Subgroups — C₃×#2968 #84832

Order 384.

372 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	C₃×#529	C₂	1	1	yes	maximal, characteristic
2	C₃×#446	C₂	1	2	yes	maximal, complemented
3	C₃×#435	C₂	1	2	yes	maximal, complemented
4	C₃×#435	C₂	1	2	yes	maximal, complemented
5	C₃×C₈×D₄	C₂	1	2	yes	maximal, complemented
6	C₃×#433	C₂	1	2	yes	maximal, complemented
7	C₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#405	C₂	1	2	yes	maximal, complemented
9	C₂×C₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
10	#2968	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₃×M₁₆	C₂²	1	2	yes	—
12	C₂²×C₃×C₈	C₂²	1	2	yes	—
13	C₂²×C₃×C₈	C₂²	1	2	yes	—
14	C₃×#126	C₄	1	2	yes	—
15	C₃×#125	C₄	1	1	yes	characteristic
16	C₃×#124	C₄	1	1	yes	characteristic
17	C₃×#123	C₄	1	2	yes	—
18	C₃×C₂²⋊Q₈	C₄	1	1	yes	characteristic
19	C₃×#121	C₄	1	1	yes	characteristic
20	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
21	C₃×C₄×D₄	C₂²	1	2	yes	—
22	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
23	C₃×#117	C₂²	1	2	yes	—
24	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
25	C₃×C₄⋊C₈	C₂²	1	2	yes	—
26	C₃×C₄⋊C₈	C₂²	1	2	yes	—
27	C₃×C₄⋊C₈	C₂²	1	2	yes	—
28	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
29	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
30	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
31	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
32	C₃×#97	C₂²	1	1	yes	characteristic
33	C₃×#97	C₂²	1	1	yes	characteristic
34	C₃×#97	C₂²	1	2	yes	—
35	C₃×#97	C₂²	1	2	yes	—
36	C₃×C₄×C₈	C₂²	1	2	yes	—
37	C₃×C₄×C₈	C₂²	1	2	yes	—
38	#529	C₂×C₃	1	1	yes	characteristic
39	#446	C₂×C₃	1	2	yes	complemented
40	#435	C₂×C₃	1	2	yes	complemented
41	#435	C₂×C₃	1	2	yes	complemented
42	C₈×D₄	C₂×C₃	1	2	yes	complemented
43	#433	C₂×C₃	1	2	yes	complemented
44	#431	C₂×C₃	1	1	yes	characteristic, complemented
45	#405	C₂×C₃	1	2	yes	complemented
46	C₂×#97	C₂×C₃	1	1	yes	characteristic, complemented
47	C₂×C₃×Q₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₃×D₄	C₂×C₄	1	2	yes	—
49	C₂×C₃×D₄	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
53	C₂²×C₃×C₄	C₂³	1	2	yes	—
54	C₃×M₁₆	—	2	4	no	—
55	C₂×C₃×C₈	—	2	4	no	—
56	C₂×C₃×C₈	—	2	4	no	—
57	C₂×C₃×C₈	C₂³	1	2	yes	—
58	C₂×C₃×C₈	C₂³	1	2	yes	—
59	C₂×C₃×C₈	C₂³	1	2	yes	—
60	C₂×C₃×C₈	C₂³	1	2	yes	—
61	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
62	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
63	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
64	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
65	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
66	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
67	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
68	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
69	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
70	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
71	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
72	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
73	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
74	C₃×C₄²	C₂³	1	1	yes	characteristic
75	C₃×C₄²	C₂³	1	2	yes	—
76	C₃×C₄²	C₂³	1	1	yes	characteristic
77	C₂×M₁₆	C₂²×C₃	1	2	yes	—
78	C₂²×C₈	C₂²×C₃	1	2	yes	—
79	C₂²×C₈	C₂²×C₃	1	2	yes	—
80	#126	C₃×C₄	1	2	yes	—
81	#125	C₃×C₄	1	1	yes	characteristic
82	#124	C₃×C₄	1	1	yes	characteristic
83	#123	C₃×C₄	1	2	yes	—
84	C₂²⋊Q₈	C₃×C₄	1	1	yes	characteristic
85	#121	C₃×C₄	1	1	yes	characteristic
86	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
87	C₄×D₄	C₂²×C₃	1	2	yes	—
88	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
89	#117	C₂²×C₃	1	2	yes	—
90	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
91	C₄⋊C₈	C₂²×C₃	1	2	yes	—
92	C₄⋊C₈	C₂²×C₃	1	2	yes	—
93	C₄⋊C₈	C₂²×C₃	1	2	yes	—
94	C₂²⋊C₈	C₂²×C₃	1	1	yes	characteristic
95	C₂²⋊C₈	C₂²×C₃	1	1	yes	characteristic
96	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
97	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
98	#97	C₂²×C₃	1	1	yes	characteristic
99	#97	C₂²×C₃	1	1	yes	characteristic
100	#97	C₂²×C₃	1	2	yes	—
101	#97	C₂²×C₃	1	2	yes	—
102	C₄×C₈	C₂²×C₃	1	2	yes	—
103	C₄×C₈	C₂²×C₃	1	2	yes	—
104	C₂³×C₃	C₂²×C₄	1	2	yes	—
105	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
106	C₃×Q₈	—	4	1	no	—
107	C₃×D₄	—	2	4	no	—
108	C₃×D₄	—	4	1	no	—
109	C₂×C₃×C₄	—	2	2	no	—
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	—	2	2	no	—
