Subgroups — C₂×C₃×C₄×C₄⋊C₄ #84133

Order 384.

928 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂³×C₃×C₄	C₂²
commutator	C₂	C₂²×C₃×C₄²
Frattini, cosocle	C₂³	C₂⁴×C₃
socle, prime radical	C₂⁴×C₃	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃×C₄×C₄⋊C₄	C₂	direct (N × H)
6. C₂×C₄×C₄⋊C₄	C₃	direct (N × H)
11. C₂×C₃×C₄⋊C₄	C₄	direct (N × H)
14. C₂×C₃×C₄²	C₄	semidirect
21. C₄×C₄⋊C₄	C₂×C₃	direct (N × H)
38. C₃×C₄⋊C₄	C₂×C₄	direct (N × H)
40. C₃×C₄²	C₂×C₄	semidirect
45. C₂×C₄⋊C₄	C₃×C₄	direct (N × H)
48. C₂×C₄²	C₃×C₄	semidirect
63. C₂×C₃×C₄	C₄²	semidirect
64. C₂×C₃×C₄	C₄⋊C₄	direct (N × H)
87. C₄⋊C₄	C₂×C₃×C₄	direct (N × H)
89. C₄²	C₂×C₃×C₄	semidirect
102. C₃×C₄	C₂×C₄²	semidirect
103. C₃×C₄	C₂×C₄⋊C₄	direct (N × H)
115. C₂×C₄	C₃×C₄²	semidirect
116. C₂×C₄	C₃×C₄⋊C₄	direct (N × H)
124. C₂×C₃	C₄×C₄⋊C₄	direct (N × H)
142. C₄	C₂×C₃×C₄²	semidirect
143. C₄	C₂×C₃×C₄⋊C₄	direct (N × H)
144. C₃	C₂×C₄×C₄⋊C₄	direct (N × H)
145. C₂	C₃×C₄×C₄⋊C₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄⋊C₄	2
C₂²×C₃×C₄²	2
C₃×C₄×C₄⋊C₄	2
C₂×C₃×#95	2
C₂×C₄×C₄⋊C₄	3

Subgroups — C2×C3×C4×C4⋊C4 #84133

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄×C₄⋊C₄ #84133