Subgroups — C₃×#3109 #84973

Order 384.

488 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#481	C₂	1	2	yes	maximal, complemented
4	C₃×#476	C₂	1	2	yes	maximal, complemented
5	C₃×#459	C₂	1	2	yes	maximal, complemented
6	C₃×#447	C₂	1	2	yes	maximal, complemented
7	C₃×#442	C₂	1	4	yes	maximal, complemented
8	C₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
9	#3109	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₃×#41	—	2	1	no	—
11	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
12	C₂³×C₃×C₄	—	2	1	no	—
13	C₂×C₃×D₈	C₂²	1	4	yes	—
14	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
15	C₃×#127	C₂²	1	1	yes	characteristic, complemented
16	C₃×#124	—	2	1	no	—
17	C₃×#121	—	2	1	no	—
18	C₃×#121	C₂²	1	2	yes	—
19	C₃×C₄×D₄	—	2	1	no	—
20	C₃×C₄×D₄	—	2	1	no	—
21	C₃×C₄×D₄	—	2	1	no	—
22	C₃×C₄×D₄	C₂²	1	2	yes	—
23	C₃×C₄×D₄	C₂²	1	2	yes	—
24	C₃×C₄×D₄	C₂²	1	2	yes	complemented
25	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
27	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
28	C₃×#106	C₂²	1	4	yes	complemented
29	C₃×C₄⋊C₈	C₂²	1	2	yes	complemented
30	C₃×#102	C₂²	1	4	yes	—
31	C₃×#102	C₂²	1	4	yes	complemented
32	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₃×#97	C₂²	1	2	yes	complemented
34	#532	C₂×C₃	1	1	yes	characteristic, complemented
35	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic, complemented
36	#481	C₂×C₃	1	2	yes	complemented
37	#476	C₂×C₃	1	2	yes	complemented
38	#459	C₂×C₃	1	2	yes	complemented
39	#447	C₂×C₃	1	2	yes	complemented
40	#442	C₂×C₃	1	4	yes	complemented
41	#432	C₂×C₃	1	1	yes	characteristic, complemented
42	C₂⁴×C₃	—	2	1	no	—
43	C₃×#41	—	4	4	no	—
44	C₂×C₃×Q₈	—	2	1	no	—
45	C₂×C₃×D₄	—	2	4	no	—
46	C₂×C₃×D₄	—	2	2	no	—
47	C₂×C₃×D₄	—	2	1	no	—
48	C₂×C₃×D₄	—	2	1	no	—
49	C₂×C₃×D₄	—	2	1	no	—
50	C₂×C₃×D₄	C₂³	1	2	yes	—
51	C₂×C₃×D₄	C₂³	1	2	yes	—
52	C₂²×C₃×C₄	—	2	1	no	—
53	C₂²×C₃×C₄	—	2	2	no	—
54	C₂²×C₃×C₄	—	2	2	no	—
55	C₂²×C₃×C₄	—	2	2	no	—
56	C₂²×C₃×C₄	—	4	4	no	—
57	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
58	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
60	C₃×D₈	—	2	8	no	—
61	C₂×C₃×C₈	C₂³	1	4	yes	—
62	C₃×C₄⋊C₄	—	2	1	no	—
63	C₃×C₄⋊C₄	—	2	1	no	—
64	C₃×C₄⋊C₄	C₂³	1	2	yes	—
65	C₃×C₄⋊C₄	C₂³	1	2	yes	—
66	C₃×C₂²⋊C₄	—	2	2	no	—
67	C₃×C₂²⋊C₄	—	2	2	no	—
68	C₃×C₂²⋊C₄	—	2	2	no	—
69	C₃×C₂²⋊C₄	—	2	2	no	—
70	C₃×C₄²	D₄	1	1	yes	characteristic
71	C₃×C₄²	D₄	1	1	yes	characteristic
72	C₃×C₄²	C₂³	1	1	yes	characteristic
73	C₃×C₄²	C₂³	1	1	yes	characteristic
74	C₂×#41	—	2	1	no	—
75	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
76	C₂³×C₄	—	2	1	no	—
77	C₂×D₈	C₂²×C₃	1	4	yes	—
78	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
79	#127	C₂²×C₃	1	1	yes	characteristic, complemented
80	#124	—	2	1	no	—
