Subgroups — C₂×C₃×#380 #84142

Order 384.

1232 conjugacy classes of subgroups.

The subgroup lattice has 188 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	C₂²×C₃×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
3	C₂²×C₃×C₂²⋊C₄	C₂	1	2	yes	maximal
4	C₃×#380	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#95	C₂	1	2	yes	maximal, complemented
6	C₂×#380	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂⁵×C₃	C₂²	1	1	yes	characteristic
8	C₂³×C₃×C₄	—	2	8	no	—
9	C₂³×C₃×C₄	C₄	1	8	yes	complemented
10	C₂³×C₃×C₄	C₂²	1	4	yes	—
11	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
12	C₂³×C₃×C₄	C₂²	1	4	yes	—
13	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄⋊C₄	—	2	8	no	—
15	C₂×C₃×C₄⋊C₄	C₂²	1	8	yes	complemented
16	C₂×C₃×C₂²⋊C₄	—	2	8	no	—
17	C₂×C₃×C₂²⋊C₄	C₂²	1	8	yes	—
18	C₃×#95	C₂²	1	8	yes	complemented
19	C₂⁴×C₄	C₂×C₃	1	1	yes	characteristic
20	C₂²×C₄⋊C₄	C₂×C₃	1	2	yes	complemented
21	C₂²×C₂²⋊C₄	C₂×C₃	1	2	yes	—
22	#380	C₂×C₃	1	8	yes	direct product factor
23	C₂×#95	C₂×C₃	1	2	yes	complemented
24	C₂⁴×C₃	—	2	4	no	—
25	C₂⁴×C₃	—	2	4	no	—
26	C₂⁴×C₃	C₂×C₄	1	4	yes	—
27	C₂⁴×C₃	D₄	1	4	yes	—
28	C₂⁴×C₃	Q₈	1	4	yes	—
29	C₂⁴×C₃	C₂³	1	2	yes	—
30	C₂⁴×C₃	C₂³	1	1	yes	characteristic
31	C₂²×C₃×C₄	—	2	4	no	—
32	C₂²×C₃×C₄	—	2	32	no	—
33	C₂²×C₃×C₄	—	2	16	no	—
34	C₂²×C₃×C₄	—	2	8	no	—
35	C₂²×C₃×C₄	—	2	8	no	—
36	C₂²×C₃×C₄	—	2	8	no	—
37	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
38	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
39	C₂²×C₃×C₄	C₂×C₄	1	16	yes	complemented
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	2	yes	—
44	C₃×C₄⋊C₄	—	2	16	no	—
45	C₃×C₂²⋊C₄	—	2	16	no	—
46	C₂⁵	C₂²×C₃	1	1	yes	characteristic
47	C₂³×C₄	—	2	8	no	—
48	C₂³×C₄	C₃×C₄	1	8	yes	complemented
49	C₂³×C₄	C₂²×C₃	1	4	yes	—
50	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
51	C₂³×C₄	C₂²×C₃	1	4	yes	—
52	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₄⋊C₄	—	2	8	no	—
54	C₂×C₄⋊C₄	C₂²×C₃	1	8	yes	complemented
55	C₂×C₂²⋊C₄	—	2	8	no	—
56	C₂×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
57	#95	C₂²×C₃	1	8	yes	complemented
58	C₂³×C₃	—	2	8	no	—
59	C₂³×C₃	—	2	8	no	—
60	C₂³×C₃	—	2	16	no	—
61	C₂³×C₃	—	2	16	no	—
62	C₂³×C₃	—	2	4	no	—
63	C₂³×C₃	—	2	4	no	—
64	C₂³×C₃	C₄⋊C₄	1	16	yes	complemented
65	C₂³×C₃	C₂²×C₄	1	4	yes	—
66	C₂³×C₃	C₂²×C₄	1	2	yes	—
67	C₂³×C₃	C₂×D₄	1	4	yes	—
68	C₂³×C₃	C₂×D₄	1	4	yes	—
69	C₂³×C₃	C₂×D₄	1	2	yes	—
