Subgroups — C₃×#33768 #10652375

Order 768.

552 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3059	C₂	1	2	yes	maximal
2	C₃×#2237	C₂	1	4	yes	maximal, complemented
3	C₃×#2219	C₂	1	2	yes	maximal, complemented
4	C₃×#2216	C₂	1	2	yes	maximal, complemented
5	C₃×#2188	C₂	1	4	yes	maximal, complemented
6	C₃×#2113	C₂	1	1	yes	maximal, characteristic
7	#33768	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂²×C₃×Q₁₆	C₂²	1	2	yes	—
9	C₂×C₃×#131	C₂²	1	1	yes	characteristic
10	C₃×#536	—	2	2	no	—
11	C₂×C₃×Q₃₂	C₂²	1	2	yes	—
12	C₂×C₃×Q₃₂	C₂²	1	2	yes	—
13	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₃×#470	—	2	2	no	—
16	C₃×#470	—	2	2	no	—
17	C₃×#467	C₂²	1	4	yes	—
18	C₃×#467	C₂²	1	4	yes	—
19	C₃×#427	C₂²	1	2	yes	—
20	C₃×#419	—	2	2	no	—
21	C₃×#367	C₂²	1	2	yes	—
22	C₃×#366	C₂²	1	2	yes	—
23	C₃×#358	C₂²	1	4	yes	—
24	C₃×#358	C₂²	1	4	yes	—
25	C₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
26	#3059	C₂×C₃	1	2	yes	—
27	#2237	C₂×C₃	1	4	yes	complemented
28	#2219	C₂×C₃	1	2	yes	complemented
29	#2216	C₂×C₃	1	2	yes	complemented
30	#2188	C₂×C₃	1	4	yes	complemented
31	#2113	C₂×C₃	1	1	yes	characteristic
32	C₂×C₃×#41	D₄	1	1	yes	characteristic
33	C₂²×C₃×Q₈	—	2	2	no	—
34	C₂×C₃×Q₁₆	—	2	2	no	—
35	C₂×C₃×Q₁₆	—	4	2	no	—
36	C₂×C₃×Q₁₆	—	4	2	no	—
37	C₂×C₃×Q₁₆	C₂³	1	4	yes	—
38	C₃×#131	—	4	1	no	—
39	C₃×#131	—	4	1	no	—
40	C₂×C₃×M₁₆	D₄	1	2	yes	—
41	C₂×C₃×M₁₆	D₄	1	1	yes	characteristic
42	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₃×C₈	C₂³	1	2	yes	—
44	C₃×C₄⋊Q₈	—	4	2	no	—
45	C₃×#124	—	4	2	no	—
46	C₃×#123	—	4	2	no	—
47	C₃×#123	—	4	2	no	—
48	C₃×C₂²⋊Q₈	—	2	4	no	—
49	C₃×C₂²⋊Q₈	—	2	4	no	—
50	C₃×#117	—	2	2	no	—
51	C₃×Q₃₂	—	4	2	no	—
52	C₃×Q₃₂	—	4	2	no	—
53	C₃×M₃₂	—	4	1	no	—
54	C₂×C₃×C₁₆	—	4	1	no	—
55	C₂×C₃×C₁₆	C₂³	1	2	yes	—
56	C₂×C₃×C₁₆	C₂³	1	2	yes	—
57	C₃×#107	C₂³	1	4	yes	—
58	C₃×#106	—	2	2	no	—
59	C₃×#103	—	2	4	no	—
60	C₃×#103	—	2	4	no	—
61	C₂²×Q₁₆	C₂²×C₃	1	2	yes	—
62	C₂×#131	C₂²×C₃	1	1	yes	characteristic
63	#536	—	2	2	no	—
64	C₂×Q₃₂	C₂²×C₃	1	2	yes	—
65	C₂×Q₃₂	C₂²×C₃	1	2	yes	—
66	C₂×M₃₂	C₂²×C₃	1	1	yes	characteristic
67	C₂²×C₁₆	C₂²×C₃	1	1	yes	characteristic
68	#470	—	2	2	no	—
69	#470	—	2	2	no	—
70	#467	C₂²×C₃	1	4	yes	—
71	#467	C₂²×C₃	1	4	yes	—
72	#427	C₂²×C₃	1	2	yes	—
73	#419	—	2	2	no	—
74	#367	C₂²×C₃	1	2	yes	—
75	#366	C₂²×C₃	1	2	yes	—
76	#358	C₂²×C₃	1	4	yes	—
77	#358	C₂²×C₃	1	4	yes	—
78	C₂²⋊C₁₆	C₂²×C₃	1	4	yes	—
79	C₃×#41	—	4	1	no	—
80	C₃×#41	—	4	1	no	—
81	C₂×C₃×Q₈	—	2	4	no	—
82	C₂×C₃×Q₈	—	4	4	no	—
83	C₂×C₃×Q₈	—	4	4	no	—
84	C₂×C₃×Q₈	—	4	2	no	—
