Subgroups — C₂²×#4683 #11664781

Order 768.

1966 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#870	C₂	1	2	yes	maximal
2	C₂²×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×#818	C₂	1	2	yes	maximal
4	C₂²×#817	C₂	1	2	yes	maximal
5	C₂×#4683	C₂	1	24	yes	maximal, direct product factor
6	C₂²×#499	—	3	1	no	maximal, Sylow
7	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
8	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊Dic₃	C₂²	1	4	yes	—
11	C₂²×Dic₃⋊C₄	—	2	4	no	—
12	C₂×#870	—	2	12	no	—
13	C₂×#870	C₂²	1	24	yes	—
14	C₂×C₃×C₄×C₈	C₂²	1	12	yes	—
15	C₂×#818	C₂²	1	24	yes	—
16	C₂×#817	C₂²	1	24	yes	—
17	#4683	C₂²	1	64	yes	direct product factor
18	C₂²×#125	—	3	2	no	—
19	C₂×#499	—	3	24	no	—
20	C₂²×#107	—	3	2	no	—
21	C₂²×#106	—	3	2	no	—
22	C₂²×C₄×C₈	D₃	1	1	yes	characteristic
23	C₂³×C₃×C₄	D₄	1	2	yes	—
24	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
25	C₂³×Dic₃	—	2	4	no	—
26	C₂²×C₃×C₈	Q₈	1	8	yes	—
27	C₂²×C₃×C₈	Q₈	1	8	yes	—
28	C₂²×C₃×C₈	C₂³	1	6	yes	—
29	C₂²×C₃×C₈	C₂³	1	6	yes	—
30	C₂×C₃×C₄²	D₄	1	6	yes	—
31	C₂×C₃×C₄²	C₂³	1	6	yes	—
32	C₂×C₄⋊Dic₃	—	2	12	no	—
33	C₂×C₄⋊Dic₃	C₂³	1	24	yes	—
34	C₂×Dic₃⋊C₄	—	2	48	no	—
35	#870	—	2	48	no	—
36	#870	C₂³	1	32	yes	—
37	C₃×C₄×C₈	C₂³	1	16	yes	—
38	#818	C₂³	1	32	yes	—
39	#817	C₂³	1	32	yes	—
40	C₂³×C₈	C₂×D₃	1	1	yes	characteristic
41	C₂³×C₈	C₂×D₃	1	1	yes	characteristic
42	C₂×#125	—	3	24	no	—
43	C₂×#125	—	6	12	no	—
44	C₂²×C₄⋊C₄	—	3	4	no	—
45	C₂²×C₄⋊C₄	—	6	4	no	—
46	C₂²×C₄²	C₂×D₃	1	1	yes	characteristic
47	#499	—	3	64	no	—
48	C₂×#107	—	3	24	no	—
49	C₂×#106	—	3	24	no	—
50	C₂×C₄×C₈	C₂×D₃	1	12	yes	—
51	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₃×C₄	—	2	8	no	—
53	C₂²×C₃×C₄	C₂×D₄	1	12	yes	—
54	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
55	C₂²×C₃×C₄	C₂×Q₈	1	4	yes	—
56	C₂²×C₃×C₄	C₂×Q₈	1	4	yes	—
57	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
58	C₂²×Dic₃	—	2	24	no	—
59	C₂²×Dic₃	—	4	16	no	—
60	C₂×C₃×C₈	C₂×Q₈	1	24	yes	—
61	C₂×C₃×C₈	C₂×Q₈	1	24	yes	—
62	C₂×C₃×C₈	C₂⁴	1	4	yes	—
63	C₂×C₃×C₈	C₂⁴	1	4	yes	—
64	C₃×C₄²	C₂×D₄	1	12	yes	—
65	C₃×C₄²	C₂⁴	1	4	yes	—
66	C₄⋊Dic₃	—	2	24	no	—
67	C₄⋊Dic₃	C₂⁴	1	16	yes	—
68	Dic₃⋊C₄	—	2	64	no	—
69	C₂³×C₄	—	6	4	no	—
70	C₂³×C₄	D₁₂	1	2	yes	—
71	C₂³×C₄	C₂²×D₃	1	1	yes	characteristic
72	C₂²×C₈	Dic₆	1	8	yes	—
73	C₂²×C₈	Dic₆	1	8	yes	—
74	C₂²×C₈	C₂²×D₃	1	6	yes	—
75	C₂²×C₈	C₂²×D₃	1	6	yes	—
76	#125	—	3	32	no	—
77	#125	—	6	48	no	—
78	C₂×C₄⋊C₄	—	3	24	no	—
79	C₂×C₄⋊C₄	—	6	12	no	—
80	C₂×C₄⋊C₄	—	6	48	no	—
81	C₂×C₄²	D₁₂	1	6	yes	—
82	C₂×C₄²	C₂²×D₃	1	6	yes	—
83	#107	—	3	32	no	—
84	#106	—	3	32	no	—
85	C₄×C₈	C₂²×D₃	1	16	yes	—
86	C₂³×C₃	C₄⋊Q₈	1	4	yes	—
87	C₂³×C₃	#135	1	8	yes	—
88	C₂³×C₃	C₂²×D₄	1	3	yes	—
89	C₂×C₃×C₄	—	2	24	no	—
90	C₂×C₃×C₄	C₄⋊Q₈	1	12	yes	—
