Subgroups — C₂²×#64157 #11918456

Order 1152.

4512 conjugacy classes of subgroups.

The subgroup lattice has 153 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×#64157	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#3748	C₂	1	3	yes	maximal, complemented
4	C₂³×C₃⋊(C₃⋊C₈)	C₂	1	3	yes	maximal, complemented
5	C₂²×#948	—	3	4	no	maximal
6	C₂×C₃²×#41	C₄	1	12	yes	—
7	C₂×C₃²×#41	C₂²	1	12	yes	—
8	C₂²×C₃²×Q₈	C₄	1	1	yes	characteristic
9	C₂²×C₃²×D₄	C₄	1	3	yes	—
10	C₂³×C₃²×C₄	C₂²	1	3	yes	—
11	#64157	C₂²	1	64	yes	direct product factor
12	C₂×#3748	C₂²	1	36	yes	complemented
13	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	3	yes	—
15	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	36	yes	complemented
16	C₂²×C₃×#41	D₃	1	4	yes	—
17	C₂×#948	—	3	96	no	—
18	C₂²×#208	—	3	12	no	—
19	C₂³×C₃⋊C₈	—	3	12	no	—
20	C₂⁴×C₃²	C₂×C₄	1	3	yes	—
21	C₃²×#41	C₂×C₄	1	48	yes	—
22	C₃²×#41	C₂³	1	16	yes	—
23	C₂×C₃²×Q₈	C₂×C₄	1	12	yes	—
24	C₂×C₃²×D₄	C₂×C₄	1	36	yes	—
25	C₂²×C₃²×C₄	C₂×C₄	1	3	yes	—
26	C₂²×C₃²×C₄	C₂×C₄	1	18	yes	—
27	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
28	C₂²×C₃²×C₄	C₂³	1	18	yes	—
29	#3748	C₂³	1	48	yes	complemented
30	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	6	yes	—
31	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	18	yes	—
32	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	48	yes	complemented
33	C₂²×#131	—	9	1	no	Sylow
34	C₂×C₃×#41	Dic₃	1	48	yes	—
35	C₂×C₃×#41	C₂×D₃	1	48	yes	—
36	C₂²×C₃×Q₈	Dic₃	1	4	yes	—
37	C₂²×C₃×D₄	Dic₃	1	12	yes	—
38	C₂³×C₃×C₄	C₂×D₃	1	12	yes	—
39	#948	—	3	256	no	—
40	C₂×#208	—	3	144	no	—
41	C₂²×C₃⋊C₈	—	3	4	no	—
42	C₂²×C₃⋊C₈	—	3	12	no	—
43	C₂²×C₃⋊C₈	—	3	144	no	—
44	C₂³×C₃²	—	2	12	no	—
45	C₂³×C₃²	C₂²×C₄	1	18	yes	—
46	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
47	C₃²×Q₈	C₂²×C₄	1	16	yes	—
48	C₃²×D₄	C₂²×C₄	1	48	yes	—
49	C₂×C₃²×C₄	C₂²×C₄	1	3	yes	—
50	C₂×C₃²×C₄	C₂²×C₄	1	18	yes	—
51	C₂×C₃²×C₄	C₂²×C₄	1	36	yes	—
52	C₂×C₃²×C₄	C₂⁴	1	3	yes	—
53	C₂×C₃²×C₄	C₂⁴	1	12	yes	—
