Subgroups — C₃×#2280 #84144

Order 384.

1088 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	C₂⁴×C₃×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#117	C₂	1	2	yes	maximal, complemented
3	C₃×#381	C₂	1	2	yes	maximal, complemented
4	C₃×#380	C₂	1	4	yes	maximal, complemented
5	C₃×#379	C₂	1	2	yes	maximal
6	C₃×C₄×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
7	#2280	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂⁵×C₃	C₂²	1	1	yes	characteristic
9	C₂³×C₃×C₄	—	2	8	no	—
10	C₂³×C₃×C₄	C₄	1	8	yes	complemented
11	C₂³×C₃×C₄	C₂²	1	2	yes	—
12	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
13	C₂³×C₃×C₄	C₂²	1	2	yes	—
14	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
15	C₃×#117	—	2	8	no	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	complemented
17	C₂×C₃×C₂²⋊C₄	C₂²	1	8	yes	—
18	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
19	C₂×C₃×C₄²	C₂²	1	4	yes	complemented
20	C₃×#95	C₂²	1	8	yes	complemented
21	C₂⁴×C₄	C₂×C₃	1	1	yes	characteristic
22	C₂×#117	C₂×C₃	1	2	yes	complemented
23	#381	C₂×C₃	1	2	yes	complemented
24	#380	C₂×C₃	1	4	yes	complemented
25	#379	C₂×C₃	1	2	yes	—
26	C₄×C₂²⋊C₄	C₂×C₃	1	4	yes	complemented
27	C₂⁴×C₃	—	2	4	no	—
28	C₂⁴×C₃	—	2	8	no	—
29	C₂⁴×C₃	C₂×C₄	1	4	yes	—
30	C₂⁴×C₃	C₂³	1	1	yes	characteristic
31	C₂⁴×C₃	C₂³	1	2	yes	—
32	C₂²×C₃×C₄	—	2	4	no	—
33	C₂²×C₃×C₄	—	2	32	no	—
34	C₂²×C₃×C₄	—	2	8	no	—
35	C₂²×C₃×C₄	—	2	8	no	—
36	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
37	C₂²×C₃×C₄	C₂×C₄	1	8	yes	—
38	C₂²×C₃×C₄	C₂×C₄	1	8	yes	—
39	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
40	C₂²×C₃×C₄	D₄	1	8	yes	—
41	C₂²×C₃×C₄	C₂³	1	8	yes	—
42	C₂²×C₃×C₄	C₂³	1	2	yes	—
43	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
44	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
45	C₃×C₄⋊C₄	—	2	8	no	—
46	C₃×C₂²⋊C₄	—	2	8	no	—
47	C₃×C₂²⋊C₄	—	2	16	no	—
48	C₃×C₄²	—	2	8	no	—
49	C₂⁵	C₂²×C₃	1	1	yes	characteristic
50	C₂³×C₄	—	2	8	no	—
51	C₂³×C₄	C₃×C₄	1	8	yes	complemented
52	C₂³×C₄	C₂²×C₃	1	2	yes	—
53	C₂³×C₄	C₂²×C₃	1	2	yes	—
54	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
55	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
56	#117	—	2	8	no	—
57	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	complemented
58	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
59	C₂×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
60	C₂×C₄²	C₂²×C₃	1	4	yes	complemented
61	#95	C₂²×C₃	1	8	yes	complemented
62	C₂³×C₃	—	2	32	no	—
63	C₂³×C₃	—	2	8	no	—
64	C₂³×C₃	—	2	16	no	—
65	C₂³×C₃	—	2	4	no	—
66	C₂³×C₃	—	2	4	no	—
67	C₂³×C₃	—	2	4	no	—
68	C₂³×C₃	C₂²×C₄	1	2	yes	—
69	C₂³×C₃	C₂²×C₄	1	4	yes	—
70	C₂³×C₃	C₂×D₄	1	4	yes	—
71	C₂³×C₃	#41	1	8	yes	—
72	C₂³×C₃	C₂⁴	1	1	yes	characteristic
73	C₂×C₃×C₄	—	2	8	no	—
74	C₂×C₃×C₄	—	2	4	no	—
75	C₂×C₃×C₄	—	2	16	no	—
76	C₂×C₃×C₄	—	2	16	no	—
77	C₂×C₃×C₄	—	2	8	no	—
78	C₂×C₃×C₄	—	2	16	no	—
79	C₂×C₃×C₄	C₂²⋊C₄	1	16	yes	complemented
80	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
81	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
82	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
83	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
84	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
85	C₂⁴	—	2	4	no	—
86	C₂⁴	—	2	8	no	—
87	C₂⁴	C₂×C₃×C₄	1	4	yes	—
88	C₂⁴	C₂³×C₃	1	1	yes	characteristic
89	C₂⁴	C₂³×C₃	1	2	yes	—
90	C₂²×C₄	—	2	4	no	—
91	C₂²×C₄	—	2	32	no	—
92	C₂²×C₄	—	2	8	no	—
93	C₂²×C₄	—	2	8	no	—
94	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
