Subgroups — C₃×#1797 #72512

Order 384.

400 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×C₂²⋊C₈	C₂	1	2	yes	maximal
4	C₃×#336	C₂	1	2	yes	maximal, complemented
5	#1797	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂³×C₃×C₄	C₄	1	1	yes	characteristic
7	C₂³×C₃×C₄	C₄	1	1	yes	characteristic
8	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₈	C₂²	1	4	yes	—
10	C₂×C₃×C₄²	—	2	4	no	—
11	C₂×C₃×C₄²	C₄	1	1	yes	characteristic
12	C₂×C₃×C₄²	C₄	1	1	yes	characteristic
13	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
15	C₃×C₄⋊C₈	—	2	4	no	—
16	C₃×C₂²⋊C₈	—	2	4	no	—
17	C₂²×C₄²	C₂×C₃	1	1	yes	characteristic
18	C₂×C₄⋊C₈	C₂×C₃	1	2	yes	complemented
19	C₂×C₂²⋊C₈	C₂×C₃	1	2	yes	—
20	#336	C₂×C₃	1	2	yes	complemented
21	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₃×C₄	—	2	2	no	—
23	C₂²×C₃×C₄	—	2	2	no	—
24	C₂²×C₃×C₄	—	2	2	no	—
25	C₂²×C₃×C₄	—	2	2	no	—
26	C₂²×C₃×C₄	—	2	2	no	—
27	C₂²×C₃×C₄	—	2	2	no	—
28	C₂²×C₃×C₄	—	2	2	no	—
29	C₂²×C₃×C₄	C₈	1	4	yes	complemented
30	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₃×C₄	D₄	1	2	yes	—
36	C₂²×C₃×C₄	Q₈	1	2	yes	—
37	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
38	C₂×C₃×C₈	—	2	4	no	—
39	C₂×C₃×C₈	—	2	4	no	—
40	C₂×C₃×C₈	—	2	4	no	—
41	C₃×C₄²	—	2	2	no	—
42	C₃×C₄²	—	2	2	no	—
43	C₃×C₄²	—	2	2	no	—
44	C₃×C₄²	D₄	1	4	yes	—
45	C₂³×C₄	C₃×C₄	1	1	yes	characteristic
46	C₂³×C₄	C₃×C₄	1	1	yes	characteristic
47	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
48	C₂²×C₈	C₂²×C₃	1	4	yes	—
49	C₂×C₄²	—	2	4	no	—
50	C₂×C₄²	C₃×C₄	1	1	yes	characteristic
51	C₂×C₄²	C₃×C₄	1	1	yes	characteristic
52	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
54	C₄⋊C₈	—	2	4	no	—
55	C₂²⋊C₈	—	2	4	no	—
56	C₂³×C₃	—	2	1	no	—
57	C₂³×C₃	—	2	1	no	—
58	C₂³×C₃	—	2	1	no	—
59	C₂³×C₃	—	2	1	no	—
60	C₂³×C₃	C₄⋊C₄	1	2	yes	—
61	C₂³×C₃	C₂×C₈	1	2	yes	—
62	C₂³×C₃	M₁₆	1	2	yes	—
63	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₃×C₄	—	2	8	no	—
65	C₂×C₃×C₄	—	2	1	no	—
66	C₂×C₃×C₄	—	2	1	no	—
67	C₂×C₃×C₄	—	2	1	no	—
68	C₂×C₃×C₄	—	2	1	no	—
69	C₂×C₃×C₄	—	2	1	no	—
70	C₂×C₃×C₄	—	2	1	no	—
71	C₂×C₃×C₄	—	2	1	no	—
72	C₂×C₃×C₄	—	2	1	no	—
73	C₂×C₃×C₄	—	2	8	no	—
74	C₂×C₃×C₄	—	2	4	no	—
75	C₂×C₃×C₄	—	2	4	no	—
76	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
77	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
78	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
79	C₂×C₃×C₄	C₂×C₈	1	2	yes	—
80	C₂×C₃×C₄	C₂×C₈	1	2	yes	—
81	C₂×C₃×C₄	M₁₆	1	2	yes	—
82	C₂×C₃×C₄	M₁₆	1	2	yes	—
83	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
85	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₃×C₄	#41	1	2	yes	—
87	C₃×C₈	—	4	4	no	—
88	C₂⁴	C₂×C₃×C₄	1	1	yes	characteristic
89	C₂²×C₄	—	2	2	no	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	—	2	2	no	—
92	C₂²×C₄	—	2	2	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	C₃×C₈	1	4	yes	complemented
97	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
98	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
99	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
100	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
101	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
102	C₂²×C₄	C₃×D₄	1	2	yes	—
103	C₂²×C₄	C₃×Q₈	1	2	yes	—
104	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
105	C₂×C₈	—	2	4	no	—
106	C₂×C₈	—	2	4	no	—
107	C₂×C₈	—	2	4	no	—
108	C₄²	—	2	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	—	2	2	no	—
111	C₄²	C₃×D₄	1	4	yes	—
112	C₂²×C₃	—	2	2	no	—
