Subgroups — C₃×#3336 #85200

Order 384.

342 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#552	C₂	1	2	yes	maximal
2	C₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#485	C₂	1	2	yes	maximal, complemented
4	C₃×#483	C₂	1	2	yes	maximal, complemented
5	C₃×#479	C₂	1	2	yes	maximal, complemented
6	C₃×#467	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#467	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×C₈×D₄	C₂	1	1	yes	maximal, characteristic
10	C₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
12	#3336	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
16	C₃×#126	—	2	2	no	—
17	C₃×#125	—	2	2	no	—
18	C₃×#125	C₂²	1	2	yes	—
19	C₃×#123	—	2	2	no	—
20	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₃×C₄×Q₈	C₂²	1	2	yes	—
23	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
24	C₃×#117	—	2	2	no	—
25	C₃×#117	C₂²	1	2	yes	—
26	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
27	C₃×#107	C₂²	1	1	yes	characteristic
28	C₃×#107	C₂²	1	1	yes	characteristic
29	C₃×#107	C₂²	1	2	yes	—
30	C₃×#107	C₂²	1	1	yes	characteristic
31	C₃×#106	C₂²	1	2	yes	—
32	C₃×#106	C₂²	1	2	yes	—
33	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₃×#103	C₂²	1	2	yes	—
35	C₃×#103	C₂²	1	2	yes	—
36	C₃×#103	C₂²	1	2	yes	—
37	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
38	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
39	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
40	#552	C₂×C₃	1	2	yes	—
41	#499	C₂×C₃	1	1	yes	characteristic, complemented
42	#485	C₂×C₃	1	2	yes	complemented
43	#483	C₂×C₃	1	2	yes	complemented
44	#479	C₂×C₃	1	2	yes	complemented
45	#467	C₂×C₃	1	1	yes	characteristic, complemented
46	#467	C₂×C₃	1	1	yes	characteristic, complemented
47	C₄×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
48	C₈×D₄	C₂×C₃	1	1	yes	characteristic
49	#427	C₂×C₃	1	1	yes	characteristic, complemented
50	C₂×#107	C₂×C₃	1	1	yes	characteristic, complemented
51	C₂×C₃×Q₈	C₂³	1	2	yes	—
52	C₂×C₃×D₄	D₄	1	1	yes	characteristic
53	C₂²×C₃×C₄	—	2	2	no	—
54	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
55	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
56	C₃×Q₁₆	—	2	2	no	—
57	C₂×C₃×C₈	—	2	2	no	—
58	C₂×C₃×C₈	—	2	2	no	—
59	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
60	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
61	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
62	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
63	C₃×C₄⋊C₄	—	2	2	no	—
64	C₃×C₄⋊C₄	—	2	2	no	—
65	C₃×C₄⋊C₄	—	2	2	no	—
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	D₄	1	1	yes	characteristic
70	C₃×C₄⋊C₄	C₂³	1	2	yes	—
71	C₃×C₄⋊C₄	C₂³	1	2	yes	—
72	C₃×C₄⋊C₄	C₂³	1	2	yes	—
73	C₃×C₂²⋊C₄	—	2	2	no	—
74	C₃×C₂²⋊C₄	—	2	2	no	—
75	C₃×C₂²⋊C₄	—	2	2	no	—
76	C₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
77	C₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
78	C₃×C₄²	—	2	2	no	—
79	C₃×C₄²	—	2	2	no	—
80	C₃×C₄²	C₂³	1	1	yes	characteristic
81	C₂×Q₁₆	C₂²×C₃	1	1	yes	characteristic
82	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
83	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
84	#126	—	2	2	no	—
85	#125	—	2	2	no	—
86	#125	C₂²×C₃	1	2	yes	—
87	#123	—	2	2	no	—
88	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
89	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
90	C₄×Q₈	C₂²×C₃	1	2	yes	—
91	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
92	#117	—	2	2	no	—
93	#117	C₂²×C₃	1	2	yes	—
94	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
95	#107	C₂²×C₃	1	1	yes	characteristic
96	#107	C₂²×C₃	1	1	yes	characteristic
97	#107	C₂²×C₃	1	2	yes	—
98	#107	C₂²×C₃	1	1	yes	characteristic
99	#106	C₂²×C₃	1	2	yes	—
100	#106	C₂²×C₃	1	2	yes	—
101	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
102	#103	C₂²×C₃	1	2	yes	—
103	#103	C₂²×C₃	1	2	yes	—
104	#103	C₂²×C₃	1	2	yes	—
105	C₂²⋊C₈	C₂²×C₃	1	1	yes	characteristic
