Subgroups — C₂³×C₃×#137 #10681689

Order 768.

11166 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×#41	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₃×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×#137	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₃×Q₁₆	C₂	1	2	yes	maximal, complemented
5	C₂³×C₃×SD₁₆	C₂	1	2	yes	maximal, complemented
6	C₂³×C₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×#137	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂²×C₃×#41	—	2	14	no	—
9	C₂²×C₃×#41	C₂²	1	28	yes	—
10	C₂³×C₃×Q₈	—	2	14	no	—
11	C₂³×C₃×Q₈	—	2	1	no	—
12	C₂³×C₃×Q₈	C₂²	1	2	yes	—
13	C₂³×C₃×Q₈	C₂²	1	1	yes	characteristic
14	C₂³×C₃×Q₈	C₂²	1	28	yes	complemented
15	C₂³×C₃×D₄	—	2	1	no	—
16	C₂³×C₃×D₄	C₂²	1	1	yes	characteristic
17	C₂⁴×C₃×C₄	—	2	1	no	—
18	C₂⁴×C₃×C₄	—	2	1	no	—
19	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
20	C₂×C₃×#137	C₂²	1	448	yes	direct product factor
21	C₂²×C₃×Q₁₆	C₂²	1	56	yes	complemented
22	C₂²×C₃×SD₁₆	C₂²	1	56	yes	complemented
23	C₂²×C₃×M₁₆	C₂²	1	28	yes	complemented
24	C₂³×C₃×C₈	C₂²	1	2	yes	—
25	C₂³×#41	C₂×C₃	1	1	yes	characteristic
26	C₂⁴×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂²×#137	C₂×C₃	1	56	yes	direct product factor
28	C₂³×Q₁₆	C₂×C₃	1	2	yes	complemented
29	C₂³×SD₁₆	C₂×C₃	1	2	yes	complemented
30	C₂³×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂⁵×C₃	—	2	1	no	—
32	C₂⁵×C₃	D₄	1	1	yes	characteristic
33	C₂×C₃×#41	—	2	168	no	—
34	C₂×C₃×#41	C₂³	1	112	yes	—
35	C₂²×C₃×Q₈	—	2	14	no	—
36	C₂²×C₃×Q₈	—	2	28	no	—
37	C₂²×C₃×Q₈	—	2	168	no	—
38	C₂²×C₃×Q₈	—	2	7	no	—
39	C₂²×C₃×Q₈	C₂³	1	28	yes	—
40	C₂²×C₃×Q₈	C₂³	1	14	yes	—
41	C₂²×C₃×Q₈	C₂³	1	112	yes	complemented
42	C₂²×C₃×D₄	—	2	28	no	—
43	C₂²×C₃×D₄	—	2	7	no	—
44	C₂²×C₃×D₄	C₂³	1	14	yes	—
45	C₂³×C₃×C₄	—	2	28	no	—
46	C₂³×C₃×C₄	—	2	1	no	—
47	C₂³×C₃×C₄	—	2	2	no	—
48	C₂³×C₃×C₄	—	2	28	no	—
49	C₂³×C₃×C₄	D₄	1	14	yes	—
50	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
51	C₂³×C₃×C₄	C₂³	1	14	yes	—
52	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
53	C₃×#137	C₂³	1	512	yes	direct product factor
54	C₂×C₃×Q₁₆	C₂³	1	224	yes	complemented
55	C₂×C₃×SD₁₆	C₂³	1	224	yes	complemented
56	C₂×C₃×M₁₆	C₂³	1	112	yes	complemented
57	C₂²×C₃×C₈	C₂³	1	28	yes	—
58	C₂²×#41	—	2	14	no	—
59	C₂²×#41	C₂²×C₃	1	28	yes	—
60	C₂³×Q₈	—	2	1	no	—
61	C₂³×Q₈	—	2	14	no	—
62	C₂³×Q₈	C₂²×C₃	1	2	yes	—
