Subgroups — C₂²×#5819 #11708952

Order 768.

11118 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×#5819	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#948	C₂	1	8	yes	maximal, complemented
4	C₂³×#208	C₂	1	6	yes	maximal, complemented
5	C₂²×#568	—	3	1	no	maximal, Sylow
6	C₂²×C₃×#41	C₄	1	4	yes	—
7	C₂²×C₃×#41	C₄	1	24	yes	—
8	C₂²×C₃×#41	C₂²	1	4	yes	—
9	C₂²×C₃×#41	C₂²	1	24	yes	—
10	C₂³×C₃×Q₈	C₄	1	1	yes	characteristic
11	C₂³×C₃×D₄	C₄	1	3	yes	—
12	C₂⁴×C₃×C₄	C₂²	1	3	yes	—
13	#5819	C₂²	1	256	yes	direct product factor
14	C₂×#948	C₂²	1	192	yes	complemented
15	C₂²×#208	C₂²	1	4	yes	—
16	C₂²×#208	C₂²	1	12	yes	complemented
17	C₂²×#208	C₂²	1	144	yes	complemented
18	C₂³×C₃⋊C₈	C₂²	1	12	yes	complemented
19	C₂³×#41	D₃	1	1	yes	characteristic
20	C₂×#568	—	3	48	no	—
21	C₂²×#131	—	3	8	no	—
22	C₂³×M₁₆	—	3	6	no	—
23	C₂⁵×C₃	C₂×C₄	1	3	yes	—
24	C₂×C₃×#41	C₂×C₄	1	48	yes	—
25	C₂×C₃×#41	C₂×C₄	1	96	yes	—
26	C₂×C₃×#41	C₂×C₄	1	192	yes	—
27	C₂×C₃×#41	C₂³	1	48	yes	—
28	C₂×C₃×#41	C₂³	1	64	yes	—
29	C₂²×C₃×Q₈	C₂×C₄	1	4	yes	—
30	C₂²×C₃×Q₈	C₂×C₄	1	24	yes	—
31	C₂²×C₃×D₄	C₂×C₄	1	12	yes	—
32	C₂²×C₃×D₄	C₂×C₄	1	72	yes	—
33	C₂³×C₃×C₄	C₂×C₄	1	3	yes	—
34	C₂³×C₃×C₄	C₂×C₄	1	6	yes	—
35	C₂³×C₃×C₄	C₂×C₄	1	36	yes	—
36	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
37	C₂³×C₃×C₄	C₂³	1	6	yes	—
38	C₂³×C₃×C₄	C₂³	1	36	yes	—
39	#948	C₂³	1	512	yes	complemented
40	C₂×#208	C₂³	1	48	yes	—
41	C₂×#208	C₂³	1	144	yes	complemented
42	C₂×#208	C₂³	1	384	yes	complemented
43	C₂²×C₃⋊C₈	C₂³	1	8	yes	—
44	C₂²×C₃⋊C₈	C₂³	1	144	yes	complemented
45	C₂²×#41	Dic₃	1	4	yes	—
46	C₂²×#41	Dic₃	1	24	yes	—
47	C₂²×#41	C₂×D₃	1	4	yes	—
48	C₂²×#41	C₂×D₃	1	24	yes	—
49	C₂³×Q₈	Dic₃	1	1	yes	characteristic
50	C₂³×D₄	Dic₃	1	3	yes	—
51	C₂⁴×C₄	C₂×D₃	1	3	yes	—
52	#568	—	3	256	no	—
53	C₂×#131	—	3	192	no	—
54	C₂²×M₁₆	—	3	4	no	—
55	C₂²×M₁₆	—	3	12	no	—
56	C₂²×M₁₆	—	3	144	no	—
57	C₂³×C₈	—	3	12	no	—
58	C₂⁴×C₃	—	4	12	no	—
59	C₂⁴×C₃	C₂²×C₄	1	1	yes	characteristic
60	C₂⁴×C₃	C₂²×C₄	1	6	yes	—
61	C₂⁴×C₃	C₂²×C₄	1	36	yes	—
62	C₃×#41	C₂²×C₄	1	192	yes	—
63	C₃×#41	C₂²×C₄	1	256	yes	—
64	C₃×#41	C₂⁴	1	64	yes	—
65	C₂×C₃×Q₈	C₂²×C₄	1	48	yes	—
66	C₂×C₃×Q₈	C₂²×C₄	1	64	yes	—
67	C₂×C₃×D₄	C₂²×C₄	1	144	yes	—
68	C₂×C₃×D₄	C₂²×C₄	1	192	yes	—
69	C₂²×C₃×C₄	C₂²×C₄	1	6	yes	—
70	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₃×C₄	C₂²×C₄	1	36	yes	—
72	C₂²×C₃×C₄	C₂²×C₄	1	6	yes	—
73	C₂²×C₃×C₄	C₂²×C₄	1	36	yes	—
74	C₂²×C₃×C₄	C₂²×C₄	1	72	yes	—
