Subgroups — C₃×#3044 #84908

Order 384.

458 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#137	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#536	C₂	1	1	yes	maximal, characteristic
4	C₃×#519	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#464	C₂	1	4	yes	maximal, complemented
6	C₃×#462	C₂	1	4	yes	maximal, complemented
7	C₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#419	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#416	C₂	1	1	yes	maximal, characteristic
10	#3044	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₃×#41	C₂²	1	1	yes	characteristic
12	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
14	C₃×#137	—	2	4	no	—
15	C₂×C₃×Q₁₆	—	2	4	no	—
16	C₂×C₃×Q₁₆	C₂²	1	4	yes	—
17	C₂×C₃×Q₁₆	C₂²	1	2	yes	—
18	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
19	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
20	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
21	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
22	C₃×#124	C₂²	1	2	yes	—
23	C₃×#123	—	2	2	no	—
24	C₃×C₂²⋊Q₈	—	2	2	no	—
25	C₃×C₄×Q₈	—	2	2	no	—
26	C₃×C₄×Q₈	C₂²	1	4	yes	—
27	C₃×#117	—	2	2	no	—
28	C₃×#117	C₂²	1	1	yes	characteristic
29	C₃×#117	C₂²	1	1	yes	characteristic
30	C₃×C₄⋊C₈	C₂²	1	4	yes	—
31	C₃×#103	C₂²	1	2	yes	—
32	C₃×#103	C₂²	1	4	yes	—
33	C₃×#102	C₂²	1	2	yes	—
34	C₂×#137	C₂×C₃	1	1	yes	characteristic
35	C₂²×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
36	#536	C₂×C₃	1	1	yes	characteristic
37	#519	C₂×C₃	1	1	yes	characteristic, complemented
38	#464	C₂×C₃	1	4	yes	complemented
39	#462	C₂×C₃	1	4	yes	complemented
40	#424	C₂×C₃	1	1	yes	characteristic, complemented
41	#419	C₂×C₃	1	1	yes	characteristic, complemented
42	#416	C₂×C₃	1	1	yes	characteristic
43	C₃×#41	—	2	2	no	—
44	C₃×#41	—	2	2	no	—
45	C₂×C₃×Q₈	—	2	1	no	—
46	C₂×C₃×Q₈	—	2	1	no	—
47	C₂×C₃×Q₈	—	2	2	no	—
48	C₂×C₃×Q₈	—	2	2	no	—
49	C₂×C₃×Q₈	—	2	2	no	—
50	C₂×C₃×Q₈	D₄	1	4	yes	—
51	C₂×C₃×Q₈	C₂³	1	2	yes	—
52	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
53	C₂×C₃×Q₈	C₂³	1	2	yes	—
54	C₂×C₃×D₄	—	2	1	no	—
55	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
56	C₂²×C₃×C₄	—	2	1	no	—
57	C₂²×C₃×C₄	—	2	1	no	—
58	C₂²×C₃×C₄	—	2	1	no	—
59	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
60	C₃×Q₁₆	—	2	8	no	—
61	C₃×Q₁₆	—	2	4	no	—
62	C₃×SD₁₆	—	2	4	no	—
63	C₃×M₁₆	—	2	2	no	—
64	C₂×C₃×C₈	—	2	2	no	—
65	C₂×C₃×C₈	C₂³	1	2	yes	—
66	C₂×C₃×C₈	C₂³	1	2	yes	—
67	C₃×C₄⋊C₄	—	2	4	no	—
68	C₃×C₄⋊C₄	—	2	4	no	—
69	C₃×C₄⋊C₄	D₄	1	2	yes	—
70	C₃×C₄⋊C₄	C₂³	1	2	yes	—
71	C₃×C₂²⋊C₄	—	2	1	no	—
72	C₃×C₂²⋊C₄	—	2	4	no	—
73	C₃×C₂²⋊C₄	D₄	1	2	yes	—
74	C₃×C₄²	—	2	4	no	—
75	C₃×C₄²	—	2	1	no	—
76	C₃×C₄²	C₂³	1	2	yes	—
77	C₂×#41	C₂²×C₃	1	1	yes	characteristic
78	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
79	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
80	#137	—	2	4	no	—
81	C₂×Q₁₆	—	2	4	no	—
82	C₂×Q₁₆	C₂²×C₃	1	4	yes	—
83	C₂×Q₁₆	C₂²×C₃	1	2	yes	—
84	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
85	C₂×M₁₆	C₂²×C₃	1	1	yes	characteristic
86	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
87	C₄⋊Q₈	C₂²×C₃	1	2	yes	—
88	#124	C₂²×C₃	1	2	yes	—
89	#123	—	2	2	no	—
90	C₂²⋊Q₈	—	2	2	no	—
91	C₄×Q₈	—	2	2	no	—
92	C₄×Q₈	C₂²×C₃	1	4	yes	—
93	#117	—	2	2	no	—
94	#117	C₂²×C₃	1	1	yes	characteristic
95	#117	C₂²×C₃	1	1	yes	characteristic
96	C₄⋊C₈	C₂²×C₃	1	4	yes	—
97	#103	C₂²×C₃	1	2	yes	—
98	#103	C₂²×C₃	1	4	yes	—
99	#102	C₂²×C₃	1	2	yes	—
100	C₂³×C₃	—	2	1	no	—
101	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
102	C₃×Q₈	—	2	4	no	—
103	C₃×Q₈	—	2	2	no	—
104	C₃×Q₈	—	2	2	no	—
105	C₃×Q₈	—	2	1	no	—
106	C₃×Q₈	—	2	2	no	—
107	C₃×Q₈	—	2	1	no	—
108	C₃×Q₈	—	4	2	no	—
109	C₃×Q₈	C₂×D₄	1	4	yes	—
