Subgroups — C₂²×#4924 #11658799

Order 768.

3438 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	C₂³×C₃×C₄²	C₂	1	1	yes	maximal, characteristic
2	C₂³×#208	C₂	1	2	yes	maximal
3	C₂×#4924	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄⋊(C₃⋊C₈)	C₂	1	4	yes	maximal, complemented
5	C₂²×#423	—	3	1	no	maximal, Sylow
6	C₂⁴×C₃×C₄	C₄	1	2	yes	—
7	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₄²	C₄	1	1	yes	characteristic
9	C₂²×C₃×C₄²	C₄	1	12	yes	—
10	C₂²×C₃×C₄²	C₄	1	1	yes	characteristic
11	C₂²×C₃×C₄²	C₂²	1	2	yes	—
12	C₂²×C₃×C₄²	C₂²	1	12	yes	—
13	C₂²×#208	—	2	16	no	—
14	C₂²×#208	C₂²	1	24	yes	—
15	C₂³×C₃⋊C₈	C₂²	1	4	yes	—
16	#4924	C₂²	1	64	yes	direct product factor
17	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	48	yes	complemented
18	C₂³×M₁₆	—	3	2	no	—
19	C₂³×C₄²	D₃	1	1	yes	characteristic
20	C₂×#423	—	3	24	no	—
21	C₂²×C₄⋊C₈	—	3	4	no	—
22	C₂⁵×C₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₃×C₄	—	2	16	no	—
24	C₂³×C₃×C₄	C₂×C₄	1	2	yes	—
25	C₂³×C₃×C₄	C₂×C₄	1	24	yes	—
26	C₂³×C₃×C₄	C₂×C₄	1	6	yes	—
27	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₃×C₄	C₂×C₄	1	2	yes	—
29	C₂³×C₃×C₄	D₄	1	8	yes	—
30	C₂³×C₃×C₄	Q₈	1	8	yes	—
31	C₂³×C₃×C₄	C₂³	1	6	yes	—
32	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
33	C₂×C₃×C₄²	C₂×C₄	1	12	yes	—
34	C₂×C₃×C₄²	C₂×C₄	1	48	yes	—
35	C₂×C₃×C₄²	C₂×C₄	1	12	yes	—
36	C₂×C₃×C₄²	C₂×C₄	1	12	yes	—
37	C₂×C₃×C₄²	C₂³	1	12	yes	—
38	C₂×C₃×C₄²	C₂³	1	16	yes	—
39	C₂×#208	—	2	96	no	—
40	C₂×#208	C₂³	1	32	yes	—
41	C₂²×C₃⋊C₈	—	2	16	no	—
42	C₂²×C₃⋊C₈	C₂³	1	24	yes	—
43	C₄⋊(C₃⋊C₈)	C₂³	1	64	yes	complemented
44	C₂⁴×C₄	Dic₃	1	2	yes	—
45	C₂⁴×C₄	C₂×D₃	1	1	yes	characteristic
46	C₂²×M₁₆	—	3	24	no	—
47	C₂²×M₁₆	—	6	16	no	—
48	C₂³×C₈	—	3	4	no	—
49	C₂²×C₄²	Dic₃	1	1	yes	characteristic
50	C₂²×C₄²	Dic₃	1	12	yes	—
51	C₂²×C₄²	Dic₃	1	1	yes	characteristic
52	C₂²×C₄²	C₂×D₃	1	12	yes	—
53	C₂²×C₄²	C₂×D₃	1	2	yes	—
54	#423	—	3	64	no	—
55	C₂×C₄⋊C₈	—	3	48	no	—