70	C₂³×C₃	C₂×Q₈	1	4	yes	—
71	C₂³×C₃	C₂×Q₈	1	2	yes	—
72	C₂³×C₃	#41	1	4	yes	—
73	C₂³×C₃	C₂⁴	1	1	yes	characteristic
74	C₂×C₃×C₄	—	2	8	no	—
75	C₂×C₃×C₄	—	2	4	no	—
76	C₂×C₃×C₄	—	2	32	no	—
77	C₂×C₃×C₄	—	2	8	no	—
78	C₂×C₃×C₄	—	2	8	no	—
79	C₂×C₃×C₄	—	2	8	no	—
80	C₂×C₃×C₄	—	4	16	no	—
81	C₂×C₃×C₄	C₂²⋊C₄	1	16	yes	complemented
82	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
83	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
84	C₂×C₃×C₄	C₂×D₄	1	8	yes	—
85	C₂⁴	—	2	4	no	—
86	C₂⁴	—	2	4	no	—
87	C₂⁴	C₂×C₃×C₄	1	4	yes	—
88	C₂⁴	C₃×D₄	1	4	yes	—
89	C₂⁴	C₃×Q₈	1	4	yes	—
90	C₂⁴	C₂³×C₃	1	2	yes	—
91	C₂⁴	C₂³×C₃	1	1	yes	characteristic
92	C₂²×C₄	—	2	4	no	—
93	C₂²×C₄	—	2	32	no	—
94	C₂²×C₄	—	2	16	no	—
95	C₂²×C₄	—	2	8	no	—
96	C₂²×C₄	—	2	8	no	—
97	C₂²×C₄	—	2	8	no	—
98	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
99	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
100	C₂²×C₄	C₂×C₃×C₄	1	16	yes	complemented
101	C₂²×C₄	C₃×D₄	1	8	yes	—
102	C₂²×C₄	C₂³×C₃	1	8	yes	—
103	C₂²×C₄	C₂³×C₃	1	2	yes	—
104	C₂²×C₄	C₂³×C₃	1	2	yes	—
105	C₄⋊C₄	—	2	16	no	—
106	C₂²⋊C₄	—	2	16	no	—
107	C₂²×C₃	—	2	4	no	—
108	C₂²×C₃	—	2	4	no	—
109	C₂²×C₃	—	2	8	no	—
110	C₂²×C₃	—	2	8	no	—
111	C₂²×C₃	—	2	32	no	—
112	C₂²×C₃	C₂×C₂²⋊C₄	1	4	yes	—
113	C₂²×C₃	C₂×C₄⋊C₄	1	4	yes	—
114	C₂²×C₃	C₂×C₄⋊C₄	1	8	yes	complemented
115	C₂²×C₃	#117	1	4	yes	—
116	C₂²×C₃	#121	1	8	yes	—
117	C₂²×C₃	C₂²⋊Q₈	1	8	yes	—
118	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
119	C₂²×C₃	C₂²×D₄	1	2	yes	—
120	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
121	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
122	C₂²×C₃	C₂×#41	1	2	yes	—
123	C₃×C₄	—	2	4	no	—
124	C₃×C₄	—	4	8	no	—
125	C₃×C₄	C₂×C₂²⋊C₄	1	8	yes	complemented
126	C₂³	—	2	8	no	—
127	C₂³	—	2	8	no	—
128	C₂³	—	2	16	no	—
129	C₂³	—	2	16	no	—
130	C₂³	—	2	4	no	—
131	C₂³	—	2	4	no	—
132	C₂³	C₃×C₄⋊C₄	1	16	yes	complemented
133	C₂³	C₂²×C₃×C₄	1	4	yes	—
134	C₂³	C₂²×C₃×C₄	1	2	yes	—
135	C₂³	C₂×C₃×D₄	1	4	yes	—
136	C₂³	C₂×C₃×D₄	1	4	yes	—
137	C₂³	C₂×C₃×D₄	1	2	yes	—
138	C₂³	C₂×C₃×Q₈	1	4	yes	—
139	C₂³	C₂×C₃×Q₈	1	2	yes	—
140	C₂³	C₃×#41	1	4	yes	—
141	C₂³	C₂⁴×C₃	1	1	yes	characteristic
142	C₂×C₄	—	2	8	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	32	no	—
145	C₂×C₄	—	2	8	no	—
146	C₂×C₄	—	2	8	no	—
147	C₂×C₄	—	2	8	no	—
148	C₂×C₄	—	4	16	no	—
149	C₂×C₄	C₃×C₂²⋊C₄	1	16	yes	complemented
150	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
