Subgroups — C₃×#3222 #85086

Order 384.

396 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#541	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#535	C₂	1	1	yes	maximal, characteristic
3	C₃×#497	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#490	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#488	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#464	C₂	1	2	yes	maximal, complemented
8	C₃×#460	C₂	1	2	yes	maximal, complemented
9	C₃×#452	C₂	1	2	yes	maximal, complemented
10	C₃×#450	C₂	1	2	yes	maximal, complemented
11	C₃×#433	C₂	1	1	yes	maximal, characteristic, complemented
12	#3222	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×#41	C₂²	1	1	yes	characteristic
14	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
15	C₂×C₃×Q₁₆	C₂²	1	2	yes	complemented
16	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
17	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
18	C₂×C₃×SD₁₆	C₂²	1	2	yes	complemented
19	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
20	C₃×#127	C₂²	1	1	yes	characteristic
21	C₃×#127	C₂²	1	1	yes	characteristic
22	C₃×#126	—	2	2	no	—
23	C₃×#125	—	2	2	no	—
24	C₃×#125	C₂²	1	1	yes	characteristic, complemented
25	C₃×#124	C₂²	1	1	yes	characteristic
26	C₃×#124	C₂²	1	1	yes	characteristic
27	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
28	C₃×#121	—	2	2	no	—
29	C₃×#120	—	2	2	no	—
30	C₃×C₄×Q₈	C₂²	1	2	yes	complemented
31	C₃×#117	C₂²	1	1	yes	characteristic
32	C₃×C₄⋊C₈	C₂²	1	2	yes	complemented
33	C₃×#103	C₂²	1	2	yes	complemented
34	C₃×#102	C₂²	1	2	yes	—
35	C₃×#102	C₂²	1	2	yes	—
36	C₃×#102	C₂²	1	2	yes	complemented
37	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
38	C₃×#97	C₂²	1	1	yes	characteristic, complemented
39	C₃×C₄×C₈	C₂²	1	1	yes	characteristic, complemented
40	#541	C₂×C₃	1	1	yes	characteristic, complemented
41	#535	C₂×C₃	1	1	yes	characteristic
42	#497	C₂×C₃	1	1	yes	characteristic, complemented
43	#492	C₂×C₃	1	1	yes	characteristic, complemented
44	#490	C₂×C₃	1	1	yes	characteristic, complemented
45	#488	C₂×C₃	1	1	yes	characteristic, complemented
46	#464	C₂×C₃	1	2	yes	complemented
47	#460	C₂×C₃	1	2	yes	complemented
48	#452	C₂×C₃	1	2	yes	complemented
49	#450	C₂×C₃	1	2	yes	complemented
50	#433	C₂×C₃	1	1	yes	characteristic, complemented
51	C₂⁴×C₃	—	2	1	no	—
52	C₃×#41	—	4	2	no	—
53	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
54	C₂×C₃×Q₈	C₂³	1	2	yes	—
55	C₂×C₃×D₄	—	2	1	no	—
56	C₂×C₃×D₄	—	2	1	no	—
57	C₂×C₃×D₄	—	2	2	no	—
58	C₂×C₃×D₄	—	2	2	no	—
59	C₂×C₃×D₄	—	4	2	no	—
60	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
61	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
62	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
63	C₂²×C₃×C₄	—	2	1	no	—
64	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
65	C₃×Q₁₆	—	4	2	no	—
66	C₃×SD₁₆	—	4	2	no	—
67	C₃×SD₁₆	—	4	2	no	—
68	C₃×SD₁₆	—	4	2	no	—
69	C₂×C₃×C₈	C₂³	1	2	yes	—
70	C₂×C₃×C₈	C₂³	1	2	yes	—
71	C₃×C₄⋊C₄	—	2	2	no	—
72	C₃×C₄⋊C₄	—	2	2	no	—
73	C₃×C₄⋊C₄	—	2	2	no	—
74	C₃×C₄⋊C₄	—	2	2	no	—
75	C₃×C₄⋊C₄	D₄	1	2	yes	—