56	C₂⁴×C₃	—	2	8	no	—
57	C₂⁴×C₃	C₄⋊C₄	1	8	yes	—
58	C₂⁴×C₃	C₂²×C₄	1	6	yes	—
59	C₂⁴×C₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₃×C₄	—	2	8	no	—
61	C₂²×C₃×C₄	—	2	96	no	—
62	C₂²×C₃×C₄	C₄⋊C₄	1	48	yes	—
63	C₂²×C₃×C₄	C₄⋊C₄	1	8	yes	—
64	C₂²×C₃×C₄	M₁₆	1	16	yes	complemented
65	C₂²×C₃×C₄	C₂²×C₄	1	12	yes	—
66	C₂²×C₃×C₄	C₂²×C₄	1	32	yes	—
67	C₂²×C₃×C₄	C₂²×C₄	1	12	yes	—
68	C₂²×C₃×C₄	C₂²×C₄	1	6	yes	—
69	C₂²×C₃×C₄	C₂²×C₄	1	3	yes	—
70	C₂²×C₃×C₄	C₂²×C₄	1	12	yes	—
71	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
72	C₂²×C₃×C₄	C₂×D₄	1	24	yes	—
73	C₂²×C₃×C₄	C₂×Q₈	1	24	yes	—
74	C₂²×C₃×C₄	C₂×Q₈	1	4	yes	—
75	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
76	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
77	C₃×C₄²	C₂²×C₄	1	16	yes	—
78	C₃×C₄²	C₂²×C₄	1	24	yes	—
79	C₃×C₄²	C₂²×C₄	1	16	yes	—
80	C₃×C₄²	C₂⁴	1	8	yes	—
81	#208	—	2	64	no	—
82	C₂×C₃⋊C₈	—	2	48	no	—
83	C₂×C₃⋊C₈	C₂⁴	1	16	yes	—
84	C₂⁵	C₂×Dic₃	1	1	yes	characteristic
85	C₂³×C₄	—	2	16	no	—
86	C₂³×C₄	Dic₆	1	8	yes	—
87	C₂³×C₄	D₁₂	1	8	yes	—
88	C₂³×C₄	C₂×Dic₃	1	2	yes	—
89	C₂³×C₄	C₂×Dic₃	1	24	yes	—
90	C₂³×C₄	C₂×Dic₃	1	2	yes	—
91	C₂³×C₄	C₂×Dic₃	1	1	yes	characteristic
92	C₂³×C₄	C₂×Dic₃	1	6	yes	—
93	C₂³×C₄	C₂²×D₃	1	6	yes	—
94	C₂³×C₄	C₂²×D₃	1	1	yes	characteristic
95	C₂×M₁₆	—	3	32	no	—
96	C₂×M₁₆	—	6	96	no	—
97	C₂²×C₈	—	3	24	no	—
98	C₂²×C₈	—	6	16	no	—
99	C₂×C₄²	C₂×Dic₃	1	12	yes	—
100	C₂×C₄²	C₂×Dic₃	1	48	yes	—
101	C₂×C₄²	C₂×Dic₃	1	12	yes	—
102	C₂×C₄²	C₂×Dic₃	1	12	yes	—
103	C₂×C₄²	C₂²×D₃	1	12	yes	—
104	C₂×C₄²	C₂²×D₃	1	16	yes	—
105	C₄⋊C₈	—	3	64	no	—
106	C₂³×C₃	—	2	24	no	—
107	C₂³×C₃	—	2	32	no	—
108	C₂³×C₃	C₂×C₄⋊C₄	1	24	yes	—
109	C₂³×C₃	C₂×C₄⋊C₄	1	4	yes	—
110	C₂³×C₃	C₂×M₁₆	1	8	yes	—
111	C₂³×C₃	C₂³×C₄	1	4	yes	—
112	C₂³×C₃	C₂³×C₄	1	3	yes	—
113	C₂×C₃×C₄	—	2	24	no	—
114	C₂×C₃×C₄	—	2	64	no	—
115	C₂×C₃×C₄	C₂×C₄⋊C₄	1	48	yes	—
116	C₂×C₃×C₄	C₂×C₄⋊C₄	1	12	yes	—
117	C₂×C₃×C₄	C₂×C₄⋊C₄	1	24	yes	—
