Subgroups — C₂²×#4999 #11658803

Order 768.

3006 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×#117	C₂	1	1	yes	maximal, characteristic
2	C₂³×#208	C₂	1	1	yes	maximal, characteristic
3	C₂⁴×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×#4999	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₄⋊(C₃⋊C₈)	C₂	1	4	yes	maximal, complemented
6	C₂²×#424	—	3	1	no	maximal, Sylow
7	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
8	C₂×C₃×#117	C₄	1	12	yes	—
9	C₂×C₃×#117	C₂²	1	12	yes	—
10	C₂²×C₃×C₄⋊C₄	C₄	1	2	yes	—
11	C₂²×C₃×C₂²⋊C₄	C₄	1	2	yes	—
12	C₂²×C₃×C₄²	C₂²	1	2	yes	—
13	C₂²×#208	—	2	8	no	—
14	C₂²×#208	C₂²	1	12	yes	—
15	C₂³×C₃⋊C₈	—	2	8	no	—
16	C₂³×C₃⋊C₈	C₂²	1	12	yes	—
17	C₂³×C₃⋊C₈	C₂²	1	2	yes	—
18	C₂³×C₃⋊C₈	C₂²	1	2	yes	—
19	#4999	C₂²	1	64	yes	direct product factor
20	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	48	yes	complemented
21	C₂³×M₁₆	—	3	1	no	—
22	C₂⁴×C₈	—	3	1	no	—
23	C₂²×#117	D₃	1	1	yes	characteristic
24	C₂×#424	—	3	24	no	—
25	C₂²×C₄⋊C₈	—	3	4	no	—
26	C₂⁵×C₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₃×C₄	C₂×C₄	1	4	yes	—
28	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₃×C₄	C₂×C₄	1	6	yes	—
30	C₂³×C₃×C₄	D₄	1	8	yes	—
31	C₂³×C₃×C₄	Q₈	1	8	yes	—
32	C₂³×C₃×C₄	C₂³	1	6	yes	—
33	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
34	C₃×#117	C₂×C₄	1	48	yes	—
35	C₃×#117	C₂³	1	16	yes	—
36	C₂×C₃×C₄⋊C₄	C₂×C₄	1	24	yes	—
37	C₂×C₃×C₂²⋊C₄	C₂×C₄	1	24	yes	—
38	C₂×C₃×C₄²	C₂×C₄	1	12	yes	—
39	C₂×C₃×C₄²	C₂³	1	12	yes	—
40	C₂×#208	—	2	48	no	—
41	C₂×#208	C₂³	1	16	yes	—
42	C₂²×C₃⋊C₈	—	2	8	no	—
43	C₂²×C₃⋊C₈	—	2	48	no	—
44	C₂²×C₃⋊C₈	—	2	8	no	—
45	C₂²×C₃⋊C₈	C₂³	1	12	yes	—
46	C₂²×C₃⋊C₈	C₂³	1	12	yes	—
47	C₂²×C₃⋊C₈	C₂³	1	16	yes	—
48	C₄⋊(C₃⋊C₈)	C₂³	1	64	yes	complemented
49	C₂⁴×C₄	C₂×D₃	1	1	yes	characteristic
50	C₂²×M₁₆	—	3	12	no	—
51	C₂²×M₁₆	—	6	8	no	—
52	C₂³×C₈	—	3	12	no	—
53	C₂³×C₈	—	3	2	no	—
54	C₂³×C₈	—	3	2	no	—
55	C₂³×C₈	—	6	8	no	—
56	C₂×#117	Dic₃	1	12	yes	—
57	C₂×#117	C₂×D₃	1	12	yes	—
58	C₂²×C₄⋊C₄	Dic₃	1	2	yes	—
59	C₂²×C₂²⋊C₄	Dic₃	1	2	yes	—
60	C₂²×C₄²	C₂×D₃	1	2	yes	—
61	#424	—	3	64	no	—
62	C₂×C₄⋊C₈	—	3	48	no	—
63	C₂⁴×C₃	—	2	8	no	—
64	C₂⁴×C₃	C₄⋊C₄	1	8	yes	—
65	C₂⁴×C₃	C₂²×C₄	1	6	yes	—
66	C₂⁴×C₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₃×C₄	—	2	16	no	—
68	C₂²×C₃×C₄	C₄⋊C₄	1	48	yes	—
69	C₂²×C₃×C₄	C₄⋊C₄	1	8	yes	—
70	C₂²×C₃×C₄	C₂²×C₄	1	24	yes	—
71	C₂²×C₃×C₄	C₂²×C₄	1	3	yes	—
72	C₂²×C₃×C₄	C₂²×C₄	1	6	yes	—
73	C₂²×C₃×C₄	C₂²×C₄	1	12	yes	—
74	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