113	C₂²×C₃	—	2	2	no	—
114	C₂²×C₃	—	2	2	no	—
115	C₂²×C₃	—	2	2	no	—
116	C₂²×C₃	—	2	2	no	—
117	C₂²×C₃	—	2	2	no	—
118	C₂²×C₃	C₄⋊C₈	1	4	yes	complemented
119	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₃	#117	1	1	yes	characteristic
122	C₂²×C₃	C₂²×C₈	1	1	yes	characteristic
123	C₂²×C₃	C₂×M₁₆	1	1	yes	characteristic
124	C₂²×C₃	C₂×M₁₆	1	1	yes	characteristic
125	C₂²×C₃	C₂×M₁₆	1	1	yes	characteristic
126	C₃×C₄	—	2	2	no	—
127	C₃×C₄	—	2	2	no	—
128	C₃×C₄	—	2	2	no	—
129	C₃×C₄	—	2	2	no	—
130	C₃×C₄	C₂²⋊C₈	1	4	yes	complemented
131	C₃×C₄	#121	1	2	yes	—
132	C₃×C₄	C₂²⋊Q₈	1	2	yes	—
133	C₂³	—	2	1	no	—
134	C₂³	—	2	1	no	—
135	C₂³	—	2	1	no	—
136	C₂³	—	2	1	no	—
137	C₂³	C₃×C₄⋊C₄	1	2	yes	—
138	C₂³	C₂×C₃×C₈	1	2	yes	—
139	C₂³	C₃×M₁₆	1	2	yes	—
140	C₂³	C₂²×C₃×C₄	1	1	yes	characteristic
141	C₂×C₄	—	2	8	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	8	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	C₃×C₂²⋊C₄	1	2	yes	—
154	C₂×C₄	C₃×C₂²⋊C₄	1	2	yes	—
155	C₂×C₄	C₃×C₄⋊C₄	1	2	yes	—
156	C₂×C₄	C₂×C₃×C₈	1	2	yes	—
157	C₂×C₄	C₂×C₃×C₈	1	2	yes	—
158	C₂×C₄	C₃×M₁₆	1	2	yes	—
159	C₂×C₄	C₃×M₁₆	1	2	yes	—
160	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
161	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
162	C₂×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
163	C₂×C₄	C₃×#41	1	2	yes	—
164	C₈	—	4	4	no	—
165	C₂×C₃	—	2	4	no	—
166	C₂×C₃	#380	1	1	yes	characteristic
167	C₂×C₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
168	C₂×C₃	#407	1	1	yes	characteristic
169	C₂×C₃	C₂×C₄⋊C₈	1	1	yes	characteristic
170	C₂×C₃	#423	1	1	yes	characteristic
171	C₂×C₃	#431	1	1	yes	characteristic
172	C₂×C₃	#432	1	1	yes	characteristic
173	C₂²	—	2	2	no	—
174	C₂²	—	2	2	no	—
175	C₂²	—	2	2	no	—
176	C₂²	—	2	2	no	—
177	C₂²	—	2	2	no	—
178	C₂²	—	2	2	no	—
179	C₂²	C₃×C₄⋊C₈	1	4	yes	complemented
180	C₂²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
181	C₂²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
182	C₂²	C₃×#117	1	1	yes	characteristic
183	C₂²	C₂²×C₃×C₈	1	1	yes	characteristic
184	C₂²	C₂×C₃×M₁₆	1	1	yes	characteristic
185	C₂²	C₂×C₃×M₁₆	1	1	yes	characteristic
186	C₂²	C₂×C₃×M₁₆	1	1	yes	characteristic
187	C₄	—	2	2	no	—
188	C₄	—	2	2	no	—
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	C₃×C₂²⋊C₈	1	4	yes	complemented
192	C₄	C₃×#121	1	2	yes	—
193	C₄	C₃×C₂²⋊Q₈	1	2	yes	—
194	C₃	#1797	1	1	yes	characteristic, direct product factor, Sylow
195	C₂	—	2	4	no	—
196	C₂	C₃×#380	1	1	yes	characteristic
197	C₂	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
198	C₂	C₃×#407	1	1	yes	characteristic
199	C₂	C₂×C₃×C₄⋊C₈	1	1	yes	characteristic
200	C₂	C₃×#423	1	1	yes	characteristic
201	C₂	C₃×#431	1	1	yes	characteristic
202	C₂	C₃×#432	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂³
commutator	C₂²	C₂²×C₃×C₈
Frattini, cosocle	C₂²×C₄	C₂³×C₃
socle	C₂³×C₃	C₂²×C₄
prime radical	C₂⁴×C₃	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄⋊C₈	C₂	semidirect
4. C₃×#336	C₂	semidirect
5. #1797	C₃	direct (N × H)
18. C₂×C₄⋊C₈	C₂×C₃	semidirect
20. #336	C₂×C₃	semidirect
29. C₂²×C₃×C₄	C₈	semidirect
96. C₂²×C₄	C₃×C₈	semidirect
118. C₂²×C₃	C₄⋊C₈	semidirect
130. C₃×C₄	C₂²⋊C₈	semidirect
179. C₂²	C₃×C₄⋊C₈	semidirect
191. C₄	C₃×C₂²⋊C₈	semidirect
194. C₃	#1797	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄²	2
C₂×C₃×C₄⋊C₈	2
C₂×C₃×C₂²⋊C₈	2
C₃×#336	2
#1797	3

Subgroups — C3×#1797 #72512

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#1797 #72512