Subgroups — C₃×#3333 #85197

Order 384.

398 conjugacy classes of subgroups.

The subgroup lattice has 258 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×#550	C₂	1	1	yes	maximal, characteristic
3	C₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#480	C₂	1	2	yes	maximal, complemented
5	C₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#468	C₂	1	2	yes	maximal, complemented
7	C₃×#460	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#452	C₂	1	2	yes	maximal, complemented
9	C₃×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#417	C₂	1	2	yes	maximal, complemented
12	#3333	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×#41	C₂²	1	2	yes	—
14	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
16	C₂×C₃×M₁₆	C₂²	1	2	yes	—
17	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₃×#127	—	2	1	no	—
19	C₃×#127	C₂²	1	1	yes	characteristic
20	C₃×#125	—	2	1	no	—
21	C₃×C₂²⋊Q₈	—	2	2	no	—
22	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
23	C₃×#121	—	2	2	no	—
24	C₃×#121	C₂²	1	2	yes	—
25	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
26	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
27	C₃×C₄×D₄	—	2	1	no	—
28	C₃×C₄×D₄	—	2	1	no	—
29	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
30	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
31	C₃×#107	C₂²	1	1	yes	characteristic
32	C₃×#107	C₂²	1	2	yes	—
33	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₃×#103	C₂²	1	1	yes	characteristic
35	C₃×#103	C₂²	1	2	yes	—
36	C₃×#103	C₂²	1	1	yes	characteristic
37	C₃×#102	C₂²	1	2	yes	—
38	C₃×#102	C₂²	1	2	yes	—
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	#550	C₂×C₃	1	1	yes	characteristic
44	#486	C₂×C₃	1	1	yes	characteristic, complemented
45	#480	C₂×C₃	1	2	yes	complemented
46	#477	C₂×C₃	1	1	yes	characteristic, complemented
47	#468	C₂×C₃	1	2	yes	complemented
48	#460	C₂×C₃	1	1	yes	characteristic, complemented
49	#452	C₂×C₃	1	2	yes	complemented
50	C₄×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
51	#436	C₂×C₃	1	1	yes	characteristic, complemented
52	#417	C₂×C₃	1	2	yes	complemented
53	C₃×#41	—	2	4	no	—
54	C₃×#41	—	4	2	no	—
55	C₂×C₃×Q₈	—	2	1	no	—
56	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
57	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
58	C₂×C₃×D₄	—	2	2	no	—
59	C₂×C₃×D₄	—	2	2	no	—
60	C₂×C₃×D₄	—	2	1	no	—
61	C₂×C₃×D₄	—	2	1	no	—
62	C₂×C₃×D₄	D₄	1	1	yes	characteristic
63	C₂×C₃×D₄	C₂³	1	2	yes	—
64	C₂²×C₃×C₄	—	2	2	no	—
65	C₂²×C₃×C₄	—	2	2	no	—
66	C₂²×C₃×C₄	C₂³	1	2	yes	—
67	C₃×Q₁₆	—	2	2	no	—
68	C₃×SD₁₆	—	4	2	no	—
69	C₃×M₁₆	—	4	2	no	—
70	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
71	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
72	C₂×C₃×C₈	C₂³	1	2	yes	—
73	C₃×C₄⋊C₄	—	2	2	no	—
74	C₃×C₄⋊C₄	—	2	1	no	—
75	C₃×C₄⋊C₄	—	2	1	no	—
76	C₃×C₄⋊C₄	—	2	1	no	—
77	C₃×C₄⋊C₄	—	2	2	no	—
78	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
79	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
80	C₃×C₄⋊C₄	C₂³	1	2	yes	—
81	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
82	C₃×C₂²⋊C₄	—	2	2	no	—
83	C₃×C₂²⋊C₄	—	2	2	no	—
84	C₃×C₂²⋊C₄	D₄	1	2	yes	—
85	C₃×C₄²	—	2	1	no	—
86	C₃×C₄²	—	2	1	no	—
87	C₃×C₄²	C₂³	1	1	yes	characteristic
88	C₂×#41	C₂²×C₃	1	2	yes	—
89	C₂×Q₁₆	C₂²×C₃	1	1	yes	characteristic
90	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
91	C₂×M₁₆	C₂²×C₃	1	2	yes	—
92	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
93	#127	—	2	1	no	—
94	#127	C₂²×C₃	1	1	yes	characteristic
95	#125	—	2	1	no	—
96	C₂²⋊Q₈	—	2	2	no	—
97	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
98	#121	—	2	2	no	—
99	#121	C₂²×C₃	1	2	yes	—
100	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
101	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
102	C₄×D₄	—	2	1	no	—
103	C₄×D₄	—	2	1	no	—
104	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
105	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
106	#107	C₂²×C₃	1	1	yes	characteristic
107	#107	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	#103	C₂²×C₃	1	1	yes	characteristic
112	#102	C₂²×C₃	1	2	yes	—
113	#102	C₂²×C₃	1	2	yes	—
114	#102	C₂²×C₃	1	2	yes	—
115	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
