Subgroups — C₂×C₃×#419 #84758

Order 384.

752 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic
4	C₃×#419	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#103	C₂	1	4	yes	maximal, complemented
6	C₂×#419	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂²×C₃×Q₈	—	2	4	no	—
8	C₂²×C₃×Q₈	—	2	2	no	—
9	C₂²×C₃×Q₈	—	2	1	no	—
10	C₂²×C₃×Q₈	C₄	1	8	yes	complemented
11	C₂²×C₃×Q₈	C₂²	1	2	yes	—
12	C₂²×C₃×Q₈	C₂²	1	4	yes	—
13	C₂³×C₃×C₄	—	2	1	no	—
14	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
15	C₂×C₃×M₁₆	—	2	4	no	—
16	C₂×C₃×M₁₆	C₂²	1	4	yes	—
17	C₂²×C₃×C₈	C₂²	1	2	yes	—
18	C₃×#117	—	2	2	no	—
19	C₃×#117	C₂²	1	4	yes	—
20	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
22	C₂×C₃×C₄²	—	2	1	no	—
23	C₃×#103	C₂²	1	16	yes	complemented
24	C₂³×Q₈	C₂×C₃	1	1	yes	characteristic
25	C₂²×M₁₆	C₂×C₃	1	1	yes	characteristic
26	C₂×#117	C₂×C₃	1	1	yes	characteristic
27	#419	C₂×C₃	1	8	yes	direct product factor
28	C₂×#103	C₂×C₃	1	4	yes	complemented
29	C₂⁴×C₃	D₄	1	1	yes	characteristic
30	C₂×C₃×Q₈	—	2	8	no	—
31	C₂×C₃×Q₈	—	2	16	no	—
32	C₂×C₃×Q₈	—	2	4	no	—
33	C₂×C₃×Q₈	—	2	8	no	—
34	C₂×C₃×Q₈	—	2	4	no	—
35	C₂×C₃×Q₈	—	2	2	no	—
36	C₂×C₃×Q₈	C₂×C₄	1	8	yes	—
37	C₂×C₃×Q₈	C₂×C₄	1	16	yes	complemented
38	C₂×C₃×Q₈	C₂³	1	4	yes	—
39	C₂²×C₃×C₄	—	2	8	no	—
40	C₂²×C₃×C₄	—	2	4	no	—
41	C₂²×C₃×C₄	—	2	2	no	—
42	C₂²×C₃×C₄	—	2	2	no	—
43	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
44	C₂²×C₃×C₄	D₄	1	2	yes	—
45	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
46	C₂²×C₃×C₄	D₄	1	4	yes	—
47	C₂²×C₃×C₄	C₂³	1	2	yes	—
48	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
49	C₃×M₁₆	—	2	8	no	—
50	C₂×C₃×C₈	—	2	4	no	—
51	C₂×C₃×C₈	C₂³	1	4	yes	—
52	C₃×C₄⋊C₄	—	2	2	no	—
53	C₃×C₄⋊C₄	C₂³	1	4	yes	—
54	C₃×C₂²⋊C₄	—	2	4	no	—
55	C₃×C₄²	—	2	4	no	—
56	C₂²×Q₈	—	2	4	no	—
57	C₂²×Q₈	—	2	1	no	—
58	C₂²×Q₈	—	2	2	no	—
59	C₂²×Q₈	C₃×C₄	1	8	yes	complemented
60	C₂²×Q₈	C₂²×C₃	1	2	yes	—
61	C₂²×Q₈	C₂²×C₃	1	4	yes	—
62	C₂³×C₄	—	2	1	no	—
63	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
64	C₂×M₁₆	—	2	4	no	—
65	C₂×M₁₆	C₂²×C₃	1	4	yes	—
66	C₂²×C₈	C₂²×C₃	1	2	yes	—
67	#117	—	2	2	no	—
68	#117	C₂²×C₃	1	4	yes	—
69	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
70	C₂×C₂²⋊C₄	—	2	1	no	—
71	C₂×C₄²	—	2	1	no	—
72	#103	C₂²×C₃	1	16	yes	complemented
73	C₂³×C₃	—	2	4	no	—
74	C₂³×C₃	C₂²⋊C₄	1	4	yes	—
75	C₂³×C₃	C₂×D₄	1	2	yes	—
76	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
77	C₃×Q₈	—	2	8	no	—
78	C₃×Q₈	—	2	16	no	—
79	C₃×Q₈	—	2	4	no	—
80	C₃×Q₈	C₂²×C₄	1	8	yes	—
81	C₂×C₃×C₄	—	2	8	no	—
82	C₂×C₃×C₄	—	2	16	no	—
83	C₂×C₃×C₄	—	2	4	no	—
84	C₂×C₃×C₄	—	2	4	no	—
85	C₂×C₃×C₄	—	4	4	no	—
86	C₂×C₃×C₄	C₂²⋊C₄	1	8	yes	—
87	C₂×C₃×C₄	C₂²⋊C₄	1	4	yes	—
88	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
89	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
90	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
91	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
93	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
94	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
95	C₃×C₈	—	2	4	no	—
96	C₂⁴	C₃×D₄	1	1	yes	characteristic
97	C₂×Q₈	—	2	8	no	—
98	C₂×Q₈	—	2	16	no	—
99	C₂×Q₈	—	2	4	no	—
100	C₂×Q₈	—	2	8	no	—
101	C₂×Q₈	—	2	4	no	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	C₂×C₃×C₄	1	8	yes	—
104	C₂×Q₈	C₂×C₃×C₄	1	16	yes	complemented
105	C₂×Q₈	C₂³×C₃	1	4	yes	—
106	C₂²×C₄	—	2	8	no	—
107	C₂²×C₄	—	2	2	no	—
