Subgroups — C₃×#3369 #85233

Order 384.

338 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×Q₈²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#556	C₂	1	1	yes	maximal, characteristic
3	C₃×#489	C₂	1	2	yes	maximal
4	C₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#476	C₂	1	1	yes	maximal, characteristic
6	C₃×#475	C₂	1	2	yes	maximal, complemented
7	C₃×#464	C₂	1	2	yes	maximal
8	C₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#441	C₂	1	2	yes	maximal, complemented
11	C₃×C₄×SD₁₆	C₂	1	1	yes	maximal, characteristic
12	#3369	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×Q₁₆	C₂²	1	2	yes	—
14	C₂×C₃×SD₁₆	C₂²	1	1	yes	characteristic
15	C₃×C₄⋊Q₈	—	2	1	no	—
16	C₃×C₄⋊Q₈	—	2	2	no	—
17	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
19	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₃×#126	—	2	2	no	—
21	C₃×#124	C₂²	1	2	yes	—
22	C₃×C₂²⋊Q₈	—	2	1	no	—
23	C₃×C₄×Q₈	—	2	1	no	—
24	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
25	C₃×C₄×Q₈	C₂²	1	2	yes	—
26	C₃×C₄×Q₈	C₂²	1	2	yes	—
27	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
28	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
29	C₃×#117	—	2	1	no	—
30	C₃×#106	C₂²	1	1	yes	characteristic
31	C₃×#106	C₂²	1	2	yes	—
32	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₃×C₄⋊C₈	C₂²	1	2	yes	—
34	C₃×#103	C₂²	1	1	yes	characteristic
35	C₃×#103	C₂²	1	2	yes	—
36	C₃×#103	C₂²	1	2	yes	—
37	C₃×#103	C₂²	1	2	yes	—
38	C₃×#102	C₂²	1	1	yes	characteristic
39	C₃×#102	C₂²	1	2	yes	—
40	C₃×#97	C₂²	1	2	yes	—
41	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
42	Q₈²	C₂×C₃	1	1	yes	characteristic, complemented
43	#556	C₂×C₃	1	1	yes	characteristic
44	#489	C₂×C₃	1	2	yes	—
45	#487	C₂×C₃	1	1	yes	characteristic, complemented
46	#476	C₂×C₃	1	1	yes	characteristic
47	#475	C₂×C₃	1	2	yes	complemented
48	#464	C₂×C₃	1	2	yes	—
49	#462	C₂×C₃	1	1	yes	characteristic, complemented
50	#446	C₂×C₃	1	1	yes	characteristic, complemented
51	#441	C₂×C₃	1	2	yes	complemented
52	C₄×SD₁₆	C₂×C₃	1	1	yes	characteristic
53	C₂×C₃×Q₈	—	2	2	no	—
54	C₂×C₃×Q₈	—	2	1	no	—
55	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
56	C₂×C₃×Q₈	C₂³	1	2	yes	—
57	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
58	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
59	C₂²×C₃×C₄	—	2	1	no	—
60	C₃×Q₁₆	—	2	4	no	—
61	C₃×SD₁₆	—	2	2	no	—
62	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
63	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
64	C₂×C₃×C₈	C₂³	1	2	yes	—
65	C₃×C₄⋊C₄	—	2	1	no	—
66	C₃×C₄⋊C₄	—	2	2	no	—
67	C₃×C₄⋊C₄	—	2	1	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	D₄	1	1	yes	characteristic
73	C₃×C₄⋊C₄	D₄	1	2	yes	—
74	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
75	C₃×C₄⋊C₄	C₂³	1	2	yes	—
76	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
77	C₃×C₂²⋊C₄	—	2	2	no	—
78	C₃×C₂²⋊C₄	—	2	1	no	—
79	C₃×C₂²⋊C₄	—	2	2	no	—
80	C₃×C₄²	—	2	2	no	—
81	C₃×C₄²	—	2	2	no	—
82	C₃×C₄²	—	2	1	no	—
83	C₃×C₄²	C₂³	1	1	yes	characteristic
84	C₃×C₄²	C₂³	1	2	yes	—
85	C₂×Q₁₆	C₂²×C₃	1	2	yes	—
86	C₂×SD₁₆	C₂²×C₃	1	1	yes	characteristic
87	C₄⋊Q₈	—	2	1	no	—
88	C₄⋊Q₈	—	2	2	no	—
89	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
90	C₄⋊Q₈	C₂²×C₃	1	2	yes	—
91	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
92	#126	—	2	2	no	—
93	#124	C₂²×C₃	1	2	yes	—
94	C₂²⋊Q₈	—	2	1	no	—
95	C₄×Q₈	—	2	1	no	—
96	C₄×Q₈	C₂²×C₃	1	2	yes	—
97	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
98	C₄×Q₈	C₂²×C₃	1	2	yes	—
99	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
100	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
101	#117	—	2	1	no	—
102	#106	C₂²×C₃	1	1	yes	characteristic
103	#106	C₂²×C₃	1	2	yes	—
104	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
105	C₄⋊C₈	C₂²×C₃	1	2	yes	—
106	#103	C₂²×C₃	1	1	yes	characteristic
107	#103	C₂²×C₃	1	2	yes	—
108	#103	C₂²×C₃	1	2	yes	—
109	#103	C₂²×C₃	1	2	yes	—
110	#102	C₂²×C₃	1	1	yes	characteristic
111	#102	C₂²×C₃	1	2	yes	—
112	#97	C₂²×C₃	1	2	yes	—
