Subgroups — C₃×#3335 #85199

Order 384.

388 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#554	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#547	C₂	1	1	yes	maximal, characteristic
3	C₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#483	C₂	1	2	yes	maximal, complemented
5	C₃×#481	C₂	1	2	yes	maximal, complemented
6	C₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#475	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#465	C₂	1	2	yes	maximal, complemented
9	C₃×C₄×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃×#106	C₂	1	2	yes	maximal, complemented
12	#3335	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×#41	—	2	1	no	—
14	C₂×C₃×SD₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₈	C₂²	1	2	yes	—
16	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₃×#125	—	2	1	no	—
19	C₃×#123	—	2	2	no	—
20	C₃×C₂²⋊Q₈	—	2	1	no	—
21	C₃×C₂²⋊Q₈	—	2	2	no	—
22	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
23	C₃×#121	C₂²	1	2	yes	—
24	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
25	C₃×C₄×D₄	—	2	1	no	—
26	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
27	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
29	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
30	C₃×#106	C₂²	1	1	yes	characteristic
31	C₃×#106	C₂²	1	2	yes	—
32	C₃×#106	C₂²	1	2	yes	—
33	C₃×#106	C₂²	1	2	yes	—
34	C₃×#106	C₂²	1	2	yes	—
35	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	C₃×#103	C₂²	1	1	yes	characteristic
37	C₃×#103	C₂²	1	2	yes	—
38	C₃×#102	C₂²	1	1	yes	characteristic
39	C₃×#102	C₂²	1	2	yes	—
40	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
41	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
42	#554	C₂×C₃	1	1	yes	characteristic, complemented
43	#547	C₂×C₃	1	1	yes	characteristic
44	#498	C₂×C₃	1	1	yes	characteristic, complemented
45	#483	C₂×C₃	1	2	yes	complemented
46	#481	C₂×C₃	1	2	yes	complemented
47	#476	C₂×C₃	1	1	yes	characteristic, complemented
48	#475	C₂×C₃	1	1	yes	characteristic, complemented
49	#465	C₂×C₃	1	2	yes	complemented
50	C₄×SD₁₆	C₂×C₃	1	1	yes	characteristic, complemented
51	C₈×D₄	C₂×C₃	1	1	yes	characteristic, complemented
52	C₂×#106	C₂×C₃	1	2	yes	complemented
53	C₃×#41	—	4	4	no	—
54	C₂×C₃×Q₈	—	2	1	no	—
55	C₂×C₃×Q₈	—	2	1	no	—
56	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
57	C₂×C₃×D₄	—	2	2	no	—
58	C₂×C₃×D₄	D₄	1	1	yes	characteristic
59	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
60	C₂²×C₃×C₄	—	2	2	no	—
61	C₂²×C₃×C₄	—	2	2	no	—
62	C₂²×C₃×C₄	—	2	1	no	—
63	C₂²×C₃×C₄	C₂³	1	2	yes	—
64	C₃×SD₁₆	—	2	2	no	—
65	C₂×C₃×C₈	—	2	4	no	—
66	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
67	C₂×C₃×C₈	C₂³	1	2	yes	—
68	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
69	C₃×C₄⋊C₄	—	2	2	no	—
70	C₃×C₄⋊C₄	—	2	2	no	—
71	C₃×C₄⋊C₄	—	2	1	no	—
72	C₃×C₄⋊C₄	—	2	1	no	—
73	C₃×C₄⋊C₄	—	2	2	no	—
74	C₃×C₄⋊C₄	—	2	1	no	—
75	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
