Subgroups — C₃×#3231 #85095

Order 384.

326 conjugacy classes of subgroups.

The subgroup lattice has 206 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
2	C₃×#533	C₂	1	1	yes	maximal, characteristic
3	C₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#485	C₂	1	4	yes	maximal
8	C₃×#479	C₂	1	4	yes	maximal, complemented
9	C₃×#431	C₂	1	1	yes	maximal, characteristic
10	#3231	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₃×#125	—	2	1	no	—
16	C₃×#125	C₂²	1	2	yes	—
17	C₃×C₂²⋊Q₈	—	2	1	no	—
18	C₃×C₂²⋊Q₈	—	2	1	no	—
19	C₃×C₂²⋊Q₈	—	2	1	no	—
20	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₃×C₄×Q₈	—	2	1	no	—
22	C₃×C₄×Q₈	—	2	1	no	—
23	C₃×C₄×Q₈	—	2	1	no	—
24	C₃×C₄×Q₈	C₂²	1	2	yes	—
25	C₃×#117	—	2	1	no	—
26	C₃×#117	C₂²	1	2	yes	—
27	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
28	C₃×#107	C₂²	1	4	yes	—
29	C₃×#106	C₂²	1	4	yes	—
30	C₃×C₄⋊C₈	C₂²	1	2	yes	—
31	C₃×#103	C₂²	1	4	yes	—
32	C₃×#103	C₂²	1	4	yes	—
33	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
34	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
35	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
36	#533	C₂×C₃	1	1	yes	characteristic
37	#533	C₂×C₃	1	1	yes	characteristic
38	#500	C₂×C₃	1	1	yes	characteristic, complemented
39	#498	C₂×C₃	1	1	yes	characteristic, complemented
40	#487	C₂×C₃	1	1	yes	characteristic, complemented
41	#487	C₂×C₃	1	1	yes	characteristic, complemented
42	#485	C₂×C₃	1	4	yes	—
43	#479	C₂×C₃	1	4	yes	complemented
44	#431	C₂×C₃	1	1	yes	characteristic
45	C₂×C₃×Q₈	—	2	1	no	—
46	C₂×C₃×Q₈	—	2	1	no	—
47	C₂×C₃×Q₈	—	2	1	no	—
48	C₂×C₃×Q₈	C₂³	1	2	yes	—
49	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
50	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
51	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₈	C₂³	1	2	yes	—
53	C₂×C₃×C₈	C₂³	1	2	yes	—
54	C₃×C₄⋊C₄	—	2	1	no	—
55	C₃×C₄⋊C₄	—	2	1	no	—
56	C₃×C₄⋊C₄	—	2	2	no	—
57	C₃×C₄⋊C₄	—	2	2	no	—
58	C₃×C₄⋊C₄	—	2	1	no	—
59	C₃×C₄⋊C₄	—	2	1	no	—
60	C₃×C₄⋊C₄	—	2	1	no	—
61	C₃×C₄⋊C₄	—	2	2	no	—
62	C₃×C₄⋊C₄	—	2	2	no	—
63	C₃×C₄⋊C₄	C₂³	1	2	yes	—
64	C₃×C₄⋊C₄	C₂³	1	2	yes	—
65	C₃×C₄⋊C₄	C₂³	1	2	yes	—
66	C₃×C₂²⋊C₄	—	2	2	no	—
67	C₃×C₂²⋊C₄	—	2	2	no	—
68	C₃×C₄²	—	2	2	no	—
69	C₃×C₄²	—	2	2	no	—
70	C₃×C₄²	D₄	1	1	yes	characteristic
71	C₃×C₄²	D₄	1	1	yes	characteristic
72	C₃×C₄²	C₂³	1	1	yes	characteristic
73	C₃×C₄²	C₂³	1	1	yes	characteristic
74	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
75	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
76	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
77	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
78	#125	—	2	1	no	—
79	#125	C₂²×C₃	1	2	yes	—
80	C₂²⋊Q₈	—	2	1	no	—
81	C₂²⋊Q₈	—	2	1	no	—
82	C₂²⋊Q₈	—	2	1	no	—
83	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
84	C₄×Q₈	—	2	1	no	—
85	C₄×Q₈	—	2	1	no	—
86	C₄×Q₈	—	2	1	no	—
87	C₄×Q₈	C₂²×C₃	1	2	yes	—
88	#117	—	2	1	no	—
89	#117	C₂²×C₃	1	2	yes	—
90	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
91	#107	C₂²×C₃	1	4	yes	—
92	#106	C₂²×C₃	1	4	yes	—
93	C₄⋊C₈	C₂²×C₃	1	2	yes	—
94	#103	C₂²×C₃	1	4	yes	—
95	#103	C₂²×C₃	1	4	yes	—
96	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
97	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
98	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
99	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
100	C₃×Q₈	—	2	4	no	—
101	C₃×Q₈	—	4	2	no	—
102	C₃×Q₈	—	4	2	no	—
103	C₃×Q₈	—	4	2	no	—
