Subgroups — C₃×#33718 #10652325

Order 768.

814 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#510	C₂	1	4	yes	maximal, complemented
2	C₃×#3147	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#2251	C₂	1	4	yes	maximal, complemented
5	C₃×#2248	C₂	1	2	yes	maximal, complemented
6	C₃×#2247	C₂	1	2	yes	maximal, complemented
7	C₃×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	#33718	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂×C₃×#135	—	2	1	no	—
10	C₂×C₃×#135	C₂²	1	2	yes	—
11	C₂²×C₃×Q₁₆	—	2	1	no	—
12	C₂²×C₃×Q₁₆	C₂²	1	2	yes	—
13	C₃×#532	—	2	1	no	—
14	C₂×C₃×C₄⋊Q₈	—	2	1	no	—
15	C₃×#510	—	2	16	no	—
16	C₂×C₃×Q₃₂	C₂²	1	8	yes	—
17	C₂×C₃×SD₃₂	C₂²	1	8	yes	—
18	C₂×C₃×M₃₂	C₂²	1	2	yes	—
19	C₃×#495	—	2	1	no	—
20	C₃×#495	C₂²	1	2	yes	—
21	C₃×#494	—	2	1	no	—
22	C₃×#494	—	2	1	no	—
23	C₃×#494	C₂²	1	1	yes	characteristic
24	C₃×#494	C₂²	1	2	yes	—
25	C₃×#494	C₂²	1	2	yes	—
26	C₃×#493	C₂²	1	1	yes	characteristic
27	C₃×#492	—	2	1	no	—
28	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
29	C₃×#346	C₂²	1	2	yes	—
30	C₃×C₄×C₁₆	C₂²	1	2	yes	—
31	C₂×#510	C₂×C₃	1	4	yes	complemented
32	#3147	C₂×C₃	1	1	yes	characteristic
33	C₂×#494	C₂×C₃	1	1	yes	characteristic, complemented
34	#2251	C₂×C₃	1	4	yes	complemented
35	#2248	C₂×C₃	1	2	yes	complemented
36	#2247	C₂×C₃	1	2	yes	complemented
37	C₄×M₃₂	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂×C₃×#41	—	2	2	no	—
39	C₂²×C₃×Q₈	—	2	2	no	—
40	C₃×#135	—	2	8	no	—
41	C₃×#135	—	4	4	no	—
42	C₂×C₃×Q₁₆	—	2	8	no	—
43	C₂×C₃×Q₁₆	—	2	2	no	—
44	C₂×C₃×Q₁₆	—	2	2	no	—
45	C₂×C₃×Q₁₆	—	2	2	no	—
46	C₂×C₃×Q₁₆	—	2	1	no	—
47	C₂×C₃×Q₁₆	—	4	4	no	—
48	C₂×C₃×Q₁₆	C₂³	1	4	yes	—
49	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
50	C₂×C₃×SD₁₆	—	2	2	no	—
51	C₂×C₃×SD₁₆	—	2	2	no	—
52	C₂×C₃×D₈	—	2	1	no	—
53	C₂×C₃×D₈	C₂³	1	2	yes	—
54	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
55	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
56	C₃×C₄⋊Q₈	—	2	2	no	—
57	C₃×C₄⋊Q₈	—	2	2	no	—
58	C₃×C₄⋊Q₈	—	2	1	no	—
59	C₃×C₄⋊Q₈	—	4	2	no	—
60	C₃×#127	—	2	1	no	—
61	C₃×#124	—	2	2	no	—
62	C₃×#121	—	4	2	no	—
63	C₃×C₄×D₄	—	4	2	no	—
64	C₂×C₃×C₄⋊C₄	—	4	2	no	—
65	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
66	C₃×Q₃₂	—	2	16	no	—
67	C₃×SD₃₂	—	2	16	no	—
68	C₃×M₃₂	D₄	1	8	yes	complemented
69	C₂×C₃×C₁₆	C₂³	1	4	yes	—
70	C₃×C₄×C₈	D₄	1	1	yes	characteristic
71	C₃×C₄×C₈	D₄	1	1	yes	characteristic
72	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
73	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
74	C₂×#135	—	2	1	no	—
75	C₂×#135	C₂²×C₃	1	2	yes	—
76	C₂²×Q₁₆	—	2	1	no	—
77	C₂²×Q₁₆	C₂²×C₃	1	2	yes	—
78	#532	—	2	1	no	—
79	C₂×C₄⋊Q₈	—	2	1	no	—
80	#510	—	2	16	no	—
81	C₂×Q₃₂	C₂²×C₃	1	8	yes	—
82	C₂×SD₃₂	C₂²×C₃	1	8	yes	—
83	C₂×M₃₂	C₂²×C₃	1	2	yes	—
84	#495	—	2	1	no	—
85	#495	C₂²×C₃	1	2	yes	—
86	#494	—	2	1	no	—
87	#494	—	2	1	no	—
88	#494	C₂²×C₃	1	2	yes	—
89	#494	C₂²×C₃	1	2	yes	—
90	#494	C₂²×C₃	1	1	yes	characteristic
91	#493	C₂²×C₃	1	1	yes	characteristic
92	#492	—	2	1	no	—
93	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
94	#346	C₂²×C₃	1	2	yes	—
95	C₄×C₁₆	C₂²×C₃	1	2	yes	—
96	C₃×#41	—	4	4	no	—
97	C₃×#41	—	4	4	no	—
98	C₂×C₃×Q₈	—	2	2	no	—
99	C₂×C₃×Q₈	—	2	4	no	—
100	C₂×C₃×Q₈	—	4	2	no	—
101	C₂×C₃×Q₈	—	4	4	no	—
102	C₂×C₃×Q₈	—	4	4	no	—
103	C₂×C₃×D₄	—	2	2	no	—
104	C₂×C₃×D₄	—	4	2	no	—
