Subgroups — C₃×#33733 #10652340

Order 768.

602 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3169	C₂	1	2	yes	maximal, complemented
2	C₃×#2252	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#2246	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#2240	C₂	1	2	yes	maximal, complemented
5	C₃×#2215	C₂	1	4	yes	maximal, complemented
6	C₃×C₄×D₁₆	C₂	1	4	yes	maximal, complemented
7	C₂×C₃×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#33733	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₃×#532	—	2	2	no	—
10	C₂×C₃×D₁₆	C₂²	1	4	yes	—
11	C₂²×C₃×C₁₆	C₂²	1	2	yes	—
12	C₃×#500	C₂²	1	2	yes	—
13	C₃×#493	C₂²	1	2	yes	—
14	C₃×#486	—	2	2	no	—
15	C₃×#466	—	2	2	no	—
16	C₃×#466	C₂²	1	4	yes	—
17	C₃×C₄×D₈	—	2	2	no	—
18	C₃×C₄×D₈	C₂²	1	4	yes	—
19	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
20	C₃×#366	C₂²	1	4	yes	—
21	C₃×#357	C₂²	1	8	yes	—
22	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
23	C₃×C₄×C₁₆	C₂²	1	2	yes	—
24	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
25	#3169	C₂×C₃	1	2	yes	complemented
26	#2252	C₂×C₃	1	1	yes	characteristic, complemented
27	#2246	C₂×C₃	1	1	yes	characteristic, complemented
28	#2240	C₂×C₃	1	2	yes	complemented
29	#2215	C₂×C₃	1	4	yes	complemented
30	C₄×D₁₆	C₂×C₃	1	4	yes	complemented
31	C₂×C₄×C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂×C₃×#41	—	4	2	no	—
33	C₂×C₃×D₈	—	2	2	no	—
34	C₂×C₃×D₈	C₂³	1	4	yes	—
35	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
36	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
37	C₃×C₄⋊Q₈	—	2	2	no	—
38	C₃×#127	—	2	2	no	—
39	C₃×#124	—	4	2	no	—
40	C₃×#121	—	2	4	no	—
41	C₃×#121	—	4	2	no	—
42	C₃×C₄×D₄	—	2	4	no	—
43	C₃×C₄×D₄	—	4	2	no	—
44	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
45	C₃×D₁₆	—	2	8	no	—
46	C₂×C₃×C₁₆	—	2	4	no	—
47	C₂×C₃×C₁₆	C₂³	1	2	yes	—
48	C₂×C₃×C₁₆	C₂³	1	2	yes	—
49	C₃×#107	—	2	2	no	—
50	C₃×#107	C₂³	1	4	yes	—
51	C₃×#102	—	2	4	no	—
52	C₃×#102	—	2	4	no	—
53	C₃×C₄×C₈	D₄	1	1	yes	characteristic
54	C₃×C₄×C₈	D₄	1	1	yes	characteristic
55	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
56	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
57	#532	—	2	2	no	—
58	C₂×D₁₆	C₂²×C₃	1	4	yes	—
59	C₂²×C₁₆	C₂²×C₃	1	2	yes	—
60	#500	C₂²×C₃	1	2	yes	—
61	#493	C₂²×C₃	1	2	yes	—
62	#486	—	2	2	no	—
63	#466	—	2	2	no	—
64	#466	C₂²×C₃	1	4	yes	—
65	C₄×D₈	—	2	2	no	—
66	C₄×D₈	C₂²×C₃	1	4	yes	—
67	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
68	#366	C₂²×C₃	1	4	yes	—
69	#357	C₂²×C₃	1	8	yes	—
70	C₄×C₁₆	C₂²×C₃	1	2	yes	—
71	C₄×C₁₆	C₂²×C₃	1	1	yes	characteristic
72	C₄×C₁₆	C₂²×C₃	1	1	yes	characteristic
73	C₃×#41	—	8	8	no	—
74	C₂×C₃×Q₈	—	4	2	no	—
75	C₂×C₃×D₄	—	2	4	no	—
76	C₂×C₃×D₄	—	4	4	no	—
77	C₂×C₃×D₄	—	4	2	no	—
78	C₂×C₃×D₄	—	4	2	no	—
79	C₂²×C₃×C₄	—	4	4	no	—
80	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
