Subgroups — C₂²×C₃×#869 #11900719

Order 1152.

3497 conjugacy classes of subgroups.

The subgroup lattice has 207 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×Dic₆	C₂	1	2	yes	maximal
2	C₂²×C₃²×C₄²	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₄⋊Dic₃	C₂	1	4	yes	maximal
4	C₂×C₃×#869	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₃×C₄⋊Q₈	—	3	1	no	maximal
6	C₂²×#869	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
8	C₂³×C₃²×C₄	C₂²	1	2	yes	—
9	C₂³×C₃×Dic₃	C₂²	1	4	yes	—
10	C₂²×C₃×Dic₆	—	2	16	no	—
11	C₂²×C₃×Dic₆	C₂²	1	24	yes	—
12	C₂×C₃²×C₄²	C₂²	1	12	yes	—
13	C₂×C₃×C₄⋊Dic₃	C₂²	1	48	yes	—
14	C₃×#869	C₂²	1	64	yes	direct product factor
15	C₂³×C₃×Q₈	—	3	2	no	—
16	C₂³×Dic₆	C₂×C₃	1	2	yes	—
17	C₂×C₃×C₄⋊Q₈	—	3	24	no	—
18	C₂²×C₃×C₄⋊C₄	—	3	4	no	—
19	C₂²×C₃×C₄²	—	2	1	no	—
20	C₂²×C₃×C₄²	D₃	1	1	yes	characteristic
21	C₂²×C₃×C₄²	C₂×C₃	1	1	yes	characteristic
22	C₂²×C₄⋊Dic₃	C₂×C₃	1	4	yes	—
23	C₂×#869	C₂×C₃	1	24	yes	direct product factor
24	C₂⁴×C₃²	C₂³	1	1	yes	characteristic
25	C₂²×C₃²×C₄	D₄	1	8	yes	—
26	C₂²×C₃²×C₄	Q₈	1	16	yes	complemented
27	C₂²×C₃²×C₄	C₂³	1	6	yes	—
28	C₂²×C₃²×C₄	C₂³	1	12	yes	—
29	C₂²×C₃×Dic₃	—	2	16	no	—
30	C₂²×C₃×Dic₃	C₂³	1	24	yes	—
31	C₂×C₃×Dic₆	—	2	96	no	—
32	C₂×C₃×Dic₆	C₂³	1	32	yes	—
33	C₃²×C₄²	C₂³	1	16	yes	—
34	C₃×C₄⋊Dic₃	C₂³	1	64	yes	—
35	C₂²×C₄⋊Q₈	—	3	1	no	Sylow
36	C₂²×C₃×Q₈	—	3	24	no	—
37	C₂²×C₃×Q₈	—	6	16	no	—
38	C₂³×C₃×C₄	—	2	2	no	—
39	C₂³×C₃×C₄	—	2	1	no	—
40	C₂³×C₃×C₄	—	3	4	no	—
41	C₂³×C₃×C₄	C₂×D₃	1	2	yes	—
42	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
43	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
44	C₂³×C₃×C₄	C₂²×C₃	1	2	yes	—
45	C₂³×Dic₃	C₂²×C₃	1	4	yes	—
46	C₂²×Dic₆	—	2	16	no	—
47	C₂²×Dic₆	C₂²×C₃	1	24	yes	—
48	C₃×C₄⋊Q₈	—	3	64	no	—
49	C₂×C₃×C₄⋊C₄	—	3	48	no	—
50	C₂×C₃×C₄²	—	2	12	no	—
51	C₂×C₃×C₄²	C₂×D₃	1	12	yes	—
52	C₂×C₃×C₄²	C₂²×C₃	1	12	yes	—
53	C₂×C₄⋊Dic₃	C₂²×C₃	1	48	yes	—
54	#869	C₂²×C₃	1	64	yes	direct product factor
55	C₂³×C₃²	C₂×D₄	1	4	yes	—
56	C₂³×C₃²	C₂×Q₈	1	8	yes	—
57	C₂³×C₃²	C₂⁴	1	3	yes	—
58	C₂×C₃²×C₄	C₂×D₄	1	24	yes	—
59	C₂×C₃²×C₄	C₂×Q₈	1	48	yes	complemented
60	C₂×C₃²×C₄	C₂⁴	1	4	yes	—
61	C₂×C₃²×C₄	C₂⁴	1	8	yes	—
62	C₂×C₃×Dic₃	—	2	48	no	—
63	C₂×C₃×Dic₃	C₂⁴	1	16	yes	—
64	C₃×Dic₆	—	2	64	no	—
65	C₂³×Q₈	—	3	2	no	—
66	C₂×C₄⋊Q₈	—	3	24	no	—
67	C₂²×C₄⋊C₄	—	3	4	no	—
68	C₂²×C₄²	C₃×D₃	1	1	yes	characteristic
