Subgroups — C₄×#63639 #11795853

Order 1152.

588 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#63639	C₂	1	2	yes	maximal
3	#10589327	C₂	1	2	yes	maximal, complemented
4	C₄×C₃⋊(C₃⋊C₁₆)	C₂	1	2	yes	maximal, complemented
5	C₄×#812	—	3	4	no	maximal
6	C₂²×C₃²×C₈	C₄	1	1	yes	characteristic
7	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃²×C₄²	C₄	1	1	yes	characteristic
9	C₃²×C₄×C₈	C₄	1	1	yes	characteristic
10	C₃²×C₄×C₈	C₄	1	1	yes	characteristic
11	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
13	#63639	C₄	1	8	yes	direct product factor
14	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	4	yes	—
15	C₂×C₃×C₄×C₈	D₃	1	4	yes	—
16	C₂×#812	—	3	8	no	—
17	#4461	—	3	8	no	—
18	C₄×C₃⋊C₁₆	—	3	8	no	—
19	C₂²×C₃²×C₄	C₈	1	2	yes	—
20	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃²×C₈	C₈	1	4	yes	—
22	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
24	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
27	C₃²×C₄²	C₈	1	2	yes	—
28	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
29	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
30	C₃⋊(C₃⋊C₁₆)	C₂×C₄	1	8	yes	complemented
31	C₄×M₃₂	—	9	1	no	Sylow
32	C₂²×C₃×C₈	Dic₃	1	4	yes	—
33	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
34	C₂×C₃×C₄²	Dic₃	1	4	yes	—
35	C₃×C₄×C₈	Dic₃	1	4	yes	—
36	C₃×C₄×C₈	Dic₃	1	4	yes	—
37	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
38	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
39	#812	—	3	32	no	—
40	C₂×C₃⋊C₁₆	—	3	16	no	—
41	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
42	C₂×C₃²×C₄	—	2	4	no	—
43	C₂×C₃²×C₄	C₄²	1	2	yes	—
44	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
45	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
46	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
47	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
48	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
49	C₃²×C₈	C₄²	1	2	yes	—
50	C₃²×C₈	C₂×C₈	1	2	yes	—
51	C₃²×C₈	C₂×C₈	1	2	yes	—
52	C₃²×C₈	C₂²×C₄	1	2	yes	—
53	C₂×M₃₂	—	9	2	no	—
54	C₂×C₄×C₈	C₃⋊D₃	1	1	yes	characteristic
55	#346	—	9	2	no	—
56	C₄×C₁₆	—	9	2	no	—
57	C₂²×C₃×C₄	C₃⋊C₈	1	8	yes	—
58	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
59	C₂×C₃×C₈	C₃⋊C₈	1	16	yes	—
60	C₂×C₃×C₈	C₄×D₃	1	16	yes	—
61	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
62	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
63	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
64	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
65	C₃×C₄²	C₃⋊C₈	1	8	yes	—
66	C₃×C₄²	C₂×Dic₃	1	4	yes	—
67	C₃×C₄²	C₂×Dic₃	1	4	yes	—
68	C₃⋊C₁₆	—	3	32	no	—
69	C₂²×C₃²	—	2	2	no	—
70	C₂²×C₃²	C₄×C₈	1	2	yes	—
71	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
72	C₃²×C₄	—	2	2	no	—
73	C₃²×C₄	C₄×C₈	1	2	yes	—
74	C₃²×C₄	M₃₂	1	4	yes	complemented
75	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
76	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
77	C₂²×C₈	C₃⋊Dic₃	1	1	yes	characteristic
78	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
79	C₂×C₄²	C₃⋊Dic₃	1	1	yes	characteristic
80	M₃₂	—	9	8	no	—
81	C₂×C₁₆	—	9	4	no	—
82	C₄×C₈	C₃⋊Dic₃	1	1	yes	characteristic
83	C₄×C₈	C₃⋊Dic₃	1	1	yes	characteristic
84	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
