Subgroups — C₃×#3252 #85116

Order 384.

348 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#529	C₂	1	1	yes	maximal, characteristic
3	C₃×#479	C₂	1	2	yes	maximal, complemented
4	C₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#470	C₂	1	2	yes	maximal, complemented
7	C₃×#468	C₂	1	2	yes	maximal
8	C₃×#464	C₂	1	2	yes	maximal, complemented
9	C₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
12	#3252	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×Q₁₆	C₂²	1	2	yes	—
14	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
15	C₂×C₃×M₁₆	C₂²	1	2	yes	—
16	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₃×#126	—	2	1	no	—
19	C₃×#125	—	2	1	no	—
20	C₃×#125	C₂²	1	1	yes	characteristic
21	C₃×#124	C₂²	1	1	yes	characteristic
22	C₃×#123	—	2	1	no	—
23	C₃×C₂²⋊Q₈	—	2	1	no	—
24	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
25	C₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
26	C₃×#121	C₂²	1	1	yes	characteristic
27	C₃×C₄×Q₈	—	2	1	no	—
28	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
29	C₃×C₄×Q₈	C₂²	1	2	yes	—
30	C₃×C₄×D₄	—	2	1	no	—
31	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
32	C₃×#117	—	2	1	no	—
33	C₃×#117	—	2	1	no	—
34	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
35	C₃×#107	C₂²	1	2	yes	—
36	C₃×#106	C₂²	1	2	yes	—
37	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
39	C₃×C₄⋊C₈	C₂²	1	2	yes	—
40	C₃×#103	C₂²	1	2	yes	—
41	C₃×#103	C₂²	1	2	yes	—
42	C₃×#103	C₂²	1	2	yes	—
43	C₃×#102	C₂²	1	2	yes	—
44	#533	C₂×C₃	1	1	yes	characteristic, complemented
45	#529	C₂×C₃	1	1	yes	characteristic
46	#479	C₂×C₃	1	2	yes	complemented
47	#477	C₂×C₃	1	1	yes	characteristic, complemented
48	#476	C₂×C₃	1	1	yes	characteristic, complemented
49	#470	C₂×C₃	1	2	yes	complemented
50	#468	C₂×C₃	1	2	yes	—
51	#464	C₂×C₃	1	2	yes	complemented
52	#462	C₂×C₃	1	1	yes	characteristic, complemented
53	#461	C₂×C₃	1	1	yes	characteristic, complemented
54	#423	C₂×C₃	1	1	yes	characteristic, complemented
55	C₂×C₃×Q₈	—	2	1	no	—
56	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
57	C₂×C₃×Q₈	C₂³	1	2	yes	—
58	C₂×C₃×D₄	—	2	1	no	—
59	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
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	C₃×Q₁₆	—	4	2	no	—
65	C₃×SD₁₆	—	4	2	no	—
66	C₃×M₁₆	—	4	2	no	—
67	C₂×C₃×C₈	C₂³	1	2	yes	—
68	C₂×C₃×C₈	C₂³	1	2	yes	—
69	C₃×C₄⋊C₄	—	2	1	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	2	no	—
76	C₃×C₄⋊C₄	—	2	2	no	—
77	C₃×C₄⋊C₄	—	2	1	no	—
78	C₃×C₄⋊C₄	C₂³	1	2	yes	—
79	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
80	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
81	C₃×C₂²⋊C₄	—	2	2	no	—
82	C₃×C₂²⋊C₄	—	2	1	no	—
83	C₃×C₂²⋊C₄	—	2	1	no	—
84	C₃×C₂²⋊C₄	—	2	1	no	—
85	C₃×C₂²⋊C₄	—	2	1	no	—
86	C₃×C₂²⋊C₄	—	2	1	no	—
87	C₃×C₄²	—	2	1	no	—
88	C₃×C₄²	—	2	2	no	—
89	C₃×C₄²	D₄	1	1	yes	characteristic
90	C₃×C₄²	D₄	1	1	yes	characteristic
91	C₃×C₄²	C₂³	1	1	yes	characteristic
92	C₃×C₄²	C₂³	1	1	yes	characteristic
93	C₂×Q₁₆	C₂²×C₃	1	2	yes	—
94	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
95	C₂×M₁₆	C₂²×C₃	1	2	yes	—
96	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
97	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
98	#126	—	2	1	no	—
99	#125	—	2	1	no	—
100	#125	C₂²×C₃	1	1	yes	characteristic
101	#124	C₂²×C₃	1	1	yes	characteristic
102	#123	—	2	1	no	—
103	C₂²⋊Q₈	—	2	1	no	—
104	C₂²⋊Q₈	C₂²×C₃	1	1	yes	characteristic
105	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
106	#121	C₂²×C₃	1	1	yes	characteristic
107	C₄×Q₈	—	2	1	no	—
108	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
109	C₄×Q₈	C₂²×C₃	1	2	yes	—
110	C₄×D₄	—	2	1	no	—
111	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
112	#117	—	2	1	no	—
113	#117	—	2	1	no	—
114	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
115	#107	C₂²×C₃	1	2	yes	—
116	#106	C₂²×C₃	1	2	yes	—
117	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
118	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
119	C₄⋊C₈	C₂²×C₃	1	2	yes	—
120	#103	C₂²×C₃	1	2	yes	—
121	#103	C₂²×C₃	1	2	yes	—
122	#103	C₂²×C₃	1	2	yes	—
123	#102	C₂²×C₃	1	2	yes	—
124	C₂³×C₃	—	2	1	no	—
125	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
