Subgroups — C₃×#20348 #10638955

Order 768.

2158 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#520	C₂	1	2	yes	maximal
3	C₃×#1736	C₂	1	6	yes	maximal, complemented
4	C₃×#1735	C₂	1	3	yes	maximal, complemented
5	C₂×C₃×#340	C₂	1	3	yes	maximal, complemented
6	#20348	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂²×C₃×#41	C₂²	1	1	yes	characteristic
8	C₂×C₃×#131	—	2	8	no	—
9	C₂×C₃×#131	C₄	1	8	yes	complemented
10	C₂²×C₃×M₁₆	C₂²	1	3	yes	—
11	C₂³×C₃×C₈	C₂²	1	3	yes	—
12	C₃×#520	—	2	8	no	—
13	C₃×#520	C₄	1	16	yes	complemented
14	C₂×C₃×C₄×Q₈	—	2	2	no	—
15	C₂×C₃×C₄×D₄	—	2	6	no	—
16	C₂×C₃×#117	C₂²	1	6	yes	—
17	C₂²×C₃×C₄⋊C₄	C₂²	1	6	yes	—
18	C₃×#340	C₂²	1	3	yes	—
19	C₃×#340	C₂²	1	6	yes	—
20	C₃×#340	C₂²	1	1	yes	characteristic
21	C₃×#340	C₂²	1	6	yes	—
22	C₂²×#131	C₂×C₃	1	1	yes	characteristic
23	C₂×#520	C₂×C₃	1	2	yes	—
24	#1736	C₂×C₃	1	6	yes	complemented
25	#1735	C₂×C₃	1	3	yes	complemented
26	C₂×#340	C₂×C₃	1	3	yes	complemented
27	C₂×C₃×#41	C₂×C₄	1	8	yes	—
28	C₂×C₃×#41	C₂×C₄	1	4	yes	—
29	C₂×C₃×#41	D₄	1	8	yes	—
30	C₂×C₃×#41	Q₈	1	4	yes	—
31	C₂²×C₃×Q₈	D₄	1	1	yes	characteristic
32	C₂²×C₃×D₄	D₄	1	3	yes	—
33	C₂³×C₃×C₄	—	2	6	no	—
34	C₂³×C₃×C₄	C₂³	1	3	yes	—
35	C₃×#131	—	2	32	no	—
36	C₂×C₃×M₁₆	—	2	12	no	—
37	C₂×C₃×M₁₆	C₂×C₄	1	12	yes	complemented
38	C₂²×C₃×C₈	—	2	12	no	—
39	C₂²×C₃×C₈	C₂×C₄	1	12	yes	complemented
40	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₃×C₈	C₂³	1	3	yes	—
42	C₃×C₄×Q₈	—	2	16	no	—
43	C₃×C₄×D₄	—	2	48	no	—
44	C₃×#117	—	2	12	no	—
45	C₃×#117	C₂×C₄	1	24	yes	complemented
46	C₂×C₃×C₄⋊C₄	—	2	6	no	—
47	C₂×C₃×C₄⋊C₄	—	2	12	no	—
48	C₂×C₃×C₄⋊C₄	C₂×C₄	1	24	yes	complemented
49	C₂×C₃×C₄⋊C₄	C₂³	1	6	yes	—
50	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
51	C₂×C₃×C₂²⋊C₄	—	2	6	no	—
52	C₂×C₃×C₄²	—	2	6	no	—
53	C₂²×#41	C₂²×C₃	1	1	yes	characteristic
54	C₂×#131	—	2	8	no	—
55	C₂×#131	C₃×C₄	1	8	yes	complemented
56	C₂²×M₁₆	C₂²×C₃	1	3	yes	—
57	C₂³×C₈	C₂²×C₃	1	3	yes	—
58	#520	—	2	8	no	—
59	#520	C₃×C₄	1	16	yes	complemented
60	C₂×C₄×Q₈	—	2	2	no	—
61	C₂×C₄×D₄	—	2	6	no	—
62	C₂×#117	C₂²×C₃	1	6	yes	—
63	C₂²×C₄⋊C₄	C₂²×C₃	1	6	yes	—
64	#340	C₂²×C₃	1	3	yes	—
65	#340	C₂²×C₃	1	6	yes	—
66	#340	C₂²×C₃	1	1	yes	characteristic
67	#340	C₂²×C₃	1	6	yes	—
68	C₂⁴×C₃	C₂×D₄	1	3	yes	—
69	C₃×#41	C₄²	1	16	yes	—
70	C₃×#41	C₂²⋊C₄	1	16	yes	—
71	C₃×#41	C₄⋊C₄	1	32	yes	—
