Subgroups — C₂²×#4524 #10792538

Order 768.

3086 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄²	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₃⋊C₈	C₂	1	2	yes	maximal
3	C₂×#4524	C₂	1	12	yes	maximal, direct product factor
4	C₂²×#336	—	3	1	no	maximal, Sylow
5	C₂⁴×C₃×C₄	C₄	1	2	yes	—
6	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₄²	—	2	8	no	—
8	C₂²×C₃×C₄²	C₄	1	6	yes	—
9	C₂²×C₃×C₄²	C₂²	1	6	yes	—
10	C₂³×C₃⋊C₈	—	2	8	no	—
11	C₂³×C₃⋊C₈	—	2	8	no	—
12	C₂³×C₃⋊C₈	C₄	1	16	yes	complemented
13	C₂³×C₃⋊C₈	C₂²	1	12	yes	—
14	#4524	C₂²	1	16	yes	direct product factor
15	C₂⁴×C₈	—	3	2	no	—
16	C₂³×C₄²	D₃	1	1	yes	characteristic
17	C₂×#336	—	3	12	no	—
18	C₂⁵×C₃	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₃×C₄	—	2	8	no	—
20	C₂³×C₃×C₄	—	2	8	no	—
21	C₂³×C₃×C₄	C₈	1	16	yes	complemented
22	C₂³×C₃×C₄	C₂×C₄	1	8	yes	—
23	C₂³×C₃×C₄	C₂×C₄	1	12	yes	—
24	C₂³×C₃×C₄	C₂×C₄	1	3	yes	—
25	C₂³×C₃×C₄	D₄	1	4	yes	—
26	C₂³×C₃×C₄	D₄	1	8	yes	—
27	C₂³×C₃×C₄	Q₈	1	4	yes	—
28	C₂³×C₃×C₄	C₂³	1	3	yes	—
29	C₂×C₃×C₄²	—	2	24	no	—
30	C₂×C₃×C₄²	—	2	24	no	—
31	C₂×C₃×C₄²	C₂×C₄	1	12	yes	—
32	C₂×C₃×C₄²	C₂³	1	4	yes	—
33	C₂²×C₃⋊C₈	—	2	64	no	—
34	C₂²×C₃⋊C₈	—	2	24	no	—
35	C₂²×C₃⋊C₈	—	2	24	no	—
36	C₂²×C₃⋊C₈	C₂×C₄	1	48	yes	complemented
37	C₂²×C₃⋊C₈	C₂³	1	8	yes	—
38	C₂⁴×C₄	Dic₃	1	2	yes	—
39	C₂⁴×C₄	C₂×D₃	1	1	yes	characteristic
40	C₂³×C₈	—	3	16	no	—
41	C₂³×C₈	—	3	12	no	—
42	C₂³×C₈	—	6	8	no	—
43	C₂³×C₈	—	6	8	no	—
44	C₂²×C₄²	—	2	8	no	—
45	C₂²×C₄²	Dic₃	1	6	yes	—
46	C₂²×C₄²	C₂×D₃	1	6	yes	—
47	#336	—	3	16	no	—
48	C₂⁴×C₃	C₄²	1	4	yes	—
49	C₂⁴×C₃	C₂²⋊C₄	1	4	yes	—
50	C₂⁴×C₃	C₄⋊C₄	1	4	yes	—
51	C₂⁴×C₃	C₂×C₈	1	8	yes	—
52	C₂⁴×C₃	M₁₆	1	8	yes	—
53	C₂⁴×C₃	C₂²×C₄	1	3	yes	—
54	C₂²×C₃×C₄	—	2	64	no	—
55	C₂²×C₃×C₄	—	2	24	no	—
56	C₂²×C₃×C₄	—	2	24	no	—
57	C₂²×C₃×C₄	C₄²	1	12	yes	—
58	C₂²×C₃×C₄	C₂²⋊C₄	1	32	yes	—
59	C₂²×C₃×C₄	C₂²⋊C₄	1	12	yes	—
60	C₂²×C₃×C₄	C₄⋊C₄	1	32	yes	—
61	C₂²×C₃×C₄	C₄⋊C₄	1	12	yes	—
62	C₂²×C₃×C₄	C₂×C₈	1	48	yes	complemented
63	C₂²×C₃×C₄	C₂²×C₄	1	12	yes	—