63	C₂³×Q₈	C₂²×C₃	1	1	yes	characteristic
64	C₂³×Q₈	C₂²×C₃	1	28	yes	complemented
65	C₂³×D₄	—	2	1	no	—
66	C₂³×D₄	C₂²×C₃	1	1	yes	characteristic
67	C₂⁴×C₄	—	2	1	no	—
68	C₂⁴×C₄	—	2	1	no	—
69	C₂⁴×C₄	C₂²×C₃	1	1	yes	characteristic
70	C₂×#137	C₂²×C₃	1	448	yes	direct product factor
71	C₂²×Q₁₆	C₂²×C₃	1	56	yes	complemented
72	C₂²×SD₁₆	C₂²×C₃	1	56	yes	complemented
73	C₂²×M₁₆	C₂²×C₃	1	28	yes	complemented
74	C₂³×C₈	C₂²×C₃	1	2	yes	—
75	C₂⁴×C₃	—	2	8	no	—
76	C₂⁴×C₃	—	2	14	no	—
77	C₂⁴×C₃	—	4	8	no	—
78	C₂⁴×C₃	C₂×D₄	1	14	yes	—
79	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
80	C₃×#41	—	2	224	no	—
81	C₃×#41	C₂⁴	1	64	yes	—
82	C₂×C₃×Q₈	—	2	84	no	—
83	C₂×C₃×Q₈	—	2	112	no	—
84	C₂×C₃×Q₈	—	2	224	no	—
85	C₂×C₃×Q₈	—	2	42	no	—
86	C₂×C₃×Q₈	C₂⁴	1	56	yes	—
87	C₂×C₃×Q₈	C₂⁴	1	28	yes	—
88	C₂×C₃×Q₈	C₂⁴	1	64	yes	complemented
89	C₂×C₃×D₄	—	2	112	no	—
90	C₂×C₃×D₄	—	2	42	no	—
91	C₂×C₃×D₄	C₂⁴	1	28	yes	—
92	C₂²×C₃×C₄	—	2	14	no	—
93	C₂²×C₃×C₄	—	2	28	no	—
94	C₂²×C₃×C₄	—	2	112	no	—
95	C₂²×C₃×C₄	—	2	112	no	—
96	C₂²×C₃×C₄	C₂×D₄	1	14	yes	—
97	C₂²×C₃×C₄	C₂×D₄	1	7	yes	—
98	C₂²×C₃×C₄	C₂×D₄	1	84	yes	—
99	C₂²×C₃×C₄	C₂⁴	1	7	yes	—
100	C₂²×C₃×C₄	C₂⁴	1	28	yes	—
101	C₃×Q₁₆	C₂⁴	1	128	yes	complemented
102	C₃×SD₁₆	C₂⁴	1	128	yes	complemented
103	C₃×M₁₆	C₂⁴	1	64	yes	complemented
104	C₂×C₃×C₈	C₂⁴	1	56	yes	—
105	C₂⁵	—	2	1	no	—
106	C₂⁵	C₃×D₄	1	1	yes	characteristic
107	C₂×#41	—	2	168	no	—
108	C₂×#41	C₂³×C₃	1	112	yes	—
109	C₂²×Q₈	—	2	14	no	—
110	C₂²×Q₈	—	2	28	no	—
111	C₂²×Q₈	—	2	168	no	—
112	C₂²×Q₈	—	2	7	no	—
113	C₂²×Q₈	C₂³×C₃	1	28	yes	—
114	C₂²×Q₈	C₂³×C₃	1	14	yes	—
115	C₂²×Q₈	C₂³×C₃	1	112	yes	complemented
116	C₂²×D₄	—	2	28	no	—
117	C₂²×D₄	—	2	7	no	—
118	C₂²×D₄	C₂³×C₃	1	14	yes	—
119	C₂³×C₄	—	2	1	no	—
120	C₂³×C₄	—	2	2	no	—
121	C₂³×C₄	—	2	28	no	—
122	C₂³×C₄	—	2	28	no	—
123	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
124	C₂³×C₄	C₃×D₄	1	14	yes	—
125	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
126	C₂³×C₄	C₂³×C₃	1	14	yes	—
127	#137	C₂³×C₃	1	512	yes	direct product factor
128	C₂×Q₁₆	C₂³×C₃	1	224	yes	complemented
129	C₂×SD₁₆	C₂³×C₃	1	224	yes	complemented
130	C₂×M₁₆	C₂³×C₃	1	112	yes	complemented
131	C₂²×C₈	C₂³×C₃	1	28	yes	—
132	C₂³×C₃	—	2	56	no	—
133	C₂³×C₃	—	2	28	no	—
134	C₂³×C₃	—	4	56	no	—
135	C₂³×C₃	#137	1	8	yes	direct product factor
136	C₂³×C₃	C₂²×D₄	1	28	yes	—
137	C₂³×C₃	C₂²×D₄	1	7	yes	—