72	C₂×C₃×Q₈	C₂²⋊C₄	1	8	yes	—
73	C₂×C₃×Q₈	C₄⋊C₄	1	4	yes	—
74	C₂×C₃×D₄	C₂²⋊C₄	1	24	yes	—
75	C₂×C₃×D₄	C₄⋊C₄	1	12	yes	—
76	C₂²×C₃×C₄	—	2	2	no	—
77	C₂²×C₃×C₄	—	2	6	no	—
78	C₂²×C₃×C₄	—	2	24	no	—
79	C₂²×C₃×C₄	—	8	6	no	—
80	C₂²×C₃×C₄	—	8	6	no	—
81	C₂²×C₃×C₄	C₂²×C₄	1	6	yes	—
82	C₂²×C₃×C₄	C₂²×C₄	1	12	yes	—
83	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
84	C₂²×C₃×C₄	C₂×D₄	1	12	yes	—
85	C₂²×C₃×C₄	C₂×Q₈	1	6	yes	—
86	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
87	C₃×M₁₆	—	2	24	no	—
88	C₂×C₃×C₈	—	2	24	no	—
89	C₂×C₃×C₈	—	2	2	no	—
90	C₂×C₃×C₈	—	2	6	no	—
91	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
92	C₂×C₃×C₈	C₂²×C₄	1	6	yes	—
93	C₃×C₄⋊C₄	—	2	24	no	—
94	C₃×C₄⋊C₄	—	2	6	no	—
95	C₃×C₄⋊C₄	—	2	2	no	—
96	C₃×C₄⋊C₄	C₂²×C₄	1	12	yes	—
97	C₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
98	C₃×C₂²⋊C₄	—	2	24	no	—
99	C₃×C₄²	—	2	24	no	—
100	C₂×#41	C₂×C₃×C₄	1	8	yes	—
101	C₂×#41	C₂×C₃×C₄	1	4	yes	—
102	C₂×#41	C₃×D₄	1	8	yes	—
103	C₂×#41	C₃×Q₈	1	4	yes	—
104	C₂²×Q₈	C₃×D₄	1	1	yes	characteristic
105	C₂²×D₄	C₃×D₄	1	3	yes	—
106	C₂³×C₄	—	2	6	no	—
107	C₂³×C₄	C₂³×C₃	1	3	yes	—
108	#131	—	2	32	no	—
109	C₂×M₁₆	—	2	12	no	—
110	C₂×M₁₆	C₂×C₃×C₄	1	12	yes	complemented
111	C₂²×C₈	—	2	12	no	—
112	C₂²×C₈	C₂×C₃×C₄	1	12	yes	complemented
113	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
114	C₂²×C₈	C₂³×C₃	1	3	yes	—
115	C₄×Q₈	—	2	16	no	—
116	C₄×D₄	—	2	48	no	—
117	#117	—	2	12	no	—
118	#117	C₂×C₃×C₄	1	24	yes	complemented
119	C₂×C₄⋊C₄	—	2	6	no	—
120	C₂×C₄⋊C₄	—	2	12	no	—
121	C₂×C₄⋊C₄	C₂×C₃×C₄	1	24	yes	complemented
122	C₂×C₄⋊C₄	C₂³×C₃	1	6	yes	—
123	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
124	C₂×C₂²⋊C₄	—	2	6	no	—
125	C₂×C₄²	—	2	6	no	—
126	C₂³×C₃	—	2	3	no	—
127	C₂³×C₃	—	2	6	no	—
128	C₂³×C₃	—	2	3	no	—
129	C₂³×C₃	C₂×C₂²⋊C₄	1	12	yes	—
130	C₂³×C₃	C₂×C₄⋊C₄	1	6	yes	—
131	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
132	C₃×Q₈	#95	1	16	yes	—
133	C₃×D₄	#95	1	48	yes	—
134	C₂×C₃×C₄	—	2	4	no	—
135	C₂×C₃×C₄	—	2	12	no	—
136	C₂×C₃×C₄	—	4	6	no	—
137	C₂×C₃×C₄	—	4	2	no	—
138	C₂×C₃×C₄	—	4	6	no	—
139	C₂×C₃×C₄	—	4	2	no	—
140	C₂×C₃×C₄	—	8	24	no	—
141	C₂×C₃×C₄	C₂×C₄²	1	12	yes	—
142	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	12	yes	—
143	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	12	yes	—
144	C₂×C₃×C₄	C₂×C₄⋊C₄	1	24	yes	—