104	C₂×C₃×C₄	—	2	2	no	—
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	—	2	2	no	—
107	C₂×C₃×C₄	—	2	2	no	—
108	C₂×C₃×C₄	—	2	2	no	—
109	C₂×C₃×C₄	—	2	2	no	—
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
112	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
113	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
114	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
115	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
116	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
117	C₃×C₈	—	2	2	no	—
118	C₃×C₈	—	2	2	no	—
119	C₂×Q₈	—	2	1	no	—
120	C₂×Q₈	—	2	1	no	—
121	C₂×Q₈	—	2	1	no	—
122	C₂×Q₈	C₂³×C₃	1	2	yes	—
123	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
124	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
125	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
126	C₂×C₈	C₂³×C₃	1	2	yes	—
127	C₂×C₈	C₂³×C₃	1	2	yes	—
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	—	2	1	no	—
135	C₄⋊C₄	—	2	2	no	—
136	C₄⋊C₄	—	2	2	no	—
137	C₄⋊C₄	C₂³×C₃	1	2	yes	—
138	C₄⋊C₄	C₂³×C₃	1	2	yes	—
139	C₄⋊C₄	C₂³×C₃	1	2	yes	—
140	C₂²⋊C₄	—	2	2	no	—
141	C₂²⋊C₄	—	2	2	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	C₃×D₄	1	1	yes	characteristic
145	C₄²	C₃×D₄	1	1	yes	characteristic
146	C₄²	C₂³×C₃	1	1	yes	characteristic
147	C₄²	C₂³×C₃	1	1	yes	characteristic
148	C₂²×C₃	—	2	1	no	—
149	C₂²×C₃	—	2	1	no	—
150	C₂²×C₃	—	2	1	no	—
151	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
152	C₃×C₄	—	2	1	no	—
153	C₃×C₄	—	2	1	no	—
154	C₃×C₄	—	2	1	no	—
155	C₃×C₄	—	4	2	no	—
156	C₃×C₄	—	4	2	no	—
157	C₃×C₄	—	4	2	no	—
158	C₃×C₄	—	4	2	no	—
159	C₃×C₄	#135	1	2	yes	—
160	C₃×C₄	#137	1	2	yes	—
161	C₃×C₄	2₋¹⁺⁴	1	2	yes	—
162	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
163	Q₈	—	2	4	no	—
164	Q₈	—	4	2	no	—
165	Q₈	—	4	2	no	—
166	Q₈	—	4	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
175	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
176	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
177	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
178	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
179	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	2	2	no	—
182	C₂×C₃	—	4	1	no	—
183	C₂×C₃	#536	1	1	yes	characteristic
184	C₂×C₃	C₂×#135	1	1	yes	characteristic
185	C₂×C₃	C₂×#137	1	1	yes	characteristic
186	C₂²	—	2	1	no	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
190	C₄	—	2	1	no	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	4	2	no	—
194	C₄	—	4	2	no	—
195	C₄	—	4	2	no	—
196	C₄	—	4	2	no	—
197	C₄	C₃×#135	1	2	yes	—
198	C₄	C₃×#137	1	2	yes	—
199	C₄	C₃×2₋¹⁺⁴	1	2	yes	—
200	C₃	#3231	1	1	yes	characteristic, direct product factor, Sylow
201	C₂	—	4	1	no	—
202	C₂	C₃×#536	1	1	yes	characteristic
203	C₂	C₂×C₃×#135	1	1	yes	characteristic
204	C₂	C₂×C₃×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₂³×C₃	C₂×D₄
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
3. C₃×#500	C₂	semidirect
4. C₃×#498	C₂	semidirect
5. C₃×#487	C₂	semidirect
6. C₃×#487	C₂	semidirect
8. C₃×#479	C₂	semidirect
10. #3231	C₃	direct (N × H)
38. #500	C₂×C₃	semidirect
39. #498	C₂×C₃	semidirect
40. #487	C₂×C₃	semidirect
41. #487	C₂×C₃	semidirect
43. #479	C₂×C₃	semidirect
200. C₃	#3231	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×#500	2
C₃×#498	2
C₃×#487	2
C₃×#485	2
C₃×#479	2
C₃×#431	2
#3231	3

Subgroups — C3×#3231 #85095

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3231 #85095