Subgroups — C₃×#2546 #84410

Order 384.

540 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄⋊Q₈	C₂	1	2	yes	maximal
2	C₂×C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
4	C₃×#398	C₂	1	4	yes	maximal
5	C₃×#391	C₂	1	2	yes	maximal
6	C₃×#389	C₂	1	4	yes	maximal
7	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
8	#2546	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂²×C₃×Q₈	C₂²	1	2	yes	—
10	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
11	C₃×C₄⋊Q₈	—	2	8	no	—
12	C₃×C₄⋊Q₈	—	2	4	no	—
13	C₃×C₄×Q₈	—	2	4	no	—
14	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₃×C₄⋊C₄	C₂²	1	8	yes	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₃×C₄²	C₂²	1	2	yes	—
21	C₂×C₃×C₄²	C₂²	1	2	yes	—
22	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
23	C₃×#95	C₂²	1	2	yes	—
24	C₃×#95	C₂²	1	8	yes	—
25	C₂×C₄⋊Q₈	C₂×C₃	1	2	yes	—
26	C₂×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
27	C₂×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
28	#398	C₂×C₃	1	4	yes	—
29	#391	C₂×C₃	1	2	yes	—
30	#389	C₂×C₃	1	4	yes	—
31	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
32	C₂×C₃×Q₈	—	2	2	no	—
33	C₂×C₃×Q₈	—	2	4	no	—
34	C₂×C₃×Q₈	—	2	4	no	—
35	C₂×C₃×Q₈	—	2	2	no	—
36	C₂×C₃×Q₈	—	2	4	no	—
37	C₂×C₃×Q₈	D₄	1	4	yes	—
38	C₂²×C₃×C₄	C₂³	1	2	yes	—
39	C₂²×C₃×C₄	C₂³	1	4	yes	—
40	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
41	C₂²×C₃×C₄	C₂³	1	2	yes	—
42	C₂²×C₃×C₄	C₂³	1	4	yes	—
43	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
44	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
45	C₃×C₄⋊C₄	—	2	4	no	—
46	C₃×C₄⋊C₄	—	2	2	no	—
47	C₃×C₄⋊C₄	—	2	16	no	—
48	C₃×C₄⋊C₄	—	2	4	no	—
49	C₃×C₄⋊C₄	—	2	4	no	—
50	C₃×C₄⋊C₄	Q₈	1	8	yes	complemented
51	C₃×C₄²	—	2	4	no	—
52	C₃×C₄²	—	2	4	no	—
53	C₃×C₄²	D₄	1	4	yes	—
54	C₂²×Q₈	C₂²×C₃	1	2	yes	—
55	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
56	C₄⋊Q₈	—	2	8	no	—
57	C₄⋊Q₈	—	2	4	no	—
58	C₄×Q₈	—	2	4	no	—
59	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
61	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
62	C₂×C₄⋊C₄	C₂²×C₃	1	8	yes	—
63	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
64	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
65	C₂×C₄²	C₂²×C₃	1	2	yes	—
66	C₂×C₄²	C₂²×C₃	1	2	yes	—
67	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
68	#95	C₂²×C₃	1	2	yes	—
69	#95	C₂²×C₃	1	8	yes	—
70	C₂³×C₃	C₂⁴	1	1	yes	characteristic
71	C₃×Q₈	—	2	8	no	—
72	C₃×Q₈	—	4	8	no	—
73	C₂×C₃×C₄	—	2	4	no	—
74	C₂×C₃×C₄	—	2	2	no	—
75	C₂×C₃×C₄	—	2	2	no	—
76	C₂×C₃×C₄	—	2	2	no	—
77	C₂×C₃×C₄	—	2	4	no	—
78	C₂×C₃×C₄	—	2	2	no	—
79	C₂×C₃×C₄	—	2	4	no	—
80	C₂×C₃×C₄	—	2	4	no	—
81	C₂×C₃×C₄	—	2	4	no	—
82	C₂×C₃×C₄	—	2	4	no	—
83	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
84	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
85	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
87	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