76	C₃×C₄⋊C₄	C₂³	1	2	yes	—
77	C₃×C₂²⋊C₄	—	2	2	no	—
78	C₃×C₂²⋊C₄	—	2	2	no	—
79	C₃×C₂²⋊C₄	—	2	2	no	—
80	C₃×C₂²⋊C₄	D₄	1	2	yes	—
81	C₃×C₄²	—	2	2	no	—
82	C₃×C₄²	C₂³	1	1	yes	characteristic
83	C₃×C₄²	C₂³	1	1	yes	characteristic
84	C₂×#41	C₂²×C₃	1	1	yes	characteristic
85	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
86	C₂×Q₁₆	C₂²×C₃	1	2	yes	complemented
87	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
88	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
89	C₂×SD₁₆	C₂²×C₃	1	2	yes	complemented
90	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
91	#127	C₂²×C₃	1	1	yes	characteristic
92	#127	C₂²×C₃	1	1	yes	characteristic
93	#126	—	2	2	no	—
94	#125	—	2	2	no	—
95	#125	C₂²×C₃	1	1	yes	characteristic, complemented
96	#124	C₂²×C₃	1	1	yes	characteristic
97	#124	C₂²×C₃	1	1	yes	characteristic
98	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
99	#121	—	2	2	no	—
100	#120	—	2	2	no	—
101	C₄×Q₈	C₂²×C₃	1	2	yes	complemented
102	#117	C₂²×C₃	1	1	yes	characteristic
103	C₄⋊C₈	C₂²×C₃	1	2	yes	complemented
104	#103	C₂²×C₃	1	2	yes	complemented
105	#102	C₂²×C₃	1	2	yes	—
106	#102	C₂²×C₃	1	2	yes	—
107	#102	C₂²×C₃	1	2	yes	complemented
108	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
109	#97	C₂²×C₃	1	1	yes	characteristic, complemented
110	C₄×C₈	C₂²×C₃	1	1	yes	characteristic, complemented
111	C₂³×C₃	—	2	1	no	—
112	C₂³×C₃	—	2	1	no	—
113	C₂³×C₃	—	2	1	no	—
114	C₂³×C₃	—	4	2	no	—
115	C₂³×C₃	—	8	2	no	—
116	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
117	C₃×Q₈	—	2	2	no	—
118	C₃×Q₈	—	2	2	no	—
119	C₃×Q₈	—	2	2	no	—
120	C₃×D₄	—	2	2	no	—
121	C₃×D₄	—	2	2	no	—
122	C₃×D₄	—	2	2	no	—
123	C₃×D₄	—	4	2	no	—
124	C₃×D₄	—	4	2	no	—
125	C₃×D₄	—	8	2	no	—
126	C₃×D₄	—	8	2	no	—
127	C₂×C₃×C₄	—	2	2	no	—
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	4	2	no	—
132	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
133	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
134	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
136	C₃×C₈	—	2	2	no	—
137	C₃×C₈	—	2	2	no	—
138	C₂⁴	—	2	1	no	—
139	#41	—	4	2	no	—
140	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
141	C₂×Q₈	C₂³×C₃	1	2	yes	—
142	C₂×D₄	—	2	1	no	—
143	C₂×D₄	—	2	1	no	—
144	C₂×D₄	—	2	2	no	—
145	C₂×D₄	—	2	2	no	—
146	C₂×D₄	—	4	2	no	—
147	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
148	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
149	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
152	Q₁₆	—	4	2	no	—
153	SD₁₆	—	4	2	no	—
154	SD₁₆	—	4	2	no	—
155	SD₁₆	—	4	2	no	—
156	C₂×C₈	C₂³×C₃	1	2	yes	—
157	C₂×C₈	C₂³×C₃	1	2	yes	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	C₃×D₄	1	2	yes	—
163	C₄⋊C₄	C₂³×C₃	1	2	yes	—
164	C₂²⋊C₄	—	2	2	no	—
165	C₂²⋊C₄	—	2	2	no	—
166	C₂²⋊C₄	—	2	2	no	—
167	C₂²⋊C₄	C₃×D₄	1	2	yes	—
168	C₄²	—	2	2	no	—
169	C₄²	C₂³×C₃	1	1	yes	characteristic
170	C₄²	C₂³×C₃	1	1	yes	characteristic
171	C₂²×C₃	—	2	1	no	—
172	C₂²×C₃	—	2	2	no	—
173	C₂²×C₃	—	4	1	no	—
174	C₂²×C₃	—	4	1	no	—
175	C₂²×C₃	—	4	1	no	—
176	C₂²×C₃	—	4	2	no	—
177	C₂²×C₃	—	4	2	no	—
178	C₂²×C₃	—	4	2	no	—
179	C₂²×C₃	—	8	4	no	—