118	C₂×C₃×C₄	C₂×M₁₆	1	48	yes	complemented
119	C₂×C₃×C₄	C₂³×C₄	1	8	yes	—
120	C₂×C₃×C₄	C₂³×C₄	1	3	yes	—
121	C₂×C₃×C₄	C₂³×C₄	1	4	yes	—
122	C₂×C₃×C₄	C₂³×C₄	1	8	yes	—
123	C₂×C₃×C₄	C₂²×D₄	1	6	yes	—
124	C₂×C₃×C₄	C₂²×D₄	1	8	yes	—
125	C₂×C₃×C₄	C₂²×Q₈	1	8	yes	—
126	C₂×C₃×C₄	C₂²×Q₈	1	6	yes	—
127	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
128	C₃⋊C₈	—	2	16	no	—
129	C₂⁴	—	2	8	no	—
130	C₂⁴	C₄⋊Dic₃	1	8	yes	—
131	C₂⁴	C₂²×Dic₃	1	6	yes	—
132	C₂⁴	C₂²×Dic₃	1	1	yes	characteristic
133	C₂²×C₄	—	2	8	no	—
134	C₂²×C₄	—	2	96	no	—
135	C₂²×C₄	#208	1	16	yes	complemented
136	C₂²×C₄	C₄⋊Dic₃	1	48	yes	—
137	C₂²×C₄	C₄⋊Dic₃	1	8	yes	—
138	C₂²×C₄	C₂×Dic₆	1	24	yes	—
139	C₂²×C₄	C₂×Dic₆	1	4	yes	—
140	C₂²×C₄	C₂×D₁₂	1	4	yes	—
141	C₂²×C₄	C₂×D₁₂	1	24	yes	—
142	C₂²×C₄	C₂²×Dic₃	1	12	yes	—
143	C₂²×C₄	C₂²×Dic₃	1	32	yes	—
144	C₂²×C₄	C₂²×Dic₃	1	12	yes	—
145	C₂²×C₄	C₂²×Dic₃	1	12	yes	—
146	C₂²×C₄	C₂²×Dic₃	1	6	yes	—
147	C₂²×C₄	C₂²×Dic₃	1	3	yes	—
148	C₂²×C₄	C₂³×D₃	1	3	yes	—
149	C₂²×C₄	C₂³×D₃	1	4	yes	—
150	M₁₆	—	6	64	no	—
151	C₂×C₈	—	3	16	no	—
152	C₂×C₈	—	6	48	no	—
153	C₄²	C₂²×Dic₃	1	16	yes	—
154	C₄²	C₂²×Dic₃	1	24	yes	—
155	C₄²	C₂²×Dic₃	1	16	yes	—
156	C₄²	C₂³×D₃	1	8	yes	—
157	C₂²×C₃	—	2	48	no	—
158	C₂²×C₃	—	2	8	no	—
159	C₂²×C₃	#423	1	16	yes	complemented
160	C₂²×C₃	C₂²×C₄⋊C₄	1	8	yes	—
161	C₂²×C₃	C₂²×C₄⋊C₄	1	6	yes	—
162	C₂²×C₃	C₂²×M₁₆	1	12	yes	—
163	C₂²×C₃	C₂⁴×C₄	1	1	yes	characteristic
164	C₃×C₄	—	2	8	no	—
165	C₃×C₄	C₂²×C₄⋊C₄	1	6	yes	—
166	C₃×C₄	C₂²×C₄⋊C₄	1	8	yes	—
167	C₃×C₄	C₂²×M₁₆	1	16	yes	complemented
168	C₃×C₄	C₂³×D₄	1	1	yes	characteristic
169	C₃×C₄	C₂³×Q₈	1	1	yes	characteristic
170	C₂³	—	2	32	no	—
171	C₂³	—	2	24	no	—
172	C₂³	C₂×#208	1	8	yes	—
173	C₂³	C₂×C₄⋊Dic₃	1	24	yes	—
174	C₂³	C₂×C₄⋊Dic₃	1	4	yes	—
175	C₂³	C₂³×Dic₃	1	4	yes	—
176	C₂³	C₂³×Dic₃	1	3	yes	—
177	C₂×C₄	—	2	24	no	—
178	C₂×C₄	—	2	64	no	—
179	C₂×C₄	C₂×#208	1	48	yes	complemented
180	C₂×C₄	C₂×C₄⋊Dic₃	1	48	yes	—
181	C₂×C₄	C₂×C₄⋊Dic₃	1	24	yes	—