151	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
152	C₂×C₄	C₂×C₃×D₄	1	8	yes	—
153	C₂×C₃	—	2	8	no	—
154	C₂×C₃	#380	1	8	yes	direct product factor
155	C₂×C₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₃	C₂×#117	1	1	yes	characteristic
158	C₂×C₃	C₂×#121	1	2	yes	—
159	C₂×C₃	C₂×C₂²⋊Q₈	1	2	yes	—
160	C₂²	—	2	32	no	—
161	C₂²	—	2	4	no	—
162	C₂²	—	2	4	no	—
163	C₂²	—	2	8	no	—
164	C₂²	—	2	8	no	—
165	C₂²	C₂×C₃×C₂²⋊C₄	1	4	yes	—
166	C₂²	C₂×C₃×C₄⋊C₄	1	4	yes	—
167	C₂²	C₂×C₃×C₄⋊C₄	1	8	yes	complemented
168	C₂²	C₃×#117	1	4	yes	—
169	C₂²	C₃×#121	1	8	yes	—
170	C₂²	C₃×C₂²⋊Q₈	1	8	yes	—
171	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
172	C₂²	C₂²×C₃×D₄	1	2	yes	—
173	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
174	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
175	C₂²	C₂×C₃×#41	1	2	yes	—
176	C₄	—	2	4	no	—
177	C₄	—	4	8	no	—
178	C₄	C₂×C₃×C₂²⋊C₄	1	8	yes	complemented
179	C₃	C₂×#380	1	1	yes	characteristic, direct product factor, Sylow
180	C₂	—	2	8	no	—
181	C₂	C₃×#380	1	8	yes	direct product factor
182	C₂	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
183	C₂	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
184	C₂	C₂×C₃×#117	1	1	yes	characteristic
185	C₂	C₂×C₃×#121	1	2	yes	—
186	C₂	C₂×C₃×C₂²⋊Q₈	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃	C₂³
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂⁵×C₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₄⋊C₄	C₂	semidirect
4. C₃×#380	C₂	direct (N × H)
5. C₂×C₃×#95	C₂	semidirect
6. C₂×#380	C₃	direct (N × H)
9. C₂³×C₃×C₄	C₄	semidirect
15. C₂×C₃×C₄⋊C₄	C₂²	semidirect
18. C₃×#95	C₂²	semidirect
20. C₂²×C₄⋊C₄	C₂×C₃	semidirect
22. #380	C₂×C₃	direct (N × H)
23. C₂×#95	C₂×C₃	semidirect
39. C₂²×C₃×C₄	C₂×C₄	semidirect
48. C₂³×C₄	C₃×C₄	semidirect
54. C₂×C₄⋊C₄	C₂²×C₃	semidirect
57. #95	C₂²×C₃	semidirect
64. C₂³×C₃	C₄⋊C₄	semidirect
81. C₂×C₃×C₄	C₂²⋊C₄	semidirect
100. C₂²×C₄	C₂×C₃×C₄	semidirect
114. C₂²×C₃	C₂×C₄⋊C₄	semidirect
125. C₃×C₄	C₂×C₂²⋊C₄	semidirect
132. C₂³	C₃×C₄⋊C₄	semidirect
149. C₂×C₄	C₃×C₂²⋊C₄	semidirect
154. C₂×C₃	#380	direct (N × H)
167. C₂²	C₂×C₃×C₄⋊C₄	semidirect
178. C₄	C₂×C₃×C₂²⋊C₄	semidirect
179. C₃	C₂×#380	direct (N × H)
181. C₂	C₃×#380	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
C₂²×C₃×C₄⋊C₄	2
C₂²×C₃×C₂²⋊C₄	2
C₃×#380	2
C₂×C₃×#95	2
C₂×#380	3

Subgroups — C2×C3×#380 #84142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#380 #84142