85	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
86	C₂³×C₃	C₂×C₃⋊C₈	1	4	yes	—
87	C₂×C₃×C₄	—	2	16	no	—
88	C₂×C₃×C₄	C₂×C₃⋊C₈	1	8	yes	—
89	C₂×C₃×C₄	C₂×C₃⋊C₈	1	4	yes	—
90	C₂×C₃×C₄	C₂×C₃⋊C₈	1	8	yes	—
91	C₂×C₃×C₄	C₂×C₃⋊C₈	1	8	yes	—
92	C₂×C₃×C₄	C₄×Dic₃	1	8	yes	—
93	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
94	C₃×C₈	C₂×C₃⋊C₈	1	8	yes	—
95	C₃×C₈	C₂×C₃⋊C₈	1	8	yes	—
96	C₃×C₈	C₄×Dic₃	1	8	yes	—
97	C₃×C₈	C₂×C₄×D₃	1	8	yes	—
98	C₂×C₃²	—	2	2	no	—
99	C₂×C₃²	C₂×C₄×C₈	1	1	yes	characteristic
100	C₂×C₃²	C₂×M₃₂	1	2	yes	—
101	C₂²×C₄	C₃⋊(C₃⋊C₈)	1	2	yes	—
102	C₂²×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
103	C₂×C₈	C₃⋊(C₃⋊C₈)	1	4	yes	—
104	C₂×C₈	C₄×C₃⋊D₃	1	4	yes	—
105	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
106	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
107	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
108	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
109	C₄²	C₃⋊(C₃⋊C₈)	1	2	yes	—
110	C₄²	C₂×C₃⋊Dic₃	1	1	yes	characteristic
111	C₄²	C₂×C₃⋊Dic₃	1	1	yes	characteristic
112	C₁₆	—	9	8	no	—
113	C₂²×C₃	—	2	8	no	—
114	C₂²×C₃	C₄×C₃⋊C₈	1	8	yes	—
115	C₂²×C₃	C₂²×C₃⋊C₈	1	4	yes	—
116	C₃×C₄	—	2	8	no	—
117	C₃×C₄	C₄×C₃⋊C₈	1	8	yes	—
118	C₃×C₄	#812	1	16	yes	complemented
119	C₃×C₄	C₂²×C₃⋊C₈	1	4	yes	—
120	C₃×C₄	C₂×C₄×Dic₃	1	4	yes	—
121	C₃²	C₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
122	C₂³	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	2	yes	—
125	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	2	yes	—
126	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
127	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	2	yes	—
128	C₂×C₄	C₄×C₃⋊Dic₃	1	2	yes	—
129	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
130	C₈	C₂×C₃⋊(C₃⋊C₈)	1	2	yes	—
131	C₈	C₂×C₃⋊(C₃⋊C₈)	1	2	yes	—
132	C₈	C₄×C₃⋊Dic₃	1	2	yes	—
133	C₈	C₂×C₄×C₃⋊D₃	1	2	yes	—
134	C₂×C₃	—	2	8	no	—
135	C₂×C₃	C₂×C₄×C₃⋊C₈	1	4	yes	—
136	C₂×C₃	C₂×#812	1	8	yes	—
137	C₂²	—	2	2	no	—
138	C₂²	C₄×C₃⋊(C₃⋊C₈)	1	2	yes	—
139	C₂²	C₂²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	C₄×C₃⋊(C₃⋊C₈)	1	2	yes	—
142	C₄	#63639	1	4	yes	direct product factor
143	C₄	C₂²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
144	C₄	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
145	C₃	C₄×#812	1	4	yes	complemented
146	C₂	—	2	2	no	—
147	C₂	C₂×C₄×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
148	C₂	C₂×#63639	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂×C₃⋊D₃
commutator	C₂×C₃²	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×C₄×C₈	C₂
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂³×C₃²	C₂×C₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂×C₄×C₈	C₃⋊D₃
hypocenter	C₃²	C₄×M₃₂
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #10589327	C₂	semidirect
4. C₄×C₃⋊(C₃⋊C₁₆)	C₂	semidirect
13. #63639	C₄	direct (N × H)
30. C₃⋊(C₃⋊C₁₆)	C₂×C₄	semidirect
74. C₃²×C₄	M₃₂	semidirect
118. C₃×C₄	#812	semidirect
121. C₃²	C₄×M₃₂	semidirect
142. C₄	#63639	direct (N × H)
145. C₃	C₄×#812	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄×C₈	2
C₂×#63639	2
#10589327	2
C₄×C₃⋊(C₃⋊C₁₆)	2
C₄×#812	3

Subgroups — C4×#63639 #11795853

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#63639 #11795853