Subgroups — C₃×#2320 #84184

Order 384.

444 conjugacy classes of subgroups.

The subgroup lattice has 172 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	1	yes	maximal, characteristic
2	C₃×#391	C₂	1	2	yes	maximal
3	C₃×#389	C₂	1	2	yes	maximal
4	C₃×#387	C₂	1	4	yes	maximal
5	C₃×#383	C₂	1	2	yes	maximal
6	C₃×#382	C₂	1	2	yes	maximal
7	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
8	C₃×#376	C₂	1	1	yes	maximal, characteristic
9	#2320	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
11	C₃×C₄×Q₈	C₄	1	8	yes	complemented
12	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
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	1	yes	characteristic
16	C₂×C₃×C₄²	C₂²	1	2	yes	—
17	C₂×C₃×C₄²	C₂²	1	2	yes	—
18	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄²	C₂²	1	2	yes	—
20	C₃×#95	C₂²	1	1	yes	characteristic
21	C₃×#95	C₂²	1	1	yes	characteristic
22	C₃×#95	C₂²	1	2	yes	—
23	C₃×#95	C₂²	1	2	yes	—
24	C₃×#95	C₂²	1	4	yes	—
25	C₃×#95	C₂²	1	4	yes	—
26	C₃×#95	C₂²	1	4	yes	—
27	C₂×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
28	#391	C₂×C₃	1	2	yes	—
29	#389	C₂×C₃	1	2	yes	—
30	#387	C₂×C₃	1	4	yes	—
31	#383	C₂×C₃	1	2	yes	—
32	#382	C₂×C₃	1	2	yes	—
33	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
34	#376	C₂×C₃	1	1	yes	characteristic
35	C₂×C₃×Q₈	—	2	4	no	—
36	C₂×C₃×Q₈	C₂×C₄	1	4	yes	—
37	C₂²×C₃×C₄	C₂³	1	2	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	1	yes	characteristic
43	C₂²×C₃×C₄	C₂³	1	2	yes	—
44	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
45	C₃×C₄⋊C₄	—	2	4	no	—
46	C₃×C₄⋊C₄	—	4	4	no	—
47	C₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
48	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
49	C₃×C₄²	—	2	4	no	—
50	C₃×C₄²	—	4	2	no	—
51	C₃×C₄²	C₂×C₄	1	8	yes	—
52	C₃×C₄²	C₂×C₄	1	4	yes	—
53	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
54	C₄×Q₈	C₃×C₄	1	8	yes	complemented
55	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
56	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
57	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
58	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₄²	C₂²×C₃	1	2	yes	—
60	C₂×C₄²	C₂²×C₃	1	2	yes	—
61	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₄²	C₂²×C₃	1	2	yes	—
63	#95	C₂²×C₃	1	1	yes	characteristic
64	#95	C₂²×C₃	1	1	yes	characteristic
65	#95	C₂²×C₃	1	2	yes	—
66	#95	C₂²×C₃	1	2	yes	—
67	#95	C₂²×C₃	1	4	yes	—
68	#95	C₂²×C₃	1	4	yes	—
69	#95	C₂²×C₃	1	4	yes	—
70	C₂³×C₃	C₂⁴	1	1	yes	characteristic
71	C₃×Q₈	—	2	8	no	—
72	C₂×C₃×C₄	—	2	4	no	—
73	C₂×C₃×C₄	—	2	1	no	—
74	C₂×C₃×C₄	—	2	2	no	—
75	C₂×C₃×C₄	—	2	2	no	—
76	C₂×C₃×C₄	—	2	1	no	—
77	C₂×C₃×C₄	—	2	2	no	—
78	C₂×C₃×C₄	—	2	2	no	—
79	C₂×C₃×C₄	—	2	2	no	—
80	C₂×C₃×C₄	—	2	4	no	—
81	C₂×C₃×C₄	—	2	4	no	—
82	C₂×C₃×C₄	—	2	2	no	—
83	C₂×C₃×C₄	—	2	2	no	—
84	C₂×C₃×C₄	—	2	2	no	—
85	C₂×C₃×C₄	—	2	2	no	—
86	C₂×C₃×C₄	—	2	4	no	—
87	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
88	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
89	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
90	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
91	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
92	C₂×C₃×C₄	#41	1	4	yes	—
93	C₂×Q₈	—	2	4	no	—
94	C₂×Q₈	C₂×C₃×C₄	1	4	yes	—
95	C₂²×C₄	C₂³×C₃	1	2	yes	—
96	C₂²×C₄	C₂³×C₃	1	2	yes	—
97	C₂²×C₄	C₂³×C₃	1	4	yes	—
98	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
99	C₂²×C₄	C₂³×C₃	1	2	yes	—
100	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
101	C₂²×C₄	C₂³×C₃	1	2	yes	—
102	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	—	4	4	no	—
105	C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
106	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
107	C₄²	—	2	4	no	—
108	C₄²	—	4	2	no	—
109	C₄²	C₂×C₃×C₄	1	8	yes	—
110	C₄²	C₂×C₃×C₄	1	4	yes	—
111	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
112	C₂²×C₃	C₂×#41	1	2	yes	—
113	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
114	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
115	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
116	C₃×C₄	—	2	4	no	—
117	C₃×C₄	—	2	2	no	—
118	C₃×C₄	—	2	2	no	—
119	C₃×C₄	—	2	4	no	—
120	C₃×C₄	—	4	2	no	—
121	C₃×C₄	—	4	2	no	—
122	C₃×C₄	—	4	4	no	—
123	C₃×C₄	#117	1	4	yes	—
124	C₂³	C₂⁴×C₃	1	1	yes	characteristic
125	Q₈	—	2	8	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
142	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
143	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
144	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
145	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
146	C₂×C₄	C₃×#41	1	4	yes	—
147	C₂×C₃	C₂×#117	1	1	yes	characteristic
148	C₂×C₃	#519	1	1	yes	characteristic
149	C₂×C₃	#520	1	1	yes	characteristic
150	C₂×C₃	#541	1	2	yes	—
151	C₂×C₃	#542	1	2	yes	—
152	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
153	C₂²	C₂×C₃×#41	1	2	yes	—
154	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
155	C₂²	C₃×2₋¹⁺⁴	1	2	yes	—
156	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
157	C₄	—	2	4	no	—
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	2	4	no	—
161	C₄	—	4	2	no	—
162	C₄	—	4	2	no	—
163	C₄	—	4	4	no	—
164	C₄	C₃×#117	1	4	yes	—
165	C₃	#2320	1	1	yes	characteristic, direct product factor, Sylow
166	C₂	C₂×C₃×#117	1	1	yes	characteristic
167	C₂	C₃×#519	1	1	yes	characteristic
168	C₂	C₃×#520	1	1	yes	characteristic
169	C₂	C₃×#541	1	2	yes	—
170	C₂	C₃×#542	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
9. #2320	C₃	direct (N × H)
11. C₃×C₄×Q₈	C₄	semidirect
54. C₄×Q₈	C₃×C₄	semidirect
165. C₃	#2320	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
C₃×#391	2
C₃×#389	2
C₃×#387	2
C₃×#383	2
C₃×#382	2
C₃×C₄×C₄⋊C₄	2
C₃×#376	2
#2320	3

Subgroups — C3×#2320 #84184

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2320 #84184