180	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
181	C₃×C₄	—	2	1	no	—
182	C₃×C₄	—	2	2	no	—
183	C₃×C₄	—	2	2	no	—
184	C₃×C₄	—	4	1	no	—
185	C₃×C₄	—	4	2	no	—
186	C₃×C₄	—	4	2	no	—
187	C₃×C₄	H(5,𝔽₂)	1	2	yes	—
188	C₂³	—	2	1	no	—
189	C₂³	—	2	1	no	—
190	C₂³	—	2	1	no	—
191	C₂³	—	4	2	no	—
192	C₂³	—	8	2	no	—
193	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
194	Q₈	—	2	2	no	—
195	Q₈	—	2	2	no	—
196	Q₈	—	2	2	no	—
197	D₄	—	2	2	no	—
198	D₄	—	2	2	no	—
199	D₄	—	2	2	no	—
200	D₄	—	4	2	no	—
201	D₄	—	4	2	no	—
202	D₄	—	8	2	no	—
203	D₄	—	8	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
210	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
211	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
212	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
213	C₈	—	2	2	no	—
214	C₈	—	2	2	no	—
215	C₂×C₃	—	4	1	no	—
216	C₂×C₃	—	8	1	no	—
217	C₂×C₃	—	8	1	no	—
218	C₂×C₃	—	8	1	no	—
219	C₂×C₃	#535	1	1	yes	characteristic
220	C₂×C₃	#577	1	2	yes	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	4	2	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	—	4	1	no	—
227	C₂²	—	4	1	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	8	4	no	—
230	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	1	no	—
234	C₄	—	4	2	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	2	no	—
237	C₄	C₃×H(5,𝔽₂)	1	2	yes	—
238	C₃	#3222	1	1	yes	characteristic, direct product factor, Sylow
239	C₂	—	4	1	no	—
240	C₂	—	8	1	no	—
241	C₂	—	8	1	no	—
242	C₂	—	8	1	no	—
243	C₂	C₃×#535	1	1	yes	characteristic
244	C₂	C₃×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₃×#535	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#541	C₂	semidirect
3. C₃×#497	C₂	semidirect
4. C₃×#492	C₂	semidirect
5. C₃×#490	C₂	semidirect
6. C₃×#488	C₂	semidirect
7. C₃×#464	C₂	semidirect
8. C₃×#460	C₂	semidirect
9. C₃×#452	C₂	semidirect
10. C₃×#450	C₂	semidirect
11. C₃×#433	C₂	semidirect
12. #3222	C₃	direct (N × H)
15. C₂×C₃×Q₁₆	C₂²	semidirect
18. C₂×C₃×SD₁₆	C₂²	semidirect
19. C₃×C₄⋊Q₈	C₂²	semidirect
24. C₃×#125	C₂²	semidirect
30. C₃×C₄×Q₈	C₂²	semidirect
32. C₃×C₄⋊C₈	C₂²	semidirect
33. C₃×#103	C₂²	semidirect
36. C₃×#102	C₂²	semidirect
38. C₃×#97	C₂²	semidirect
39. C₃×C₄×C₈	C₂²	semidirect
40. #541	C₂×C₃	semidirect
42. #497	C₂×C₃	semidirect
43. #492	C₂×C₃	semidirect
44. #490	C₂×C₃	semidirect
45. #488	C₂×C₃	semidirect
46. #464	C₂×C₃	semidirect
47. #460	C₂×C₃	semidirect
48. #452	C₂×C₃	semidirect
49. #450	C₂×C₃	semidirect
50. #433	C₂×C₃	semidirect
86. C₂×Q₁₆	C₂²×C₃	semidirect
89. C₂×SD₁₆	C₂²×C₃	semidirect
90. C₄⋊Q₈	C₂²×C₃	semidirect
95. #125	C₂²×C₃	semidirect
101. C₄×Q₈	C₂²×C₃	semidirect
103. C₄⋊C₈	C₂²×C₃	semidirect
104. #103	C₂²×C₃	semidirect
107. #102	C₂²×C₃	semidirect
109. #97	C₂²×C₃	semidirect
110. C₄×C₈	C₂²×C₃	semidirect
238. C₃	#3222	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×#541	2
C₃×#535	2
C₃×#497	2
C₃×#492	2
C₃×#490	2
C₃×#488	2
C₃×#464	2
C₃×#460	2
C₃×#452	2
C₃×#450	2
C₃×#433	2
#3222	3

Subgroups — C3×#3222 #85086

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3222 #85086