Subgroups — C₂×C₃×#2231 #10652243

Order 768.

526 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#443	C₂	1	1	yes	maximal, characteristic
3	C₃×#2231	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×#2231	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×#125	—	2	1	no	—
11	C₂×C₃×M₃₂	C₂²	1	2	yes	—
12	C₃×#499	—	2	2	no	—
13	C₃×#499	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×#107	—	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×#499	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂×#443	C₂×C₃	1	1	yes	characteristic
26	#2231	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	C₃×#135	—	2	1	no	—
32	C₃×#135	—	2	1	no	—
33	C₃×#135	Q₈	1	2	yes	—
34	C₃×#135	C₂³	1	2	yes	—
35	C₂×C₃×Q₁₆	Q₈	1	1	yes	characteristic
36	C₂×C₃×SD₁₆	—	2	1	no	—
37	C₂×C₃×D₈	Q₈	1	1	yes	characteristic
38	C₃×#131	—	2	4	no	—
39	C₃×#131	—	4	2	no	—
40	C₂×C₃×M₁₆	—	2	1	no	—
41	C₂×C₃×M₁₆	—	4	1	no	—
42	C₂²×C₃×C₈	—	4	1	no	—
43	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
44	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
45	C₃×#125	—	2	4	no	—
46	C₃×#125	—	4	2	no	—
47	C₂×C₃×C₄⋊C₄	—	2	2	no	—
48	C₂×C₃×C₄⋊C₄	—	4	2	no	—
49	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
50	C₃×M₃₂	—	2	2	no	—
51	C₃×M₃₂	C₂³	1	4	yes	—
52	C₂×C₃×C₁₆	—	2	2	no	—
53	C₃×#108	—	2	1	no	—
54	C₃×#108	C₂³	1	2	yes	—
55	C₃×#107	—	2	4	no	—
56	C₃×#106	—	2	2	no	—
57	C₃×#106	C₂³	1	4	yes	—
58	C₃×#104	—	2	4	no	—
59	C₃×C₄×C₈	D₄	1	2	yes	—
60	C₃×C₄×C₈	C₂³	1	2	yes	—
61	C₂×#135	C₂²×C₃	1	1	yes	characteristic
62	C₂×#131	—	2	1	no	—
63	C₂×#125	—	2	1	no	—
64	C₂×M₃₂	C₂²×C₃	1	2	yes	—
65	#499	—	2	2	no	—
66	#499	C₂²×C₃	1	4	yes	complemented
67	#443	—	2	2	no	—
68	#443	C₂²×C₃	1	4	yes	—
69	C₂×#108	C₂²×C₃	1	1	yes	characteristic
70	C₂×#107	—	2	1	no	—
71	C₂×#106	C₂²×C₃	1	2	yes	—
72	C₂×#104	—	2	1	no	—
73	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
74	#365	C₂²×C₃	1	8	yes	complemented
75	#364	C₂²×C₃	1	4	yes	complemented
76	#360	C₂²×C₃	1	8	yes	—
77	C₃×#41	—	2	2	no	—
78	C₃×#41	—	2	2	no	—
79	C₃×#41	—	4	1	no	—
80	C₃×#41	—	4	1	no	—
81	C₂×C₃×Q₈	—	2	1	no	—
82	C₂×C₃×D₄	—	2	1	no	—
83	C₂×C₃×D₄	—	4	1	no	—
84	C₂²×C₃×C₄	—	2	1	no	—
85	C₂²×C₃×C₄	—	4	1	no	—
86	C₂²×C₃×C₄	—	4	2	no	—
87	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
88	C₃×Q₁₆	—	2	1	no	—
89	C₃×Q₁₆	C₂×Q₈	1	2	yes	—
90	C₃×SD₁₆	—	2	2	no	—
91	C₃×SD₁₆	—	2	2	no	—
92	C₃×D₈	—	2	1	no	—
93	C₃×D₈	C₂×Q₈	1	2	yes	—
94	C₃×M₁₆	—	2	2	no	—
95	C₃×M₁₆	—	4	1	no	—
96	C₃×M₁₆	—	4	4	no	—
97	C₂×C₃×C₈	—	4	1	no	—
98	C₂×C₃×C₈	—	4	4	no	—
99	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
100	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
101	C₂×C₃×C₈	SD₁₆	1	2	yes	—
102	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
104	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
105	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
106	C₂×C₃×C₈	#41	1	1	yes	characteristic
107	C₂×C₃×C₈	#41	1	1	yes	characteristic
