Subgroups — C₂²×#5024 #11659545

Order 768.

2222 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#5024	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊(C₃⋊C₈)	C₂	1	2	yes	maximal
4	C₂²×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	C₂²×#803	C₂	1	2	yes	maximal
6	C₂²×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₂²×#446	—	3	1	no	maximal, Sylow
8	C₂³×C₃×Q₈	C₄	1	1	yes	characteristic
9	C₂×C₃×C₄×Q₈	C₄	1	12	yes	—
10	C₂×C₃×C₄×Q₈	C₂²	1	12	yes	—
11	C₂²×C₃×C₄⋊C₄	C₄	1	2	yes	—
12	C₂²×C₃×C₄⋊C₄	C₄	1	1	yes	characteristic
13	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄²	C₂²	1	2	yes	—
15	C₂³×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂³×C₃⋊C₈	C₂²	1	2	yes	—
17	C₂³×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	#5024	C₂²	1	64	yes	direct product factor
19	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	24	yes	—
20	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	12	yes	—
21	C₂×#803	C₂²	1	24	yes	—
22	C₂×C₄×C₃⋊C₈	C₂²	1	12	yes	—
23	C₂²×C₄×Q₈	D₃	1	1	yes	characteristic
24	C₂×#446	—	3	24	no	—
25	C₂²×C₄⋊C₈	—	3	2	no	—
26	C₂²×C₄⋊C₈	—	3	1	no	—
27	C₂²×#97	—	3	2	no	—
28	C₂²×C₄×C₈	—	3	1	no	—
29	C₂²×C₃×Q₈	—	2	8	no	—
30	C₂²×C₃×Q₈	C₂×C₄	1	12	yes	—
31	C₂³×C₃×C₄	C₂×C₄	1	2	yes	—
32	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₃×C₄	C₂×C₄	1	2	yes	—
34	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
36	C₃×C₄×Q₈	C₂×C₄	1	48	yes	—
37	C₃×C₄×Q₈	C₂³	1	16	yes	—
38	C₂×C₃×C₄⋊C₄	C₂×C₄	1	24	yes	—
39	C₂×C₃×C₄⋊C₄	C₂×C₄	1	12	yes	—
40	C₂×C₃×C₄²	C₂×C₄	1	12	yes	—
41	C₂×C₃×C₄²	C₂×C₄	1	6	yes	—
42	C₂×C₃×C₄²	C₂³	1	6	yes	—
43	C₂×C₃×C₄²	C₂³	1	12	yes	—
44	C₂²×C₃⋊C₈	—	2	4	no	—
45	C₂²×C₃⋊C₈	—	2	8	no	—
46	C₂²×C₃⋊C₈	Q₈	1	8	yes	complemented
47	C₂²×C₃⋊C₈	C₂³	1	6	yes	—
48	C₂²×C₃⋊C₈	C₂³	1	12	yes	—
49	C₂²×C₃⋊C₈	C₂³	1	6	yes	—
50	C₄⋊(C₃⋊C₈)	C₂³	1	32	yes	—
51	C₄⋊(C₃⋊C₈)	C₂³	1	16	yes	—
52	#803	C₂³	1	32	yes	—
53	C₄×C₃⋊C₈	C₂³	1	16	yes	—