69	C₂⁴×C₃	—	2	1	no	—
70	C₂⁴×C₃	C₂²×D₃	1	1	yes	characteristic
71	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
72	C₂×C₃×Q₈	—	3	32	no	—
73	C₂×C₃×Q₈	—	6	96	no	—
74	C₂²×C₃×C₄	—	2	16	no	—
75	C₂²×C₃×C₄	—	2	8	no	—
76	C₂²×C₃×C₄	—	2	12	no	—
77	C₂²×C₃×C₄	—	2	6	no	—
78	C₂²×C₃×C₄	—	3	24	no	—
79	C₂²×C₃×C₄	—	6	16	no	—
80	C₂²×C₃×C₄	Dic₆	1	16	yes	complemented
81	C₂²×C₃×C₄	D₁₂	1	8	yes	—
82	C₂²×C₃×C₄	C₃×D₄	1	8	yes	—
83	C₂²×C₃×C₄	C₃×Q₈	1	16	yes	complemented
84	C₂²×C₃×C₄	C₂²×D₃	1	12	yes	—
85	C₂²×C₃×C₄	C₂²×D₃	1	6	yes	—
86	C₂²×C₃×C₄	C₂³×C₃	1	6	yes	—
87	C₂²×C₃×C₄	C₂³×C₃	1	12	yes	—
88	C₂²×Dic₃	—	2	16	no	—
89	C₂²×Dic₃	C₂³×C₃	1	24	yes	—
90	C₂×Dic₆	—	2	96	no	—
91	C₂×Dic₆	C₂³×C₃	1	32	yes	—
92	C₃×C₄⋊C₄	—	3	64	no	—
93	C₃×C₄²	—	2	16	no	—
94	C₃×C₄²	C₂²×D₃	1	16	yes	—
95	C₃×C₄²	C₂³×C₃	1	16	yes	—
96	C₄⋊Dic₃	C₂³×C₃	1	64	yes	—
97	C₂²×C₃²	C₄⋊Q₈	1	16	yes	complemented
98	C₂²×C₃²	C₂²×D₄	1	6	yes	—
99	C₂²×C₃²	C₂²×Q₈	1	12	yes	—
100	C₂²×C₃²	C₂⁵	1	1	yes	characteristic
101	C₃²×C₄	C₂²×D₄	1	8	yes	—
102	C₃²×C₄	C₂²×Q₈	1	16	yes	complemented
103	C₃×Dic₃	—	2	16	no	—
104	C₂²×Q₈	—	3	24	no	—
105	C₂²×Q₈	—	6	16	no	—
106	C₂³×C₄	—	3	4	no	—
107	C₂³×C₄	C₂×C₃×D₃	1	2	yes	—
108	C₂³×C₄	C₂×C₃×D₃	1	1	yes	characteristic
109	C₄⋊Q₈	—	3	64	no	—
110	C₂×C₄⋊C₄	—	3	48	no	—
111	C₂×C₄²	C₂×C₃×D₃	1	12	yes	—
112	C₂³×C₃	—	2	3	no	—
113	C₂³×C₃	—	2	8	no	—
114	C₂³×C₃	—	2	4	no	—
115	C₂³×C₃	C₂×Dic₆	1	8	yes	—
116	C₂³×C₃	C₂×D₁₂	1	4	yes	—
117	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
118	C₂³×C₃	C₂×C₃×Q₈	1	8	yes	—
119	C₂³×C₃	C₂³×D₃	1	3	yes	—
120	C₂³×C₃	C₂⁴×C₃	1	3	yes	—
121	C₃×Q₈	—	6	64	no	—
122	C₂×C₃×C₄	—	2	8	no	—
123	C₂×C₃×C₄	—	2	4	no	—
124	C₂×C₃×C₄	—	2	48	no	—
125	C₂×C₃×C₄	—	2	24	no	—
126	C₂×C₃×C₄	—	3	16	no	—
127	C₂×C₃×C₄	—	6	48	no	—
128	C₂×C₃×C₄	C₂×Dic₆	1	48	yes	complemented
129	C₂×C₃×C₄	C₂×D₁₂	1	24	yes	—
130	C₂×C₃×C₄	C₂×C₃×D₄	1	24	yes	—
131	C₂×C₃×C₄	C₂×C₃×Q₈	1	48	yes	complemented
132	C₂×C₃×C₄	C₂³×D₃	1	8	yes	—
133	C₂×C₃×C₄	C₂³×D₃	1	4	yes	—
134	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
135	C₂×C₃×C₄	C₂⁴×C₃	1	8	yes	—
136	C₂×Dic₃	—	2	48	no	—
137	C₂×Dic₃	C₂⁴×C₃	1	16	yes	—
138	Dic₆	—	2	64	no	—
139	C₂×C₃²	C₂×C₄⋊Q₈	1	12	yes	complemented
140	C₂×C₃²	C₂³×D₄	1	1	yes	characteristic
141	C₂×C₃²	C₂³×Q₈	1	2	yes	—
142	C₂⁴	C₂²×C₃×D₃	1	1	yes	characteristic
143	C₂×Q₈	—	3	32	no	—
144	C₂×Q₈	—	6	96	no	—
