Subgroups — C₃×#2584 #84448

Order 384.

468 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#125	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₃×#398	C₂	1	2	yes	maximal
4	C₃×#395	C₂	1	2	yes	maximal
5	C₃×#389	C₂	1	2	yes	maximal
6	C₃×#389	C₂	1	4	yes	maximal
7	C₃×#389	C₂	1	2	yes	maximal
8	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
9	#2584	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
11	C₃×#125	—	2	4	no	—
12	C₃×C₄×Q₈	—	2	4	no	—
13	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
14	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
21	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₄²	C₂²	1	2	yes	—
23	C₂×C₃×C₄²	C₂²	1	2	yes	—
24	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
25	C₃×#95	C₂²	1	2	yes	—
26	C₃×#95	C₂²	1	4	yes	—
27	C₃×#95	C₂²	1	4	yes	—
28	C₂×#125	C₂×C₃	1	1	yes	characteristic
29	C₂×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
30	#398	C₂×C₃	1	2	yes	—
31	#395	C₂×C₃	1	2	yes	—
32	#389	C₂×C₃	1	2	yes	—
33	#389	C₂×C₃	1	4	yes	—
34	#389	C₂×C₃	1	2	yes	—
35	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
36	C₂×C₃×Q₈	—	2	2	no	—
37	C₂×C₃×Q₈	—	2	4	no	—
38	C₂²×C₃×C₄	C₂³	1	2	yes	—
39	C₂²×C₃×C₄	C₂³	1	2	yes	—
40	C₂²×C₃×C₄	C₂³	1	2	yes	—
41	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
42	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
43	C₂²×C₃×C₄	C₂³	1	4	yes	—
44	C₂²×C₃×C₄	C₂³	1	2	yes	—
45	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
46	C₃×C₄⋊C₄	—	2	4	no	—
47	C₃×C₄⋊C₄	—	2	2	no	—
48	C₃×C₄⋊C₄	—	2	4	no	—
49	C₃×C₄⋊C₄	—	2	4	no	—
50	C₃×C₄⋊C₄	—	2	2	no	—
51	C₃×C₄⋊C₄	—	2	2	no	—
52	C₃×C₄⋊C₄	—	2	8	no	—
53	C₃×C₄⋊C₄	—	4	4	no	—
54	C₃×C₄⋊C₄	Q₈	1	8	yes	complemented
55	C₃×C₄²	—	2	4	no	—
56	C₃×C₄²	—	4	2	no	—
57	C₃×C₄²	D₄	1	4	yes	—
58	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
59	#125	—	2	4	no	—
60	C₄×Q₈	—	2	4	no	—
61	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
62	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
64	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
65	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
67	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
68	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
69	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
70	C₂×C₄²	C₂²×C₃	1	2	yes	—
71	C₂×C₄²	C₂²×C₃	1	2	yes	—
72	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
73	#95	C₂²×C₃	1	2	yes	—
74	#95	C₂²×C₃	1	4	yes	—
75	#95	C₂²×C₃	1	4	yes	—
76	C₂³×C₃	C₂⁴	1	1	yes	characteristic
77	C₃×Q₈	—	2	8	no	—
78	C₂×C₃×C₄	—	2	2	no	—
79	C₂×C₃×C₄	—	2	4	no	—
80	C₂×C₃×C₄	—	2	2	no	—
81	C₂×C₃×C₄	—	2	2	no	—
82	C₂×C₃×C₄	—	2	1	no	—
83	C₂×C₃×C₄	—	2	4	no	—
84	C₂×C₃×C₄	—	2	1	no	—
85	C₂×C₃×C₄	—	2	2	no	—
86	C₂×C₃×C₄	—	2	2	no	—
87	C₂×C₃×C₄	—	2	2	no	—
88	C₂×C₃×C₄	—	2	4	no	—
89	C₂×C₃×C₄	—	2	2	no	—
90	C₂×C₃×C₄	—	2	2	no	—
91	C₂×C₃×C₄	—	2	2	no	—
92	C₂×C₃×C₄	—	2	2	no	—
93	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
94	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
95	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
96	C₂×C₃×C₄	C₂×Q₈	1	8	yes	—
97	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
98	C₂×C₃×C₄	#41	1	4	yes	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	4	no	—
101	C₂²×C₄	C₂³×C₃	1	2	yes	—
102	C₂²×C₄	C₂³×C₃	1	2	yes	—
103	C₂²×C₄	C₂³×C₃	1	2	yes	—
104	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
105	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
106	C₂²×C₄	C₂³×C₃	1	4	yes	—
107	C₂²×C₄	C₂³×C₃	1	2	yes	—
108	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
109	C₄⋊C₄	—	2	4	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	4	no	—
112	C₄⋊C₄	—	2	4	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	2	8	no	—
116	C₄⋊C₄	—	4	4	no	—
117	C₄⋊C₄	C₃×Q₈	1	8	yes	complemented
118	C₄²	—	2	4	no	—
119	C₄²	—	4	2	no	—
120	C₄²	C₃×D₄	1	4	yes	—
121	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
122	C₂²×C₃	C₂²×Q₈	1	2	yes	—
123	C₂²×C₃	C₂×#41	1	2	yes	—
124	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
125	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
126	C₃×C₄	—	2	2	no	—
127	C₃×C₄	—	2	4	no	—
128	C₃×C₄	—	2	2	no	—
129	C₃×C₄	—	2	8	no	—
130	C₃×C₄	—	4	2	no	—
131	C₃×C₄	—	4	2	no	—
132	C₃×C₄	—	4	2	no	—
133	C₃×C₄	C₄⋊Q₈	1	4	yes	—
134	C₂³	C₂⁴×C₃	1	1	yes	characteristic
135	Q₈	—	2	8	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
152	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
153	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
154	C₂×C₄	C₂×C₃×Q₈	1	8	yes	—
155	C₂×C₄	C₂×C₃×Q₈	1	4	yes	—
156	C₂×C₄	C₃×#41	1	4	yes	—
157	C₂×C₃	#525	1	1	yes	characteristic
158	C₂×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
159	C₂×C₃	#537	1	1	yes	characteristic
160	C₂×C₃	#554	1	2	yes	—
161	C₂×C₃	#557	1	2	yes	—
162	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
163	C₂²	C₂²×C₃×Q₈	1	2	yes	—
164	C₂²	C₂×C₃×#41	1	2	yes	—
165	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
166	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	2	4	no	—
169	C₄	—	2	2	no	—
170	C₄	—	2	8	no	—
171	C₄	—	4	2	no	—
172	C₄	—	4	2	no	—
173	C₄	—	4	2	no	—
174	C₄	C₃×C₄⋊Q₈	1	4	yes	—
175	C₃	#2584	1	1	yes	characteristic, direct product factor, Sylow
176	C₂	C₃×#525	1	1	yes	characteristic
177	C₂	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
178	C₂	C₃×#537	1	1	yes	characteristic
179	C₂	C₃×#554	1	2	yes	—
180	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
9. #2584	C₃	direct (N × H)
54. C₃×C₄⋊C₄	Q₈	semidirect
117. C₄⋊C₄	C₃×Q₈	semidirect
175. C₃	#2584	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×#125	2
C₂×C₃×C₄×Q₈	2
C₃×#398	2
C₃×#395	2
C₃×#389	2
C₃×C₄×C₄⋊C₄	2
#2584	3

Subgroups — C3×#2584 #84448

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2584 #84448