Subgroups — C₃×#3373 #85237

Order 384.

384 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×#559	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#489	C₂	1	2	yes	maximal, complemented
4	C₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#476	C₂	1	2	yes	maximal, complemented
6	C₃×#475	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#463	C₂	1	2	yes	maximal, complemented
8	C₃×#459	C₂	1	1	yes	maximal, characteristic
9	C₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#442	C₂	1	2	yes	maximal, complemented
11	C₃×C₄×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#3373	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₃×Q₈	—	2	1	no	—
14	C₂×C₃×SD₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₃×D₈	C₂²	1	2	yes	—
16	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
18	C₃×#127	C₂²	1	1	yes	characteristic
19	C₃×#124	—	2	1	no	—
20	C₃×#124	C₂²	1	2	yes	—
21	C₃×#123	—	2	2	no	—
22	C₃×C₂²⋊Q₈	—	2	1	no	—
23	C₃×C₂²⋊Q₈	—	2	2	no	—
24	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
25	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
26	C₃×C₄×D₄	—	2	1	no	—
27	C₃×C₄×D₄	C₂²	1	2	yes	—
28	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
29	C₃×C₄×D₄	C₂²	1	2	yes	—
30	C₃×#106	C₂²	1	1	yes	characteristic
31	C₃×#106	C₂²	1	2	yes	—
32	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₃×C₄⋊C₈	C₂²	1	2	yes	—
34	C₃×#103	C₂²	1	1	yes	characteristic
35	C₃×#103	C₂²	1	2	yes	—
36	C₃×#102	C₂²	1	1	yes	characteristic
37	C₃×#102	C₂²	1	2	yes	—
38	C₃×#102	C₂²	1	2	yes	—
39	C₃×#102	C₂²	1	2	yes	—
40	C₃×#97	C₂²	1	2	yes	—
41	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
42	#559	C₂×C₃	1	1	yes	characteristic, complemented
43	D₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
44	#489	C₂×C₃	1	2	yes	complemented
45	#486	C₂×C₃	1	1	yes	characteristic, complemented
46	#476	C₂×C₃	1	2	yes	complemented
47	#475	C₂×C₃	1	1	yes	characteristic, complemented
48	#463	C₂×C₃	1	2	yes	complemented
49	#459	C₂×C₃	1	1	yes	characteristic
50	#446	C₂×C₃	1	1	yes	characteristic, complemented
51	#442	C₂×C₃	1	2	yes	complemented
52	C₄×SD₁₆	C₂×C₃	1	1	yes	characteristic, complemented
53	C₂×C₃×Q₈	—	2	2	no	—
54	C₂×C₃×Q₈	—	2	1	no	—
55	C₂×C₃×Q₈	—	4	4	no	—
56	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
57	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
58	C₂×C₃×D₄	—	2	1	no	—
59	C₂×C₃×D₄	—	2	1	no	—
60	C₂×C₃×D₄	C₂³	1	2	yes	—
61	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
62	C₂²×C₃×C₄	—	2	2	no	—
63	C₂²×C₃×C₄	—	2	1	no	—
64	C₂²×C₃×C₄	—	2	2	no	—
65	C₃×SD₁₆	—	2	2	no	—
66	C₃×D₈	—	2	4	no	—
67	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
68	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
69	C₂×C₃×C₈	C₂³	1	2	yes	—
70	C₃×C₄⋊C₄	—	2	1	no	—
71	C₃×C₄⋊C₄	—	2	2	no	—
72	C₃×C₄⋊C₄	—	2	1	no	—
73	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
74	C₃×C₄⋊C₄	D₄	1	2	yes	—
75	C₃×C₄⋊C₄	C₂³	1	2	yes	—
