Subgroups — C₃×#3354 #85218

Order 384.

386 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#555	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#547	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#482	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×#474	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#469	C₂	1	2	yes	maximal, complemented
9	C₃×C₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#428	C₂	1	2	yes	maximal, complemented
12	#3354	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×#41	—	2	1	no	—
14	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×M₁₆	C₂²	1	2	yes	—
16	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₃×#124	—	2	2	no	—
19	C₃×#123	—	2	2	no	—
20	C₃×C₂²⋊Q₈	—	2	1	no	—
21	C₃×#121	—	2	1	no	—
22	C₃×#121	C₂²	1	2	yes	—
23	C₃×#121	C₂²	1	2	yes	—
24	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
25	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
26	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
27	C₃×#117	—	2	1	no	—
28	C₃×#117	C₂²	1	2	yes	—
29	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
30	C₃×#107	C₂²	1	1	yes	characteristic
31	C₃×#107	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×#102	C₂²	1	1	yes	characteristic
37	C₃×#102	C₂²	1	1	yes	characteristic
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	#555	C₂×C₃	1	1	yes	characteristic, complemented
43	#547	C₂×C₃	1	1	yes	characteristic, complemented
44	#498	C₂×C₃	1	1	yes	characteristic, complemented
45	#482	C₂×C₃	1	2	yes	complemented
46	#481	C₂×C₃	1	2	yes	complemented
47	#476	C₂×C₃	1	1	yes	characteristic, complemented
48	#474	C₂×C₃	1	1	yes	characteristic, complemented
49	#469	C₂×C₃	1	2	yes	complemented
50	C₄×D₈	C₂×C₃	1	1	yes	characteristic, complemented
51	#436	C₂×C₃	1	1	yes	characteristic, complemented
52	#428	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×D₄	—	2	2	no	—
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	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
61	C₂²×C₃×C₄	—	2	2	no	—
62	C₂²×C₃×C₄	—	2	1	no	—
63	C₂²×C₃×C₄	—	2	1	no	—
64	C₂²×C₃×C₄	C₂³	1	2	yes	—
65	C₃×D₈	—	2	2	no	—
66	C₃×M₁₆	—	2	4	no	—
67	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
68	C₂×C₃×C₈	C₂³	1	2	yes	—
69	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
74	C₃×C₄⋊C₄	C₂³	1	2	yes	—
75	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
76	C₃×C₄⋊C₄	C₂³	1	2	yes	—
77	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
78	C₃×C₂²⋊C₄	—	2	2	no	—
79	C₃×C₂²⋊C₄	—	2	2	no	—
80	C₃×C₂²⋊C₄	—	2	1	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	2	no	—
85	C₃×C₄²	C₂³	1	1	yes	characteristic
86	C₂×#41	—	2	1	no	—
87	C₂×D₈	C₂²×C₃	1	1	yes	characteristic
88	C₂×M₁₆	C₂²×C₃	1	2	yes	—
89	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
90	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
91	#124	—	2	2	no	—
92	#123	—	2	2	no	—
93	C₂²⋊Q₈	—	2	1	no	—
94	#121	—	2	1	no	—
95	#121	C₂²×C₃	1	2	yes	—
96	#121	C₂²×C₃	1	2	yes	—
97	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
98	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
99	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
100	#117	—	2	1	no	—
101	#117	C₂²×C₃	1	2	yes	—
102	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
103	#107	C₂²×C₃	1	1	yes	characteristic
104	#107	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	#102	C₂²×C₃	1	1	yes	characteristic
110	#102	C₂²×C₃	1	1	yes	characteristic
111	#102	C₂²×C₃	1	2	yes	—
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	—	2	1	no	—
117	C₂³×C₃	C₂×D₄	1	2	yes	—
118	C₃×Q₈	—	4	2	no	—
119	C₃×Q₈	—	8	1	no	—
120	C₃×D₄	—	2	2	no	—
121	C₃×D₄	—	2	1	no	—
122	C₃×D₄	—	2	1	no	—
123	C₃×D₄	—	2	1	no	—
124	C₃×D₄	—	2	1	no	—
125	C₃×D₄	—	4	4	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	1	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	4	4	no	—
134	C₂×C₃×C₄	—	4	2	no	—
135	C₂×C₃×C₄	—	4	2	no	—
136	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
137	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