108	C₂²×C₄	—	2	4	no	—
109	C₂²×C₄	—	2	2	no	—
110	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
111	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
112	C₂²×C₄	C₃×D₄	1	2	yes	—
113	C₂²×C₄	C₃×D₄	1	4	yes	—
114	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
115	C₂²×C₄	C₂³×C₃	1	2	yes	—
116	M₁₆	—	2	8	no	—
117	C₂×C₈	—	2	4	no	—
118	C₂×C₈	C₂³×C₃	1	4	yes	—
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	C₂³×C₃	1	4	yes	—
121	C₂²⋊C₄	—	2	4	no	—
122	C₄²	—	2	4	no	—
123	C₂²×C₃	—	2	4	no	—
124	C₂²×C₃	—	2	8	no	—
125	C₂²×C₃	C₂×C₂²⋊C₄	1	4	yes	—
126	C₂²×C₃	C₂×C₂²⋊C₄	1	2	yes	—
127	C₂²×C₃	#137	1	4	yes	—
128	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
129	C₃×C₄	—	2	8	no	—
130	C₃×C₄	—	4	4	no	—
131	C₃×C₄	C₂×C₂²⋊C₄	1	2	yes	—
132	C₃×C₄	C₂×C₂²⋊C₄	1	4	yes	—
133	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
134	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
135	C₂³	—	2	4	no	—
136	C₂³	C₃×C₂²⋊C₄	1	4	yes	—
137	C₂³	C₂×C₃×D₄	1	2	yes	—
138	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
139	Q₈	—	2	8	no	—
140	Q₈	—	2	16	no	—
141	Q₈	—	2	4	no	—
142	Q₈	C₂²×C₃×C₄	1	8	yes	—
143	C₂×C₄	—	2	8	no	—
144	C₂×C₄	—	2	16	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	4	4	no	—
148	C₂×C₄	C₃×C₂²⋊C₄	1	8	yes	—
149	C₂×C₄	C₃×C₂²⋊C₄	1	4	yes	—
150	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
151	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
152	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
153	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
154	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
155	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
156	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
157	C₈	—	2	4	no	—
158	C₂×C₃	—	2	4	no	—
159	C₂×C₃	#419	1	4	yes	direct product factor
160	C₂×C₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
161	C₂×C₃	C₂×#137	1	2	yes	—
162	C₂²	—	2	4	no	—
163	C₂²	—	2	8	no	—
164	C₂²	C₂×C₃×C₂²⋊C₄	1	4	yes	—
165	C₂²	C₂×C₃×C₂²⋊C₄	1	2	yes	—
166	C₂²	C₃×#137	1	4	yes	—
167	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
168	C₄	—	2	8	no	—
169	C₄	—	4	4	no	—
170	C₄	C₂×C₃×C₂²⋊C₄	1	2	yes	—
171	C₄	C₂×C₃×C₂²⋊C₄	1	4	yes	—
172	C₄	C₂³×C₃×C₄	1	1	yes	characteristic
173	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
174	C₃	C₂×#419	1	1	yes	characteristic, direct product factor, Sylow
175	C₂	—	2	4	no	—
176	C₂	C₃×#419	1	4	yes	direct product factor
177	C₂	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
178	C₂	C₂×C₃×#137	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂×D₄
commutator	C₄	C₂³×C₃×C₄
Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₂⁴×C₃	D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂³×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃×#419	C₂	direct (N × H)
5. C₂×C₃×#103	C₂	semidirect
6. C₂×#419	C₃	direct (N × H)
10. C₂²×C₃×Q₈	C₄	semidirect
23. C₃×#103	C₂²	semidirect
27. #419	C₂×C₃	direct (N × H)
28. C₂×#103	C₂×C₃	semidirect
37. C₂×C₃×Q₈	C₂×C₄	semidirect
59. C₂²×Q₈	C₃×C₄	semidirect
72. #103	C₂²×C₃	semidirect
104. C₂×Q₈	C₂×C₃×C₄	semidirect
159. C₂×C₃	#419	direct (N × H)
174. C₃	C₂×#419	direct (N × H)
176. C₂	C₃×#419	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×Q₈	2
C₂²×C₃×M₁₆	2
C₂×C₃×#117	2
C₃×#419	2
C₂×C₃×#103	2
C₂×#419	3

Subgroups — C2×C3×#419 #84758

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#419 #84758