108	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
109	C₃×C₄⋊C₄	—	2	4	no	—
110	C₃×C₄⋊C₄	—	4	8	no	—
111	C₃×C₄⋊C₄	—	4	2	no	—
112	C₃×C₄²	—	2	1	no	—
113	C₃×C₄²	C₂×D₄	1	2	yes	—
114	C₃×C₁₆	—	2	4	no	—
115	C₂×#41	—	2	1	no	—
116	#135	—	2	1	no	—
117	#135	—	2	1	no	—
118	#135	C₃×Q₈	1	2	yes	—
119	#135	C₂³×C₃	1	2	yes	—
120	C₂×Q₁₆	C₃×Q₈	1	1	yes	characteristic
121	C₂×SD₁₆	—	2	1	no	—
122	C₂×D₈	C₃×Q₈	1	1	yes	characteristic
123	#131	—	2	4	no	—
124	#131	—	4	2	no	—
125	C₂×M₁₆	—	2	1	no	—
126	C₂×M₁₆	—	4	1	no	—
127	C₂²×C₈	—	4	1	no	—
128	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
129	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
130	#125	—	2	4	no	—
131	#125	—	4	2	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	4	2	no	—
134	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
135	M₃₂	—	2	2	no	—
136	M₃₂	C₂³×C₃	1	4	yes	—
137	C₂×C₁₆	—	2	2	no	—
138	#108	—	2	1	no	—
139	#108	C₂³×C₃	1	2	yes	—
140	#107	—	2	4	no	—
141	#106	—	2	2	no	—
142	#106	C₂³×C₃	1	4	yes	—
143	#104	—	2	4	no	—
144	C₄×C₈	C₃×D₄	1	2	yes	—
145	C₄×C₈	C₂³×C₃	1	2	yes	—
146	C₂³×C₃	—	4	1	no	—
147	C₂³×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₃×Q₈	—	2	2	no	—
149	C₃×Q₈	—	4	1	no	—
150	C₃×D₄	—	2	2	no	—
151	C₃×D₄	—	4	2	no	—
152	C₃×D₄	—	4	2	no	—
153	C₃×D₄	—	4	1	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	4	2	no	—
156	C₂×C₃×C₄	—	4	1	no	—
157	C₂×C₃×C₄	—	4	2	no	—
158	C₂×C₃×C₄	—	4	2	no	—
159	C₂×C₃×C₄	—	4	4	no	—
160	C₂×C₃×C₄	—	8	4	no	—
161	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
165	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
167	C₃×C₈	—	4	2	no	—
168	C₃×C₈	C₂²⋊Q₈	1	1	yes	characteristic
169	C₃×C₈	C₂²⋊Q₈	1	1	yes	characteristic
170	C₃×C₈	C₂×SD₁₆	1	2	yes	—
171	C₃×C₈	C₂×SD₁₆	1	2	yes	—
172	C₃×C₈	C₂²×Q₈	1	1	yes	characteristic
173	C₃×C₈	C₂×#41	1	1	yes	characteristic
174	#41	—	2	2	no	—
175	#41	—	2	2	no	—
176	#41	—	4	1	no	—
177	#41	—	4	1	no	—
178	C₂×Q₈	—	2	1	no	—
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	—	4	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	4	1	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
185	Q₁₆	—	2	1	no	—
186	Q₁₆	C₂×C₃×Q₈	1	2	yes	—
187	SD₁₆	—	2	2	no	—
188	SD₁₆	—	2	2	no	—
189	D₈	—	2	1	no	—
190	D₈	C₂×C₃×Q₈	1	2	yes	—
191	M₁₆	—	2	2	no	—
192	M₁₆	—	4	1	no	—
193	M₁₆	—	4	4	no	—
194	C₂×C₈	—	4	1	no	—
195	C₂×C₈	—	4	4	no	—
196	C₂×C₈	C₃×SD₁₆	1	1	yes	characteristic
197	C₂×C₈	C₃×SD₁₆	1	1	yes	characteristic
198	C₂×C₈	C₃×SD₁₆	1	2	yes	—
199	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
200	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
201	C₂×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
202	C₂×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
203	C₂×C₈	C₃×#41	1	1	yes	characteristic
204	C₂×C₈	C₃×#41	1	1	yes	characteristic
205	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	4	8	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄²	—	2	1	no	—
210	C₄²	C₂×C₃×D₄	1	2	yes	—