64	C₂²×C₃×C₄	C₂²×C₄	1	8	yes	—
65	C₂²×C₃×C₄	C₂²×C₄	1	3	yes	—
66	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
67	C₂²×C₃×C₄	C₂×D₄	1	12	yes	—
68	C₂²×C₃×C₄	C₂×Q₈	1	6	yes	—
69	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
70	C₃×C₄²	—	2	24	no	—
71	C₃×C₄²	—	2	8	no	—
72	C₂×C₃⋊C₈	—	2	96	no	—
73	C₂×C₃⋊C₈	—	2	8	no	—
74	C₂×C₃⋊C₈	—	2	8	no	—
75	C₂×C₃⋊C₈	C₂²×C₄	1	16	yes	complemented
76	C₂⁵	C₂×Dic₃	1	1	yes	characteristic
77	C₂³×C₄	—	2	8	no	—
78	C₂³×C₄	—	2	8	no	—
79	C₂³×C₄	C₃⋊C₈	1	16	yes	complemented
80	C₂³×C₄	Dic₆	1	4	yes	—
81	C₂³×C₄	C₄×D₃	1	8	yes	—
82	C₂³×C₄	D₁₂	1	4	yes	—
83	C₂³×C₄	C₂×Dic₃	1	12	yes	—
84	C₂³×C₄	C₂×Dic₃	1	3	yes	—
85	C₂³×C₄	#67	1	8	yes	—
86	C₂³×C₄	C₂²×D₃	1	3	yes	—
87	C₂²×C₈	—	3	48	no	—
88	C₂²×C₈	—	3	8	no	—
89	C₂²×C₈	—	6	64	no	—
90	C₂²×C₈	—	6	24	no	—
91	C₂²×C₈	—	6	24	no	—
92	C₂×C₄²	—	2	24	no	—
93	C₂×C₄²	—	2	24	no	—
94	C₂×C₄²	C₂×Dic₃	1	12	yes	—
95	C₂×C₄²	C₂²×D₃	1	4	yes	—
96	C₂³×C₃	#95	1	16	yes	—
97	C₂³×C₃	C₄×C₈	1	16	yes	—
98	C₂³×C₃	#97	1	16	yes	—
99	C₂³×C₃	C₂²⋊C₈	1	32	yes	—
100	C₂³×C₃	C₄⋊C₈	1	32	yes	—
101	C₂³×C₃	C₂×C₄²	1	6	yes	—
102	C₂³×C₃	C₂×C₂²⋊C₄	1	6	yes	—
103	C₂³×C₃	C₂×C₄⋊C₄	1	6	yes	—
104	C₂³×C₃	C₂²×C₈	1	12	yes	—
105	C₂³×C₃	C₂×M₁₆	1	12	yes	—
106	C₂³×C₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₃×C₄	—	2	8	no	—
108	C₂×C₃×C₄	—	2	8	no	—
109	C₂×C₃×C₄	—	2	96	no	—
110	C₂×C₃×C₄	#95	1	48	yes	—
111	C₂×C₃×C₄	C₂×C₄²	1	6	yes	—
112	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	24	yes	—
113	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	6	yes	—
114	C₂×C₃×C₄	C₂×C₄⋊C₄	1	24	yes	—
115	C₂×C₃×C₄	C₂×C₄⋊C₄	1	6	yes	—
116	C₂×C₃×C₄	C₂²×C₈	1	16	yes	complemented
117	C₂×C₃×C₄	C₂³×C₄	1	2	yes	—
118	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
120	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
121	C₃⋊C₈	—	2	16	no	—
122	C₂⁴	C₂×C₃⋊C₈	1	8	yes	—
123	C₂⁴	#208	1	8	yes	—
124	C₂⁴	C₄×Dic₃	1	4	yes	—
125	C₂⁴	C₄⋊Dic₃	1	4	yes	—
126	C₂⁴	#217	1	4	yes	—
127	C₂⁴	C₂²×Dic₃	1	3	yes	—
