D₄	1	2	yes	—
149	C₄×C₈	C₂³×C₃	1	2	yes	—
150	C₂³×C₃	—	4	1	no	—
151	C₂³×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₃×Q₈	—	2	2	no	—
153	C₃×Q₈	—	4	1	no	—
154	C₃×Q₈	—	8	8	no	—
155	C₃×D₄	—	2	2	no	—
156	C₃×D₄	—	4	2	no	—
157	C₃×D₄	—	4	2	no	—
158	C₃×D₄	—	4	1	no	—
159	C₂×C₃×C₄	—	2	4	no	—
160	C₂×C₃×C₄	—	2	2	no	—
161	C₂×C₃×C₄	—	4	1	no	—
162	C₂×C₃×C₄	—	4	2	no	—
163	C₂×C₃×C₄	—	4	2	no	—
164	C₂×C₃×C₄	—	4	4	no	—
165	C₂×C₃×C₄	—	8	4	no	—
166	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
170	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
172	C₃×C₈	—	4	2	no	—
173	C₃×C₈	C₂²⋊Q₈	1	1	yes	characteristic
174	C₃×C₈	C₂²⋊Q₈	1	1	yes	characteristic
175	C₃×C₈	C₂×SD₁₆	1	2	yes	—
176	C₃×C₈	C₂×SD₁₆	1	2	yes	—
177	C₃×C₈	C₂²×Q₈	1	1	yes	characteristic
178	C₃×C₈	C₂×#41	1	1	yes	characteristic
179	#41	—	2	2	no	—
180	#41	—	2	2	no	—
181	#41	—	4	1	no	—
182	#41	—	4	1	no	—
183	C₂×Q₈	—	2	1	no	—
184	C₂×Q₈	—	4	4	no	—
185	C₂×Q₈	—	8	4	no	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	—	4	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	4	1	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
192	Q₁₆	—	2	1	no	—
193	Q₁₆	C₂×C₃×Q₈	1	2	yes	complemented
194	SD₁₆	—	2	2	no	—
195	SD₁₆	—	2	2	no	—
196	D₈	—	2	1	no	—
197	D₈	C₂×C₃×Q₈	1	2	yes	complemented
198	M₁₆	—	2	2	no	—
199	M₁₆	—	4	1	no	—
200	M₁₆	—	4	4	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	4	4	no	—
203	C₂×C₈	C₃×SD₁₆	1	1	yes	characteristic
204	C₂×C₈	C₃×SD₁₆	1	1	yes	characteristic
205	C₂×C₈	C₃×SD₁₆	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	C₂×C₃×Q₈	1	1	yes	characteristic
209	C₂×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
210	C₂×C₈	C₃×#41	1	1	yes	characteristic
211	C₂×C₈	C₃×#41	1	1	yes	characteristic
212	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄²	—	2	1	no	—
217	C₄²	C₂×C₃×D₄	1	2	yes	—
218	C₁₆	—	2	4	no	—
219	C₂²×C₃	—	2	2	no	—
220	C₂²×C₃	—	4	2	no	—
221	C₂²×C₃	—	8	2	no	—
222	C₂²×C₃	#476	1	1	yes	characteristic
223	C₂²×C₃	#476	1	1	yes	characteristic
224	C₂²×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
225	C₃×C₄	—	2	4	no	—
226	C₃×C₄	—	4	2	no	—
227	C₃×C₄	—	8	4	no	—
228	C₃×C₄	#476	1	1	yes	characteristic
229	C₃×C₄	#476	1	1	yes	characteristic
230	C₃×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
231	C₃×C₄	C₂²×SD₁₆	1	1	yes	characteristic
232	C₂³	—	4	1	no	—
233	C₂³	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
234	Q₈	—	2	2	no	—
235	Q₈	—	4	1	no	—
236	Q₈	—	8	8	no	—
237	D₄	—	2	2	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	2	no	—
240	D₄	—	4	1	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	8	4	no	—
248	C₂×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
249	C₂×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
250	C₂×C₄	C₂×C₃×SD₁₆	1	1	yes	characteristic
251	C₂×C₄	C₂×C₃×SD₁₆	1	1	yes	characteristic
252	C₂×C₄	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
253	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
254	C₈	—	4	2	no	—
255	C₈	C₃×C₂²⋊Q₈	1	1	yes	characteristic
256	C₈	C₃×C₂²⋊Q₈	1	1	yes	characteristic
257	C₈	C₂×C₃×SD₁₆	1	2	yes	—
258	C₈	C₂×C₃×SD₁₆	1	2	yes	—
259	C₈	C₂²×C₃×Q₈	1	1	yes	characteristic
260	C₈	C₂×C₃×#41	1	1	yes	characteristic
261	C₂×C₃	—	2	1	no	—
262	C₂×C₃	—	2	1	no	—
263	C₂×C₃	—	8	2	no	—
264	C₂×C₃	#2230	1	2	yes	direct product factor
265	C₂×C₃	C₂×#476	1	1	yes	characteristic
266	C₂²	—	2	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	8	2	no	—
269	C₂²	C₃×#476	1	1	yes	characteristic
270	C₂²	C₃×#476	1	1	yes	characteristic
271	C₂²	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
272	C₄	—	2	4	no	—
273	C₄	—	4	2	no	—
274	C₄	—	8	4	no	—
275	C₄	C₃×#476	1	1	yes	characteristic
276	C₄	C₃×#476	1	1	yes	characteristic
277	C₄	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
278	C₄	C₂²×C₃×SD₁₆	1	1	yes	characteristic
279	C₃	C₂×#2230	1	1	yes	characteristic, direct product factor, Sylow
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	8	2	no	—
283	C₂	C₃×#2230	1	2	yes	direct product factor
284	C₂	C₂×C₃×#476	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	#476
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
prime radical	C₂×C₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#498	C₂	semidirect
3. C₃×#2230	C₂	direct (N × H)
4. C₂×C₃×#365	C₂	semidirect
5. C₂×C₃×#364	C₂	semidirect
7. C₂×#2230	C₃	direct (N × H)
13. C₃×#498	C₂²	semidirect
21. C₃×#365	C₂²	semidirect
22. C₃×#364	C₂²	semidirect
24. C₂×#498	C₂×C₃	semidirect
26. #2230	C₂×C₃	direct (N × H)
27. C₂×#365	C₂×C₃	semidirect
28. C₂×#364	C₂×C₃	semidirect
34. C₃×#135	Q₈	semidirect
36. C₂×C₃×Q₁₆	Q₈	semidirect
38. C₂×C₃×D₈	Q₈	semidirect
67. #498	C₂²×C₃	semidirect
75. #365	C₂²×C₃	semidirect
76. #364	C₂²×C₃	semidirect
92. C₃×Q₁₆	C₂×Q₈	semidirect
96. C₃×D₈	C₂×Q₈	semidirect
122. #135	C₃×Q₈	semidirect
124. C₂×Q₁₆	C₃×Q₈	semidirect
126. C₂×D₈	C₃×Q₈	semidirect
193. Q₁₆	C₂×C₃×Q₈	semidirect
197. D₈	C₂×C₃×Q₈	semidirect
264. C₂×C₃	#2230	direct (N × H)
279. C₃	C₂×#2230	direct (N × H)
283. C₂	C₃×#2230	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×C₃×#498	2
C₂×C₃×#443	2
C₃×#2230	2
C₂×C₃×#365	2
C₂×C₃×#364	2
C₂×C₃×#360	2
C₂×#2230	3

Subgroups — C2×C3×#2230 #10652242

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#2230 #10652242