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	—	4	1	no	—
114	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
120	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
121	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
122	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
123	C₃×C₈	—	2	2	no	—
124	C₃×C₈	—	2	2	no	—
125	C₃×C₈	—	2	2	no	—
126	C₃×C₈	#41	1	4	yes	—
127	C₂×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
128	C₂×D₄	C₂×C₃×C₄	1	2	yes	—
129	C₂×D₄	C₂×C₃×C₄	1	1	yes	characteristic
130	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
132	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
133	C₂²×C₄	C₂³×C₃	1	2	yes	—
134	M₁₆	—	2	4	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	C₂³×C₃	1	2	yes	—
138	C₂×C₈	C₂³×C₃	1	2	yes	—
139	C₂×C₈	C₂³×C₃	1	2	yes	—
140	C₂×C₈	C₂³×C₃	1	2	yes	—
141	C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
142	C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
143	C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
144	C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
145	C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
146	C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
147	C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
148	C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
149	C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
150	C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
151	C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
152	C₄²	C₂×C₃×C₄	1	1	yes	characteristic
153	C₄²	C₂×C₃×C₄	1	1	yes	characteristic
154	C₄²	C₂³×C₃	1	1	yes	characteristic
155	C₄²	C₂³×C₃	1	2	yes	—
156	C₄²	C₂³×C₃	1	1	yes	characteristic
157	C₂²×C₃	—	2	1	no	—
158	C₂²×C₃	—	2	1	no	—
159	C₂²×C₃	—	2	2	no	—
160	C₂²×C₃	—	2	2	no	—
161	C₂²×C₃	—	2	2	no	—
162	C₂²×C₃	#131	1	2	yes	—
163	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
164	C₃×C₄	—	2	1	no	—
165	C₃×C₄	—	2	2	no	—
166	C₃×C₄	—	2	1	no	—
167	C₃×C₄	—	2	1	no	—
168	C₃×C₄	—	2	2	no	—
169	C₃×C₄	—	2	1	no	—
170	C₃×C₄	—	2	1	no	—
171	C₃×C₄	—	2	2	no	—
172	C₃×C₄	C₂×#41	1	2	yes	—
173	C₂³	C₂²×C₃×C₄	1	2	yes	—
174	C₂³	C₂²×C₃×C₄	1	1	yes	characteristic
175	Q₈	—	4	1	no	—
176	D₄	—	2	4	no	—
177	D₄	—	4	1	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	4	1	no	—
183	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
184	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
185	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
186	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
187	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
188	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
189	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
190	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
191	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	—	2	2	no	—
194	C₈	—	2	2	no	—
195	C₈	C₃×#41	1	4	yes	—
196	C₂×C₃	—	2	2	no	—
197	C₂×C₃	—	4	2	no	—
198	C₂×C₃	C₄×#41	1	1	yes	characteristic
199	C₂×C₃	C₂×#131	1	1	yes	characteristic
200	C₂×C₃	#568	1	1	yes	characteristic
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	2	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	2	2	no	—
206	C₂²	C₃×#131	1	2	yes	—
207	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	2	2	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	2	no	—
216	C₄	C₂×C₃×#41	1	2	yes	—
217	C₃	#2968	1	1	yes	characteristic, direct product factor, Sylow
218	C₂	—	2	2	no	—
219	C₂	—	4	2	no	—
220	C₂	C₃×C₄×#41	1	1	yes	characteristic
221	C₂	C₂×C₃×#131	1	1	yes	characteristic
222	C₂	C₃×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂⁴
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂³×C₄
prime radical	C₃×#121	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#446	C₂	semidirect
3. C₃×#435	C₂	semidirect
4. C₃×#435	C₂	semidirect
5. C₃×C₈×D₄	C₂	semidirect
6. C₃×#433	C₂	semidirect
7. C₃×#431	C₂	semidirect
8. C₃×#405	C₂	semidirect
9. C₂×C₃×#97	C₂	semidirect
10. #2968	C₃	direct (N × H)
39. #446	C₂×C₃	semidirect
40. #435	C₂×C₃	semidirect
41. #435	C₂×C₃	semidirect
42. C₈×D₄	C₂×C₃	semidirect
43. #433	C₂×C₃	semidirect
44. #431	C₂×C₃	semidirect
45. #405	C₂×C₃	semidirect
46. C₂×#97	C₂×C₃	semidirect
217. C₃	#2968	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×#529	2
C₃×#446	2
C₃×#435	2
C₃×C₈×D₄	2
C₃×#433	2
C₃×#431	2
C₃×#405	2
C₂×C₃×#97	2
#2968	3

Subgroups — C3×#2968 #84832

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2968 #84832