145	C₂²×C₄	—	3	24	no	—
146	C₂²×C₄	—	6	16	no	—
147	C₂²×C₄	C₃×Dic₆	1	16	yes	complemented
148	C₂²×C₄	C₃×D₁₂	1	8	yes	—
149	C₂²×C₄	C₂²×C₃×D₃	1	12	yes	—
150	C₂²×C₄	C₂²×C₃×D₃	1	6	yes	—
151	C₄⋊C₄	—	3	64	no	—
152	C₄²	C₂²×C₃×D₃	1	16	yes	—
153	C₂²×C₃	—	2	12	no	—
154	C₂²×C₃	—	2	6	no	—
155	C₂²×C₃	—	2	1	no	—
156	C₂²×C₃	—	2	16	no	—
157	C₂²×C₃	#869	1	16	yes	direct product factor
158	C₂²×C₃	C₃×C₄⋊Q₈	1	16	yes	complemented
159	C₂²×C₃	C₂²×Dic₆	1	12	yes	—
160	C₂²×C₃	C₂²×D₁₂	1	6	yes	—
161	C₂²×C₃	C₂²×C₃×D₄	1	6	yes	—
162	C₂²×C₃	C₂²×C₃×Q₈	1	12	yes	—
163	C₂²×C₃	C₂⁴×D₃	1	1	yes	characteristic
164	C₂²×C₃	C₂⁵×C₃	1	1	yes	characteristic
165	C₃×C₄	—	2	16	no	—
166	C₃×C₄	—	2	8	no	—
167	C₃×C₄	—	6	16	no	—
168	C₃×C₄	C₂²×Dic₆	1	16	yes	complemented
169	C₃×C₄	C₂²×D₁₂	1	8	yes	—
170	C₃×C₄	C₂²×C₃×D₄	1	8	yes	—
171	C₃×C₄	C₂²×C₃×Q₈	1	16	yes	complemented
172	Dic₃	—	2	16	no	—
173	C₃²	C₂²×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
174	C₂³	C₂×C₃×Dic₆	1	8	yes	—
175	C₂³	C₂×C₃×D₁₂	1	4	yes	—
176	C₂³	C₂³×C₃×D₃	1	3	yes	—
177	Q₈	—	6	64	no	—
178	C₂×C₄	—	3	16	no	—
179	C₂×C₄	—	6	48	no	—
180	C₂×C₄	C₂×C₃×Dic₆	1	48	yes	complemented
181	C₂×C₄	C₂×C₃×D₁₂	1	24	yes	—
182	C₂×C₄	C₂³×C₃×D₃	1	8	yes	—
183	C₂×C₄	C₂³×C₃×D₃	1	4	yes	—
184	C₂×C₃	—	2	2	no	—
185	C₂×C₃	—	2	1	no	—
186	C₂×C₃	—	2	12	no	—
187	C₂×C₃	C₂×#869	1	12	yes	direct product factor
188	C₂×C₃	C₂×C₃×C₄⋊Q₈	1	12	yes	complemented
189	C₂×C₃	C₂³×Dic₆	1	2	yes	—
190	C₂×C₃	C₂³×D₁₂	1	1	yes	characteristic
191	C₂×C₃	C₂³×C₃×D₄	1	1	yes	characteristic
192	C₂×C₃	C₂³×C₃×Q₈	1	2	yes	—
193	C₂²	C₃×#869	1	16	yes	direct product factor
194	C₂²	C₂²×C₃×Dic₆	1	12	yes	—
195	C₂²	C₂²×C₃×D₁₂	1	6	yes	—
196	C₂²	C₂⁴×C₃×D₃	1	1	yes	characteristic
197	C₄	—	6	16	no	—
198	C₄	C₂²×C₃×Dic₆	1	16	yes	complemented
199	C₄	C₂²×C₃×D₁₂	1	8	yes	—
200	C₃	—	2	1	no	—
201	C₃	C₂²×#869	1	1	yes	characteristic, direct product factor
202	C₃	C₂²×C₃×C₄⋊Q₈	1	1	yes	characteristic, complemented
203	C₂	C₂×C₃×#869	1	12	yes	direct product factor
204	C₂	C₂³×C₃×Dic₆	1	2	yes	—
205	C₂	C₂³×C₃×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₃	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁵×C₃
Frattini	C₂²	C₂⁴×C₃×D₃
Fitting	C₂²×C₃²×C₄²	C₂
socle, prime radical	C₂⁴×C₃²	C₂³
hypercenter	C₂²×C₃×C₄²	D₃
hypocenter	C₃	C₂²×C₃×C₄⋊Q₈
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴×C₃ ◁ C₂²×C₃×C₄²

