Subgroups — C₃×C₄×#117 #84134

Order 384.

752 conjugacy classes of subgroups.

The subgroup lattice has 136 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	2	yes	maximal
2	C₂²×C₃×C₄²	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₄⋊C₄	C₂	1	4	yes	maximal, complemented
4	C₃×C₄×C₂²⋊C₄	C₂	1	4	yes	maximal
5	C₃×#376	C₂	1	2	yes	maximal, complemented
6	C₃×C₄³	C₂	1	2	yes	maximal, complemented
7	C₄×#117	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	C₄	1	16	yes	direct product factor
11	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
13	C₂×C₃×C₄²	C₄	1	8	yes	complemented
14	C₂×C₃×C₄²	C₂²	1	8	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	C₃×#95	C₂²	1	8	yes	—
20	C₂×#117	C₂×C₃	1	2	yes	—
21	C₂²×C₄²	C₂×C₃	1	1	yes	characteristic
22	C₄×C₄⋊C₄	C₂×C₃	1	4	yes	complemented
23	C₄×C₂²⋊C₄	C₂×C₃	1	4	yes	—
24	#376	C₂×C₃	1	2	yes	complemented
25	C₄³	C₂×C₃	1	2	yes	complemented
26	C₂⁴×C₃	C₂³	1	1	yes	characteristic
27	C₂²×C₃×C₄	—	2	8	no	—
28	C₂²×C₃×C₄	C₂×C₄	1	8	yes	—
29	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
30	C₂²×C₃×C₄	C₂×C₄	1	8	yes	—
31	C₂²×C₃×C₄	C₂³	1	2	yes	—
32	C₂²×C₃×C₄	C₂³	1	8	yes	—
33	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
34	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
35	C₂²×C₃×C₄	C₂³	1	2	yes	—
36	C₃×C₄⋊C₄	C₂×C₄	1	16	yes	complemented
37	C₃×C₂²⋊C₄	C₂×C₄	1	16	yes	—
38	C₃×C₄²	—	2	16	no	—
39	C₃×C₄²	C₂×C₄	1	4	yes	—
40	C₃×C₄²	C₂×C₄	1	8	yes	—
41	C₃×C₄²	C₂×C₄	1	4	yes	—
42	C₃×C₄²	C₂×C₄	1	16	yes	complemented
43	C₂³×C₄	C₂²×C₃	1	2	yes	—
44	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
45	#117	C₃×C₄	1	16	yes	direct product factor
46	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
47	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
48	C₂×C₄²	C₃×C₄	1	8	yes	complemented
49	C₂×C₄²	C₂²×C₃	1	8	yes	—
50	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₄²	C₂²×C₃	1	2	yes	—
53	C₂×C₄²	C₂²×C₃	1	4	yes	—
54	#95	C₂²×C₃	1	8	yes	—
55	C₂³×C₃	—	2	4	no	—
56	C₂³×C₃	C₂²×C₄	1	4	yes	—
57	C₂³×C₃	C₂²×C₄	1	2	yes	—
58	C₂³×C₃	C₂⁴	1	1	yes	characteristic
59	C₂×C₃×C₄	—	2	16	no	—
60	C₂×C₃×C₄	—	2	4	no	—
61	C₂×C₃×C₄	—	2	16	no	—
62	C₂×C₃×C₄	C₄²	1	16	yes	complemented
63	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
64	C₂×C₃×C₄	C₂²×C₄	1	16	yes	—
65	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
66	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
67	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
68	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
69	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
70	C₂×C₃×C₄	#41	1	8	yes	—
71	C₂⁴	C₂³×C₃	1	1	yes	characteristic
72	C₂²×C₄	—	2	8	no	—
73	C₂²×C₄	C₂×C₃×C₄	1	8	yes	—
74	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
75	C₂²×C₄	C₂×C₃×C₄	1	8	yes	—
76	C₂²×C₄	C₂³×C₃	1	2	yes	—
77	C₂²×C₄	C₂³×C₃	1	8	yes	—
78	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
79	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
80	C₂²×C₄	C₂³×C₃	1	2	yes	—
81	C₄⋊C₄	C₂×C₃×C₄	1	16	yes	complemented
82	C₂²⋊C₄	C₂×C₃×C₄	1	16	yes	—
83	C₄²	—	2	16	no	—
84	C₄²	C₂×C₃×C₄	1	4	yes	—
85	C₄²	C₂×C₃×C₄	1	8	yes	—
86	C₄²	C₂×C₃×C₄	1	4	yes	—
87	C₄²	C₂×C₃×C₄	1	16	yes	complemented
88	C₂²×C₃	—	2	4	no	—
89	C₂²×C₃	—	2	8	no	—
90	C₂²×C₃	C₂×C₄²	1	4	yes	—
91	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
92	C₂²×C₃	C₂³×C₄	1	2	yes	—
93	C₂²×C₃	C₂×#41	1	4	yes	—
94	C₃×C₄	—	2	4	no	—
95	C₃×C₄	—	2	16	no	—
96	C₃×C₄	C₂×C₄²	1	4	yes	—
97	C₃×C₄	C₂×C₄²	1	4	yes	—
98	C₃×C₄	#117	1	8	yes	direct product factor
99	C₂³	—	2	4	no	—
100	C₂³	C₂²×C₃×C₄	1	4	yes	—
101	C₂³	C₂²×C₃×C₄	1	2	yes	—
102	C₂³	C₂⁴×C₃	1	1	yes	characteristic
103	C₂×C₄	—	2	16	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	16	no	—
106	C₂×C₄	C₃×C₄²	1	16	yes	complemented
107	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
108	C₂×C₄	C₂²×C₃×C₄	1	16	yes	—
109	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
110	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
111	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
112	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
113	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
114	C₂×C₄	C₃×#41	1	8	yes	—
115	C₂×C₃	—	2	4	no	—
116	C₂×C₃	C₂²×C₄²	1	1	yes	characteristic
117	C₂×C₃	C₂×#117	1	2	yes	—
118	C₂×C₃	C₄×#41	1	4	yes	—
119	C₂²	—	2	4	no	—
120	C₂²	—	2	8	no	—
121	C₂²	C₂×C₃×C₄²	1	4	yes	—
122	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
123	C₂²	C₂³×C₃×C₄	1	2	yes	—
124	C₂²	C₂×C₃×#41	1	4	yes	—
125	C₄	—	2	4	no	—
126	C₄	—	2	16	no	—
127	C₄	C₂×C₃×C₄²	1	4	yes	—
128	C₄	C₂×C₃×C₄²	1	4	yes	—
129	C₄	C₃×#117	1	8	yes	direct product factor
130	C₃	C₄×#117	1	1	yes	characteristic, direct product factor, Sylow
131	C₂	—	2	4	no	—
132	C₂	C₂²×C₃×C₄²	1	1	yes	characteristic
133	C₂	C₂×C₃×#117	1	2	yes	—
134	C₂	C₃×C₄×#41	1	4	yes	—