76	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
77	C₃×C₄⋊C₄	C₂³	1	2	yes	—
78	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
79	C₃×C₄⋊C₄	C₂³	1	2	yes	—
80	C₃×C₂²⋊C₄	—	2	2	no	—
81	C₃×C₂²⋊C₄	—	2	1	no	—
82	C₃×C₂²⋊C₄	—	2	2	no	—
83	C₃×C₂²⋊C₄	D₄	1	2	yes	—
84	C₃×C₄²	—	2	1	no	—
85	C₃×C₄²	C₂³	1	1	yes	characteristic
86	C₂×#41	—	2	1	no	—
87	C₂×SD₁₆	C₂²×C₃	1	1	yes	characteristic
88	C₂²×C₈	C₂²×C₃	1	2	yes	—
89	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
90	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
91	#125	—	2	1	no	—
92	#123	—	2	2	no	—
93	C₂²⋊Q₈	—	2	1	no	—
94	C₂²⋊Q₈	—	2	2	no	—
95	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
96	#121	C₂²×C₃	1	2	yes	—
97	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
98	C₄×D₄	—	2	1	no	—
99	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
100	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
101	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
102	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
103	#106	C₂²×C₃	1	1	yes	characteristic
104	#106	C₂²×C₃	1	2	yes	—
105	#106	C₂²×C₃	1	2	yes	—
106	#106	C₂²×C₃	1	2	yes	—
107	#106	C₂²×C₃	1	2	yes	—
108	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
109	#103	C₂²×C₃	1	1	yes	characteristic
110	#103	C₂²×C₃	1	2	yes	—
111	#102	C₂²×C₃	1	1	yes	characteristic
112	#102	C₂²×C₃	1	2	yes	—
113	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
114	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
115	C₂³×C₃	—	2	1	no	—
116	C₂³×C₃	C₂×D₄	1	2	yes	—
117	C₃×Q₈	—	2	1	no	—
118	C₃×Q₈	—	2	1	no	—
119	C₃×Q₈	—	4	2	no	—
120	C₃×Q₈	—	8	1	no	—
121	C₃×D₄	—	2	1	no	—
122	C₃×D₄	—	2	1	no	—
123	C₃×D₄	—	4	4	no	—
124	C₃×D₄	SD₁₆	1	4	yes	complemented
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	4	4	no	—
133	C₂×C₃×C₄	—	4	2	no	—
134	C₂×C₃×C₄	—	8	2	no	—
135	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
137	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
139	C₃×C₈	—	2	2	no	—
140	C₃×C₈	—	2	1	no	—
141	C₃×C₈	#41	1	2	yes	complemented
142	#41	—	4	4	no	—
143	C₂×Q₈	—	2	1	no	—
144	C₂×Q₈	—	2	1	no	—
145	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
146	C₂×D₄	—	2	2	no	—
147	C₂×D₄	C₃×D₄	1	1	yes	characteristic
148	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	—	2	1	no	—
152	C₂²×C₄	C₂³×C₃	1	2	yes	—
153	SD₁₆	—	2	2	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
156	C₂×C₈	C₂³×C₃	1	2	yes	—
157	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	1	no	—
161	C₄⋊C₄	—	2	1	no	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
165	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
166	C₄⋊C₄	C₂³×C₃	1	2	yes	—
167	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
168	C₄⋊C₄	C₂³×C₃	1	2	yes	—
169	C₂²⋊C₄	—	2	2	no	—
170	C₂²⋊C₄	—	2	1	no	—
171	C₂²⋊C₄	—	2	2	no	—
172	C₂²⋊C₄	C₃×D₄	1	2	yes	—
173	C₄²	—	2	1	no	—
174	C₄²	C₂³×C₃	1	1	yes	characteristic
175	C₂²×C₃	—	2	2	no	—
176	C₂²×C₃	—	2	2	no	—