81	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
82	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
83	C₃×D₈	—	2	8	no	—
84	C₃×D₈	—	4	4	no	—
85	C₂×C₃×C₈	—	2	2	no	—
86	C₂×C₃×C₈	—	2	2	no	—
87	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
91	C₃×C₄⋊C₄	—	2	4	no	—
92	C₃×C₄⋊C₄	—	4	2	no	—
93	C₃×C₂²⋊C₄	—	4	4	no	—
94	C₃×C₂²⋊C₄	—	4	4	no	—
95	C₃×C₄²	D₈	1	1	yes	characteristic
96	C₃×C₄²	D₈	1	1	yes	characteristic
97	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
98	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
99	C₃×C₁₆	—	2	2	no	—
100	C₃×C₁₆	#41	1	4	yes	complemented
101	C₂×#41	—	4	2	no	—
102	C₂×D₈	—	2	2	no	—
103	C₂×D₈	C₂³×C₃	1	4	yes	—
104	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
105	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
106	C₄⋊Q₈	—	2	2	no	—
107	#127	—	2	2	no	—
108	#124	—	4	2	no	—
109	#121	—	2	4	no	—
110	#121	—	4	2	no	—
111	C₄×D₄	—	2	4	no	—
112	C₄×D₄	—	4	2	no	—
113	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
114	D₁₆	—	2	8	no	—
115	C₂×C₁₆	—	2	4	no	—
116	C₂×C₁₆	C₂³×C₃	1	2	yes	—
117	C₂×C₁₆	C₂³×C₃	1	2	yes	—
118	#107	—	2	2	no	—
119	#107	C₂³×C₃	1	4	yes	—
120	#102	—	2	4	no	—
121	#102	—	2	4	no	—
122	C₄×C₈	C₃×D₄	1	1	yes	characteristic
123	C₄×C₈	C₃×D₄	1	1	yes	characteristic
124	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
125	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
126	C₂³×C₃	—	4	4	no	—
127	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
128	C₃×Q₈	—	8	4	no	—
129	C₃×D₄	—	4	4	no	—
130	C₃×D₄	—	4	4	no	—
131	C₃×D₄	—	8	8	no	—
132	C₃×D₄	—	8	4	no	—
133	C₃×D₄	—	8	4	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	4	4	no	—
138	C₂×C₃×C₄	—	8	8	no	—
139	C₂×C₃×C₄	D₁₆	1	4	yes	complemented
140	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
144	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
146	C₃×C₈	—	2	1	no	—
147	C₃×C₈	—	2	1	no	—
148	C₃×C₈	#135	1	2	yes	—
149	C₃×C₈	C₂×#41	1	2	yes	—
150	#41	—	8	8	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×D₄	—	2	4	no	—
153	C₂×D₄	—	4	4	no	—
154	C₂×D₄	—	4	2	no	—
155	C₂×D₄	—	4	2	no	—
156	C₂²×C₄	—	4	4	no	—
157	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
158	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
160	D₈	—	2	8	no	—
161	D₈	—	4	4	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
165	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
166	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
167	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₂²⋊C₄	—	4	4	no	—
171	C₂²⋊C₄	—	4	4	no	—
172	C₄²	C₃×D₈	1	1	yes	characteristic
173	C₄²	C₃×D₈	1	1	yes	characteristic
174	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
175	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
176	C₁₆	—	2	2	no	—
177	C₁₆	C₃×#41	1	4	yes	complemented
178	C₂²×C₃	—	2	1	no	—