81	#121	—	2	1	no	—
82	#121	C₂²×C₃	1	2	yes	—
83	C₄×D₄	—	2	1	no	—
84	C₄×D₄	—	2	1	no	—
85	C₄×D₄	—	2	1	no	—
86	C₄×D₄	C₂²×C₃	1	2	yes	—
87	C₄×D₄	C₂²×C₃	1	2	yes	—
88	C₄×D₄	C₂²×C₃	1	2	yes	complemented
89	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
90	C₂×C₂²⋊C₄	—	2	1	no	—
91	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
92	#106	C₂²×C₃	1	4	yes	complemented
93	C₄⋊C₈	C₂²×C₃	1	2	yes	complemented
94	#102	C₂²×C₃	1	4	yes	—
95	#102	C₂²×C₃	1	4	yes	complemented
96	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
97	#97	C₂²×C₃	1	2	yes	complemented
98	C₂³×C₃	—	2	4	no	—
99	C₂³×C₃	—	2	2	no	—
100	C₂³×C₃	—	2	2	no	—
101	C₂³×C₃	—	4	2	no	—
102	C₂³×C₃	—	4	2	no	—
103	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
104	C₃×Q₈	—	4	2	no	—
105	C₃×D₄	—	2	4	no	—
106	C₃×D₄	—	2	2	no	—
107	C₃×D₄	—	2	4	no	—
108	C₃×D₄	—	4	2	no	—
109	C₃×D₄	—	4	4	no	—
110	C₃×D₄	—	4	2	no	—
111	C₃×D₄	#41	1	4	yes	complemented
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	—	2	2	no	—
114	C₂×C₃×C₄	—	2	2	no	—
115	C₂×C₃×C₄	—	2	2	no	—
116	C₂×C₃×C₄	—	2	2	no	—
117	C₂×C₃×C₄	—	4	4	no	—
118	C₂×C₃×C₄	—	4	4	no	—
119	C₂×C₃×C₄	—	4	2	no	—
120	C₂×C₃×C₄	—	4	4	no	—
121	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
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	4	no	—
128	C₂⁴	—	2	1	no	—
129	#41	—	4	4	no	—
130	C₂×Q₈	—	2	1	no	—
131	C₂×D₄	—	2	4	no	—
132	C₂×D₄	—	2	2	no	—
133	C₂×D₄	—	2	1	no	—
134	C₂×D₄	—	2	1	no	—
135	C₂×D₄	—	2	1	no	—
136	C₂×D₄	C₂³×C₃	1	2	yes	—
137	C₂×D₄	C₂³×C₃	1	2	yes	—
138	C₂²×C₄	—	2	1	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	—	4	4	no	—
143	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
144	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
145	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
146	D₈	—	2	8	no	—
147	C₂×C₈	C₂³×C₃	1	4	yes	—
148	C₄⋊C₄	—	2	1	no	—
149	C₄⋊C₄	—	2	1	no	—
150	C₄⋊C₄	C₂³×C₃	1	2	yes	—
151	C₄⋊C₄	C₂³×C₃	1	2	yes	—
152	C₂²⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	2	2	no	—
154	C₂²⋊C₄	—	2	2	no	—
155	C₂²⋊C₄	—	2	2	no	—
156	C₄²	C₃×D₄	1	1	yes	characteristic
157	C₄²	C₃×D₄	1	1	yes	characteristic
158	C₄²	C₂³×C₃	1	1	yes	characteristic
159	C₄²	C₂³×C₃	1	1	yes	characteristic
160	C₂²×C₃	—	2	1	no	—
161	C₂²×C₃	—	2	1	no	—
162	C₂²×C₃	—	2	4	no	—
163	C₂²×C₃	—	4	8	no	—
164	C₂²×C₃	—	4	2	no	—
165	C₂²×C₃	—	4	2	no	—
166	C₂²×C₃	—	4	2	no	—
167	C₂²×C₃	—	4	2	no	—
168	C₂²×C₃	—	4	2	no	—
169	C₂²×C₃	AGL(1,ℤ/8)	1	2	yes	—
170	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
171	C₃×C₄	—	2	1	no	—
172	C₃×C₄	—	2	1	no	—
173	C₃×C₄	—	2	1	no	—
174	C₃×C₄	—	2	1	no	—
175	C₃×C₄	—	4	2	no	—
176	C₃×C₄	—	4	2	no	—
177	C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
178	C₃×C₄	C₂×#41	1	2	yes	—
179	C₂³	—	2	4	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	4	2	no	—