	subgroup	G/H
center	C₂×C₃×C₄²	C₂²
commutator	C₂	C₂²×C₃×C₄²
Frattini, cosocle	C₂³	C₂⁴×C₃
socle	C₂³×C₃	C₂⁴
prime radical	C₂⁴×C₃	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×C₄⋊C₄	C₂	semidirect
5. C₃×#376	C₂	semidirect
6. C₃×C₄³	C₂	semidirect
7. C₄×#117	C₃	direct (N × H)
10. C₃×#117	C₄	direct (N × H)
13. C₂×C₃×C₄²	C₄	semidirect
22. C₄×C₄⋊C₄	C₂×C₃	semidirect
24. #376	C₂×C₃	semidirect
25. C₄³	C₂×C₃	semidirect
36. C₃×C₄⋊C₄	C₂×C₄	semidirect
42. C₃×C₄²	C₂×C₄	semidirect
45. #117	C₃×C₄	direct (N × H)
48. C₂×C₄²	C₃×C₄	semidirect
62. C₂×C₃×C₄	C₄²	semidirect
81. C₄⋊C₄	C₂×C₃×C₄	semidirect
87. C₄²	C₂×C₃×C₄	semidirect
98. C₃×C₄	#117	direct (N × H)
106. C₂×C₄	C₃×C₄²	semidirect
129. C₄	C₃×#117	direct (N × H)
130. C₃	C₄×#117	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×C₄×C₄⋊C₄	2
C₃×C₄×C₂²⋊C₄	2
C₃×#376	2
C₃×C₄³	2
C₄×#117	3

Subgroups — C3×C4×#117 #84134

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×#117 #84134