179	C₂²×C₃	—	2	1	no	—
180	C₂²×C₃	—	8	4	no	—
181	C₂²×C₃	—	8	4	no	—
182	C₂²×C₃	—	8	4	no	—
183	C₂²×C₃	C₂×D₁₆	1	2	yes	—
184	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
185	C₃×C₄	—	2	1	no	—
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	—	8	4	no	—
188	C₃×C₄	C₂×D₁₆	1	2	yes	—
189	C₃×C₄	C₂×D₁₆	1	2	yes	—
190	C₃×C₄	#508	1	2	yes	—
191	C₃×C₄	#532	1	1	yes	characteristic
192	C₃×C₄	C₂×#135	1	1	yes	characteristic
193	C₂³	—	4	4	no	—
194	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
195	Q₈	—	8	4	no	—
196	D₄	—	4	4	no	—
197	D₄	—	4	4	no	—
198	D₄	—	8	8	no	—
199	D₄	—	8	4	no	—
200	D₄	—	8	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	—	8	8	no	—
206	C₂×C₄	C₃×D₁₆	1	4	yes	complemented
207	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
208	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
209	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
210	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
211	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
212	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
213	C₈	—	2	1	no	—
214	C₈	—	2	1	no	—
215	C₈	C₃×#135	1	2	yes	—
216	C₈	C₂×C₃×#41	1	2	yes	—
217	C₂×C₃	—	2	2	no	—
218	C₂×C₃	—	16	4	no	—
219	C₂×C₃	#3169	1	1	yes	characteristic
220	C₂×C₃	C₂²×D₁₆	1	1	yes	characteristic
221	C₂×C₃	C₂×#508	1	1	yes	characteristic
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	8	4	no	—
225	C₂²	—	8	4	no	—
226	C₂²	—	8	4	no	—
227	C₂²	C₂×C₃×D₁₆	1	2	yes	—
228	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	8	4	no	—
232	C₄	C₂×C₃×D₁₆	1	2	yes	—
233	C₄	C₂×C₃×D₁₆	1	2	yes	—
234	C₄	C₃×#508	1	2	yes	—
235	C₄	C₃×#532	1	1	yes	characteristic
236	C₄	C₂×C₃×#135	1	1	yes	characteristic
237	C₃	#33733	1	1	yes	characteristic, direct product factor, Sylow
238	C₂	—	2	2	no	—
239	C₂	—	16	4	no	—
240	C₂	C₃×#3169	1	1	yes	characteristic
241	C₂	C₂²×C₃×D₁₆	1	1	yes	characteristic
242	C₂	C₂×C₃×#508	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
socle	C₂²×C₃	C₂²×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#3169	C₂	semidirect
2. C₃×#2252	C₂	semidirect
3. C₃×#2246	C₂	semidirect
4. C₃×#2240	C₂	semidirect
5. C₃×#2215	C₂	semidirect
6. C₃×C₄×D₁₆	C₂	semidirect
7. C₂×C₃×C₄×C₁₆	C₂	semidirect
8. #33733	C₃	direct (N × H)
25. #3169	C₂×C₃	semidirect
26. #2252	C₂×C₃	semidirect
27. #2246	C₂×C₃	semidirect
28. #2240	C₂×C₃	semidirect
29. #2215	C₂×C₃	semidirect
30. C₄×D₁₆	C₂×C₃	semidirect
31. C₂×C₄×C₁₆	C₂×C₃	semidirect
100. C₃×C₁₆	#41	semidirect
139. C₂×C₃×C₄	D₁₆	semidirect
177. C₁₆	C₃×#41	semidirect
206. C₂×C₄	C₃×D₁₆	semidirect
237. C₃	#33733	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×#3169	2
C₃×#2252	2
C₃×#2246	2
C₃×#2240	2
C₃×#2215	2
C₃×C₄×D₁₆	2
C₂×C₃×C₄×C₁₆	2
#33733	3

Subgroups — C3×#33733 #10652340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33733 #10652340