110	C₃×D₄	—	2	1	no	—
111	C₃×D₄	—	2	1	no	—
112	C₃×D₄	—	4	2	no	—
113	C₂×C₃×C₄	—	2	4	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	—	2	1	no	—
118	C₂×C₃×C₄	—	4	2	no	—
119	C₂×C₃×C₄	—	4	2	no	—
120	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
121	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
122	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
123	C₂×C₃×C₄	#41	1	2	yes	—
124	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
125	C₃×C₈	—	2	2	no	—
126	C₃×C₈	—	2	2	no	—
127	#41	—	2	2	no	—
128	#41	—	2	2	no	—
129	C₂×Q₈	—	2	1	no	—
130	C₂×Q₈	—	2	1	no	—
131	C₂×Q₈	—	2	2	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂×Q₈	—	2	2	no	—
134	C₂×Q₈	C₃×D₄	1	4	yes	—
135	C₂×Q₈	C₂³×C₃	1	2	yes	—
136	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
137	C₂×Q₈	C₂³×C₃	1	2	yes	—
138	C₂×D₄	—	2	1	no	—
139	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	2	1	no	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
144	Q₁₆	—	2	8	no	—
145	Q₁₆	—	2	4	no	—
146	SD₁₆	—	2	4	no	—
147	M₁₆	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	C₂³×C₃	1	2	yes	—
150	C₂×C₈	C₂³×C₃	1	2	yes	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	C₃×D₄	1	2	yes	—
154	C₄⋊C₄	C₂³×C₃	1	2	yes	—
155	C₂²⋊C₄	—	2	1	no	—
156	C₂²⋊C₄	—	2	4	no	—
157	C₂²⋊C₄	C₃×D₄	1	2	yes	—
158	C₄²	—	2	4	no	—
159	C₄²	—	2	1	no	—
160	C₄²	C₂³×C₃	1	2	yes	—
161	C₂²×C₃	—	2	2	no	—
162	C₂²×C₃	—	4	1	no	—
163	C₂²×C₃	—	4	2	no	—
164	C₂²×C₃	#121	1	2	yes	—
165	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
166	C₃×C₄	—	2	4	no	—
167	C₃×C₄	—	2	4	no	—
168	C₃×C₄	—	4	1	no	—
169	C₃×C₄	—	4	2	no	—
170	C₃×C₄	—	4	2	no	—
171	C₃×C₄	#121	1	2	yes	—
172	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
173	C₃×C₄	C₂×#41	1	1	yes	characteristic
174	C₂³	—	2	1	no	—
175	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
176	Q₈	—	2	4	no	—
177	Q₈	—	2	2	no	—
178	Q₈	—	2	2	no	—
179	Q₈	—	2	1	no	—
180	Q₈	—	2	2	no	—
181	Q₈	—	2	1	no	—
182	Q₈	—	4	2	no	—
183	Q₈	C₂×C₃×D₄	1	4	yes	—
184	D₄	—	2	1	no	—
185	D₄	—	2	1	no	—
186	D₄	—	4	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
195	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
196	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
197	C₂×C₄	C₃×#41	1	2	yes	—
198	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
199	C₈	—	2	2	no	—
200	C₈	—	2	2	no	—
201	C₂×C₃	—	2	2	no	—
202	C₂×C₃	—	8	1	no	—
203	C₂×C₃	C₂×#121	1	1	yes	characteristic
204	C₂×C₃	#578	1	2	yes	—
205	C₂²	—	2	2	no	—
206	C₂²	—	4	2	no	—
207	C₂²	—	4	1	no	—
208	C₂²	C₃×#121	1	2	yes	—
209	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
210	C₄	—	2	4	no	—
211	C₄	—	2	4	no	—
212	C₄	—	4	2	no	—
213	C₄	—	4	1	no	—
214	C₄	—	4	2	no	—
215	C₄	C₃×#121	1	2	yes	—
216	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
217	C₄	C₂×C₃×#41	1	1	yes	characteristic
218	C₃	#3044	1	1	yes	characteristic, direct product factor, Sylow
219	C₂	—	2	2	no	—
220	C₂	—	8	1	no	—
221	C₂	C₂×C₃×#121	1	1	yes	characteristic
222	C₂	C₃×#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₂×C₃×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×Q₁₆	C₂	semidirect
4. C₃×#519	C₂	semidirect
5. C₃×#464	C₂	semidirect
6. C₃×#462	C₂	semidirect
7. C₃×#424	C₂	semidirect
8. C₃×#419	C₂	semidirect
10. #3044	C₃	direct (N × H)
35. C₂²×Q₁₆	C₂×C₃	semidirect
37. #519	C₂×C₃	semidirect
38. #464	C₂×C₃	semidirect
39. #462	C₂×C₃	semidirect
40. #424	C₂×C₃	semidirect
41. #419	C₂×C₃	semidirect
218. C₃	#3044	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×#137	2
C₂²×C₃×Q₁₆	2
C₃×#536	2
C₃×#519	2
C₃×#464	2
C₃×#462	2
C₃×#424	2
C₃×#419	2
C₃×#416	2
#3044	3

Subgroups — C3×#3044 #84908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3044 #84908