106	C₂²⋊C₈	C₂²×C₃	1	1	yes	characteristic
107	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
108	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
109	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
110	C₃×Q₈	—	2	2	no	—
111	C₃×Q₈	—	2	2	no	—
112	C₃×D₄	—	2	2	no	—
113	C₂×C₃×C₄	—	2	1	no	—
114	C₂×C₃×C₄	—	2	1	no	—
115	C₂×C₃×C₄	—	2	1	no	—
116	C₂×C₃×C₄	—	2	1	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	2	2	no	—
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	—	4	4	no	—
122	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
127	C₃×C₈	—	2	1	no	—
128	C₃×C₈	—	2	1	no	—
129	C₃×C₈	—	2	1	no	—
130	C₃×C₈	#41	1	2	yes	—
131	C₂×Q₈	C₂³×C₃	1	2	yes	—
132	C₂×D₄	C₃×D₄	1	1	yes	characteristic
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
135	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
136	Q₁₆	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
140	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
141	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
142	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
150	C₄⋊C₄	C₂³×C₃	1	2	yes	—
151	C₄⋊C₄	C₂³×C₃	1	2	yes	—
152	C₄⋊C₄	C₂³×C₃	1	2	yes	—
153	C₂²⋊C₄	—	2	2	no	—
154	C₂²⋊C₄	—	2	2	no	—
155	C₂²⋊C₄	—	2	2	no	—
156	C₂²⋊C₄	C₃×D₄	1	1	yes	characteristic
157	C₂²⋊C₄	C₃×D₄	1	1	yes	characteristic
158	C₄²	—	2	2	no	—
159	C₄²	—	2	2	no	—
160	C₄²	C₂³×C₃	1	1	yes	characteristic
161	C₂²×C₃	—	2	1	no	—
162	C₂²×C₃	—	2	1	no	—
163	C₂²×C₃	—	2	1	no	—
164	C₂²×C₃	—	2	1	no	—
165	C₂²×C₃	—	2	1	no	—
166	C₂²×C₃	#135	1	2	yes	—
167	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
168	C₃×C₄	—	2	1	no	—
169	C₃×C₄	—	2	1	no	—
170	C₃×C₄	—	2	1	no	—
171	C₃×C₄	—	2	1	no	—
172	C₃×C₄	—	4	2	no	—
173	C₃×C₄	—	4	2	no	—
174	C₃×C₄	—	4	2	no	—
175	C₃×C₄	—	4	2	no	—
176	C₃×C₄	C₂×#41	1	1	yes	characteristic
177	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
178	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
179	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
180	Q₈	—	2	2	no	—
181	Q₈	—	2	2	no	—
182	D₄	—	2	2	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
193	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
194	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
195	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
196	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
197	C₈	—	2	1	no	—
198	C₈	—	2	1	no	—
199	C₈	—	2	1	no	—
200	C₈	C₃×#41	1	2	yes	—
201	C₂×C₃	—	2	2	no	—
202	C₂×C₃	—	4	1	no	—
203	C₂×C₃	#547	1	1	yes	characteristic
204	C₂×C₃	C₂×#135	1	1	yes	characteristic
205	C₂×C₃	#578	1	1	yes	characteristic
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	C₃×#135	1	2	yes	—
212	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	4	2	no	—
218	C₄	—	4	2	no	—
219	C₄	—	4	2	no	—
220	C₄	—	4	2	no	—
221	C₄	C₂×C₃×#41	1	1	yes	characteristic
222	C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
223	C₃	#3336	1	1	yes	characteristic, direct product factor, Sylow
224	C₂	—	2	2	no	—
225	C₂	—	4	1	no	—
226	C₂	C₃×#547	1	1	yes	characteristic
227	C₂	C₂×C₃×#135	1	1	yes	characteristic
228	C₂	C₃×#578	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃×#499	C₂	semidirect
3. C₃×#485	C₂	semidirect
4. C₃×#483	C₂	semidirect
5. C₃×#479	C₂	semidirect
6. C₃×#467	C₂	semidirect
7. C₃×#467	C₂	semidirect
8. C₃×C₄×Q₁₆	C₂	semidirect
10. C₃×#427	C₂	semidirect
11. C₂×C₃×#107	C₂	semidirect
12. #3336	C₃	direct (N × H)
41. #499	C₂×C₃	semidirect
42. #485	C₂×C₃	semidirect
43. #483	C₂×C₃	semidirect
44. #479	C₂×C₃	semidirect
45. #467	C₂×C₃	semidirect
46. #467	C₂×C₃	semidirect
47. C₄×Q₁₆	C₂×C₃	semidirect
49. #427	C₂×C₃	semidirect
50. C₂×#107	C₂×C₃	semidirect
223. C₃	#3336	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×#552	2
C₃×#499	2
C₃×#485	2
C₃×#483	2
C₃×#479	2
C₃×#467	2
C₃×C₄×Q₁₆	2
C₃×C₈×D₄	2
C₃×#427	2
C₂×C₃×#107	2
#3336	3

Subgroups — C3×#3336 #85200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3336 #85200