128	C₂²×C₄	—	2	64	no	—
129	C₂²×C₄	—	2	24	no	—
130	C₂²×C₄	—	2	24	no	—
131	C₂²×C₄	C₂×C₃⋊C₈	1	48	yes	complemented
132	C₂²×C₄	C₄×Dic₃	1	12	yes	—
133	C₂²×C₄	Dic₃⋊C₄	1	32	yes	—
134	C₂²×C₄	C₄⋊Dic₃	1	12	yes	—
135	C₂²×C₄	(C₂×D₃)⋊C₄	1	32	yes	—
136	C₂²×C₄	#217	1	12	yes	—
137	C₂²×C₄	C₂×Dic₆	1	6	yes	—
138	C₂²×C₄	C₂×C₄×D₃	1	12	yes	—
139	C₂²×C₄	C₂×D₁₂	1	6	yes	—
140	C₂²×C₄	C₂²×Dic₃	1	8	yes	—
141	C₂²×C₄	C₂²×Dic₃	1	3	yes	—
142	C₂²×C₄	C₂×#67	1	12	yes	—
143	C₂²×C₄	C₂³×D₃	1	1	yes	characteristic
144	C₂×C₈	—	3	16	no	—
145	C₂×C₈	—	6	96	no	—
146	C₂×C₈	—	6	8	no	—
147	C₂×C₈	—	6	8	no	—
148	C₄²	—	2	24	no	—
149	C₄²	—	2	8	no	—
150	C₂²×C₃	#336	1	64	yes	complemented
151	C₂²×C₃	C₂×#95	1	12	yes	—
152	C₂²×C₃	C₂×C₄×C₈	1	12	yes	—
153	C₂²×C₃	C₂×#97	1	12	yes	—
154	C₂²×C₃	C₂×C₂²⋊C₈	1	24	yes	—
155	C₂²×C₃	C₂×C₄⋊C₈	1	24	yes	—
156	C₂²×C₃	C₂²×C₄²	1	1	yes	characteristic
157	C₂²×C₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
158	C₂²×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
159	C₂²×C₃	C₂³×C₈	1	2	yes	—
160	C₂²×C₃	C₂²×M₁₆	1	2	yes	—
161	C₃×C₄	—	2	16	no	—
162	C₃×C₄	C₂×#95	1	12	yes	—
163	C₃×C₄	C₂²×C₂²⋊C₄	1	2	yes	—
164	C₃×C₄	C₂²×C₄⋊C₄	1	2	yes	—
165	C₂³	C₄×C₃⋊C₈	1	16	yes	—
166	C₂³	#803	1	16	yes	—
167	C₂³	C₄⋊(C₃⋊C₈)	1	32	yes	—
168	C₂³	#830	1	32	yes	—
169	C₂³	C₃⋊#95	1	16	yes	—
170	C₂³	C₂²×C₃⋊C₈	1	12	yes	—
171	C₂³	C₂×#208	1	12	yes	—
172	C₂³	C₂×C₄×Dic₃	1	6	yes	—
173	C₂³	C₂×C₄⋊Dic₃	1	6	yes	—
174	C₂³	C₂×#217	1	6	yes	—
175	C₂³	C₂³×Dic₃	1	1	yes	characteristic
176	C₂×C₄	—	2	96	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	C₃⋊#95	1	48	yes	—
180	C₂×C₄	C₂²×C₃⋊C₈	1	16	yes	complemented
181	C₂×C₄	C₂×C₄×Dic₃	1	6	yes	—
182	C₂×C₄	C₂×Dic₃⋊C₄	1	24	yes	—
183	C₂×C₄	C₂×C₄⋊Dic₃	1	6	yes	—
184	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	24	yes	—
185	C₂×C₄	C₂×#217	1	6	yes	—
186	C₂×C₄	C₂²×Dic₆	1	1	yes	characteristic
187	C₂×C₄	C₂²×C₄×D₃	1	2	yes	—
188	C₂×C₄	C₂²×D₁₂	1	1	yes	characteristic
189	C₂×C₄	C₂²×#67	1	2	yes	—
190	C₈	—	6	16	no	—
