Subgroups — C₃×#3232 #85096

Order 384.

334 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#529	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#499	C₂	1	2	yes	maximal, complemented
4	C₃×#488	C₂	1	2	yes	maximal, complemented
5	C₃×#484	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#483	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#481	C₂	1	1	yes	maximal, characteristic
8	C₃×#480	C₂	1	1	yes	maximal, characteristic
9	C₃×#479	C₂	1	2	yes	maximal, complemented
10	C₃×#478	C₂	1	2	yes	maximal, complemented
11	C₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
12	#3232	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₃×#126	—	2	1	no	—
14	C₃×#125	—	2	1	no	—
15	C₃×#125	—	2	1	no	—
16	C₃×#125	C₂²	1	2	yes	—
17	C₃×#125	C₂²	1	2	yes	—
18	C₃×#125	C₂²	1	1	yes	characteristic
19	C₃×#124	C₂²	1	1	yes	characteristic
20	C₃×#123	—	2	1	no	—
21	C₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
22	C₃×#121	C₂²	1	1	yes	characteristic
23	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
24	C₃×C₄×D₄	—	2	1	no	—
25	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
26	C₃×#117	—	2	1	no	—
27	C₂×C₃×C₄⋊C₄	—	2	1	no	—
28	C₂×C₃×C₄⋊C₄	—	2	1	no	—
29	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
31	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
32	C₃×#107	C₂²	1	2	yes	—
33	C₃×#107	C₂²	1	2	yes	—
34	C₃×#106	C₂²	1	2	yes	—
35	C₃×#106	C₂²	1	2	yes	—
36	C₃×C₄⋊C₈	C₂²	1	2	yes	—
37	C₃×#103	C₂²	1	2	yes	—
38	C₃×#103	C₂²	1	2	yes	—
39	C₃×#102	C₂²	1	2	yes	—
40	C₃×#102	C₂²	1	2	yes	—
41	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
42	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
43	C₃×C₄×C₈	C₂²	1	2	yes	—
44	#529	C₂×C₃	1	1	yes	characteristic
45	C₂×#125	C₂×C₃	1	1	yes	characteristic, complemented
46	#499	C₂×C₃	1	2	yes	complemented
47	#488	C₂×C₃	1	2	yes	complemented
48	#484	C₂×C₃	1	1	yes	characteristic, complemented
49	#483	C₂×C₃	1	1	yes	characteristic, complemented
50	#481	C₂×C₃	1	1	yes	characteristic
51	#480	C₂×C₃	1	1	yes	characteristic
52	#479	C₂×C₃	1	2	yes	complemented
53	#478	C₂×C₃	1	2	yes	complemented
54	#431	C₂×C₃	1	1	yes	characteristic, complemented
55	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
56	C₂×C₃×D₄	—	2	1	no	—
57	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
58	C₂²×C₃×C₄	—	2	1	no	—
59	C₂²×C₃×C₄	—	2	1	no	—
60	C₂²×C₃×C₄	—	2	1	no	—
61	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
62	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
63	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
64	C₂×C₃×C₈	C₂³	1	2	yes	—
65	C₂×C₃×C₈	C₂³	1	2	yes	—
66	C₃×C₄⋊C₄	—	2	2	no	—
67	C₃×C₄⋊C₄	—	2	2	no	—
68	C₃×C₄⋊C₄	—	2	2	no	—
69	C₃×C₄⋊C₄	—	2	2	no	—
70	C₃×C₄⋊C₄	—	2	1	no	—
71	C₃×C₄⋊C₄	—	2	1	no	—
72	C₃×C₄⋊C₄	—	2	1	no	—
73	C₃×C₄⋊C₄	—	2	1	no	—
74	C₃×C₄⋊C₄	—	2	1	no	—
75	C₃×C₄⋊C₄	—	2	1	no	—
76	C₃×C₄⋊C₄	C₂³	1	2	yes	—
77	C₃×C₄⋊C₄	C₂³	1	2	yes	—
78	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
79	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
80	C₃×C₂²⋊C₄	—	2	1	no	—
81	C₃×C₂²⋊C₄	—	2	1	no	—
82	C₃×C₂²⋊C₄	—	2	1	no	—
83	C₃×C₂²⋊C₄	—	2	1	no	—
84	C₃×C₂²⋊C₄	—	2	1	no	—
85	C₃×C₄²	—	2	1	no	—
86	C₃×C₄²	D₄	1	1	yes	characteristic
87	C₃×C₄²	D₄	1	1	yes	characteristic
88	C₃×C₄²	C₂³	1	1	yes	characteristic
89	C₃×C₄²	C₂³	1	1	yes	characteristic
90	#126	—	2	1	no	—
91	#125	—	2	1	no	—
92	#125	—	2	1	no	—
93	#125	C₂²×C₃	1	2	yes	—
94	#125	C₂²×C₃	1	2	yes	—
95	#125	C₂²×C₃	1	1	yes	characteristic
96	#124	C₂²×C₃	1	1	yes	characteristic
97	#123	—	2	1	no	—
98	C₂²⋊Q₈	C₂²×C₃	1	1	yes	characteristic
99	#121	C₂²×C₃	1	1	yes	characteristic
100	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
101	C₄×D₄	—	2	1	no	—
102	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
103	#117	—	2	1	no	—
104	C₂×C₄⋊C₄	—	2	1	no	—
105	C₂×C₄⋊C₄	—	2	1	no	—
106	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
107	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
108	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
109	#107	C₂²×C₃	1	2	yes	—
110	#107	C₂²×C₃	1	2	yes	—
111	#106	C₂²×C₃	1	2	yes	—