54	C₂³×Q₈	Dic₃	1	1	yes	characteristic
55	C₂³×C₈	—	3	1	no	—
56	C₂³×C₈	—	3	1	no	—
57	C₂³×C₈	—	3	2	no	—
58	C₂×C₄×Q₈	Dic₃	1	12	yes	—
59	C₂×C₄×Q₈	C₂×D₃	1	12	yes	—
60	C₂²×C₄⋊C₄	Dic₃	1	2	yes	—
61	C₂²×C₄⋊C₄	Dic₃	1	1	yes	characteristic
62	C₂²×C₄²	C₂×D₃	1	1	yes	characteristic
63	C₂²×C₄²	C₂×D₃	1	2	yes	—
64	#446	—	3	64	no	—
65	C₂×C₄⋊C₈	—	3	12	no	—
66	C₂×C₄⋊C₈	—	3	24	no	—
67	C₂×#97	—	3	24	no	—
68	C₂×C₄×C₈	—	3	12	no	—
69	C₂⁴×C₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₃×Q₈	—	2	48	no	—
71	C₂×C₃×Q₈	C₂²×C₄	1	16	yes	—
72	C₂²×C₃×C₄	—	2	8	no	—
73	C₂²×C₃×C₄	—	2	4	no	—
74	C₂²×C₃×C₄	—	2	8	no	—
75	C₂²×C₃×C₄	M₁₆	1	8	yes	—
76	C₂²×C₃×C₄	C₂²×C₄	1	6	yes	—
77	C₂²×C₃×C₄	C₂²×C₄	1	12	yes	—
78	C₂²×C₃×C₄	C₂²×C₄	1	12	yes	—
79	C₂²×C₃×C₄	C₂²×C₄	1	3	yes	—
80	C₂²×C₃×C₄	C₂²×C₄	1	6	yes	—
81	C₂²×C₃×C₄	C₂×Q₈	1	4	yes	—
82	C₂²×C₃×C₄	#41	1	4	yes	—
83	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
84	C₃×C₄⋊C₄	C₂²×C₄	1	32	yes	—
85	C₃×C₄⋊C₄	C₂²×C₄	1	16	yes	—
86	C₃×C₄²	C₂²×C₄	1	24	yes	—
87	C₃×C₄²	C₂²×C₄	1	12	yes	—
88	C₃×C₄²	C₂⁴	1	4	yes	—
89	C₃×C₄²	C₂⁴	1	8	yes	—
90	C₂×C₃⋊C₈	—	2	12	no	—
91	C₂×C₃⋊C₈	—	2	24	no	—
92	C₂×C₃⋊C₈	C₂×Q₈	1	24	yes	complemented
93	C₂×C₃⋊C₈	C₂⁴	1	4	yes	—
94	C₂×C₃⋊C₈	C₂⁴	1	8	yes	—
95	C₂×C₃⋊C₈	C₂⁴	1	4	yes	—
96	C₂²×Q₈	—	2	8	no	—
97	C₂²×Q₈	C₂×Dic₃	1	12	yes	—
98	C₂³×C₄	C₂×Dic₃	1	1	yes	characteristic
99	C₂³×C₄	C₂×Dic₃	1	2	yes	—
100	C₂³×C₄	C₂×Dic₃	1	2	yes	—
101	C₂³×C₄	C₂×Dic₃	1	1	yes	characteristic
102	C₂³×C₄	C₂²×D₃	1	1	yes	characteristic
103	C₂²×C₈	—	3	8	no	—
104	C₂²×C₈	—	3	6	no	—
105	C₂²×C₈	—	3	6	no	—
106	C₂²×C₈	—	3	12	no	—
107	C₂²×C₈	—	6	4	no	—
108	C₂²×C₈	—	6	8	no	—
109	C₄×Q₈	C₂×Dic₃	1	48	yes	—
110	C₄×Q₈	C₂²×D₃	1	16	yes	—
111	C₂×C₄⋊C₄	C₂×Dic₃	1	24	yes	—
112	C₂×C₄⋊C₄	C₂×Dic₃	1	12	yes	—
113	C₂×C₄²	C₂×Dic₃	1	12	yes	—
114	C₂×C₄²	C₂×Dic₃	1	6	yes	—
115	C₂×C₄²	C₂²×D₃	1	6	yes	—
116	C₂×C₄²	C₂²×D₃	1	12	yes	—
117	C₄⋊C₈	—	3	16	no	—
118	C₄⋊C₈	—	3	32	no	—
119	#97	—	3	32	no	—