75	C₂²×C₃×C₄	C₂²×C₄	1	144	yes	—
76	C₂²×C₃×C₄	C₂⁴	1	36	yes	—
77	C₂²×C₃×C₄	C₂⁴	1	6	yes	—
78	C₂²×C₃×C₄	C₂⁴	1	48	yes	—
79	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
80	#208	C₂⁴	1	64	yes	—
81	#208	C₂⁴	1	192	yes	complemented
82	C₂×C₃⋊C₈	C₂⁴	1	48	yes	—
83	C₂×C₃⋊C₈	C₂⁴	1	192	yes	complemented
84	C₂⁵	C₂×Dic₃	1	3	yes	—
85	C₂×#41	C₂×Dic₃	1	48	yes	—
86	C₂×#41	C₂×Dic₃	1	96	yes	—
87	C₂×#41	C₂×Dic₃	1	192	yes	—
88	C₂×#41	C₂²×D₃	1	48	yes	—
89	C₂×#41	C₂²×D₃	1	64	yes	—
90	C₂²×Q₈	C₂×Dic₃	1	4	yes	—
91	C₂²×Q₈	C₂×Dic₃	1	24	yes	—
92	C₂²×D₄	C₂×Dic₃	1	12	yes	—
93	C₂²×D₄	C₂×Dic₃	1	72	yes	—
94	C₂³×C₄	C₂×Dic₃	1	3	yes	—
95	C₂³×C₄	C₂×Dic₃	1	6	yes	—
96	C₂³×C₄	C₂×Dic₃	1	36	yes	—
97	C₂³×C₄	C₂²×D₃	1	36	yes	—
98	C₂³×C₄	C₂²×D₃	1	6	yes	—
99	C₂³×C₄	C₂²×D₃	1	1	yes	characteristic
100	#131	—	3	512	no	—
101	C₂×M₁₆	—	3	48	no	—
102	C₂×M₁₆	—	3	144	no	—
103	C₂×M₁₆	—	3	384	no	—
104	C₂²×C₈	—	3	144	no	—
105	C₂²×C₈	—	3	8	no	—
106	C₂³×C₃	—	2	4	no	—
107	C₂³×C₃	—	2	24	no	—
108	C₂³×C₃	—	4	72	no	—
109	C₂³×C₃	C₂³×C₄	1	6	yes	—
110	C₂³×C₃	C₂³×C₄	1	36	yes	—
111	C₂³×C₃	C₂³×C₄	1	48	yes	—
112	C₂³×C₃	C₂³×C₄	1	1	yes	characteristic
113	C₃×Q₈	C₂³×C₄	1	64	yes	—
114	C₃×D₄	C₂³×C₄	1	192	yes	—
115	C₂×C₃×C₄	C₂³×C₄	1	12	yes	—
116	C₂×C₃×C₄	C₂³×C₄	1	6	yes	—
117	C₂×C₃×C₄	C₂³×C₄	1	3	yes	—
118	C₂×C₃×C₄	C₂³×C₄	1	48	yes	—
119	C₂×C₃×C₄	C₂³×C₄	1	36	yes	—
120	C₂×C₃×C₄	C₂³×C₄	1	72	yes	—
121	C₂×C₃×C₄	C₂³×C₄	1	96	yes	—
122	C₂×C₃×C₄	C₂⁵	1	3	yes	—
123	C₂×C₃×C₄	C₂⁵	1	24	yes	—
124	C₂×C₃×C₄	C₂⁵	1	4	yes	—
125	C₃⋊C₈	C₂⁵	1	32	yes	—
126	C₂⁴	—	4	12	no	—
127	C₂⁴	C₂²×Dic₃	1	1	yes	characteristic
128	C₂⁴	C₂²×Dic₃	1	6	yes	—
129	C₂⁴	C₂²×Dic₃	1	36	yes	—
130	#41	C₂²×Dic₃	1	192	yes	—
131	#41	C₂²×Dic₃	1	256	yes	—
132	#41	C₂³×D₃	1	64	yes	—
133	C₂×Q₈	C₂²×Dic₃	1	48	yes	—
134	C₂×Q₈	C₂²×Dic₃	1	64	yes	—
135	C₂×D₄	C₂²×Dic₃	1	144	yes	—
136	C₂×D₄	C₂²×Dic₃	1	192	yes	—
137	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
138	C₂²×C₄	C₂²×Dic₃	1	6	yes	—
139	C₂²×C₄	C₂²×Dic₃	1	36	yes	—
140	C₂²×C₄	C₂²×Dic₃	1	6	yes	—
141	C₂²×C₄	C₂²×Dic₃	1	72	yes	—
142	C₂²×C₄	C₂²×Dic₃	1	144	yes	—
143	C₂²×C₄	C₂²×Dic₃	1	36	yes	—
144	C₂²×C₄	C₂³×D₃	1	36	yes	—
145	C₂²×C₄	C₂³×D₃	1	6	yes	—
146	C₂²×C₄	C₂³×D₃	1	48	yes	—
147	C₂²×C₄	C₂³×D₃	1	1	yes	characteristic
148	M₁₆	—	3	64	no	—
149	M₁₆	—	3	192	no	—
150	C₂×C₈	—	3	48	no	—
151	C₂×C₈	—	3	192	no	—
152	C₂²×C₃	—	2	12	no	—
153	C₂²×C₃	—	2	72	no	—
154	C₂²×C₃	—	4	48	no	—