75	C₂²×C₃×C₄	C₂×D₄	1	24	yes	—
76	C₂²×C₃×C₄	C₂×Q₈	1	24	yes	—
77	C₂²×C₃×C₄	C₂×Q₈	1	4	yes	—
78	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
79	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
80	C₃×C₄⋊C₄	C₂²×C₄	1	32	yes	—
81	C₃×C₂²⋊C₄	C₂²×C₄	1	32	yes	—
82	C₃×C₄²	C₂²×C₄	1	24	yes	—
83	C₃×C₄²	C₂⁴	1	8	yes	—
84	#208	—	2	32	no	—
85	C₂×C₃⋊C₈	—	2	24	no	—
86	C₂×C₃⋊C₈	—	2	32	no	—
87	C₂×C₃⋊C₈	—	2	24	no	—
88	C₂×C₃⋊C₈	C₂⁴	1	8	yes	—
89	C₂×C₃⋊C₈	C₂⁴	1	8	yes	—
90	C₂⁵	C₂×Dic₃	1	1	yes	characteristic
91	C₂³×C₄	Dic₆	1	8	yes	—
92	C₂³×C₄	D₁₂	1	8	yes	—
93	C₂³×C₄	C₂×Dic₃	1	4	yes	—
94	C₂³×C₄	C₂×Dic₃	1	1	yes	characteristic
95	C₂³×C₄	C₂×Dic₃	1	6	yes	—
96	C₂³×C₄	C₂²×D₃	1	6	yes	—
97	C₂³×C₄	C₂²×D₃	1	1	yes	characteristic
98	C₂×M₁₆	—	3	16	no	—
99	C₂×M₁₆	—	6	48	no	—
100	C₂²×C₈	—	3	12	no	—
101	C₂²×C₈	—	3	16	no	—
102	C₂²×C₈	—	3	12	no	—
103	C₂²×C₈	—	6	8	no	—
104	C₂²×C₈	—	6	48	no	—
105	C₂²×C₈	—	6	8	no	—
106	#117	C₂×Dic₃	1	48	yes	—
107	#117	C₂²×D₃	1	16	yes	—
108	C₂×C₄⋊C₄	C₂×Dic₃	1	24	yes	—
109	C₂×C₂²⋊C₄	C₂×Dic₃	1	24	yes	—
110	C₂×C₄²	C₂×Dic₃	1	12	yes	—
111	C₂×C₄²	C₂²×D₃	1	12	yes	—
112	C₄⋊C₈	—	3	64	no	—
113	C₂³×C₃	—	2	24	no	—
114	C₂³×C₃	—	2	32	no	—
115	C₂³×C₃	C₂×C₄⋊C₄	1	24	yes	—
116	C₂³×C₃	C₂×C₄⋊C₄	1	4	yes	—
117	C₂³×C₃	#131	1	8	yes	—
118	C₂³×C₃	C₂³×C₄	1	4	yes	—
119	C₂³×C₃	C₂³×C₄	1	3	yes	—
120	C₂×C₃×C₄	—	2	48	no	—
121	C₂×C₃×C₄	C₂×C₄⋊C₄	1	48	yes	—
122	C₂×C₃×C₄	C₂×C₄⋊C₄	1	12	yes	—
123	C₂×C₃×C₄	C₂×C₄⋊C₄	1	24	yes	—
124	C₂×C₃×C₄	C₂³×C₄	1	16	yes	—
125	C₂×C₃×C₄	C₂³×C₄	1	3	yes	—
126	C₂×C₃×C₄	C₂³×C₄	1	4	yes	—
127	C₂×C₃×C₄	C₂²×D₄	1	6	yes	—
128	C₂×C₃×C₄	C₂²×D₄	1	8	yes	—
129	C₂×C₃×C₄	C₂²×Q₈	1	8	yes	—
130	C₂×C₃×C₄	C₂²×Q₈	1	6	yes	—
131	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
132	C₃⋊C₈	—	2	8	no	—
133	C₃⋊C₈	—	2	8	no	—
134	C₂⁴	—	2	8	no	—
135	C₂⁴	C₄⋊Dic₃	1	8	yes	—
136	C₂⁴	C₂²×Dic₃	1	6	yes	—
137	C₂⁴	C₂²×Dic₃	1	1	yes	characteristic
138	C₂²×C₄	—	2	16	no	—
139	C₂²×C₄	C₄⋊Dic₃	1	48	yes	—
140	C₂²×C₄	C₄⋊Dic₃	1	8	yes	—
141	C₂²×C₄	C₂×Dic₆	1	24	yes	—
142	C₂²×C₄	C₂×Dic₆	1	4	yes	—
143	C₂²×C₄	C₂×D₁₂	1	4	yes	—
144	C₂²×C₄	C₂×D₁₂	1	24	yes	—
145	C₂²×C₄	C₂²×Dic₃	1	24	yes	—
146	C₂²×C₄	C₂²×Dic₃	1	3	yes	—
147	C₂²×C₄	C₂²×Dic₃	1	6	yes	—
148	C₂²×C₄	C₂²×Dic₃	1	12	yes	—
149	C₂²×C₄	C₂³×D₃	1	3	yes	—
150	C₂²×C₄	C₂³×D₃	1	4	yes	—
151	M₁₆	—	6	32	no	—
152	C₂×C₈	—	3	8	no	—
153	C₂×C₈	—	3	8	no	—
154	C₂×C₈	—	6	24	no	—
155	C₂×C₈	—	6	32	no	—
156	C₂×C₈	—	6	24	no	—
157	C₄⋊C₄	C₂²×Dic₃	1	32	yes	—
158	C₂²⋊C₄	C₂²×Dic₃	1	32	yes	—
159	C₄²	C₂²×Dic₃	1	24	yes	—
160	C₄²	C₂³×D₃	1	8	yes	—