182	C₂×C₄	C₂×C₄⋊Dic₃	1	12	yes	—
183	C₂×C₄	C₂²×Dic₆	1	8	yes	—
184	C₂×C₄	C₂²×Dic₆	1	6	yes	—
185	C₂×C₄	C₂²×D₁₂	1	6	yes	—
186	C₂×C₄	C₂²×D₁₂	1	8	yes	—
187	C₂×C₄	C₂³×Dic₃	1	8	yes	—
188	C₂×C₄	C₂³×Dic₃	1	8	yes	—
189	C₂×C₄	C₂³×Dic₃	1	4	yes	—
190	C₂×C₄	C₂³×Dic₃	1	3	yes	—
191	C₂×C₄	C₂⁴×D₃	1	1	yes	characteristic
192	C₈	—	6	16	no	—
193	C₂×C₃	—	2	8	no	—
194	C₂×C₃	C₂×#423	1	12	yes	complemented
195	C₂×C₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
196	C₂×C₃	C₂³×M₁₆	1	2	yes	—
197	C₂²	—	2	48	no	—
198	C₂²	—	2	8	no	—
199	C₂²	#4924	1	16	yes	direct product factor
200	C₂²	C₂²×#208	1	12	yes	—
201	C₂²	C₂²×C₄⋊Dic₃	1	8	yes	—
202	C₂²	C₂²×C₄⋊Dic₃	1	6	yes	—
203	C₂²	C₂⁴×Dic₃	1	1	yes	characteristic
204	C₄	—	2	8	no	—
205	C₄	C₂²×#208	1	16	yes	complemented
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×#423	1	1	yes	characteristic, complemented, Sylow
211	C₂	—	2	8	no	—
212	C₂	C₂×#4924	1	12	yes	direct product factor
213	C₂	C₂³×#208	1	2	yes	—
214	C₂	C₂³×C₄⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₃
Fitting	C₂³×C₃×C₄²	C₂
socle	C₂⁴×C₃	C₂²×C₄
prime radical	C₂⁵×C₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂⁵
hypercenter	C₂³×C₄²	D₃
hypocenter	C₃	C₂²×#423
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
3. C₂×#4924	C₂	direct (N × H)
4. C₂²×C₄⋊(C₃⋊C₈)	C₂	semidirect
16. #4924	C₂²	direct (N × H)
17. C₂×C₄⋊(C₃⋊C₈)	C₂²	semidirect
43. C₄⋊(C₃⋊C₈)	C₂³	semidirect
64. C₂²×C₃×C₄	M₁₆	semidirect
118. C₂×C₃×C₄	C₂×M₁₆	semidirect
135. C₂²×C₄	#208	semidirect
159. C₂²×C₃	#423	semidirect
167. C₃×C₄	C₂²×M₁₆	semidirect
179. C₂×C₄	C₂×#208	semidirect
194. C₂×C₃	C₂×#423	semidirect
199. C₂²	#4924	direct (N × H)
205. C₄	C₂²×#208	semidirect
210. C₃	C₂²×#423	semidirect
212. C₂	C₂×#4924	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
C₂³×#208	2
C₂×#4924	2
C₂²×C₄⋊(C₃⋊C₈)	2
C₂²×#423	3

Subgroups — C22×#4924 #11658799

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#4924 #11658799