145	C₂×C₃×C₄	C₂×C₄⋊C₄	1	6	yes	—
146	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂³×C₄	1	2	yes	—
148	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
149	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
150	C₃×C₈	—	2	2	no	—
151	C₃×C₈	—	2	6	no	—
152	C₂⁴	C₂×C₃×D₄	1	3	yes	—
153	#41	C₃×C₄²	1	16	yes	—
154	#41	C₃×C₂²⋊C₄	1	16	yes	—
155	#41	C₃×C₄⋊C₄	1	32	yes	—
156	C₂×Q₈	C₃×C₂²⋊C₄	1	8	yes	—
157	C₂×Q₈	C₃×C₄⋊C₄	1	4	yes	—
158	C₂×D₄	C₃×C₂²⋊C₄	1	24	yes	—
159	C₂×D₄	C₃×C₄⋊C₄	1	12	yes	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	6	no	—
162	C₂²×C₄	—	2	24	no	—
163	C₂²×C₄	—	8	6	no	—
164	C₂²×C₄	—	8	6	no	—
165	C₂²×C₄	C₂²×C₃×C₄	1	6	yes	—
166	C₂²×C₄	C₂²×C₃×C₄	1	12	yes	—
167	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
168	C₂²×C₄	C₂×C₃×D₄	1	12	yes	—
169	C₂²×C₄	C₂×C₃×Q₈	1	6	yes	—
170	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
171	M₁₆	—	2	24	no	—
172	C₂×C₈	—	2	24	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	6	no	—
175	C₂×C₈	C₂²×C₃×C₄	1	2	yes	—
176	C₂×C₈	C₂²×C₃×C₄	1	6	yes	—
177	C₄⋊C₄	—	2	24	no	—
178	C₄⋊C₄	—	2	6	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	C₂²×C₃×C₄	1	12	yes	—
181	C₄⋊C₄	C₂²×C₃×C₄	1	4	yes	—
182	C₂²⋊C₄	—	2	24	no	—
183	C₄²	—	2	24	no	—
184	C₂²×C₃	—	2	12	no	—
185	C₂²×C₃	—	2	12	no	—
186	C₂²×C₃	—	2	6	no	—
187	C₂²×C₃	—	2	6	no	—
188	C₂²×C₃	C₂×#95	1	12	yes	—
189	C₂²×C₃	C₂²×C₂²⋊C₄	1	2	yes	—
190	C₂²×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
191	C₂²×C₃	#576	1	1	yes	characteristic
192	C₂²×C₃	#577	1	2	yes	—
193	C₂²×C₃	#578	1	1	yes	characteristic
194	C₃×C₄	—	4	12	no	—
195	C₃×C₄	—	4	4	no	—
196	C₃×C₄	C₂×#95	1	12	yes	—
197	C₃×C₄	C₂²×C₄²	1	1	yes	characteristic
198	C₃×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
199	C₃×C₄	C₂²×C₄⋊C₄	1	2	yes	—
200	C₂³	—	2	3	no	—
201	C₂³	—	2	6	no	—
202	C₂³	—	2	3	no	—
203	C₂³	C₂×C₃×C₂²⋊C₄	1	12	yes	—
204	C₂³	C₂×C₃×C₄⋊C₄	1	6	yes	—
205	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
206	Q₈	C₃×#95	1	16	yes	—
207	D₄	C₃×#95	1	48	yes	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	12	no	—
210	C₂×C₄	—	4	6	no	—
211	C₂×C₄	—	4	6	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	8	24	no	—
215	C₂×C₄	C₂×C₃×C₄²	1	12	yes	—
216	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	12	yes	—
217	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	12	yes	—
