Subgroups — C₃×C₄×#125 #84169

Order 384.

484 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#125	C₂	1	1	yes	maximal, characteristic
2	C₃×#389	C₂	1	2	yes	maximal
3	C₃×#387	C₂	1	4	yes	maximal
4	C₃×#382	C₂	1	1	yes	maximal, characteristic
5	C₃×C₄×C₄⋊C₄	C₂	1	4	yes	maximal
6	C₃×C₄×C₄⋊C₄	C₂	1	2	yes	maximal
7	C₃×C₄³	C₂	1	1	yes	maximal, characteristic
8	C₄×#125	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₃×#125	C₄	1	8	yes	direct product factor
10	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
13	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₃×C₄²	C₂²	1	2	yes	—
15	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄²	C₂²	1	2	yes	—
17	C₂×C₃×C₄²	C₂²	1	4	yes	—
18	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
20	C₃×#95	C₂²	1	4	yes	—
21	C₃×#95	C₂²	1	4	yes	—
22	C₃×#95	C₂²	1	4	yes	—
23	C₂×#125	C₂×C₃	1	1	yes	characteristic
24	#389	C₂×C₃	1	2	yes	—
25	#387	C₂×C₃	1	4	yes	—
26	#382	C₂×C₃	1	1	yes	characteristic
27	C₄×C₄⋊C₄	C₂×C₃	1	4	yes	—
28	C₄×C₄⋊C₄	C₂×C₃	1	2	yes	—
29	C₄³	C₂×C₃	1	1	yes	characteristic
30	C₂²×C₃×C₄	C₂³	1	4	yes	—
31	C₂²×C₃×C₄	C₂³	1	2	yes	—
32	C₂²×C₃×C₄	C₂³	1	4	yes	—
33	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
34	C₂²×C₃×C₄	C₂³	1	2	yes	—
35	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
36	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
37	C₃×C₄⋊C₄	—	2	8	no	—
38	C₃×C₄⋊C₄	—	2	4	no	—
39	C₃×C₄⋊C₄	C₂×C₄	1	16	yes	—
40	C₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
41	C₃×C₄²	—	2	4	no	—
42	C₃×C₄²	—	2	2	no	—
43	C₃×C₄²	—	2	4	no	—
44	C₃×C₄²	—	2	8	no	—
45	C₃×C₄²	C₂×C₄	1	4	yes	—
46	C₃×C₄²	Q₈	1	4	yes	—
47	#125	C₃×C₄	1	8	yes	direct product factor
48	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
49	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
50	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
51	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
52	C₂×C₄²	C₂²×C₃	1	2	yes	—
53	C₂×C₄²	C₂²×C₃	1	2	yes	—
54	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₄²	C₂²×C₃	1	4	yes	—
56	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
58	#95	C₂²×C₃	1	4	yes	—
59	#95	C₂²×C₃	1	4	yes	—
60	#95	C₂²×C₃	1	4	yes	—
61	C₂³×C₃	C₂⁴	1	1	yes	characteristic
62	C₂×C₃×C₄	—	2	2	no	—
63	C₂×C₃×C₄	—	2	4	no	—
64	C₂×C₃×C₄	—	2	2	no	—
65	C₂×C₃×C₄	—	2	2	no	—
66	C₂×C₃×C₄	—	2	2	no	—
67	C₂×C₃×C₄	—	2	2	no	—
68	C₂×C₃×C₄	—	2	4	no	—
69	C₂×C₃×C₄	—	2	4	no	—
70	C₂×C₃×C₄	—	2	4	no	—
71	C₂×C₃×C₄	—	2	4	no	—
72	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
73	C₂×C₃×C₄	C₂²×C₄	1	8	yes	—
74	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
75	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
76	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
77	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
78	C₂×C₃×C₄	#41	1	4	yes	—
79	C₂×C₃×C₄	#41	1	2	yes	—
80	C₂×C₃×C₄	#41	1	4	yes	—
81	C₂²×C₄	C₂³×C₃	1	4	yes	—
82	C₂²×C₄	C₂³×C₃	1	2	yes	—
83	C₂²×C₄	C₂³×C₃	1	4	yes	—
84	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
85	C₂²×C₄	C₂³×C₃	1	2	yes	—
86	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
87	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
88	C₄⋊C₄	—	2	8	no	—
89	C₄⋊C₄	—	2	4	no	—
90	C₄⋊C₄	C₂×C₃×C₄	1	16	yes	—
91	C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
92	C₄²	—	2	4	no	—
93	C₄²	—	2	2	no	—
94	C₄²	—	2	4	no	—
95	C₄²	—	2	8	no	—
96	C₄²	C₂×C₃×C₄	1	4	yes	—
97	C₄²	C₃×Q₈	1	4	yes	—
98	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
99	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
100	C₂²×C₃	C₂×#41	1	1	yes	characteristic
101	C₂²×C₃	C₂×#41	1	2	yes	—
102	C₂²×C₃	C₂×#41	1	2	yes	—
103	C₃×C₄	—	2	4	no	—
104	C₃×C₄	—	2	8	no	—
105	C₃×C₄	—	2	4	no	—
106	C₃×C₄	—	2	2	no	—
107	C₃×C₄	—	4	4	no	—
108	C₃×C₄	C₄×Q₈	1	4	yes	—
109	C₃×C₄	#125	1	4	yes	direct product factor
110	C₂³	C₂⁴×C₃	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
122	C₂×C₄	C₂²×C₃×C₄	1	8	yes	—
123	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
124	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
125	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
126	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
127	C₂×C₄	C₃×#41	1	4	yes	—
128	C₂×C₄	C₃×#41	1	2	yes	—
129	C₂×C₄	C₃×#41	1	4	yes	—
130	C₂×C₃	C₂×C₄×Q₈	1	1	yes	characteristic
131	C₂×C₃	C₄×#41	1	2	yes	—
132	C₂×C₃	C₂×#125	1	1	yes	characteristic
133	C₂×C₃	#529	1	2	yes	—
134	C₂×C₃	#533	1	1	yes	characteristic
135	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
136	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
137	C₂²	C₂×C₃×#41	1	1	yes	characteristic
138	C₂²	C₂×C₃×#41	1	2	yes	—
139	C₂²	C₂×C₃×#41	1	2	yes	—
140	C₄	—	2	4	no	—
141	C₄	—	2	8	no	—
142	C₄	—	2	4	no	—
143	C₄	—	2	2	no	—
144	C₄	—	4	4	no	—
145	C₄	C₃×C₄×Q₈	1	4	yes	—
146	C₄	C₃×#125	1	4	yes	direct product factor
147	C₃	C₄×#125	1	1	yes	characteristic, direct product factor, Sylow
148	C₂	C₂×C₃×C₄×Q₈	1	1	yes	characteristic
149	C₂	C₃×C₄×#41	1	2	yes	—
150	C₂	C₂×C₃×#125	1	1	yes	characteristic
151	C₂	C₃×#529	1	2	yes	—
152	C₂	C₃×#533	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂³
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂³	C₂⁴×C₃
socle, prime radical	C₂³×C₃	C₂⁴
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
8. C₄×#125	C₃	direct (N × H)
9. C₃×#125	C₄	direct (N × H)
47. #125	C₃×C₄	direct (N × H)
109. C₃×C₄	#125	direct (N × H)
146. C₄	C₃×#125	direct (N × H)
147. C₃	C₄×#125	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×#125	2
C₃×#389	2
C₃×#387	2
C₃×#382	2
C₃×C₄×C₄⋊C₄	2
C₃×C₄³	2
C₄×#125	3

Subgroups — C3×C4×#125 #84169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×#125 #84169