116	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
117	C₂³×C₃	—	2	2	no	—
118	C₂³×C₃	C₂×D₄	1	2	yes	—
119	C₃×Q₈	—	2	1	no	—
120	C₃×Q₈	—	2	1	no	—
121	C₃×Q₈	—	2	1	no	—
122	C₃×Q₈	—	8	1	no	—
123	C₃×Q₈	#41	1	2	yes	—
124	C₃×D₄	—	2	2	no	—
125	C₃×D₄	—	2	2	no	—
126	C₃×D₄	—	2	2	no	—
127	C₃×D₄	—	4	4	no	—
128	C₃×D₄	—	4	2	no	—
129	C₃×D₄	—	8	1	no	—
130	C₃×D₄	—	8	2	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	—	4	4	no	—
139	C₂×C₃×C₄	—	8	2	no	—
140	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
142	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
144	C₃×C₈	—	2	1	no	—
145	C₃×C₈	—	2	1	no	—
146	C₃×C₈	—	2	2	no	—
147	#41	—	2	4	no	—
148	#41	—	4	2	no	—
149	C₂×Q₈	—	2	1	no	—
150	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
151	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
152	C₂×D₄	—	2	2	no	—
153	C₂×D₄	—	2	2	no	—
154	C₂×D₄	—	2	1	no	—
155	C₂×D₄	—	2	1	no	—
156	C₂×D₄	C₃×D₄	1	1	yes	characteristic
157	C₂×D₄	C₂³×C₃	1	2	yes	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	C₂³×C₃	1	2	yes	—
161	Q₁₆	—	2	2	no	—
162	SD₁₆	—	4	2	no	—
163	M₁₆	—	4	2	no	—
164	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
165	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
166	C₂×C₈	C₂³×C₃	1	2	yes	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
173	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
174	C₄⋊C₄	C₂³×C₃	1	2	yes	—
175	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
176	C₂²⋊C₄	—	2	2	no	—
177	C₂²⋊C₄	—	2	2	no	—
178	C₂²⋊C₄	C₃×D₄	1	2	yes	—
179	C₄²	—	2	1	no	—
180	C₄²	—	2	1	no	—
181	C₄²	C₂³×C₃	1	1	yes	characteristic
182	C₂²×C₃	—	2	2	no	—
183	C₂²×C₃	—	2	2	no	—
184	C₂²×C₃	—	2	2	no	—
185	C₂²×C₃	—	4	2	no	—
186	C₂²×C₃	—	4	2	no	—
187	C₂²×C₃	—	4	2	no	—
188	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
189	C₃×C₄	—	2	2	no	—
190	C₃×C₄	—	2	2	no	—
191	C₃×C₄	—	2	1	no	—
192	C₃×C₄	—	4	1	no	—
193	C₃×C₄	—	4	1	no	—
194	C₃×C₄	—	4	2	no	—
195	C₃×C₄	—	4	1	no	—
196	C₃×C₄	#137	1	2	yes	—
197	C₃×C₄	C₂×#41	1	1	yes	characteristic
198	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
199	C₂³	—	2	2	no	—
200	C₂³	C₂×C₃×D₄	1	2	yes	—
201	Q₈	—	2	1	no	—
202	Q₈	—	2	1	no	—
203	Q₈	—	2	1	no	—
204	Q₈	—	8	1	no	—
205	Q₈	C₃×#41	1	2	yes	—
206	D₄	—	2	2	no	—
207	D₄	—	2	2	no	—
208	D₄	—	2	2	no	—
209	D₄	—	4	4	no	—
210	D₄	—	4	2	no	—
211	D₄	—	8	1	no	—
212	D₄	—	8	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	8	2	no	—
222	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
223	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
224	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
225	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
226	C₈	—	2	1	no	—
227	C₈	—	2	1	no	—
228	C₈	—	2	2	no	—
229	C₂×C₃	—	4	2	no	—
230	C₂×C₃	—	8	2	no	—
231	C₂×C₃	#546	1	1	yes	characteristic
232	C₂×C₃	C₂×#137	1	1	yes	characteristic
233	C₂×C₃	#576	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	4	1	no	—
245	C₄	—	4	2	no	—
246	C₄	—	4	1	no	—
247	C₄	—	4	1	no	—
248	C₄	C₃×#137	1	2	yes	—
249	C₄	C₂×C₃×#41	1	1	yes	characteristic
250	C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
251	C₃	#3333	1	1	yes	characteristic, direct product factor, Sylow
252	C₂	—	4	2	no	—
253	C₂	—	8	2	no	—
254	C₂	C₃×#546	1	1	yes	characteristic
255	C₂	C₂×C₃×#137	1	1	yes	characteristic
256	C₂	C₃×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₃×#550	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×#486	C₂	semidirect
4. C₃×#480	C₂	semidirect
5. C₃×#477	C₂	semidirect
6. C₃×#468	C₂	semidirect
7. C₃×#460	C₂	semidirect
8. C₃×#452	C₂	semidirect
9. C₃×C₄×Q₁₆	C₂	semidirect
10. C₃×#436	C₂	semidirect
11. C₃×#417	C₂	semidirect
12. #3333	C₃	direct (N × H)
42. #554	C₂×C₃	semidirect
44. #486	C₂×C₃	semidirect
45. #480	C₂×C₃	semidirect
46. #477	C₂×C₃	semidirect
47. #468	C₂×C₃	semidirect
48. #460	C₂×C₃	semidirect
49. #452	C₂×C₃	semidirect
50. C₄×Q₁₆	C₂×C₃	semidirect
51. #436	C₂×C₃	semidirect
52. #417	C₂×C₃	semidirect
251. C₃	#3333	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×#550	2
C₃×#486	2
C₃×#480	2
C₃×#477	2
C₃×#468	2
C₃×#460	2
C₃×#452	2
C₃×C₄×Q₁₆	2
C₃×#436	2
C₃×#417	2
#3333	3

Subgroups — C3×#3333 #85197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3333 #85197