161	C₂²×C₃	—	2	48	no	—
162	C₂²×C₃	—	2	8	no	—
163	C₂²×C₃	#424	1	16	yes	complemented
164	C₂²×C₃	C₂²×C₄⋊C₄	1	8	yes	—
165	C₂²×C₃	C₂²×C₄⋊C₄	1	6	yes	—
166	C₂²×C₃	C₂×#131	1	12	yes	—
167	C₂²×C₃	C₂⁴×C₄	1	1	yes	characteristic
168	C₃×C₄	—	2	16	no	—
169	C₃×C₄	C₂²×C₄⋊C₄	1	6	yes	—
170	C₃×C₄	C₂²×C₄⋊C₄	1	8	yes	—
171	C₃×C₄	C₂³×D₄	1	1	yes	characteristic
172	C₃×C₄	C₂³×Q₈	1	1	yes	characteristic
173	C₂³	—	2	32	no	—
174	C₂³	—	2	24	no	—
175	C₂³	C₂×C₄⋊Dic₃	1	24	yes	—
176	C₂³	C₂×C₄⋊Dic₃	1	4	yes	—
177	C₂³	#948	1	8	yes	—
178	C₂³	C₂³×Dic₃	1	4	yes	—
179	C₂³	C₂³×Dic₃	1	3	yes	—
180	C₂×C₄	—	2	48	no	—
181	C₂×C₄	C₂×C₄⋊Dic₃	1	48	yes	—
182	C₂×C₄	C₂×C₄⋊Dic₃	1	24	yes	—
183	C₂×C₄	C₂×C₄⋊Dic₃	1	12	yes	—
184	C₂×C₄	C₂²×Dic₆	1	8	yes	—
185	C₂×C₄	C₂²×Dic₆	1	6	yes	—
186	C₂×C₄	C₂²×D₁₂	1	6	yes	—
187	C₂×C₄	C₂²×D₁₂	1	8	yes	—
188	C₂×C₄	C₂³×Dic₃	1	16	yes	—
189	C₂×C₄	C₂³×Dic₃	1	3	yes	—
190	C₂×C₄	C₂³×Dic₃	1	4	yes	—
191	C₂×C₄	C₂⁴×D₃	1	1	yes	characteristic
192	C₈	—	6	8	no	—
193	C₈	—	6	8	no	—
194	C₂×C₃	—	2	8	no	—
195	C₂×C₃	C₂×#424	1	12	yes	complemented
196	C₂×C₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
197	C₂×C₃	C₂²×#131	1	2	yes	—
198	C₂²	—	2	48	no	—
199	C₂²	—	2	8	no	—
200	C₂²	#4999	1	16	yes	direct product factor
201	C₂²	C₂²×C₄⋊Dic₃	1	8	yes	—
202	C₂²	C₂²×C₄⋊Dic₃	1	6	yes	—
203	C₂²	C₂×#948	1	12	yes	—
204	C₂²	C₂⁴×Dic₃	1	1	yes	characteristic
205	C₄	—	2	16	no	—
206	C₄	C₂²×C₄⋊Dic₃	1	6	yes	—
207	C₄	C₂²×C₄⋊Dic₃	1	8	yes	—
208	C₄	C₂³×Dic₆	1	1	yes	characteristic
209	C₄	C₂³×D₁₂	1	1	yes	characteristic
210	C₃	C₂²×#424	1	1	yes	characteristic, complemented, Sylow
211	C₂	—	2	8	no	—
212	C₂	C₂×#4999	1	12	yes	direct product factor
213	C₂	C₂³×C₄⋊Dic₃	1	1	yes	characteristic
214	C₂	C₂²×#948	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₃
Fitting	C₂²×C₃×#117	C₂
socle	C₂⁴×C₃	C₂²×C₄
prime radical	C₂⁵×C₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂⁵
hypercenter	C₂²×#117	D₃
hypocenter	C₃	C₂²×#424
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
4. C₂×#4999	C₂	direct (N × H)
5. C₂²×C₄⋊(C₃⋊C₈)	C₂	semidirect
19. #4999	C₂²	direct (N × H)
20. C₂×C₄⋊(C₃⋊C₈)	C₂²	semidirect
48. C₄⋊(C₃⋊C₈)	C₂³	semidirect
163. C₂²×C₃	#424	semidirect
195. C₂×C₃	C₂×#424	semidirect
200. C₂²	#4999	direct (N × H)
210. C₃	C₂²×#424	semidirect
212. C₂	C₂×#4999	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×C₃×#117	2
C₂³×#208	2
C₂⁴×C₃⋊C₈	2
C₂×#4999	2
C₂²×C₄⋊(C₃⋊C₈)	2
C₂²×#424	3

Subgroups — C22×#4999 #11658803

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#4999 #11658803