85	C₂×C₃×Q₈	D₈	1	1	yes	characteristic
86	C₂×C₃×D₄	D₈	1	1	yes	characteristic
87	C₂×C₃×D₄	D₈	1	2	yes	—
88	C₂²×C₃×C₄	—	4	2	no	—
89	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
90	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
91	C₃×Q₁₆	—	2	4	no	—
92	C₃×Q₁₆	—	2	4	no	—
93	C₃×Q₁₆	—	4	4	no	—
94	C₃×M₁₆	—	4	2	no	—
95	C₃×M₁₆	—	4	1	no	—
96	C₂×C₃×C₈	—	2	1	no	—
97	C₂×C₃×C₈	—	2	1	no	—
98	C₂×C₃×C₈	—	2	2	no	—
99	C₂×C₃×C₈	—	2	2	no	—
100	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
102	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
103	C₃×C₄⋊C₄	—	2	4	no	—
104	C₃×C₄⋊C₄	—	4	4	no	—
105	C₃×C₄⋊C₄	—	4	4	no	—
106	C₃×C₂²⋊C₄	—	4	2	no	—
107	C₃×C₂²⋊C₄	—	4	4	no	—
108	C₃×C₂²⋊C₄	—	4	4	no	—
109	C₃×C₄²	—	4	2	no	—
110	C₃×C₁₆	—	2	2	no	—
111	C₃×C₁₆	—	2	2	no	—
112	C₂×#41	C₃×D₄	1	1	yes	characteristic
113	C₂²×Q₈	—	2	2	no	—
114	C₂×Q₁₆	—	2	2	no	—
115	C₂×Q₁₆	—	4	2	no	—
116	C₂×Q₁₆	—	4	2	no	—
117	C₂×Q₁₆	C₂³×C₃	1	4	yes	—
118	#131	—	4	1	no	—
119	#131	—	4	1	no	—
120	C₂×M₁₆	C₃×D₄	1	2	yes	—
121	C₂×M₁₆	C₃×D₄	1	1	yes	characteristic
122	C₂²×C₈	C₂³×C₃	1	2	yes	—
123	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
124	C₄⋊Q₈	—	4	2	no	—
125	#124	—	4	2	no	—
126	#123	—	4	2	no	—
127	#123	—	4	2	no	—
128	C₂²⋊Q₈	—	2	4	no	—
129	C₂²⋊Q₈	—	2	4	no	—
130	#117	—	2	2	no	—
131	Q₃₂	—	4	2	no	—
132	Q₃₂	—	4	2	no	—
133	M₃₂	—	4	1	no	—
134	C₂×C₁₆	—	4	1	no	—
135	C₂×C₁₆	C₂³×C₃	1	2	yes	—
136	C₂×C₁₆	C₂³×C₃	1	2	yes	—
137	#107	C₂³×C₃	1	4	yes	—
138	#106	—	2	2	no	—
139	#103	—	2	4	no	—
140	#103	—	2	4	no	—
141	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
142	C₂³×C₃	C₂×D₈	1	2	yes	—
143	C₃×Q₈	—	4	1	no	—
144	C₃×Q₈	—	4	4	no	—
145	C₃×Q₈	—	4	4	no	—
146	C₃×Q₈	—	8	4	no	—
147	C₃×D₄	—	4	1	no	—
148	C₃×D₄	—	4	2	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	1	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	—	4	4	no	—
154	C₂×C₃×C₄	—	4	4	no	—
155	C₂×C₃×C₄	—	8	4	no	—
156	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
157	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
159	C₃×C₈	—	2	1	no	—
160	C₃×C₈	—	2	2	no	—
161	C₃×C₈	H(5,𝔽₂)	1	1	yes	characteristic
162	C₃×C₈	H(5,𝔽₂)	1	1	yes	characteristic
163	#41	—	4	1	no	—
164	#41	—	4	1	no	—
165	C₂×Q₈	—	2	4	no	—
166	C₂×Q₈	—	4	4	no	—
167	C₂×Q₈	—	4	4	no	—
168	C₂×Q₈	—	4	2	no	—
169	C₂×Q₈	C₃×D₈	1	1	yes	characteristic
170	C₂×D₄	C₃×D₈	1	1	yes	characteristic
171	C₂×D₄	C₃×D₈	1	2	yes	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
174	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
175	Q₁₆	—	2	4	no	—