177	C₂²×C₃	—	4	1	no	—
178	C₂²×C₃	—	4	1	no	—
179	C₂²×C₃	—	4	1	no	—
180	C₂²×C₃	C₂×SD₁₆	1	4	yes	—
181	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
182	C₃×C₄	—	2	2	no	—
183	C₃×C₄	—	2	1	no	—
184	C₃×C₄	—	4	1	no	—
185	C₃×C₄	—	4	1	no	—
186	C₃×C₄	—	4	2	no	—
187	C₃×C₄	—	4	1	no	—
188	C₃×C₄	—	4	2	no	—
189	C₃×C₄	C₂×SD₁₆	1	2	yes	—
190	C₃×C₄	C₂×#41	1	1	yes	characteristic
191	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
192	C₂³	—	2	1	no	—
193	C₂³	C₂×C₃×D₄	1	2	yes	—
194	Q₈	—	2	1	no	—
195	Q₈	—	2	1	no	—
196	Q₈	—	4	2	no	—
197	Q₈	—	8	1	no	—
198	D₄	—	2	1	no	—
199	D₄	—	2	1	no	—
200	D₄	—	4	4	no	—
201	D₄	C₃×SD₁₆	1	4	yes	complemented
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	8	2	no	—
212	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
213	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
214	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
215	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
216	C₈	—	2	2	no	—
217	C₈	—	2	1	no	—
218	C₈	C₃×#41	1	2	yes	complemented
219	C₂×C₃	—	2	4	no	—
220	C₂×C₃	—	8	1	no	—
221	C₂×C₃	#547	1	1	yes	characteristic
222	C₂×C₃	C₂²×SD₁₆	1	1	yes	characteristic
223	C₂×C₃	#577	1	1	yes	characteristic
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	4	1	no	—
227	C₂²	—	4	1	no	—
228	C₂²	—	4	1	no	—
229	C₂²	C₂×C₃×SD₁₆	1	4	yes	—
230	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
231	C₄	—	2	1	no	—
232	C₄	—	2	2	no	—
233	C₄	—	4	1	no	—
234	C₄	—	4	2	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	1	no	—
238	C₄	C₂×C₃×SD₁₆	1	2	yes	—
239	C₄	C₂×C₃×#41	1	1	yes	characteristic
240	C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
241	C₃	#3335	1	1	yes	characteristic, direct product factor, Sylow
242	C₂	—	2	4	no	—
243	C₂	—	8	1	no	—
244	C₂	C₃×#547	1	1	yes	characteristic
245	C₂	C₂²×C₃×SD₁₆	1	1	yes	characteristic
246	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	C₃×#547	C₂
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
1. C₃×#554	C₂	semidirect
3. C₃×#498	C₂	semidirect
4. C₃×#483	C₂	semidirect
5. C₃×#481	C₂	semidirect
6. C₃×#476	C₂	semidirect
7. C₃×#475	C₂	semidirect
8. C₃×#465	C₂	semidirect
9. C₃×C₄×SD₁₆	C₂	semidirect
10. C₃×C₈×D₄	C₂	semidirect
11. C₂×C₃×#106	C₂	semidirect
12. #3335	C₃	direct (N × H)
42. #554	C₂×C₃	semidirect
44. #498	C₂×C₃	semidirect
45. #483	C₂×C₃	semidirect
46. #481	C₂×C₃	semidirect
47. #476	C₂×C₃	semidirect
48. #475	C₂×C₃	semidirect
49. #465	C₂×C₃	semidirect
50. C₄×SD₁₆	C₂×C₃	semidirect
51. C₈×D₄	C₂×C₃	semidirect
52. C₂×#106	C₂×C₃	semidirect
124. C₃×D₄	SD₁₆	semidirect
141. C₃×C₈	#41	semidirect
201. D₄	C₃×SD₁₆	semidirect
218. C₈	C₃×#41	semidirect
241. C₃	#3335	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×#554	2
C₃×#547	2
C₃×#498	2
C₃×#483	2
C₃×#481	2
C₃×#476	2
C₃×#475	2
C₃×#465	2
C₃×C₄×SD₁₆	2
C₃×C₈×D₄	2
C₂×C₃×#106	2
#3335	3

Subgroups — C3×#3335 #85199

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3335 #85199