138	C₃×Q₈	—	2	56	no	—
139	C₃×Q₈	—	2	64	no	—
140	C₃×Q₈	—	2	28	no	—
141	C₃×Q₈	C₂⁵	1	16	yes	—
142	C₃×Q₈	C₂⁵	1	8	yes	—
143	C₃×D₄	—	2	64	no	—
144	C₃×D₄	—	2	28	no	—
145	C₃×D₄	C₂⁵	1	8	yes	—
146	C₂×C₃×C₄	—	2	28	no	—
147	C₂×C₃×C₄	—	2	56	no	—
148	C₂×C₃×C₄	—	2	64	no	—
149	C₂×C₃×C₄	—	2	64	no	—
150	C₂×C₃×C₄	C₂²×D₄	1	28	yes	—
151	C₂×C₃×C₄	C₂²×D₄	1	21	yes	—
152	C₂×C₃×C₄	C₂²×D₄	1	56	yes	—
153	C₂×C₃×C₄	C₂⁵	1	7	yes	—
154	C₂×C₃×C₄	C₂⁵	1	8	yes	—
155	C₃×C₈	C₂⁵	1	16	yes	—
156	C₂⁴	—	2	8	no	—
157	C₂⁴	—	2	14	no	—
158	C₂⁴	—	4	8	no	—
159	C₂⁴	C₂×C₃×D₄	1	14	yes	—
160	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
161	#41	—	2	224	no	—
162	#41	C₂⁴×C₃	1	64	yes	—
163	C₂×Q₈	—	2	84	no	—
164	C₂×Q₈	—	2	112	no	—
165	C₂×Q₈	—	2	224	no	—
166	C₂×Q₈	—	2	42	no	—
167	C₂×Q₈	C₂⁴×C₃	1	56	yes	—
168	C₂×Q₈	C₂⁴×C₃	1	28	yes	—
169	C₂×Q₈	C₂⁴×C₃	1	64	yes	complemented
170	C₂×D₄	—	2	112	no	—
171	C₂×D₄	—	2	42	no	—
172	C₂×D₄	C₂⁴×C₃	1	28	yes	—
173	C₂²×C₄	—	2	14	no	—
174	C₂²×C₄	—	2	28	no	—
175	C₂²×C₄	—	2	112	no	—
176	C₂²×C₄	—	2	112	no	—
177	C₂²×C₄	C₂×C₃×D₄	1	14	yes	—
178	C₂²×C₄	C₂×C₃×D₄	1	7	yes	—
179	C₂²×C₄	C₂×C₃×D₄	1	84	yes	—
180	C₂²×C₄	C₂⁴×C₃	1	7	yes	—
181	C₂²×C₄	C₂⁴×C₃	1	28	yes	—
182	Q₁₆	C₂⁴×C₃	1	128	yes	complemented
183	SD₁₆	C₂⁴×C₃	1	128	yes	complemented
184	M₁₆	C₂⁴×C₃	1	64	yes	complemented
185	C₂×C₈	C₂⁴×C₃	1	56	yes	—
186	C₂²×C₃	—	2	56	no	—
187	C₂²×C₃	—	2	8	no	—
188	C₂²×C₃	—	4	56	no	—
189	C₂²×C₃	C₂×#137	1	28	yes	direct product factor
190	C₂²×C₃	C₂³×D₄	1	8	yes	—
191	C₂²×C₃	C₂³×D₄	1	7	yes	—
192	C₃×C₄	—	2	8	no	—
193	C₃×C₄	—	2	16	no	—
194	C₃×C₄	C₂³×D₄	1	8	yes	—
195	C₃×C₄	C₂³×D₄	1	7	yes	—
196	C₃×C₄	C₂⁶	1	1	yes	characteristic
197	C₂³	—	2	56	no	—
198	C₂³	—	2	28	no	—
199	C₂³	—	4	56	no	—
200	C₂³	C₃×#137	1	8	yes	direct product factor
201	C₂³	C₂²×C₃×D₄	1	28	yes	—
202	C₂³	C₂²×C₃×D₄	1	7	yes	—
203	Q₈	—	2	56	no	—
204	Q₈	—	2	64	no	—
205	Q₈	—	2	28	no	—
206	Q₈	C₂⁵×C₃	1	16	yes	—
207	Q₈	C₂⁵×C₃	1	8	yes	—
208	D₄	—	2	64	no	—
209	D₄	—	2	28	no	—
210	D₄	C₂⁵×C₃	1	8	yes	—
211	C₂×C₄	—	2	28	no	—
212	C₂×C₄	—	2	56	no	—
213	C₂×C₄	—	2	64	no	—
214	C₂×C₄	—	2	64	no	—
215	C₂×C₄	C₂²×C₃×D₄	1	28	yes	—
216	C₂×C₄	C₂²×C₃×D₄	1	21	yes	—
217	C₂×C₄	C₂²×C₃×D₄	1	56	yes	—
218	C₂×C₄	C₂⁵×C₃	1	7	yes	—
219	C₂×C₄	C₂⁵×C₃	1	8	yes	—
220	C₈	C₂⁵×C₃	1	16	yes	—
221	C₂×C₃	—	2	8	no	—
222	C₂×C₃	—	4	8	no	—
223	C₂×C₃	C₂²×#137	1	14	yes	direct product factor