120	C₄×C₈	—	3	16	no	—
121	C₂³×C₃	C₄×Q₈	1	4	yes	—
122	C₂³×C₃	C₂×M₁₆	1	4	yes	—
123	C₂³×C₃	#131	1	4	yes	—
124	C₂³×C₃	C₂³×C₄	1	3	yes	—
125	C₃×Q₈	—	2	32	no	—
126	C₂×C₃×C₄	—	2	24	no	—
127	C₂×C₃×C₄	—	2	12	no	—
128	C₂×C₃×C₄	—	2	24	no	—
129	C₂×C₃×C₄	C₄×Q₈	1	12	yes	—
130	C₂×C₃×C₄	C₂×M₁₆	1	24	yes	—
131	C₂×C₃×C₄	C₂³×C₄	1	4	yes	—
132	C₂×C₃×C₄	C₂³×C₄	1	8	yes	—
133	C₂×C₃×C₄	C₂³×C₄	1	8	yes	—
134	C₂×C₃×C₄	C₂³×C₄	1	3	yes	—
135	C₂×C₃×C₄	C₂³×C₄	1	4	yes	—
136	C₂×C₃×C₄	C₂²×Q₈	1	6	yes	—
137	C₂×C₃×C₄	C₂×#41	1	6	yes	—
138	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
139	C₃⋊C₈	—	2	4	no	—
140	C₃⋊C₈	—	2	8	no	—
141	C₃⋊C₈	C₂²×Q₈	1	8	yes	complemented
142	C₂⁴	C₂²×Dic₃	1	1	yes	characteristic
143	C₂×Q₈	—	2	48	no	—
144	C₂×Q₈	C₂²×Dic₃	1	16	yes	—
145	C₂²×C₄	—	2	8	no	—
146	C₂²×C₄	—	2	4	no	—
147	C₂²×C₄	—	2	8	no	—
148	C₂²×C₄	#208	1	8	yes	—
149	C₂²×C₄	D₃×Q₈	1	4	yes	—
150	C₂²×C₄	#239	1	4	yes	—
151	C₂²×C₄	C₂²×Dic₃	1	6	yes	—
152	C₂²×C₄	C₂²×Dic₃	1	12	yes	—
153	C₂²×C₄	C₂²×Dic₃	1	12	yes	—
154	C₂²×C₄	C₂²×Dic₃	1	6	yes	—
155	C₂²×C₄	C₂²×Dic₃	1	3	yes	—
156	C₂²×C₄	C₂³×D₃	1	3	yes	—
157	C₂×C₈	—	3	24	no	—
158	C₂×C₈	—	3	4	no	—
159	C₂×C₈	—	3	4	no	—
160	C₂×C₈	—	3	8	no	—
161	C₂×C₈	—	6	12	no	—
162	C₂×C₈	—	6	24	no	—
163	C₄⋊C₄	C₂²×Dic₃	1	32	yes	—
164	C₄⋊C₄	C₂²×Dic₃	1	16	yes	—
165	C₄²	C₂²×Dic₃	1	24	yes	—
166	C₄²	C₂²×Dic₃	1	12	yes	—
167	C₄²	C₂³×D₃	1	4	yes	—
168	C₄²	C₂³×D₃	1	8	yes	—
169	C₂²×C₃	#446	1	16	yes	complemented
170	C₂²×C₃	C₂×C₄×Q₈	1	6	yes	—
171	C₂²×C₃	C₂²×M₁₆	1	6	yes	—
172	C₂²×C₃	C₂×#131	1	6	yes	—
173	C₂²×C₃	C₂⁴×C₄	1	1	yes	characteristic
174	C₃×C₄	—	2	8	no	—
175	C₃×C₄	—	2	4	no	—
176	C₃×C₄	—	2	8	no	—
177	C₃×C₄	C₂×C₄×Q₈	1	6	yes	—
178	C₃×C₄	C₂²×M₁₆	1	8	yes	—
179	C₃×C₄	C₂³×Q₈	1	1	yes	characteristic
180	C₃×C₄	C₂²×#41	1	1	yes	characteristic
181	C₂³	C₂×#208	1	4	yes	—
182	C₂³	Q₈×Dic₃	1	4	yes	—
183	C₂³	#948	1	4	yes	—
184	C₂³	C₂³×Dic₃	1	3	yes	—
185	Q₈	—	2	32	no	—
186	C₂×C₄	—	2	24	no	—
187	C₂×C₄	—	2	12	no	—