112	#106	C₂²×C₃	1	2	yes	—
113	C₄⋊C₈	C₂²×C₃	1	2	yes	—
114	#103	C₂²×C₃	1	2	yes	—
115	#103	C₂²×C₃	1	2	yes	—
116	#102	C₂²×C₃	1	2	yes	—
117	#102	C₂²×C₃	1	2	yes	—
118	C₂²⋊C₈	C₂²×C₃	1	1	yes	characteristic
119	C₂²⋊C₈	C₂²×C₃	1	1	yes	characteristic
120	C₄×C₈	C₂²×C₃	1	2	yes	—
121	C₂³×C₃	—	2	1	no	—
122	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
123	C₃×Q₈	—	2	2	no	—
124	C₃×D₄	—	2	2	no	—
125	C₃×D₄	—	4	2	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	2	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	4	1	no	—
134	C₂×C₃×C₄	—	4	2	no	—
135	C₂×C₃×C₄	—	4	2	no	—
136	C₂×C₃×C₄	—	4	2	no	—
137	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
143	C₃×C₈	—	2	2	no	—
144	C₃×C₈	—	2	2	no	—
145	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
146	C₂×D₄	—	2	1	no	—
147	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
152	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
153	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
154	C₂×C₈	C₂³×C₃	1	2	yes	—
155	C₂×C₈	C₂³×C₃	1	2	yes	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	1	no	—
161	C₄⋊C₄	—	2	1	no	—
162	C₄⋊C₄	—	2	1	no	—
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	C₂³×C₃	1	2	yes	—
167	C₄⋊C₄	C₂³×C₃	1	2	yes	—
168	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
169	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
170	C₂²⋊C₄	—	2	1	no	—
171	C₂²⋊C₄	—	2	1	no	—
172	C₂²⋊C₄	—	2	1	no	—
173	C₂²⋊C₄	—	2	1	no	—
174	C₂²⋊C₄	—	2	1	no	—
175	C₄²	—	2	1	no	—
176	C₄²	C₃×D₄	1	1	yes	characteristic
177	C₄²	C₃×D₄	1	1	yes	characteristic
178	C₄²	C₂³×C₃	1	1	yes	characteristic
179	C₄²	C₂³×C₃	1	1	yes	characteristic
180	C₂²×C₃	—	2	1	no	—
181	C₂²×C₃	—	2	1	no	—
182	C₂²×C₃	—	4	1	no	—
183	C₂²×C₃	—	4	1	no	—
184	C₂²×C₃	—	4	1	no	—
185	C₂²×C₃	#135	1	2	yes	—
186	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
187	C₃×C₄	—	2	1	no	—
188	C₃×C₄	—	2	1	no	—
189	C₃×C₄	—	2	1	no	—
190	C₃×C₄	—	2	1	no	—
191	C₃×C₄	—	2	1	no	—
192	C₃×C₄	—	4	1	no	—
193	C₃×C₄	—	4	2	no	—
194	C₃×C₄	—	4	1	no	—
195	C₃×C₄	—	4	1	no	—
196	C₃×C₄	—	4	2	no	—
197	C₃×C₄	2₋¹⁺⁴	1	2	yes	—
198	C₂³	—	2	1	no	—
199	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
200	Q₈	—	2	2	no	—
201	D₄	—	2	2	no	—
202	D₄	—	4	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
215	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
216	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
217	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
218	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
219	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₂×C₃	—	2	2	no	—
223	C₂×C₃	—	8	1	no	—
224	C₂×C₃	#536	1	1	yes	characteristic
225	C₂×C₃	C₂×#135	1	1	yes	characteristic
226	C₂×C₃	#575	1	1	yes	characteristic
227	C₂²	—	2	1	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	4	1	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	4	1	no	—
232	C₂²	C₃×#135	1	2	yes	—
233	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	1	no	—
241	C₄	—	4	1	no	—
242	C₄	—	4	2	no	—
243	C₄	—	4	1	no	—
244	C₄	C₃×2₋¹⁺⁴	1	2	yes	—
245	C₃	#3232	1	1	yes	characteristic, direct product factor, Sylow
246	C₂	—	2	2	no	—
247	C₂	—	8	1	no	—
248	C₂	C₃×#536	1	1	yes	characteristic
249	C₂	C₂×C₃×#135	1	1	yes	characteristic
250	C₂	C₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₃×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#125	C₂	semidirect
3. C₃×#499	C₂	semidirect
4. C₃×#488	C₂	semidirect
5. C₃×#484	C₂	semidirect
6. C₃×#483	C₂	semidirect
9. C₃×#479	C₂	semidirect
10. C₃×#478	C₂	semidirect
11. C₃×#431	C₂	semidirect
12. #3232	C₃	direct (N × H)
45. C₂×#125	C₂×C₃	semidirect
46. #499	C₂×C₃	semidirect
47. #488	C₂×C₃	semidirect
48. #484	C₂×C₃	semidirect
49. #483	C₂×C₃	semidirect
52. #479	C₂×C₃	semidirect
53. #478	C₂×C₃	semidirect
54. #431	C₂×C₃	semidirect
245. C₃	#3232	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×#529	2
C₂×C₃×#125	2
C₃×#499	2
C₃×#488	2
C₃×#484	2
C₃×#483	2
C₃×#481	2
C₃×#480	2
C₃×#479	2
C₃×#478	2
C₃×#431	2
#3232	3

Subgroups — C3×#3232 #85096

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3232 #85096