95	C₂²×C₄	C₂×C₃×C₄	1	8	yes	—
96	C₂²×C₄	C₂×C₃×C₄	1	8	yes	—
97	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
98	C₂²×C₄	C₃×D₄	1	8	yes	—
99	C₂²×C₄	C₂³×C₃	1	8	yes	—
100	C₂²×C₄	C₂³×C₃	1	2	yes	—
101	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
102	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
103	C₄⋊C₄	—	2	8	no	—
104	C₂²⋊C₄	—	2	8	no	—
105	C₂²⋊C₄	—	2	16	no	—
106	C₄²	—	2	8	no	—
107	C₂²×C₃	—	2	32	no	—
108	C₂²×C₃	—	2	4	no	—
109	C₂²×C₃	—	2	4	no	—
110	C₂²×C₃	—	2	4	no	—
111	C₂²×C₃	—	2	8	no	—
112	C₂²×C₃	—	2	16	no	—
113	C₂²×C₃	C₂×C₂²⋊C₄	1	4	yes	—
114	C₂²×C₃	#117	1	8	yes	complemented
115	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
116	C₂²×C₃	C₂²×D₄	1	2	yes	—
117	C₂²×C₃	C₂×#41	1	4	yes	—
118	C₃×C₄	—	2	4	no	—
119	C₃×C₄	—	4	8	no	—
120	C₃×C₄	C₂×C₂²⋊C₄	1	4	yes	—
121	C₃×C₄	C₂×C₂²⋊C₄	1	4	yes	—
122	C₂³	—	2	32	no	—
123	C₂³	—	2	8	no	—
124	C₂³	—	2	16	no	—
125	C₂³	—	2	4	no	—
126	C₂³	—	2	4	no	—
127	C₂³	—	2	4	no	—
128	C₂³	C₂²×C₃×C₄	1	2	yes	—
129	C₂³	C₂²×C₃×C₄	1	4	yes	—
130	C₂³	C₂×C₃×D₄	1	4	yes	—
131	C₂³	C₃×#41	1	8	yes	—
132	C₂³	C₂⁴×C₃	1	1	yes	characteristic
133	C₂×C₄	—	2	8	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	16	no	—
136	C₂×C₄	—	2	16	no	—
137	C₂×C₄	—	2	8	no	—
138	C₂×C₄	—	2	16	no	—
139	C₂×C₄	C₃×C₂²⋊C₄	1	16	yes	complemented
140	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
141	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
142	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
143	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
144	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
145	C₂×C₃	—	2	4	no	—
146	C₂×C₃	—	2	8	no	—
147	C₂×C₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₃	C₂×#117	1	2	yes	—
149	C₂×C₃	#525	1	4	yes	—
150	C₂²	—	2	32	no	—
151	C₂²	—	2	4	no	—
152	C₂²	—	2	4	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	2	8	no	—
155	C₂²	—	2	16	no	—
156	C₂²	C₂×C₃×C₂²⋊C₄	1	4	yes	—
157	C₂²	C₃×#117	1	8	yes	complemented
158	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
159	C₂²	C₂²×C₃×D₄	1	2	yes	—
160	C₂²	C₂×C₃×#41	1	4	yes	—
161	C₄	—	2	4	no	—
162	C₄	—	4	8	no	—
163	C₄	C₂×C₃×C₂²⋊C₄	1	4	yes	—
164	C₄	C₂×C₃×C₂²⋊C₄	1	4	yes	—
165	C₃	#2280	1	1	yes	characteristic, direct product factor, Sylow
166	C₂	—	2	4	no	—
167	C₂	—	2	8	no	—
168	C₂	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
169	C₂	C₂×C₃×#117	1	2	yes	—
170	C₂	C₃×#525	1	4	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂³
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂³	C₂⁴×C₃
socle	C₂³×C₃	C₂⁴
prime radical	C₂⁵×C₃	C₂²
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×#117	C₂	semidirect
3. C₃×#381	C₂	semidirect
4. C₃×#380	C₂	semidirect
6. C₃×C₄×C₂²⋊C₄	C₂	semidirect
7. #2280	C₃	direct (N × H)
10. C₂³×C₃×C₄	C₄	semidirect
16. C₂×C₃×C₄⋊C₄	C₂²	semidirect
19. C₂×C₃×C₄²	C₂²	semidirect
20. C₃×#95	C₂²	semidirect
22. C₂×#117	C₂×C₃	semidirect
23. #381	C₂×C₃	semidirect
24. #380	C₂×C₃	semidirect
26. C₄×C₂²⋊C₄	C₂×C₃	semidirect
51. C₂³×C₄	C₃×C₄	semidirect
57. C₂×C₄⋊C₄	C₂²×C₃	semidirect
60. C₂×C₄²	C₂²×C₃	semidirect
61. #95	C₂²×C₃	semidirect
79. C₂×C₃×C₄	C₂²⋊C₄	semidirect
114. C₂²×C₃	#117	semidirect
139. C₂×C₄	C₃×C₂²⋊C₄	semidirect
157. C₂²	C₃×#117	semidirect
165. C₃	#2280	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃×C₄	2
C₂×C₃×#117	2
C₃×#381	2
C₃×#380	2
C₃×#379	2
C₃×C₄×C₂²⋊C₄	2
#2280	3

Subgroups — C3×#2280 #84144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2280 #84144