76	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
77	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
78	C₃×C₂²⋊C₄	—	2	2	no	—
79	C₃×C₂²⋊C₄	—	2	2	no	—
80	C₃×C₂²⋊C₄	—	2	1	no	—
81	C₃×C₂²⋊C₄	—	2	2	no	—
82	C₃×C₂²⋊C₄	—	2	2	no	—
83	C₃×C₄²	—	2	1	no	—
84	C₃×C₄²	C₂³	1	1	yes	characteristic
85	C₃×C₄²	C₂³	1	2	yes	—
86	C₂²×Q₈	—	2	1	no	—
87	C₂×SD₁₆	C₂²×C₃	1	1	yes	characteristic
88	C₂×D₈	C₂²×C₃	1	2	yes	—
89	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
90	C₄⋊Q₈	C₂²×C₃	1	2	yes	—
91	#127	C₂²×C₃	1	1	yes	characteristic
92	#124	—	2	1	no	—
93	#124	C₂²×C₃	1	2	yes	—
94	#123	—	2	2	no	—
95	C₂²⋊Q₈	—	2	1	no	—
96	C₂²⋊Q₈	—	2	2	no	—
97	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
98	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
99	C₄×D₄	—	2	1	no	—
100	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
101	C₄×D₄	C₂²×C₃	1	2	yes	—
102	C₄×D₄	C₂²×C₃	1	2	yes	—
103	#106	C₂²×C₃	1	1	yes	characteristic
104	#106	C₂²×C₃	1	2	yes	—
105	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
106	C₄⋊C₈	C₂²×C₃	1	2	yes	—
107	#103	C₂²×C₃	1	1	yes	characteristic
108	#103	C₂²×C₃	1	2	yes	—
109	#102	C₂²×C₃	1	1	yes	characteristic
110	#102	C₂²×C₃	1	2	yes	—
111	#102	C₂²×C₃	1	2	yes	—
112	#102	C₂²×C₃	1	2	yes	—
113	#97	C₂²×C₃	1	2	yes	—
114	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
115	C₂³×C₃	—	2	1	no	—
116	C₂³×C₃	—	2	2	no	—
117	C₃×Q₈	—	2	2	no	—
118	C₃×Q₈	—	2	1	no	—
119	C₃×Q₈	—	2	1	no	—
120	C₃×Q₈	—	4	4	no	—
121	C₃×Q₈	—	4	2	no	—
122	C₃×D₄	—	2	2	no	—
123	C₃×D₄	—	2	1	no	—
124	C₃×D₄	—	2	2	no	—
125	C₃×D₄	—	4	2	no	—
126	C₃×D₄	—	8	1	no	—
127	C₃×D₄	#41	1	2	yes	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	4	4	no	—
133	C₂×C₃×C₄	—	4	2	no	—
134	C₂×C₃×C₄	—	8	2	no	—
135	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
136	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
137	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
140	C₃×C₈	—	2	2	no	—
141	C₃×C₈	—	2	1	no	—
142	C₃×C₈	—	2	1	no	—
143	C₂×Q₈	—	2	2	no	—
144	C₂×Q₈	—	2	1	no	—
145	C₂×Q₈	—	4	4	no	—
146	C₂×Q₈	C₃×D₄	1	1	yes	characteristic
147	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
148	C₂×D₄	—	2	1	no	—
149	C₂×D₄	—	2	1	no	—
150	C₂×D₄	C₂³×C₃	1	2	yes	—
151	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	—	2	1	no	—
154	C₂²×C₄	—	2	2	no	—
155	SD₁₆	—	2	2	no	—
156	D₈	—	2	4	no	—
157	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
158	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
159	C₂×C₈	C₂³×C₃	1	2	yes	—
160	C₄⋊C₄	—	2	1	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	2	1	no	—
163	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
164	C₄⋊C₄	C₃×D₄	1	2	yes	—
165	C₄⋊C₄	C₂³×C₃	1	2	yes	—
166	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
167	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
168	C₂²⋊C₄	—	2	2	no	—
169	C₂²⋊C₄	—	2	2	no	—
170	C₂²⋊C₄	—	2	1	no	—
171	C₂²⋊C₄	—	2	2	no	—
172	C₂²⋊C₄	—	2	2	no	—
173	C₄²	—	2	1	no	—
174	C₄²	C₂³×C₃	1	1	yes	characteristic
