Finite Group Databasebeta

← C₃×#1681 #72396

Subgroups — C₃×#1681 #72396

Order 384.

182 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#459	C₂	1	1	yes	maximal, characteristic
4	C₃×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#331	C₂	1	1	yes	maximal, characteristic, complemented
8	#1681	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂×C₃×SD₁₆	—	2	1	no	—
10	C₂×C₃×D₈	—	2	1	no	—
11	C₂²×C₃×C₈	—	2	1	no	—
12	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₃×#127	C₂²	1	1	yes	characteristic
14	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
15	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₃×#103	—	2	1	no	—
19	C₃×#102	—	2	1	no	—
20	C₃×#102	—	2	1	no	—
21	C₃×#102	—	2	1	no	—
22	C₃×C₂²⋊C₈	—	2	1	no	—
23	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
24	#486	C₂×C₃	1	1	yes	characteristic, complemented
25	#461	C₂×C₃	1	1	yes	characteristic, complemented
26	#459	C₂×C₃	1	1	yes	characteristic
27	C₈×D₄	C₂×C₃	1	1	yes	characteristic, complemented
28	#335	C₂×C₃	1	1	yes	characteristic, complemented
29	#332	C₂×C₃	1	1	yes	characteristic, complemented
30	#331	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×C₃×Q₈	—	2	1	no	—
32	C₂×C₃×D₄	—	2	1	no	—
33	C₂×C₃×D₄	—	2	1	no	—
34	C₂×C₃×D₄	—	2	1	no	—
35	C₂×C₃×D₄	D₄	1	1	yes	characteristic
36	C₂²×C₃×C₄	—	2	1	no	—
37	C₃×SD₁₆	—	4	2	no	—
38	C₃×D₈	—	4	2	no	—
39	C₂×C₃×C₈	—	2	1	no	—
40	C₂×C₃×C₈	—	2	1	no	—
41	C₂×C₃×C₈	—	2	1	no	—
42	C₂×C₃×C₈	—	4	2	no	—
43	C₂×C₃×C₈	D₄	1	1	yes	characteristic
44	C₂×C₃×C₈	D₄	1	1	yes	characteristic
45	C₃×C₄⋊C₄	—	2	1	no	—
46	C₃×C₄⋊C₄	—	2	1	no	—
47	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
48	C₃×C₂²⋊C₄	—	2	1	no	—
49	C₃×C₄²	C₂³	1	1	yes	characteristic
50	C₂×SD₁₆	—	2	1	no	—
51	C₂×D₈	—	2	1	no	—
52	C₂²×C₈	—	2	1	no	—
53	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
54	#127	C₂²×C₃	1	1	yes	characteristic
55	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
56	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
57	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
58	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
59	#103	—	2	1	no	—
60	#102	—	2	1	no	—
61	#102	—	2	1	no	—
62	#102	—	2	1	no	—
63	C₂²⋊C₈	—	2	1	no	—
64	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
65	C₂³×C₃	—	2	1	no	—
66	C₂³×C₃	—	4	1	no	—
67	C₃×Q₈	—	4	1	no	—
68	C₃×Q₈	—	4	1	no	—
69	C₃×D₄	—	2	1	no	—
70	C₃×D₄	—	4	1	no	—
71	C₃×D₄	—	4	1	no	—
72	C₃×D₄	—	4	1	no	—
73	C₃×D₄	—	4	1	no	—
74	C₃×D₄	—	8	1	no	—
75	C₃×D₄	D₈	1	2	yes	—
76	C₂×C₃×C₄	—	2	1	no	—
77	C₂×C₃×C₄	—	4	1	no	—
78	C₂×C₃×C₄	—	4	1	no	—
79	C₂×C₃×C₄	—	4	1	no	—
80	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄	#41	1	1	yes	characteristic
83	C₃×C₈	—	2	1	no	—
84	C₃×C₈	—	2	1	no	—