191	C₂×C₃	C₂×#336	1	24	yes	complemented
192	C₂×C₃	C₂²×#95	1	1	yes	characteristic
193	C₂×C₃	C₂²×C₄×C₈	1	1	yes	characteristic
194	C₂×C₃	C₂²×#97	1	1	yes	characteristic
195	C₂×C₃	C₂²×C₂²⋊C₈	1	2	yes	—
196	C₂×C₃	C₂²×C₄⋊C₈	1	2	yes	—
197	C₂²	#4524	1	64	yes	direct product factor
198	C₂²	C₂×C₄×C₃⋊C₈	1	12	yes	—
199	C₂²	C₂×#803	1	12	yes	—
200	C₂²	C₂×C₄⋊(C₃⋊C₈)	1	24	yes	—
201	C₂²	C₂×#830	1	24	yes	—
202	C₂²	C₂×C₃⋊#95	1	12	yes	—
203	C₂²	C₂³×C₃⋊C₈	1	2	yes	—
204	C₂²	C₂²×#208	1	2	yes	—
205	C₂²	C₂²×C₄×Dic₃	1	1	yes	characteristic
206	C₂²	C₂²×C₄⋊Dic₃	1	1	yes	characteristic
207	C₂²	C₂²×#217	1	1	yes	characteristic
208	C₄	—	2	16	no	—
209	C₄	C₂×C₃⋊#95	1	12	yes	—
210	C₄	C₂²×Dic₃⋊C₄	1	2	yes	—
211	C₄	C₂²×(C₂×D₃)⋊C₄	1	2	yes	—
212	C₃	C₂²×#336	1	1	yes	characteristic, complemented, Sylow
213	C₂	C₂×#4524	1	24	yes	direct product factor
214	C₂	C₂²×C₄×C₃⋊C₈	1	1	yes	characteristic
215	C₂	C₂²×#803	1	1	yes	characteristic
216	C₂	C₂²×C₄⋊(C₃⋊C₈)	1	2	yes	—
217	C₂	C₂²×#830	1	2	yes	—
218	C₂	C₂²×C₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₃
commutator	C₂×C₃	C₂²×C₄×C₈
Frattini	C₂²×C₄	C₂³×D₃
Fitting	C₂³×C₃×C₄²	C₂
socle, prime radical	C₂⁵×C₃	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂⁴
hypercenter	C₂³×C₄²	D₃
hypocenter	C₃	C₂²×#336
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
3. C₂×#4524	C₂	direct (N × H)
12. C₂³×C₃⋊C₈	C₄	semidirect
14. #4524	C₂²	direct (N × H)
21. C₂³×C₃×C₄	C₈	semidirect
36. C₂²×C₃⋊C₈	C₂×C₄	semidirect
62. C₂²×C₃×C₄	C₂×C₈	semidirect
75. C₂×C₃⋊C₈	C₂²×C₄	semidirect
79. C₂³×C₄	C₃⋊C₈	semidirect
116. C₂×C₃×C₄	C₂²×C₈	semidirect
131. C₂²×C₄	C₂×C₃⋊C₈	semidirect
150. C₂²×C₃	#336	semidirect
180. C₂×C₄	C₂²×C₃⋊C₈	semidirect
191. C₂×C₃	C₂×#336	semidirect
197. C₂²	#4524	direct (N × H)
212. C₃	C₂²×#336	semidirect
213. C₂	C₂×#4524	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×C₄²	2
C₂⁴×C₃⋊C₈	2
C₂×#4524	2
C₂²×#336	3

Subgroups — C22×#4524 #10792538

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#4524 #10792538