176	Q₁₆	—	2	4	no	—
177	Q₁₆	—	4	4	no	—
178	M₁₆	—	4	1	no	—
179	M₁₆	—	4	2	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
185	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
186	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₂²⋊C₄	—	4	2	no	—
191	C₂²⋊C₄	—	4	4	no	—
192	C₂²⋊C₄	—	4	4	no	—
193	C₄²	—	4	2	no	—
194	C₁₆	—	2	2	no	—
195	C₁₆	—	2	2	no	—
196	C₂²×C₃	—	2	2	no	—
197	C₂²×C₃	—	2	4	no	—
198	C₂²×C₃	—	2	2	no	—
199	C₂²×C₃	—	2	1	no	—
200	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
201	C₃×C₄	—	2	1	no	—
202	C₃×C₄	—	2	2	no	—
203	C₃×C₄	—	8	4	no	—
204	C₃×C₄	—	8	4	no	—
205	C₃×C₄	#534	1	1	yes	characteristic
206	C₃×C₄	#575	1	1	yes	characteristic
207	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
208	C₂³	C₂×C₃×D₈	1	2	yes	—
209	Q₈	—	4	1	no	—
210	Q₈	—	4	4	no	—
211	Q₈	—	4	4	no	—
212	Q₈	—	8	4	no	—
213	D₄	—	4	2	no	—
214	D₄	—	4	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	8	4	no	—
222	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
223	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
224	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
225	C₈	—	2	2	no	—
226	C₈	—	2	1	no	—
227	C₈	C₃×H(5,𝔽₂)	1	1	yes	characteristic
228	C₈	C₃×H(5,𝔽₂)	1	1	yes	characteristic
229	C₂×C₃	—	4	2	no	—
230	C₂×C₃	—	4	1	no	—
231	C₂×C₃	#3186	1	1	yes	characteristic
232	C₂×C₃	#3417	1	2	yes	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	4	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	1	no	—
237	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
238	C₄	—	2	2	no	—
239	C₄	—	2	1	no	—
240	C₄	—	8	4	no	—
241	C₄	—	8	4	no	—
242	C₄	C₃×#534	1	1	yes	characteristic
243	C₄	C₃×#575	1	1	yes	characteristic
244	C₃	#33768	1	1	yes	characteristic, direct product factor, Sylow
245	C₂	—	4	2	no	—
246	C₂	—	4	1	no	—
247	C₂	C₃×#3186	1	1	yes	characteristic
248	C₂	C₃×#3417	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
prime radical	C₂×C₃×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×#41 ◁ C₂×C₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#2237	C₂	semidirect
3. C₃×#2219	C₂	semidirect
4. C₃×#2216	C₂	semidirect
5. C₃×#2188	C₂	semidirect
7. #33768	C₃	direct (N × H)
27. #2237	C₂×C₃	semidirect
28. #2219	C₂×C₃	semidirect
29. #2216	C₂×C₃	semidirect
30. #2188	C₂×C₃	semidirect
244. C₃	#33768	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#3059	2
C₃×#2237	2
C₃×#2219	2
C₃×#2216	2
C₃×#2188	2
C₃×#2113	2
#33768	3

Subgroups — C3×#33768 #10652375

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33768 #10652375