Subgroups — C₃×#2296 #84160

Order 384.

656 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₄²	C₂	1	1	yes	maximal, characteristic
4	C₃×#389	C₂	1	4	yes	maximal, complemented
5	C₃×#385	C₂	1	4	yes	maximal
6	C₃×C₄×C₄⋊C₄	C₂	1	4	yes	maximal, complemented
7	#2296	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂³×C₃×C₄	C₂²	1	2	yes	—
9	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₃×#117	—	2	4	no	—
11	C₃×#117	C₄	1	8	yes	complemented
12	C₂×C₃×C₄⋊C₄	C₂²	1	8	yes	—
13	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
16	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
17	C₂×C₃×C₄²	—	2	4	no	—
18	C₂×C₃×C₄²	C₂²	1	2	yes	—
19	C₂×C₃×C₄²	C₂²	1	2	yes	—
20	C₂×C₃×C₄²	C₂²	1	2	yes	—
21	C₂×C₃×C₄²	C₂²	1	2	yes	—
22	C₃×#95	C₂²	1	8	yes	—
23	C₂×#117	C₂×C₃	1	1	yes	characteristic
24	C₂×#117	C₂×C₃	1	1	yes	characteristic
25	C₂²×C₄²	C₂×C₃	1	1	yes	characteristic
26	#389	C₂×C₃	1	4	yes	complemented
27	#385	C₂×C₃	1	4	yes	—
28	C₄×C₄⋊C₄	C₂×C₃	1	4	yes	complemented
29	C₂⁴×C₃	C₂³	1	1	yes	characteristic
30	C₂²×C₃×C₄	—	2	4	no	—
31	C₂²×C₃×C₄	—	2	8	no	—
32	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
33	C₂²×C₃×C₄	D₄	1	4	yes	—
34	C₂²×C₃×C₄	Q₈	1	4	yes	—
35	C₂²×C₃×C₄	C₂³	1	4	yes	—
36	C₂²×C₃×C₄	C₂³	1	4	yes	—
37	C₂²×C₃×C₄	C₂³	1	4	yes	—
38	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
39	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
40	C₃×C₄⋊C₄	—	2	4	no	—
41	C₃×C₄⋊C₄	—	2	16	no	—
42	C₃×C₄⋊C₄	C₂×C₄	1	8	yes	complemented
43	C₃×C₂²⋊C₄	—	2	4	no	—
44	C₃×C₂²⋊C₄	C₂×C₄	1	8	yes	—
45	C₃×C₄²	—	2	4	no	—
46	C₃×C₄²	—	2	4	no	—
47	C₃×C₄²	—	2	4	no	—
48	C₃×C₄²	C₂×C₄	1	8	yes	complemented
49	C₂³×C₄	C₂²×C₃	1	2	yes	—
50	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
51	#117	—	2	4	no	—
52	#117	C₃×C₄	1	8	yes	complemented
53	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
54	C₂×C₄⋊C₄	C₂²×C₃	1	8	yes	—
55	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
56	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
57	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
58	C₂×C₄²	—	2	4	no	—
59	C₂×C₄²	C₂²×C₃	1	2	yes	—
60	C₂×C₄²	C₂²×C₃	1	2	yes	—
61	C₂×C₄²	C₂²×C₃	1	2	yes	—
62	C₂×C₄²	C₂²×C₃	1	2	yes	—
63	#95	C₂²×C₃	1	8	yes	—
64	C₂³×C₃	—	2	4	no	—
65	C₂³×C₃	C₂²×C₄	1	2	yes	—
66	C₂³×C₃	C₂×D₄	1	2	yes	—
67	C₂³×C₃	C₂×Q₈	1	2	yes	—
68	C₂³×C₃	C₂⁴	1	1	yes	characteristic
69	C₂×C₃×C₄	—	2	16	no	—
70	C₂×C₃×C₄	—	2	4	no	—
71	C₂×C₃×C₄	—	2	8	no	—
72	C₂×C₃×C₄	—	2	4	no	—
73	C₂×C₃×C₄	—	2	4	no	—
74	C₂×C₃×C₄	—	2	8	no	—
75	C₂×C₃×C₄	C₄⋊C₄	1	16	yes	complemented
76	C₂×C₃×C₄	C₂²×C₄	1	8	yes	—
77	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
78	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
79	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
80	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
81	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
82	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
83	C₂×C₃×C₄	#41	1	8	yes	—
84	C₂⁴	C₂³×C₃	1	1	yes	characteristic
85	C₂²×C₄	—	2	4	no	—
86	C₂²×C₄	—	2	8	no	—
87	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
88	C₂²×C₄	C₃×D₄	1	4	yes	—
89	C₂²×C₄	C₃×Q₈	1	4	yes	—
90	C₂²×C₄	C₂³×C₃	1	4	yes	—