140	C₃×C₈	—	2	2	no	—
141	C₃×C₈	—	2	1	no	—
142	C₃×C₈	#41	1	2	yes	complemented
143	#41	—	4	4	no	—
144	C₂×Q₈	—	2	1	no	—
145	C₂×Q₈	—	2	1	no	—
146	C₂×D₄	—	2	2	no	—
147	C₂×D₄	—	2	2	no	—
148	C₂×D₄	C₃×D₄	1	1	yes	characteristic
149	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
150	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	—	2	1	no	—
153	C₂²×C₄	—	2	1	no	—
154	C₂²×C₄	C₂³×C₃	1	2	yes	—
155	D₈	—	2	2	no	—
156	M₁₆	—	2	4	no	—
157	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
158	C₂×C₈	C₂³×C₃	1	2	yes	—
159	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	1	no	—
162	C₄⋊C₄	—	2	1	no	—
163	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
164	C₄⋊C₄	C₂³×C₃	1	2	yes	—
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	C₂²⋊C₄	—	2	2	no	—
169	C₂²⋊C₄	—	2	2	no	—
170	C₂²⋊C₄	—	2	1	no	—
171	C₂²⋊C₄	—	2	1	no	—
172	C₂²⋊C₄	—	2	2	no	—
173	C₂²⋊C₄	C₃×D₄	1	2	yes	—
174	C₄²	—	2	2	no	—
175	C₄²	C₂³×C₃	1	1	yes	characteristic
176	C₂²×C₃	—	2	2	no	—
177	C₂²×C₃	—	2	2	no	—
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	—	4	1	no	—
180	C₂²×C₃	—	4	1	no	—
181	C₂²×C₃	—	4	1	no	—
182	C₂²×C₃	—	4	1	no	—
183	C₂²×C₃	—	4	1	no	—
184	C₂²×C₃	—	4	1	no	—
185	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
186	C₃×C₄	—	2	2	no	—
187	C₃×C₄	—	2	1	no	—
188	C₃×C₄	—	4	1	no	—
189	C₃×C₄	—	4	1	no	—
190	C₃×C₄	—	4	2	no	—
191	C₃×C₄	—	4	2	no	—
192	C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
193	C₃×C₄	C₂×#41	1	1	yes	characteristic
194	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
195	C₂³	—	2	1	no	—
196	C₂³	—	2	1	no	—
197	C₂³	C₂×C₃×D₄	1	2	yes	—
198	Q₈	—	4	2	no	—
199	Q₈	—	8	1	no	—
200	D₄	—	2	2	no	—
201	D₄	—	2	1	no	—
202	D₄	—	2	1	no	—
203	D₄	—	2	1	no	—
204	D₄	—	2	1	no	—
205	D₄	—	4	4	no	—
206	D₄	—	8	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
217	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
218	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
219	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	2	1	no	—
222	C₈	C₃×#41	1	2	yes	complemented
223	C₂×C₃	—	4	2	no	—
224	C₂×C₃	—	8	1	no	—
225	C₂×C₃	—	8	1	no	—
226	C₂×C₃	#547	1	1	yes	characteristic
227	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
228	C₂×C₃	#577	1	1	yes	characteristic
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	4	1	no	—
233	C₂²	—	4	1	no	—
234	C₂²	—	4	1	no	—
235	C₂²	—	4	1	no	—
236	C₂²	—	4	1	no	—
237	C₂²	—	4	1	no	—
238	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
239	C₄	—	2	1	no	—
240	C₄	—	2	2	no	—
241	C₄	—	4	1	no	—
242	C₄	—	4	2	no	—
243	C₄	—	4	1	no	—
244	C₄	—	4	2	no	—
245	C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
246	C₄	C₂×C₃×#41	1	1	yes	characteristic
247	C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
248	C₃	#3354	1	1	yes	characteristic, direct product factor, Sylow
249	C₂	—	4	2	no	—
250	C₂	—	8	1	no	—
251	C₂	—	8	1	no	—
252	C₂	C₃×#547	1	1	yes	characteristic
253	C₂	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
254	C₂	C₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
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×#555	C₂	semidirect
2. C₃×#547	C₂	semidirect
3. C₃×#498	C₂	semidirect
4. C₃×#482	C₂	semidirect
5. C₃×#481	C₂	semidirect
6. C₃×#476	C₂	semidirect
7. C₃×#474	C₂	semidirect
8. C₃×#469	C₂	semidirect
9. C₃×C₄×D₈	C₂	semidirect
10. C₃×#436	C₂	semidirect
11. C₃×#428	C₂	semidirect
12. #3354	C₃	direct (N × H)
42. #555	C₂×C₃	semidirect
43. #547	C₂×C₃	semidirect
44. #498	C₂×C₃	semidirect
45. #482	C₂×C₃	semidirect
46. #481	C₂×C₃	semidirect
47. #476	C₂×C₃	semidirect
48. #474	C₂×C₃	semidirect
49. #469	C₂×C₃	semidirect
50. C₄×D₈	C₂×C₃	semidirect
51. #436	C₂×C₃	semidirect
52. #428	C₂×C₃	semidirect
142. C₃×C₈	#41	semidirect
222. C₈	C₃×#41	semidirect
248. C₃	#3354	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×#555	2
C₃×#547	2
C₃×#498	2
C₃×#482	2
C₃×#481	2
C₃×#476	2
C₃×#474	2
C₃×#469	2
C₃×C₄×D₈	2
C₃×#436	2
C₃×#428	2
#3354	3

Subgroups — C3×#3354 #85218

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3354 #85218