175	C₄²	C₂³×C₃	1	2	yes	—
176	C₂²×C₃	—	2	2	no	—
177	C₂²×C₃	—	4	1	no	—
178	C₂²×C₃	—	4	1	no	—
179	C₂²×C₃	—	4	2	no	—
180	C₂²×C₃	—	4	2	no	—
181	C₂²×C₃	—	4	2	no	—
182	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
183	C₃×C₄	—	2	2	no	—
184	C₃×C₄	—	2	2	no	—
185	C₃×C₄	—	2	1	no	—
186	C₃×C₄	—	4	1	no	—
187	C₃×C₄	—	4	2	no	—
188	C₃×C₄	—	4	1	no	—
189	C₃×C₄	—	4	1	no	—
190	C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
191	C₃×C₄	C₂×#41	1	1	yes	characteristic
192	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
193	C₂³	—	2	1	no	—
194	C₂³	—	2	2	no	—
195	Q₈	—	2	2	no	—
196	Q₈	—	2	1	no	—
197	Q₈	—	2	1	no	—
198	Q₈	—	4	4	no	—
199	Q₈	—	4	2	no	—
200	D₄	—	2	2	no	—
201	D₄	—	2	1	no	—
202	D₄	—	2	2	no	—
203	D₄	—	4	2	no	—
204	D₄	—	8	1	no	—
205	D₄	C₃×#41	1	2	yes	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	8	2	no	—
213	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
214	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
215	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
216	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
217	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₂×C₃	—	4	2	no	—
222	C₂×C₃	—	8	2	no	—
223	C₂×C₃	#548	1	1	yes	characteristic
224	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
225	C₂×C₃	#577	1	1	yes	characteristic
226	C₂²	—	2	2	no	—
227	C₂²	—	4	1	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	2	no	—
230	C₂²	—	4	2	no	—
231	C₂²	—	4	2	no	—
232	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	1	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	1	no	—
238	C₄	—	4	1	no	—
239	C₄	—	4	1	no	—
240	C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
241	C₄	C₂×C₃×#41	1	1	yes	characteristic
242	C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
243	C₃	#3373	1	1	yes	characteristic, direct product factor, Sylow
244	C₂	—	4	2	no	—
245	C₂	—	8	2	no	—
246	C₂	C₃×#548	1	1	yes	characteristic
247	C₂	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
248	C₂	C₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₃×#459	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#559	C₂	semidirect
2. C₃×D₄×Q₈	C₂	semidirect
3. C₃×#489	C₂	semidirect
4. C₃×#486	C₂	semidirect
5. C₃×#476	C₂	semidirect
6. C₃×#475	C₂	semidirect
7. C₃×#463	C₂	semidirect
9. C₃×#446	C₂	semidirect
10. C₃×#442	C₂	semidirect
11. C₃×C₄×SD₁₆	C₂	semidirect
12. #3373	C₃	direct (N × H)
42. #559	C₂×C₃	semidirect
43. D₄×Q₈	C₂×C₃	semidirect
44. #489	C₂×C₃	semidirect
45. #486	C₂×C₃	semidirect
46. #476	C₂×C₃	semidirect
47. #475	C₂×C₃	semidirect
48. #463	C₂×C₃	semidirect
50. #446	C₂×C₃	semidirect
51. #442	C₂×C₃	semidirect
52. C₄×SD₁₆	C₂×C₃	semidirect
243. C₃	#3373	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×#559	2
C₃×D₄×Q₈	2
C₃×#489	2
C₃×#486	2
C₃×#476	2
C₃×#475	2
C₃×#463	2
C₃×#459	2
C₃×#446	2
C₃×#442	2
C₃×C₄×SD₁₆	2
#3373	3

Subgroups — C3×#3373 #85237

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3373 #85237