126	C₃×Q₈	—	2	2	no	—
127	C₃×Q₈	—	2	1	no	—
128	C₃×Q₈	—	2	1	no	—
129	C₃×Q₈	—	2	2	no	—
130	C₃×Q₈	—	4	2	no	—
131	C₃×D₄	—	2	1	no	—
132	C₃×D₄	—	2	1	no	—
133	C₃×D₄	—	4	2	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	4	1	no	—
143	C₂×C₃×C₄	—	4	2	no	—
144	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
146	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
150	C₃×C₈	—	2	2	no	—
151	C₃×C₈	—	2	2	no	—
152	C₂×Q₈	—	2	1	no	—
153	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
154	C₂×Q₈	C₂³×C₃	1	2	yes	—
155	C₂×D₄	—	2	1	no	—
156	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
159	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
160	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
161	Q₁₆	—	4	2	no	—
162	SD₁₆	—	4	2	no	—
163	M₁₆	—	4	2	no	—
164	C₂×C₈	C₂³×C₃	1	2	yes	—
165	C₂×C₈	C₂³×C₃	1	2	yes	—
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	—	2	1	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	1	no	—
175	C₄⋊C₄	C₂³×C₃	1	2	yes	—
176	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
177	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
178	C₂²⋊C₄	—	2	2	no	—
179	C₂²⋊C₄	—	2	1	no	—
180	C₂²⋊C₄	—	2	1	no	—
181	C₂²⋊C₄	—	2	1	no	—
182	C₂²⋊C₄	—	2	1	no	—
183	C₂²⋊C₄	—	2	1	no	—
184	C₄²	—	2	1	no	—
185	C₄²	—	2	2	no	—
186	C₄²	C₃×D₄	1	1	yes	characteristic
187	C₄²	C₃×D₄	1	1	yes	characteristic
188	C₄²	C₂³×C₃	1	1	yes	characteristic
189	C₄²	C₂³×C₃	1	1	yes	characteristic
190	C₂²×C₃	—	2	1	no	—
191	C₂²×C₃	—	2	1	no	—
192	C₂²×C₃	—	2	1	no	—
193	C₂²×C₃	—	4	1	no	—
194	C₂²×C₃	—	4	1	no	—
195	C₂²×C₃	—	4	1	no	—
196	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
197	C₃×C₄	—	2	1	no	—
198	C₃×C₄	—	2	1	no	—
199	C₃×C₄	—	2	1	no	—
200	C₃×C₄	—	2	1	no	—
201	C₃×C₄	—	4	1	no	—
202	C₃×C₄	—	4	2	no	—
203	C₃×C₄	—	4	1	no	—
204	C₃×C₄	—	4	1	no	—
205	C₃×C₄	—	4	2	no	—
206	C₃×C₄	#137	1	2	yes	—
207	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
208	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
209	C₂³	—	2	1	no	—
210	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
211	Q₈	—	2	2	no	—
212	Q₈	—	2	1	no	—
213	Q₈	—	2	1	no	—
214	Q₈	—	2	2	no	—
215	Q₈	—	4	2	no	—
216	D₄	—	2	1	no	—
217	D₄	—	2	1	no	—
218	D₄	—	4	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
230	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
231	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
232	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
233	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
234	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	—	2	2	no	—
237	C₂×C₃	—	4	1	no	—
238	C₂×C₃	—	8	1	no	—
239	C₂×C₃	#537	1	1	yes	characteristic
240	C₂×C₃	C₂×#137	1	1	yes	characteristic
241	C₂×C₃	#575	1	1	yes	characteristic
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	4	1	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	1	no	—
248	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	1	no	—
254	C₄	—	4	1	no	—
255	C₄	—	4	2	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	2	no	—
258	C₄	C₃×#137	1	2	yes	—
259	C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
260	C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
261	C₃	#3252	1	1	yes	characteristic, direct product factor, Sylow
262	C₂	—	4	1	no	—
263	C₂	—	8	1	no	—
264	C₂	C₃×#537	1	1	yes	characteristic
265	C₂	C₂×C₃×#137	1	1	yes	characteristic
266	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
1. C₃×#533	C₂	semidirect
3. C₃×#479	C₂	semidirect
4. C₃×#477	C₂	semidirect
5. C₃×#476	C₂	semidirect
6. C₃×#470	C₂	semidirect
8. C₃×#464	C₂	semidirect
9. C₃×#462	C₂	semidirect
10. C₃×#461	C₂	semidirect
11. C₃×#423	C₂	semidirect
12. #3252	C₃	direct (N × H)
44. #533	C₂×C₃	semidirect
46. #479	C₂×C₃	semidirect
47. #477	C₂×C₃	semidirect
48. #476	C₂×C₃	semidirect
49. #470	C₂×C₃	semidirect
51. #464	C₂×C₃	semidirect
52. #462	C₂×C₃	semidirect
53. #461	C₂×C₃	semidirect
54. #423	C₂×C₃	semidirect
261. C₃	#3252	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×#533	2
C₃×#529	2
C₃×#479	2
C₃×#477	2
C₃×#476	2
C₃×#470	2
C₃×#468	2
C₃×#464	2
C₃×#462	2
C₃×#461	2
C₃×#423	2
#3252	3

Subgroups — C3×#3252 #85116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3252 #85116