105	C₂×C₃×D₄	—	4	2	no	—
106	C₂²×C₃×C₄	—	4	2	no	—
107	C₂²×C₃×C₄	—	4	2	no	—
108	C₂²×C₃×C₄	D₈	1	2	yes	—
109	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
110	C₃×Q₁₆	—	2	8	no	—
111	C₃×Q₁₆	—	2	4	no	—
112	C₃×Q₁₆	—	4	4	no	—
113	C₃×Q₁₆	—	4	4	no	—
114	C₃×Q₁₆	—	4	4	no	—
115	C₃×Q₁₆	—	4	2	no	—
116	C₃×SD₁₆	—	4	4	no	—
117	C₃×SD₁₆	—	4	4	no	—
118	C₃×D₈	—	2	4	no	—
119	C₃×D₈	—	4	2	no	—
120	C₂×C₃×C₈	D₈	1	4	yes	—
121	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
122	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
123	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
124	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
125	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
126	C₃×C₄⋊C₄	—	4	4	no	—
127	C₃×C₄⋊C₄	—	4	4	no	—
128	C₃×C₄⋊C₄	—	4	2	no	—
129	C₃×C₂²⋊C₄	—	4	4	no	—
130	C₃×C₄²	D₈	1	2	yes	—
131	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
132	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
133	C₃×C₁₆	C₂×D₄	1	8	yes	—
134	C₂×#41	—	2	2	no	—
135	C₂²×Q₈	—	2	2	no	—
136	#135	—	2	8	no	—
137	#135	—	4	4	no	—
138	C₂×Q₁₆	—	2	8	no	—
139	C₂×Q₁₆	—	2	2	no	—
140	C₂×Q₁₆	—	2	2	no	—
141	C₂×Q₁₆	—	2	2	no	—
142	C₂×Q₁₆	—	2	1	no	—
143	C₂×Q₁₆	—	4	4	no	—
144	C₂×Q₁₆	C₂³×C₃	1	4	yes	—
145	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
146	C₂×SD₁₆	—	2	2	no	—
147	C₂×SD₁₆	—	2	2	no	—
148	C₂×D₈	—	2	1	no	—
149	C₂×D₈	C₂³×C₃	1	2	yes	—
150	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
151	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
152	C₄⋊Q₈	—	2	2	no	—
153	C₄⋊Q₈	—	2	2	no	—
154	C₄⋊Q₈	—	2	1	no	—
155	C₄⋊Q₈	—	4	2	no	—
156	#127	—	2	1	no	—
157	#124	—	2	2	no	—
158	#121	—	4	2	no	—
159	C₄×D₄	—	4	2	no	—
160	C₂×C₄⋊C₄	—	4	2	no	—
161	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
162	Q₃₂	—	2	16	no	—
163	SD₃₂	—	2	16	no	—
164	M₃₂	C₃×D₄	1	8	yes	complemented
165	C₂×C₁₆	C₂³×C₃	1	4	yes	—
166	C₄×C₈	C₃×D₄	1	1	yes	characteristic
167	C₄×C₈	C₃×D₄	1	1	yes	characteristic
168	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
169	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
170	C₂³×C₃	—	4	2	no	—
171	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
172	C₃×Q₈	—	4	2	no	—
173	C₃×Q₈	—	4	4	no	—
174	C₃×Q₈	—	4	2	no	—
175	C₃×Q₈	—	4	4	no	—
176	C₃×Q₈	—	8	4	no	—
177	C₃×D₄	—	4	2	no	—
178	C₃×D₄	—	4	2	no	—
179	C₃×D₄	—	8	4	no	—
180	C₃×D₄	—	8	4	no	—
181	C₂×C₃×C₄	—	2	4	no	—
182	C₂×C₃×C₄	—	4	2	no	—
183	C₂×C₃×C₄	—	4	4	no	—
184	C₂×C₃×C₄	—	8	4	no	—
185	C₂×C₃×C₄	—	8	4	no	—
186	C₂×C₃×C₄	#127	1	2	yes	—
187	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
188	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
189	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
191	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
192	C₃×C₈	#127	1	2	yes	—
193	C₃×C₈	C₂×D₈	1	2	yes	—
194	C₃×C₈	C₂×D₈	1	2	yes	—
195	C₃×C₈	C₂²×D₄	1	2	yes	—
196	#41	—	4	4	no	—
197	#41	—	4	4	no	—
198	C₂×Q₈	—	2	2	no	—
199	C₂×Q₈	—	2	4	no	—
200	C₂×Q₈	—	4	2	no	—
201	C₂×Q₈	—	4	4	no	—
202	C₂×Q₈	—	4	4	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂×D₄	—	4	2	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	C₃×D₈	1	2	yes	—
209	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
210	Q₁₆	—	2	8	no	—
211	Q₁₆	—	2	4	no	—
212	Q₁₆	—	4	4	no	—
213	Q₁₆	—	4	4	no	—
214	Q₁₆	—	4	4	no	—
215	Q₁₆	—	4	2	no	—