218	C₂×C₄	C₂×C₃×C₄⋊C₄	1	6	yes	—
219	C₂×C₄	C₂×C₃×C₄⋊C₄	1	24	yes	—
220	C₂×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
221	C₂×C₄	C₂³×C₃×C₄	1	2	yes	—
222	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
223	C₂×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	2	6	no	—
226	C₂×C₃	—	2	12	no	—
227	C₂×C₃	C₂²×#95	1	1	yes	characteristic
228	C₂×C₃	#2897	1	2	yes	—
229	C₂×C₃	#2898	1	2	yes	—
230	C₂×C₃	#2912	1	1	yes	characteristic
231	C₂×C₃	#2913	1	1	yes	characteristic
232	C₂²	—	2	12	no	—
233	C₂²	—	2	12	no	—
234	C₂²	—	2	6	no	—
235	C₂²	—	2	6	no	—
236	C₂²	C₂×C₃×#95	1	12	yes	—
237	C₂²	C₂²×C₃×C₂²⋊C₄	1	2	yes	—
238	C₂²	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
239	C₂²	C₃×#576	1	1	yes	characteristic
240	C₂²	C₃×#577	1	2	yes	—
241	C₂²	C₃×#578	1	1	yes	characteristic
242	C₄	—	4	12	no	—
243	C₄	—	4	4	no	—
244	C₄	C₂×C₃×#95	1	12	yes	—
245	C₄	C₂²×C₃×C₄²	1	1	yes	characteristic
246	C₄	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
247	C₄	C₂²×C₃×C₄⋊C₄	1	2	yes	—
248	C₃	#20348	1	1	yes	characteristic, direct product factor, Sylow
249	C₂	—	2	12	no	—
250	C₂	C₂²×C₃×#95	1	1	yes	characteristic
251	C₂	C₃×#2897	1	2	yes	—
252	C₂	C₃×#2898	1	2	yes	—
253	C₂	C₃×#2912	1	1	yes	characteristic
254	C₂	C₃×#2913	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂²×D₄
commutator	C₄	C₂²×C₃×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
prime radical	C₂²×C₃×#41	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×#41 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×#1736	C₂	semidirect
4. C₃×#1735	C₂	semidirect
5. C₂×C₃×#340	C₂	semidirect
6. #20348	C₃	direct (N × H)
9. C₂×C₃×#131	C₄	semidirect
13. C₃×#520	C₄	semidirect
24. #1736	C₂×C₃	semidirect
25. #1735	C₂×C₃	semidirect
26. C₂×#340	C₂×C₃	semidirect
37. C₂×C₃×M₁₆	C₂×C₄	semidirect
39. C₂²×C₃×C₈	C₂×C₄	semidirect
45. C₃×#117	C₂×C₄	semidirect
48. C₂×C₃×C₄⋊C₄	C₂×C₄	semidirect
55. C₂×#131	C₃×C₄	semidirect
59. #520	C₃×C₄	semidirect
110. C₂×M₁₆	C₂×C₃×C₄	semidirect
112. C₂²×C₈	C₂×C₃×C₄	semidirect
118. #117	C₂×C₃×C₄	semidirect
121. C₂×C₄⋊C₄	C₂×C₃×C₄	semidirect
248. C₃	#20348	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×#131	2
C₂×C₃×#520	2
C₃×#1736	2
C₃×#1735	2
C₂×C₃×#340	2
#20348	3

Subgroups — C3×#20348 #10638955

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#20348 #10638955