113	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
114	C₂³×C₃	—	2	1	no	—
115	C₃×Q₈	—	2	2	no	—
116	C₃×Q₈	—	2	1	no	—
117	C₃×Q₈	—	2	2	no	—
118	C₃×Q₈	—	2	2	no	—
119	C₃×Q₈	—	8	2	no	—
120	C₃×Q₈	—	8	1	no	—
121	C₃×Q₈	#41	1	2	yes	—
122	C₃×D₄	—	2	1	no	—
123	C₃×D₄	—	2	1	no	—
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	—	2	1	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	—	4	2	no	—
130	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
131	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
132	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
135	C₃×C₈	—	2	1	no	—
136	C₃×C₈	—	2	2	no	—
137	C₃×C₈	—	2	1	no	—
138	C₂×Q₈	—	2	2	no	—
139	C₂×Q₈	—	2	1	no	—
140	C₂×Q₈	C₃×D₄	1	1	yes	characteristic
141	C₂×Q₈	C₂³×C₃	1	2	yes	—
142	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
143	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
144	C₂²×C₄	—	2	1	no	—
145	Q₁₆	—	2	4	no	—
146	SD₁₆	—	2	2	no	—
147	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
148	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
149	C₂×C₈	C₂³×C₃	1	2	yes	—
150	C₄⋊C₄	—	2	1	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	1	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	1	no	—
156	C₄⋊C₄	—	2	1	no	—
157	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
158	C₄⋊C₄	C₃×D₄	1	2	yes	—
159	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
160	C₄⋊C₄	C₂³×C₃	1	2	yes	—
161	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
162	C₂²⋊C₄	—	2	2	no	—
163	C₂²⋊C₄	—	2	1	no	—
164	C₂²⋊C₄	—	2	2	no	—
165	C₄²	—	2	2	no	—
166	C₄²	—	2	2	no	—
167	C₄²	—	2	1	no	—
168	C₄²	C₂³×C₃	1	1	yes	characteristic
169	C₄²	C₂³×C₃	1	2	yes	—
170	C₂²×C₃	—	4	1	no	—
171	C₂²×C₃	—	4	1	no	—
172	C₂²×C₃	—	4	1	no	—
173	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
174	C₃×C₄	—	2	2	no	—
175	C₃×C₄	—	2	2	no	—
176	C₃×C₄	—	2	2	no	—
177	C₃×C₄	—	2	1	no	—
178	C₃×C₄	—	4	2	no	—
179	C₃×C₄	—	4	1	no	—
180	C₃×C₄	—	4	2	no	—
181	C₃×C₄	—	4	1	no	—
182	C₃×C₄	#137	1	2	yes	—
183	C₃×C₄	C₂×#41	1	1	yes	characteristic
184	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
185	C₂³	—	2	1	no	—
186	Q₈	—	2	2	no	—
187	Q₈	—	2	1	no	—
188	Q₈	—	2	2	no	—
189	Q₈	—	2	2	no	—
190	Q₈	—	8	2	no	—
191	Q₈	—	8	1	no	—
192	Q₈	C₃×#41	1	2	yes	—
193	D₄	—	2	1	no	—
194	D₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
202	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
203	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
204	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
205	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	2	2	no	—
208	C₈	—	2	1	no	—
209	C₂×C₃	—	8	1	no	—
210	C₂×C₃	#548	1	1	yes	characteristic
211	C₂×C₃	C₂×#137	1	1	yes	characteristic
212	C₂×C₃	#577	1	1	yes	characteristic
213	C₂²	—	4	1	no	—
214	C₂²	—	4	1	no	—
215	C₂²	—	4	1	no	—
216	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	—	4	1	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	1	no	—
224	C₄	—	4	2	no	—
225	C₄	C₃×#137	1	2	yes	—
226	C₄	C₂×C₃×#41	1	1	yes	characteristic
227	C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
228	C₃	#3369	1	1	yes	characteristic, direct product factor, Sylow
229	C₂	—	8	1	no	—
230	C₂	C₃×#548	1	1	yes	characteristic
231	C₂	C₂×C₃×#137	1	1	yes	characteristic
232	C₂	C₃×#577	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	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×Q₈²	C₂	semidirect
4. C₃×#487	C₂	semidirect
6. C₃×#475	C₂	semidirect
8. C₃×#462	C₂	semidirect
9. C₃×#446	C₂	semidirect
10. C₃×#441	C₂	semidirect
12. #3369	C₃	direct (N × H)
42. Q₈²	C₂×C₃	semidirect
45. #487	C₂×C₃	semidirect
47. #475	C₂×C₃	semidirect
49. #462	C₂×C₃	semidirect
50. #446	C₂×C₃	semidirect
51. #441	C₂×C₃	semidirect
228. C₃	#3369	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×Q₈²	2
C₃×#556	2
C₃×#489	2
C₃×#487	2
C₃×#476	2
C₃×#475	2
C₃×#464	2
C₃×#462	2
C₃×#446	2
C₃×#441	2
C₃×C₄×SD₁₆	2
#3369	3

Subgroups — C3×#3369 #85233

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3369 #85233