54	C₃⋊(C₃⋊C₈)	C₂⁴	1	4	yes	—
55	C₃⋊(C₃⋊C₈)	C₂⁴	1	12	yes	—
56	C₂²×#41	C₃⋊D₃	1	1	yes	characteristic
57	C₂×#131	—	9	24	no	—
58	C₂²×M₁₆	—	9	3	no	—
59	C₂³×C₈	—	9	3	no	—
60	C₂⁴×C₃	C₂×Dic₃	1	12	yes	—
61	C₃×#41	C₂×Dic₃	1	192	yes	—
62	C₃×#41	C₂²×D₃	1	64	yes	—
63	C₂×C₃×Q₈	C₂×Dic₃	1	48	yes	—
64	C₂×C₃×D₄	C₂×Dic₃	1	144	yes	—
65	C₂²×C₃×C₄	C₂×Dic₃	1	12	yes	—
66	C₂²×C₃×C₄	C₂×Dic₃	1	72	yes	—
67	C₂²×C₃×C₄	C₂²×D₃	1	72	yes	—
68	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
69	#208	—	3	192	no	—
70	C₂×C₃⋊C₈	—	3	24	no	—
71	C₂×C₃⋊C₈	—	3	72	no	—
72	C₂×C₃⋊C₈	—	3	192	no	—
73	C₂²×C₃²	—	2	36	no	—
74	C₂²×C₃²	#131	1	4	yes	complemented
75	C₂²×C₃²	C₂³×C₄	1	12	yes	—
76	C₂²×C₃²	C₂³×C₄	1	3	yes	—
77	C₃²×C₄	C₂³×C₄	1	3	yes	—
78	C₃²×C₄	C₂³×C₄	1	12	yes	—
79	C₃²×C₄	C₂⁵	1	1	yes	characteristic
80	C₂×#41	C₃⋊Dic₃	1	12	yes	—
81	C₂×#41	C₂×C₃⋊D₃	1	12	yes	—
82	C₂²×Q₈	C₃⋊Dic₃	1	1	yes	characteristic
83	C₂²×D₄	C₃⋊Dic₃	1	3	yes	—
84	C₂³×C₄	C₂×C₃⋊D₃	1	3	yes	—
85	#131	—	9	64	no	—
86	C₂×M₁₆	—	9	36	no	—
87	C₂²×C₈	—	9	36	no	—
88	C₂²×C₈	—	9	1	no	—
89	C₂²×C₈	—	9	3	no	—
90	C₂³×C₃	—	2	48	no	—
91	C₂³×C₃	C₂²×Dic₃	1	72	yes	—
92	C₂³×C₃	C₂²×Dic₃	1	4	yes	—
93	C₃×Q₈	C₂²×Dic₃	1	64	yes	—
94	C₃×D₄	C₂²×Dic₃	1	192	yes	—
95	C₂×C₃×C₄	C₂²×Dic₃	1	12	yes	—
96	C₂×C₃×C₄	C₂²×Dic₃	1	72	yes	—
97	C₂×C₃×C₄	C₂²×Dic₃	1	144	yes	—
98	C₂×C₃×C₄	C₂³×D₃	1	12	yes	—
99	C₂×C₃×C₄	C₂³×D₃	1	48	yes	—
100	C₃⋊C₈	—	3	16	no	—
101	C₃⋊C₈	—	3	48	no	—
102	C₂×C₃²	—	2	12	no	—
103	C₂×C₃²	C₂×#131	1	6	yes	complemented
104	C₂×C₃²	C₂⁴×C₄	1	1	yes	characteristic
105	C₂⁴	C₂×C₃⋊Dic₃	1	3	yes	—
106	#41	C₂×C₃⋊Dic₃	1	48	yes	—
107	#41	C₂²×C₃⋊D₃	1	16	yes	—
108	C₂×Q₈	C₂×C₃⋊Dic₃	1	12	yes	—
109	C₂×D₄	C₂×C₃⋊Dic₃	1	36	yes	—
110	C₂²×C₄	C₂×C₃⋊Dic₃	1	3	yes	—
111	C₂²×C₄	C₂×C₃⋊Dic₃	1	18	yes	—
112	C₂²×C₄	C₂²×C₃⋊D₃	1	18	yes	—
113	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
114	M₁₆	—	9	48	no	—
115	C₂×C₈	—	9	6	no	—
116	C₂×C₈	—	9	18	no	—
117	C₂×C₈	—	9	48	no	—
118	C₂²×C₃	—	2	144	no	—
119	C₂²×C₃	#948	1	16	yes	complemented
120	C₂²×C₃	C₂³×Dic₃	1	48	yes	—