88	C₂×C₃×C₄	C₂×Q₈	1	8	yes	—
89	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
90	C₂×C₃×C₄	#41	1	2	yes	—
91	C₂×Q₈	—	2	2	no	—
92	C₂×Q₈	—	2	4	no	—
93	C₂×Q₈	—	2	4	no	—
94	C₂×Q₈	—	2	2	no	—
95	C₂×Q₈	—	2	4	no	—
96	C₂×Q₈	C₃×D₄	1	4	yes	—
97	C₂²×C₄	C₂³×C₃	1	2	yes	—
98	C₂²×C₄	C₂³×C₃	1	4	yes	—
99	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
100	C₂²×C₄	C₂³×C₃	1	2	yes	—
101	C₂²×C₄	C₂³×C₃	1	4	yes	—
102	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
103	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
104	C₄⋊C₄	—	2	4	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	2	16	no	—
107	C₄⋊C₄	—	2	4	no	—
108	C₄⋊C₄	—	2	4	no	—
109	C₄⋊C₄	C₃×Q₈	1	8	yes	complemented
110	C₄²	—	2	4	no	—
111	C₄²	—	2	4	no	—
112	C₄²	C₃×D₄	1	4	yes	—
113	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
114	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
115	C₂²×C₃	C₂²×Q₈	1	2	yes	—
116	C₂²×C₃	C₂×#41	1	1	yes	characteristic
117	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
118	C₃×C₄	—	2	4	no	—
119	C₃×C₄	—	2	2	no	—
120	C₃×C₄	—	2	8	no	—
121	C₃×C₄	—	4	4	no	—
122	C₃×C₄	—	4	2	no	—
123	C₃×C₄	#121	1	4	yes	—
124	C₃×C₄	C₄⋊Q₈	1	4	yes	complemented
125	C₂³	C₂⁴×C₃	1	1	yes	characteristic
126	Q₈	—	2	8	no	—
127	Q₈	—	4	8	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
139	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
140	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
141	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
142	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
143	C₂×C₄	C₂×C₃×Q₈	1	8	yes	—
144	C₂×C₄	C₂×C₃×Q₈	1	4	yes	—
145	C₂×C₄	C₃×#41	1	2	yes	—
146	C₂×C₃	C₂×#121	1	1	yes	characteristic
147	C₂×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
148	C₂×C₃	#536	1	1	yes	characteristic
149	C₂×C₃	D₄×Q₈	1	2	yes	—
150	C₂×C₃	#557	1	2	yes	—
151	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
152	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
153	C₂²	C₂²×C₃×Q₈	1	2	yes	—
154	C₂²	C₂×C₃×#41	1	1	yes	characteristic
155	C₂²	C₃×2₋¹⁺⁴	1	2	yes	—
156	C₄	—	2	4	no	—
157	C₄	—	2	2	no	—
158	C₄	—	2	8	no	—
159	C₄	—	4	4	no	—
160	C₄	—	4	2	no	—
161	C₄	C₃×#121	1	4	yes	—
162	C₄	C₃×C₄⋊Q₈	1	4	yes	complemented
163	C₃	#2546	1	1	yes	characteristic, direct product factor, Sylow
164	C₂	C₂×C₃×#121	1	1	yes	characteristic
165	C₂	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
166	C₂	C₃×#536	1	1	yes	characteristic
167	C₂	C₃×D₄×Q₈	1	2	yes	—
168	C₂	C₃×#557	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
8. #2546	C₃	direct (N × H)
50. C₃×C₄⋊C₄	Q₈	semidirect
109. C₄⋊C₄	C₃×Q₈	semidirect
124. C₃×C₄	C₄⋊Q₈	semidirect
162. C₄	C₃×C₄⋊Q₈	semidirect
163. C₃	#2546	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×C₄⋊Q₈	2
C₂×C₃×C₄×Q₈	2
C₃×#398	2
C₃×#391	2
C₃×#389	2
C₃×C₄×C₄⋊C₄	2
#2546	3

Subgroups — C3×#2546 #84410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2546 #84410