91	C₂×C₃×C₄	C₂²×D₄	1	8	yes	—
92	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
93	C₂×C₃×C₄	C₂²×Q₈	1	6	yes	—
94	C₂×C₃×C₄	C₂²×Q₈	1	6	yes	—
95	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
96	C₂×Dic₃	—	2	16	no	—
97	C₂×Dic₃	—	4	48	no	—
98	C₃×C₈	C₂²×Q₈	1	8	yes	—
99	C₃×C₈	C₂²×Q₈	1	8	yes	—
100	C₂⁴	C₂×D₁₂	1	1	yes	characteristic
101	C₂²×C₄	—	2	8	no	—
102	C₂²×C₄	—	6	24	no	—
103	C₂²×C₄	—	12	16	no	—
104	C₂²×C₄	C₂×Dic₆	1	4	yes	—
105	C₂²×C₄	C₂×Dic₆	1	4	yes	—
106	C₂²×C₄	C₂×D₁₂	1	12	yes	—
107	C₂²×C₄	C₂×D₁₂	1	3	yes	—
108	C₂²×C₄	C₂³×D₃	1	3	yes	—
109	C₂×C₈	C₂×Dic₆	1	24	yes	—
110	C₂×C₈	C₂×Dic₆	1	24	yes	—
111	C₂×C₈	C₂³×D₃	1	4	yes	—
112	C₂×C₈	C₂³×D₃	1	4	yes	—
113	C₄⋊C₄	—	3	16	no	—
114	C₄⋊C₄	—	6	24	no	—
115	C₄⋊C₄	—	6	64	no	—
116	C₄²	C₂×D₁₂	1	12	yes	—
117	C₄²	C₂³×D₃	1	4	yes	—
118	C₂²×C₃	#499	1	16	yes	complemented
119	C₂²×C₃	C₂×C₄⋊Q₈	1	6	yes	—
120	C₂²×C₃	C₂×#135	1	12	yes	—
121	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
122	C₃×C₄	—	2	8	no	—
123	C₃×C₄	C₂×C₄⋊Q₈	1	6	yes	—
124	C₃×C₄	C₂³×Q₈	1	1	yes	characteristic
125	C₃×C₄	C₂³×Q₈	1	1	yes	characteristic
126	Dic₃	—	4	16	no	—
127	C₂³	#869	1	4	yes	—
128	C₂³	#904	1	8	yes	—
129	C₂³	C₂²×D₁₂	1	3	yes	—
130	C₂×C₄	—	2	24	no	—
131	C₂×C₄	—	6	16	no	—
132	C₂×C₄	—	12	48	no	—
133	C₂×C₄	#869	1	12	yes	—
134	C₂×C₄	C₂²×Dic₆	1	6	yes	—
135	C₂×C₄	C₂²×Dic₆	1	6	yes	—
136	C₂×C₄	C₂²×D₁₂	1	8	yes	—
137	C₂×C₄	C₂²×D₁₂	1	3	yes	—
138	C₂×C₄	C₂⁴×D₃	1	1	yes	characteristic
139	C₈	C₂²×Dic₆	1	8	yes	—
140	C₈	C₂²×Dic₆	1	8	yes	—
141	C₂×C₃	C₂×#499	1	12	yes	complemented
142	C₂×C₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
143	C₂×C₃	C₂²×#135	1	2	yes	—
144	C₂²	#4683	1	16	yes	direct product factor
145	C₂²	C₂×#869	1	6	yes	—
146	C₂²	C₂×#904	1	12	yes	—
147	C₂²	C₂³×D₁₂	1	1	yes	characteristic
148	C₄	—	2	8	no	—
149	C₄	—	12	16	no	—
150	C₄	C₂×#869	1	6	yes	—
151	C₄	C₂³×Dic₆	1	1	yes	characteristic
152	C₄	C₂³×Dic₆	1	1	yes	characteristic
153	C₃	C₂²×#499	1	1	yes	characteristic, complemented, Sylow
154	C₂	C₂×#4683	1	12	yes	direct product factor
155	C₂	C₂²×#869	1	1	yes	characteristic
156	C₂	C₂²×#904	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₃
Fitting	C₂²×C₃×C₄×C₈	C₂
socle, prime radical	C₂⁴×C₃	C₂×D₄
hypercenter	C₂²×C₄×C₈	D₃
hypocenter	C₃	C₂²×#499
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
5. C₂×#4683	C₂	direct (N × H)
17. #4683	C₂²	direct (N × H)
118. C₂²×C₃	#499	semidirect
141. C₂×C₃	C₂×#499	semidirect
144. C₂²	#4683	direct (N × H)
153. C₃	C₂²×#499	semidirect
154. C₂	C₂×#4683	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×#870	2
C₂²×C₃×C₄×C₈	2
C₂²×#818	2
C₂²×#817	2
C₂×#4683	2
C₂²×#499	3

Subgroups — C22×#4683 #11664781

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#4683 #11664781