155	C₂²×C₃	#568	1	4	yes	complemented
156	C₂²×C₃	C₂⁴×C₄	1	4	yes	—
157	C₂²×C₃	C₂⁴×C₄	1	24	yes	—
158	C₂²×C₃	C₂⁴×C₄	1	3	yes	—
159	C₃×C₄	C₂⁴×C₄	1	3	yes	—
160	C₃×C₄	C₂⁴×C₄	1	4	yes	—
161	C₃×C₄	C₂⁴×C₄	1	24	yes	—
162	C₃×C₄	C₂⁶	1	1	yes	characteristic
163	C₂³	—	2	4	no	—
164	C₂³	—	2	24	no	—
165	C₂³	—	4	72	no	—
166	C₂³	C₂³×Dic₃	1	6	yes	—
167	C₂³	C₂³×Dic₃	1	36	yes	—
168	C₂³	C₂³×Dic₃	1	48	yes	—
169	C₂³	C₂³×Dic₃	1	1	yes	characteristic
170	Q₈	C₂³×Dic₃	1	64	yes	—
171	D₄	C₂³×Dic₃	1	192	yes	—
172	C₂×C₄	C₂³×Dic₃	1	12	yes	—
173	C₂×C₄	C₂³×Dic₃	1	6	yes	—
174	C₂×C₄	C₂³×Dic₃	1	3	yes	—
175	C₂×C₄	C₂³×Dic₃	1	48	yes	—
176	C₂×C₄	C₂³×Dic₃	1	36	yes	—
177	C₂×C₄	C₂³×Dic₃	1	72	yes	—
178	C₂×C₄	C₂³×Dic₃	1	96	yes	—
179	C₂×C₄	C₂⁴×D₃	1	3	yes	—
180	C₂×C₄	C₂⁴×D₃	1	24	yes	—
181	C₂×C₄	C₂⁴×D₃	1	4	yes	—
182	C₈	—	3	32	no	—
183	C₂×C₃	—	2	4	no	—
184	C₂×C₃	—	2	24	no	—
185	C₂×C₃	C₂×#568	1	6	yes	complemented
186	C₂×C₃	C₂⁵×C₄	1	1	yes	characteristic
187	C₂²	—	2	12	no	—
188	C₂²	—	2	72	no	—
189	C₂²	—	4	48	no	—
190	C₂²	#5819	1	4	yes	direct product factor
191	C₂²	C₂⁴×Dic₃	1	4	yes	—
192	C₂²	C₂⁴×Dic₃	1	24	yes	—
193	C₂²	C₂⁴×Dic₃	1	3	yes	—
194	C₄	C₂⁴×Dic₃	1	4	yes	—
195	C₄	C₂⁴×Dic₃	1	3	yes	—
196	C₄	C₂⁴×Dic₃	1	24	yes	—
197	C₄	C₂⁵×D₃	1	1	yes	characteristic
198	C₃	C₂²×#568	1	1	yes	characteristic, complemented, Sylow
199	C₂	—	2	4	no	—
200	C₂	—	2	24	no	—
201	C₂	C₂×#5819	1	6	yes	direct product factor
202	C₂	C₂⁵×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂³×D₃
commutator	C₂×C₃	C₂⁵×C₄
Frattini	C₄	C₂⁵×D₃
Fitting, prime radical	C₂³×C₃×#41	C₂
socle	C₂³×C₃	C₂³×C₄
cosocle	C₃×C₄	C₂⁶
hypercenter	C₂³×#41	D₃
hypocenter	C₃	C₂²×#568
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
2. C₂×#5819	C₂	direct (N × H)
3. C₂²×#948	C₂	semidirect
4. C₂³×#208	C₂	semidirect
13. #5819	C₂²	direct (N × H)
14. C₂×#948	C₂²	semidirect
16. C₂²×#208	C₂²	semidirect
17. C₂²×#208	C₂²	semidirect
18. C₂³×C₃⋊C₈	C₂²	semidirect
39. #948	C₂³	semidirect
41. C₂×#208	C₂³	semidirect
42. C₂×#208	C₂³	semidirect
44. C₂²×C₃⋊C₈	C₂³	semidirect
81. #208	C₂⁴	semidirect
83. C₂×C₃⋊C₈	C₂⁴	semidirect
155. C₂²×C₃	#568	semidirect
185. C₂×C₃	C₂×#568	semidirect
190. C₂²	#5819	direct (N × H)
198. C₃	C₂²×#568	semidirect
201. C₂	C₂×#5819	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×#41	2
C₂×#5819	2
C₂²×#948	2
C₂³×#208	2
C₂²×#568	3

Subgroups — C22×#5819 #11708952

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#5819 #11708952