216	SD₁₆	—	4	4	no	—
217	SD₁₆	—	4	4	no	—
218	D₈	—	2	4	no	—
219	D₈	—	4	2	no	—
220	C₂×C₈	C₃×D₈	1	4	yes	—
221	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
222	C₂×C₈	C₂×C₃×D₄	1	4	yes	—
223	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
224	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
225	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
226	C₄⋊C₄	—	4	4	no	—
227	C₄⋊C₄	—	4	4	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₂²⋊C₄	—	4	4	no	—
230	C₄²	C₃×D₈	1	2	yes	—
231	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
232	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
233	C₁₆	C₂×C₃×D₄	1	8	yes	—
234	C₂²×C₃	—	2	2	no	—
235	C₂²×C₃	—	8	2	no	—
236	C₂²×C₃	—	8	4	no	—
237	C₂²×C₃	#493	1	2	yes	—
238	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
239	C₃×C₄	—	2	2	no	—
240	C₃×C₄	—	8	2	no	—
241	C₃×C₄	—	8	4	no	—
242	C₃×C₄	#493	1	2	yes	—
243	C₃×C₄	#510	1	4	yes	complemented
244	C₃×C₄	C₂×#127	1	1	yes	characteristic
245	C₃×C₄	C₂²×D₈	1	1	yes	characteristic
246	C₂³	—	4	2	no	—
247	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
248	Q₈	—	4	2	no	—
249	Q₈	—	4	4	no	—
250	Q₈	—	4	2	no	—
251	Q₈	—	4	4	no	—
252	Q₈	—	8	4	no	—
253	D₄	—	4	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	8	4	no	—
256	D₄	—	8	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	8	4	no	—
261	C₂×C₄	—	8	4	no	—
262	C₂×C₄	C₃×#127	1	2	yes	—
263	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
264	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
265	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
266	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
267	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
268	C₈	C₃×#127	1	2	yes	—
269	C₈	C₂×C₃×D₈	1	2	yes	—
270	C₈	C₂×C₃×D₈	1	2	yes	—
271	C₈	C₂²×C₃×D₄	1	2	yes	—
272	C₂×C₃	—	2	2	no	—
273	C₂×C₃	—	16	2	no	—
274	C₂×C₃	C₂×#493	1	1	yes	characteristic
275	C₂×C₃	C₂×#510	1	2	yes	—
276	C₂²	—	2	2	no	—
277	C₂²	—	8	2	no	—
278	C₂²	—	8	4	no	—
279	C₂²	C₃×#493	1	2	yes	—
280	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	8	2	no	—
283	C₄	—	8	4	no	—
284	C₄	C₃×#493	1	2	yes	—
285	C₄	C₃×#510	1	4	yes	complemented
286	C₄	C₂×C₃×#127	1	1	yes	characteristic
287	C₄	C₂²×C₃×D₈	1	1	yes	characteristic
288	C₃	#33718	1	1	yes	characteristic, direct product factor, Sylow
289	C₂	—	2	2	no	—
290	C₂	—	16	2	no	—
291	C₂	C₂×C₃×#493	1	1	yes	characteristic
292	C₂	C₂×C₃×#510	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
prime radical	C₃×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ 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. C₂×C₃×#510	C₂	semidirect
3. C₂×C₃×#494	C₂	semidirect
4. C₃×#2251	C₂	semidirect
5. C₃×#2248	C₂	semidirect
6. C₃×#2247	C₂	semidirect
7. C₃×C₄×M₃₂	C₂	semidirect
8. #33718	C₃	direct (N × H)
31. C₂×#510	C₂×C₃	semidirect
33. C₂×#494	C₂×C₃	semidirect
34. #2251	C₂×C₃	semidirect
35. #2248	C₂×C₃	semidirect
36. #2247	C₂×C₃	semidirect
37. C₄×M₃₂	C₂×C₃	semidirect
68. C₃×M₃₂	D₄	semidirect
164. M₃₂	C₃×D₄	semidirect
243. C₃×C₄	#510	semidirect
285. C₄	C₃×#510	semidirect
288. C₃	#33718	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×#510	2
C₃×#3147	2
C₂×C₃×#494	2
C₃×#2251	2
C₃×#2248	2
C₃×#2247	2
C₃×C₄×M₃₂	2
#33718	3

Subgroups — C3×#33718 #10652325

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33718 #10652325