121	C₂²×C₃	C₂³×Dic₃	1	12	yes	—
122	C₃×C₄	C₂³×Dic₃	1	12	yes	—
123	C₃×C₄	C₂³×Dic₃	1	48	yes	—
124	C₃×C₄	C₂⁴×D₃	1	4	yes	—
125	C₃²	C₂²×#131	1	1	yes	characteristic, complemented, Sylow
126	C₂³	—	2	12	no	—
127	C₂³	C₂²×C₃⋊Dic₃	1	18	yes	—
128	C₂³	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
129	Q₈	C₂²×C₃⋊Dic₃	1	16	yes	—
130	D₄	C₂²×C₃⋊Dic₃	1	48	yes	—
131	C₂×C₄	C₂²×C₃⋊Dic₃	1	3	yes	—
132	C₂×C₄	C₂²×C₃⋊Dic₃	1	18	yes	—
133	C₂×C₄	C₂²×C₃⋊Dic₃	1	36	yes	—
134	C₂×C₄	C₂³×C₃⋊D₃	1	3	yes	—
135	C₂×C₄	C₂³×C₃⋊D₃	1	12	yes	—
136	C₈	—	9	4	no	—
137	C₈	—	9	12	no	—
138	C₂×C₃	—	2	48	no	—
139	C₂×C₃	C₂×#948	1	24	yes	complemented
140	C₂×C₃	C₂⁴×Dic₃	1	4	yes	—
141	C₂²	—	2	36	no	—
142	C₂²	#64157	1	4	yes	direct product factor
143	C₂²	C₂³×C₃⋊Dic₃	1	12	yes	—
144	C₂²	C₂³×C₃⋊Dic₃	1	3	yes	—
145	C₄	C₂³×C₃⋊Dic₃	1	3	yes	—
146	C₄	C₂³×C₃⋊Dic₃	1	12	yes	—
147	C₄	C₂⁴×C₃⋊D₃	1	1	yes	characteristic
148	C₃	C₂²×#948	1	4	yes	complemented
149	C₂	—	2	12	no	—
150	C₂	C₂×#64157	1	6	yes	direct product factor
151	C₂	C₂⁴×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×C₃⋊D₃
commutator	C₂×C₃²	C₂⁴×C₄
Frattini	C₄	C₂⁴×C₃⋊D₃
Fitting, prime radical	C₂²×C₃²×#41	C₂
socle	C₂³×C₃²	C₂²×C₄
cosocle	C₃²×C₄	C₂⁵
hypercenter	C₂²×#41	C₃⋊D₃
hypocenter	C₃²	C₂²×#131
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
2. C₂×#64157	C₂	direct (N × H)
3. C₂²×#3748	C₂	semidirect
4. C₂³×C₃⋊(C₃⋊C₈)	C₂	semidirect
11. #64157	C₂²	direct (N × H)
12. C₂×#3748	C₂²	semidirect
15. C₂²×C₃⋊(C₃⋊C₈)	C₂²	semidirect
29. #3748	C₂³	semidirect
32. C₂×C₃⋊(C₃⋊C₈)	C₂³	semidirect
74. C₂²×C₃²	#131	semidirect
103. C₂×C₃²	C₂×#131	semidirect
119. C₂²×C₃	#948	semidirect
125. C₃²	C₂²×#131	semidirect
139. C₂×C₃	C₂×#948	semidirect
142. C₂²	#64157	direct (N × H)
148. C₃	C₂²×#948	semidirect
150. C₂	C₂×#64157	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×C₃²×#41	2
C₂×#64157	2
C₂²×#3748	2
C₂³×C₃⋊(C₃⋊C₈)	2
C₂²×#948	3

Subgroups — C22×#64157 #11918456

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#64157 #11918456