224	C₂×C₃	C₂⁴×D₄	1	1	yes	characteristic
225	C₂²	—	2	56	no	—
226	C₂²	—	2	8	no	—
227	C₂²	—	4	56	no	—
228	C₂²	C₂×C₃×#137	1	28	yes	direct product factor
229	C₂²	C₂³×C₃×D₄	1	8	yes	—
230	C₂²	C₂³×C₃×D₄	1	7	yes	—
231	C₄	—	2	8	no	—
232	C₄	—	2	16	no	—
233	C₄	C₂³×C₃×D₄	1	8	yes	—
234	C₄	C₂³×C₃×D₄	1	7	yes	—
235	C₄	C₂⁶×C₃	1	1	yes	characteristic
236	C₃	C₂³×#137	1	1	yes	characteristic, direct product factor, Sylow
237	C₂	—	2	8	no	—
238	C₂	—	4	8	no	—
239	C₂	C₂²×C₃×#137	1	14	yes	direct product factor
240	C₂	C₂⁴×C₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁶×C₃
prime radical	C₂³×C₃×#41	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ C₂⁴×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₃×Q₈	C₂	semidirect
3. C₂²×C₃×#137	C₂	direct (N × H)
4. C₂³×C₃×Q₁₆	C₂	semidirect
5. C₂³×C₃×SD₁₆	C₂	semidirect
6. C₂³×C₃×M₁₆	C₂	semidirect
7. C₂³×#137	C₃	direct (N × H)
14. C₂³×C₃×Q₈	C₂²	semidirect
20. C₂×C₃×#137	C₂²	direct (N × H)
21. C₂²×C₃×Q₁₆	C₂²	semidirect
22. C₂²×C₃×SD₁₆	C₂²	semidirect
23. C₂²×C₃×M₁₆	C₂²	semidirect
26. C₂⁴×Q₈	C₂×C₃	semidirect
27. C₂²×#137	C₂×C₃	direct (N × H)
28. C₂³×Q₁₆	C₂×C₃	semidirect
29. C₂³×SD₁₆	C₂×C₃	semidirect
30. C₂³×M₁₆	C₂×C₃	semidirect
41. C₂²×C₃×Q₈	C₂³	semidirect
53. C₃×#137	C₂³	direct (N × H)
54. C₂×C₃×Q₁₆	C₂³	semidirect
55. C₂×C₃×SD₁₆	C₂³	semidirect
56. C₂×C₃×M₁₆	C₂³	semidirect
64. C₂³×Q₈	C₂²×C₃	semidirect
70. C₂×#137	C₂²×C₃	direct (N × H)
71. C₂²×Q₁₆	C₂²×C₃	semidirect
72. C₂²×SD₁₆	C₂²×C₃	semidirect
73. C₂²×M₁₆	C₂²×C₃	semidirect
88. C₂×C₃×Q₈	C₂⁴	semidirect
101. C₃×Q₁₆	C₂⁴	semidirect
102. C₃×SD₁₆	C₂⁴	semidirect
103. C₃×M₁₆	C₂⁴	semidirect
115. C₂²×Q₈	C₂³×C₃	semidirect
127. #137	C₂³×C₃	direct (N × H)
128. C₂×Q₁₆	C₂³×C₃	semidirect
129. C₂×SD₁₆	C₂³×C₃	semidirect
130. C₂×M₁₆	C₂³×C₃	semidirect
135. C₂³×C₃	#137	direct (N × H)
169. C₂×Q₈	C₂⁴×C₃	semidirect
182. Q₁₆	C₂⁴×C₃	semidirect
183. SD₁₆	C₂⁴×C₃	semidirect
184. M₁₆	C₂⁴×C₃	semidirect
189. C₂²×C₃	C₂×#137	direct (N × H)
200. C₂³	C₃×#137	direct (N × H)
223. C₂×C₃	C₂²×#137	direct (N × H)
228. C₂²	C₂×C₃×#137	direct (N × H)
236. C₃	C₂³×#137	direct (N × H)
239. C₂	C₂²×C₃×#137	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×#41	2
C₂⁴×C₃×Q₈	2
C₂²×C₃×#137	2
C₂³×C₃×Q₁₆	2
C₂³×C₃×SD₁₆	2
C₂³×C₃×M₁₆	2
C₂³×#137	3

Subgroups — C23×C3×#137 #10681689

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃×#137 #10681689