188	C₂×C₄	—	2	24	no	—
189	C₂×C₄	C₂×#208	1	24	yes	—
190	C₂×C₄	Q₈×Dic₃	1	12	yes	—
191	C₂×C₄	C₂×D₃×Q₈	1	6	yes	—
192	C₂×C₄	C₂×#239	1	6	yes	—
193	C₂×C₄	C₂³×Dic₃	1	4	yes	—
194	C₂×C₄	C₂³×Dic₃	1	8	yes	—
195	C₂×C₄	C₂³×Dic₃	1	8	yes	—
196	C₂×C₄	C₂³×Dic₃	1	4	yes	—
197	C₂×C₄	C₂³×Dic₃	1	3	yes	—
198	C₂×C₄	C₂⁴×D₃	1	1	yes	characteristic
199	C₈	—	3	8	no	—
200	C₈	—	6	4	no	—
201	C₈	—	6	8	no	—
202	C₂×C₃	C₂×#446	1	12	yes	complemented
203	C₂×C₃	C₂²×C₄×Q₈	1	1	yes	characteristic
204	C₂×C₃	C₂³×M₁₆	1	1	yes	characteristic
205	C₂×C₃	C₂²×#131	1	1	yes	characteristic
206	C₂²	#5024	1	16	yes	direct product factor
207	C₂²	C₂²×#208	1	6	yes	—
208	C₂²	C₂×Q₈×Dic₃	1	6	yes	—
209	C₂²	C₂×#948	1	6	yes	—
210	C₂²	C₂⁴×Dic₃	1	1	yes	characteristic
211	C₄	—	2	8	no	—
212	C₄	—	2	4	no	—
213	C₄	—	2	8	no	—
214	C₄	C₂²×#208	1	8	yes	—
215	C₄	C₂×Q₈×Dic₃	1	6	yes	—
216	C₄	C₂²×D₃×Q₈	1	1	yes	characteristic
217	C₄	C₂²×#239	1	1	yes	characteristic
218	C₃	C₂²×#446	1	1	yes	characteristic, complemented, Sylow
219	C₂	C₂×#5024	1	12	yes	direct product factor
220	C₂	C₂³×#208	1	1	yes	characteristic
221	C₂	C₂²×Q₈×Dic₃	1	1	yes	characteristic
222	C₂	C₂²×#948	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₃
Fitting	C₂²×C₃×C₄×Q₈	C₂
socle, prime radical	C₂⁴×C₃	C₂²×C₄
cosocle	C₂×C₃×C₄	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₃
hypocenter	C₃	C₂²×#446
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×#5024	C₂	direct (N × H)
18. #5024	C₂²	direct (N × H)
46. C₂²×C₃⋊C₈	Q₈	semidirect
92. C₂×C₃⋊C₈	C₂×Q₈	semidirect
141. C₃⋊C₈	C₂²×Q₈	semidirect
169. C₂²×C₃	#446	semidirect
202. C₂×C₃	C₂×#446	semidirect
206. C₂²	#5024	direct (N × H)
218. C₃	C₂²×#446	semidirect
219. C₂	C₂×#5024	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×C₄×Q₈	2
C₂×#5024	2
C₂²×C₄⋊(C₃⋊C₈)	2
C₂²×#803	2
C₂²×C₄×C₃⋊C₈	2
C₂²×#446	3

Subgroups — C22×#5024 #11659545

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#5024 #11659545