Complements

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

N	H ≅ G/N	type
4. C₂×C₃×#869	C₂	direct (N × H)
6. C₂²×#869	C₃	direct (N × H)
14. C₃×#869	C₂²	direct (N × H)
23. C₂×#869	C₂×C₃	direct (N × H)
26. C₂²×C₃²×C₄	Q₈	semidirect
54. #869	C₂²×C₃	direct (N × H)
59. C₂×C₃²×C₄	C₂×Q₈	semidirect
80. C₂²×C₃×C₄	Dic₆	semidirect
83. C₂²×C₃×C₄	C₃×Q₈	semidirect
97. C₂²×C₃²	C₄⋊Q₈	semidirect
102. C₃²×C₄	C₂²×Q₈	semidirect
128. C₂×C₃×C₄	C₂×Dic₆	semidirect
131. C₂×C₃×C₄	C₂×C₃×Q₈	semidirect
139. C₂×C₃²	C₂×C₄⋊Q₈	semidirect
147. C₂²×C₄	C₃×Dic₆	semidirect
157. C₂²×C₃	#869	direct (N × H)
158. C₂²×C₃	C₃×C₄⋊Q₈	semidirect
168. C₃×C₄	C₂²×Dic₆	semidirect
171. C₃×C₄	C₂²×C₃×Q₈	semidirect
173. C₃²	C₂²×C₄⋊Q₈	semidirect
180. C₂×C₄	C₂×C₃×Dic₆	semidirect
187. C₂×C₃	C₂×#869	direct (N × H)
188. C₂×C₃	C₂×C₃×C₄⋊Q₈	semidirect
193. C₂²	C₃×#869	direct (N × H)
198. C₄	C₂²×C₃×Dic₆	semidirect
201. C₃	C₂²×#869	direct (N × H)
202. C₃	C₂²×C₃×C₄⋊Q₈	semidirect
203. C₂	C₂×C₃×#869	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×Dic₆	2
C₂²×C₃²×C₄²	2
C₂²×C₃×C₄⋊Dic₃	2
C₂×C₃×#869	2
C₂²×C₃×C₄⋊Q₈	3
C₂²×#869	3

Subgroups — C22×C3×#869 #11900719

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#869 #11900719