183	C₂³	—	4	2	no	—
184	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
185	Q₈	—	4	2	no	—
186	D₄	—	2	4	no	—
187	D₄	—	2	2	no	—
188	D₄	—	2	4	no	—
189	D₄	—	4	2	no	—
190	D₄	—	4	4	no	—
191	D₄	—	4	2	no	—
192	D₄	C₃×#41	1	4	yes	complemented
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
203	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
204	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
205	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
206	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
207	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
208	C₈	—	2	4	no	—
209	C₂×C₃	—	2	2	no	—
210	C₂×C₃	—	4	4	no	—
211	C₂×C₃	—	8	2	no	—
212	C₂×C₃	#525	1	1	yes	characteristic
213	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
214	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
215	C₂²	—	2	1	no	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	4	no	—
218	C₂²	—	4	8	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	4	2	no	—
224	C₂²	C₃×AGL(1,ℤ/8)	1	2	yes	—
225	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	4	2	no	—
231	C₄	—	4	2	no	—
232	C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
233	C₄	C₂×C₃×#41	1	2	yes	—
234	C₃	#3109	1	1	yes	characteristic, direct product factor, Sylow
235	C₂	—	2	2	no	—
236	C₂	—	4	4	no	—
237	C₂	—	8	2	no	—
238	C₂	C₃×#525	1	1	yes	characteristic
239	C₂	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
240	C₂	C₂×C₃×AGL(1,ℤ/8)	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 ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#532	C₂	semidirect
2. C₂×C₃×C₄×D₄	C₂	semidirect
3. C₃×#481	C₂	semidirect
4. C₃×#476	C₂	semidirect
5. C₃×#459	C₂	semidirect
6. C₃×#447	C₂	semidirect
7. C₃×#442	C₂	semidirect
8. C₃×#432	C₂	semidirect
9. #3109	C₃	direct (N × H)
14. C₃×C₄⋊Q₈	C₂²	semidirect
15. C₃×#127	C₂²	semidirect
24. C₃×C₄×D₄	C₂²	semidirect
28. C₃×#106	C₂²	semidirect
29. C₃×C₄⋊C₈	C₂²	semidirect
31. C₃×#102	C₂²	semidirect
33. C₃×#97	C₂²	semidirect
34. #532	C₂×C₃	semidirect
35. C₂×C₄×D₄	C₂×C₃	semidirect
36. #481	C₂×C₃	semidirect
37. #476	C₂×C₃	semidirect
38. #459	C₂×C₃	semidirect
39. #447	C₂×C₃	semidirect
40. #442	C₂×C₃	semidirect
41. #432	C₂×C₃	semidirect
78. C₄⋊Q₈	C₂²×C₃	semidirect
79. #127	C₂²×C₃	semidirect
88. C₄×D₄	C₂²×C₃	semidirect
92. #106	C₂²×C₃	semidirect
93. C₄⋊C₈	C₂²×C₃	semidirect
95. #102	C₂²×C₃	semidirect
97. #97	C₂²×C₃	semidirect
111. C₃×D₄	#41	semidirect
192. D₄	C₃×#41	semidirect
234. C₃	#3109	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×#532	2
C₂×C₃×C₄×D₄	2
C₃×#481	2
C₃×#476	2
C₃×#459	2
C₃×#447	2
C₃×#442	2
C₃×#432	2
#3109	3

Subgroups — C3×#3109 #84973

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3109 #84973