85	C₃×C₈	—	4	1	no	—
86	C₃×C₈	—	4	1	no	—
87	C₃×C₈	—	4	1	no	—
88	C₂×Q₈	—	2	1	no	—
89	C₂×D₄	—	2	1	no	—
90	C₂×D₄	—	2	1	no	—
91	C₂×D₄	—	2	1	no	—
92	C₂×D₄	C₃×D₄	1	1	yes	characteristic
93	C₂²×C₄	—	2	1	no	—
94	SD₁₆	—	4	2	no	—
95	D₈	—	4	2	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	4	2	no	—
100	C₂×C₈	C₃×D₄	1	1	yes	characteristic
101	C₂×C₈	C₃×D₄	1	1	yes	characteristic
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
105	C₂²⋊C₄	—	2	1	no	—
106	C₄²	C₂³×C₃	1	1	yes	characteristic
107	C₂²×C₃	—	2	2	no	—
108	C₂²×C₃	—	4	1	no	—
109	C₂²×C₃	—	4	1	no	—
110	C₂²×C₃	—	8	1	no	—
111	C₂²×C₃	—	8	1	no	—
112	C₂²×C₃	—	8	1	no	—
113	C₂²×C₃	#121	1	1	yes	characteristic
114	C₃×C₄	—	2	1	no	—
115	C₃×C₄	—	4	1	no	—
116	C₃×C₄	—	8	1	no	—
117	C₃×C₄	C₂×D₈	1	1	yes	characteristic
118	C₃×C₄	#135	1	1	yes	characteristic
119	C₃×C₄	#135	1	1	yes	characteristic
120	C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂³	—	2	1	no	—
122	C₂³	—	4	1	no	—
123	Q₈	—	4	1	no	—
124	Q₈	—	4	1	no	—
125	D₄	—	2	1	no	—
126	D₄	—	4	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	4	1	no	—
130	D₄	—	8	1	no	—
131	D₄	C₃×D₈	1	2	yes	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
137	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
138	C₂×C₄	C₃×#41	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	—	4	1	no	—
142	C₈	—	4	1	no	—
143	C₈	—	4	1	no	—
144	C₂×C₃	—	4	2	no	—
145	C₂×C₃	—	16	1	no	—
146	C₂×C₃	#463	1	1	yes	characteristic
147	C₂×C₃	#466	1	1	yes	characteristic
148	C₂×C₃	#471	1	1	yes	characteristic
149	C₂²	—	2	2	no	—
150	C₂²	—	4	1	no	—
151	C₂²	—	4	1	no	—
152	C₂²	—	8	1	no	—
153	C₂²	—	8	1	no	—
154	C₂²	—	8	1	no	—
155	C₂²	C₃×#121	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	4	1	no	—
158	C₄	—	8	1	no	—
159	C₄	C₂×C₃×D₈	1	1	yes	characteristic
160	C₄	C₃×#135	1	1	yes	characteristic
161	C₄	C₃×#135	1	1	yes	characteristic
162	C₄	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
163	C₃	#1681	1	1	yes	characteristic, direct product factor, Sylow
164	C₂	—	4	2	no	—
165	C₂	—	16	1	no	—
166	C₂	C₃×#463	1	1	yes	characteristic
167	C₂	C₃×#466	1	1	yes	characteristic
168	C₂	C₃×#471	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₃
prime radical	C₃×#459	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#486	C₂	semidirect
2. C₃×#461	C₂	semidirect
4. C₃×C₈×D₄	C₂	semidirect
5. C₃×#335	C₂	semidirect
6. C₃×#332	C₂	semidirect
7. C₃×#331	C₂	semidirect
8. #1681	C₃	direct (N × H)
24. #486	C₂×C₃	semidirect
25. #461	C₂×C₃	semidirect
27. C₈×D₄	C₂×C₃	semidirect
28. #335	C₂×C₃	semidirect
29. #332	C₂×C₃	semidirect
30. #331	C₂×C₃	semidirect
163. C₃	#1681	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×#486	2
C₃×#461	2
C₃×#459	2
C₃×C₈×D₄	2
C₃×#335	2
C₃×#332	2
C₃×#331	2
#1681	3