211	C₁₆	—	2	4	no	—
212	C₂²×C₃	—	2	2	no	—
213	C₂²×C₃	—	4	2	no	—
214	C₂²×C₃	—	8	2	no	—
215	C₂²×C₃	#476	1	1	yes	characteristic
216	C₂²×C₃	#476	1	1	yes	characteristic
217	C₂²×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
218	C₃×C₄	—	4	2	no	—
219	C₃×C₄	—	4	2	no	—
220	C₃×C₄	—	8	4	no	—
221	C₃×C₄	#476	1	1	yes	characteristic
222	C₃×C₄	#476	1	1	yes	characteristic
223	C₃×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
224	C₃×C₄	C₂²×SD₁₆	1	1	yes	characteristic
225	C₂³	—	4	1	no	—
226	C₂³	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
227	Q₈	—	2	2	no	—
228	Q₈	—	4	1	no	—
229	D₄	—	2	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	4	2	no	—
232	D₄	—	4	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	8	4	no	—
240	C₂×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
241	C₂×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
242	C₂×C₄	C₂×C₃×SD₁₆	1	1	yes	characteristic
243	C₂×C₄	C₂×C₃×SD₁₆	1	1	yes	characteristic
244	C₂×C₄	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
245	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
246	C₈	—	4	2	no	—
247	C₈	C₃×C₂²⋊Q₈	1	1	yes	characteristic
248	C₈	C₃×C₂²⋊Q₈	1	1	yes	characteristic
249	C₈	C₂×C₃×SD₁₆	1	2	yes	—
250	C₈	C₂×C₃×SD₁₆	1	2	yes	—
251	C₈	C₂²×C₃×Q₈	1	1	yes	characteristic
252	C₈	C₂×C₃×#41	1	1	yes	characteristic
253	C₂×C₃	—	2	1	no	—
254	C₂×C₃	—	2	1	no	—
255	C₂×C₃	—	8	2	no	—
256	C₂×C₃	#2231	1	2	yes	direct product factor
257	C₂×C₃	C₂×#476	1	1	yes	characteristic
258	C₂²	—	2	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	8	2	no	—
261	C₂²	C₃×#476	1	1	yes	characteristic
262	C₂²	C₃×#476	1	1	yes	characteristic
263	C₂²	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
264	C₄	—	4	2	no	—
265	C₄	—	4	2	no	—
266	C₄	—	8	4	no	—
267	C₄	C₃×#476	1	1	yes	characteristic
268	C₄	C₃×#476	1	1	yes	characteristic
269	C₄	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
270	C₄	C₂²×C₃×SD₁₆	1	1	yes	characteristic
271	C₃	C₂×#2231	1	1	yes	characteristic, direct product factor, Sylow
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	—	8	2	no	—
275	C₂	C₃×#2231	1	2	yes	direct product factor
276	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×#499	C₂	semidirect
3. C₃×#2231	C₂	direct (N × H)
4. C₂×C₃×#365	C₂	semidirect
5. C₂×C₃×#364	C₂	semidirect
7. C₂×#2231	C₃	direct (N × H)
13. C₃×#499	C₂²	semidirect
21. C₃×#365	C₂²	semidirect
22. C₃×#364	C₂²	semidirect
24. C₂×#499	C₂×C₃	semidirect
26. #2231	C₂×C₃	direct (N × H)
27. C₂×#365	C₂×C₃	semidirect
28. C₂×#364	C₂×C₃	semidirect
66. #499	C₂²×C₃	semidirect
74. #365	C₂²×C₃	semidirect
75. #364	C₂²×C₃	semidirect
256. C₂×C₃	#2231	direct (N × H)
271. C₃	C₂×#2231	direct (N × H)
275. C₂	C₃×#2231	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×#499	2
C₂×C₃×#443	2
C₃×#2231	2
C₂×C₃×#365	2
C₂×C₃×#364	2
C₂×C₃×#360	2
C₂×#2231	3

Subgroups — C2×C3×#2231 #10652243

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#2231 #10652243