91	C₂²×C₄	C₂³×C₃	1	4	yes	—
92	C₂²×C₄	C₂³×C₃	1	4	yes	—
93	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
94	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	16	no	—
97	C₄⋊C₄	C₂×C₃×C₄	1	8	yes	complemented
98	C₂²⋊C₄	—	2	4	no	—
99	C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	—
100	C₄²	—	2	4	no	—
101	C₄²	—	2	4	no	—
102	C₄²	—	2	4	no	—
103	C₄²	C₂×C₃×C₄	1	8	yes	complemented
104	C₂²×C₃	—	2	4	no	—
105	C₂²×C₃	—	2	4	no	—
106	C₂²×C₃	—	2	4	no	—
107	C₂²×C₃	C₂×C₄⋊C₄	1	4	yes	—
108	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
109	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
110	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
111	C₂²×C₃	C₂×#41	1	4	yes	—
112	C₃×C₄	—	2	8	no	—
113	C₃×C₄	—	2	8	no	—
114	C₃×C₄	—	4	4	no	—
115	C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
116	C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
117	C₂³	—	2	4	no	—
118	C₂³	C₂²×C₃×C₄	1	2	yes	—
119	C₂³	C₂×C₃×D₄	1	2	yes	—
120	C₂³	C₂×C₃×Q₈	1	2	yes	—
121	C₂³	C₂⁴×C₃	1	1	yes	characteristic
122	C₂×C₄	—	2	16	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	2	8	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	8	no	—
128	C₂×C₄	C₃×C₄⋊C₄	1	16	yes	complemented
129	C₂×C₄	C₂²×C₃×C₄	1	8	yes	—
130	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
131	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
132	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
133	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
134	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
135	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
136	C₂×C₄	C₃×#41	1	8	yes	—
137	C₂×C₃	—	2	4	no	—
138	C₂×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₃	C₄×#41	1	2	yes	—
140	C₂×C₃	#525	1	2	yes	—
141	C₂×C₃	#533	1	2	yes	—
142	C₂²	—	2	4	no	—
143	C₂²	—	2	4	no	—
144	C₂²	—	2	4	no	—
145	C₂²	C₂×C₃×C₄⋊C₄	1	4	yes	—
146	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
147	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
148	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
149	C₂²	C₂×C₃×#41	1	4	yes	—
150	C₄	—	2	8	no	—
151	C₄	—	2	8	no	—
152	C₄	—	4	4	no	—
153	C₄	C₂×C₃×C₄⋊C₄	1	4	yes	—
154	C₄	C₂×C₃×C₄⋊C₄	1	4	yes	—
155	C₃	#2296	1	1	yes	characteristic, direct product factor, Sylow
156	C₂	—	2	4	no	—
157	C₂	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
158	C₂	C₃×C₄×#41	1	2	yes	—
159	C₂	C₃×#525	1	2	yes	—
160	C₂	C₃×#533	1	2	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	C₂⁴×C₃	C₂³
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
4. C₃×#389	C₂	semidirect
6. C₃×C₄×C₄⋊C₄	C₂	semidirect
7. #2296	C₃	direct (N × H)
11. C₃×#117	C₄	semidirect
26. #389	C₂×C₃	semidirect
28. C₄×C₄⋊C₄	C₂×C₃	semidirect
42. C₃×C₄⋊C₄	C₂×C₄	semidirect
48. C₃×C₄²	C₂×C₄	semidirect
52. #117	C₃×C₄	semidirect
75. C₂×C₃×C₄	C₄⋊C₄	semidirect
97. C₄⋊C₄	C₂×C₃×C₄	semidirect
103. C₄²	C₂×C₃×C₄	semidirect
128. C₂×C₄	C₃×C₄⋊C₄	semidirect
155. C₃	#2296	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×#117	2
C₂²×C₃×C₄²	2
C₃×#389	2
C₃×#385	2
C₃×C₄×C₄⋊C₄	2
#2296	3

Subgroups — C3×#2296 #84160

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2296 #84160