Subgroups — C₂²×C₃×#817 #11858378

Order 1152.

1369 conjugacy classes of subgroups.

The subgroup lattice has 211 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃²×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₄⋊Dic₃	C₂	1	2	yes	maximal
3	C₂×C₃×#817	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₃×#106	—	3	1	no	maximal
5	C₂²×#817	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
7	C₂³×C₃×Dic₃	—	2	2	no	—
8	C₂²×C₃²×C₈	C₄	1	8	yes	complemented
9	C₂²×C₃²×C₈	C₂²	1	6	yes	—
10	C₂×C₃×C₄⋊Dic₃	—	2	6	no	—
11	C₂×C₃×C₄⋊Dic₃	C₂²	1	12	yes	—
12	C₃×#817	C₂²	1	16	yes	direct product factor
13	C₂³×C₃×C₈	—	2	1	no	—
14	C₂³×C₃×C₈	D₃	1	1	yes	characteristic
15	C₂³×C₃×C₈	C₂×C₃	1	1	yes	characteristic
16	C₂²×C₃×C₄⋊C₄	—	3	2	no	—
17	C₂²×C₄⋊Dic₃	C₂×C₃	1	2	yes	—
18	C₂×C₃×#106	—	3	12	no	—
19	C₂×#817	C₂×C₃	1	12	yes	direct product factor
20	C₂⁴×C₃²	D₄	1	1	yes	characteristic
21	C₂²×C₃²×C₄	C₂×C₄	1	4	yes	—
22	C₂²×C₃²×C₄	D₄	1	3	yes	—
23	C₂²×C₃²×C₄	Q₈	1	4	yes	—
24	C₂²×C₃²×C₄	C₂³	1	3	yes	—
25	C₂²×C₃×Dic₃	—	2	12	no	—
26	C₂²×C₃×Dic₃	—	4	8	no	—
27	C₂×C₃²×C₈	C₂×C₄	1	24	yes	complemented
28	C₂×C₃²×C₈	C₂³	1	4	yes	—
29	C₃×C₄⋊Dic₃	—	2	12	no	—
30	C₃×C₄⋊Dic₃	C₂³	1	8	yes	—
31	C₂²×#106	—	3	1	no	Sylow
32	C₂³×C₃×C₄	—	2	1	no	—
33	C₂³×C₃×C₄	—	6	2	no	—
34	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
35	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
36	C₂³×Dic₃	—	2	2	no	—
37	C₂²×C₃×C₈	—	2	8	no	—
38	C₂²×C₃×C₈	—	2	6	no	—
39	C₂²×C₃×C₈	Dic₃	1	8	yes	complemented
40	C₂²×C₃×C₈	C₃×C₄	1	8	yes	complemented
41	C₂²×C₃×C₈	C₂×D₃	1	6	yes	—
42	C₂²×C₃×C₈	C₂²×C₃	1	6	yes	—
43	C₂×C₃×C₄⋊C₄	—	3	12	no	—
44	C₂×C₃×C₄⋊C₄	—	6	6	no	—
45	C₂×C₄⋊Dic₃	—	2	6	no	—
46	C₂×C₄⋊Dic₃	C₂²×C₃	1	12	yes	—
47	C₃×#106	—	3	16	no	—
48	#817	C₂²×C₃	1	16	yes	direct product factor
49	C₂³×C₃²	C₄⋊C₄	1	4	yes	—
50	C₂³×C₃²	SD₁₆	1	8	yes	—
51	C₂³×C₃²	C₂×D₄	1	3	yes	—
52	C₂×C₃²×C₄	C₄⋊C₄	1	12	yes	—
53	C₂×C₃²×C₄	C₂²×C₄	1	6	yes	—
54	C₂×C₃²×C₄	C₂×D₄	1	3	yes	—
55	C₂×C₃²×C₄	C₂×Q₈	1	6	yes	—
56	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
57	C₂×C₃×Dic₃	—	2	8	no	—
58	C₂×C₃×Dic₃	—	4	24	no	—
59	C₃²×C₈	C₂²×C₄	1	8	yes	complemented
60	C₂³×C₈	C₃×D₃	1	1	yes	characteristic
61	C₂²×C₄⋊C₄	—	3	2	no	—
62	C₂×#106	—	3	12	no	—
63	C₂⁴×C₃	—	2	1	no	—
64	C₂⁴×C₃	D₁₂	1	1	yes	characteristic
65	C₂⁴×C₃	C₃×D₄	1	1	yes	characteristic
66	C₂²×C₃×C₄	—	2	4	no	—
67	C₂²×C₃×C₄	—	2	3	no	—
68	C₂²×C₃×C₄	—	2	4	no	—
69	C₂²×C₃×C₄	—	2	3	no	—
70	C₂²×C₃×C₄	—	6	12	no	—
71	C₂²×C₃×C₄	—	12	8	no	—
72	C₂²×C₃×C₄	Dic₆	1	4	yes	—
73	C₂²×C₃×C₄	D₁₂	1	3	yes	—
74	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
75	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
76	C₂²×C₃×C₄	C₃×D₄	1	3	yes	—
77	C₂²×C₃×C₄	C₃×Q₈	1	4	yes	—
78	C₂²×C₃×C₄	C₂²×D₃	1	3	yes	—
79	C₂²×C₃×C₄	C₂³×C₃	1	3	yes	—
80	C₂²×Dic₃	—	2	12	no	—
81	C₂²×Dic₃	—	4	8	no	—
82	C₂×C₃×C₈	—	2	24	no	—
83	C₂×C₃×C₈	—	2	4	no	—
84	C₂×C₃×C₈	C₂×Dic₃	1	24	yes	complemented
85	C₂×C₃×C₈	C₂×C₃×C₄	1	24	yes	complemented
86	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
87	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
88	C₃×C₄⋊C₄	—	3	8	no	—
89	C₃×C₄⋊C₄	—	6	12	no	—
90	C₄⋊Dic₃	—	2	12	no	—
91	C₄⋊Dic₃	C₂³×C₃	1	8	yes	—
92	C₂²×C₃²	#106	1	16	yes	complemented
93	C₂²×C₃²	C₂×C₄⋊C₄	1	6	yes	—
94	C₂²×C₃²	C₂×SD₁₆	1	12	yes	—
95	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
96	C₃²×C₄	C₂×C₄⋊C₄	1	6	yes	—
97	C₃²×C₄	C₂³×C₄	1	1	yes	characteristic
98	C₃²×C₄	C₂²×Q₈	1	1	yes	characteristic
99	C₃×Dic₃	—	4	8	no	—
100	C₂³×C₄	—	6	2	no	—
101	C₂³×C₄	C₂×C₃×D₃	1	1	yes	characteristic
102	C₂²×C₈	C₃×Dic₃	1	8	yes	complemented
103	C₂²×C₈	C₂×C₃×D₃	1	6	yes	—
104	C₂×C₄⋊C₄	—	3	12	no	—
105	C₂×C₄⋊C₄	—	6	6	no	—
106	#106	—	3	16	no	—
107	C₂³×C₃	—	2	8	no	—
108	C₂³×C₃	—	2	4	no	—
109	C₂³×C₃	—	2	3	no	—
110	C₂³×C₃	#204	1	8	yes	—
111	C₂³×C₃	C₄⋊Dic₃	1	4	yes	—
112	C₂³×C₃	C₃×C₄⋊C₄	1	4	yes	—
113	C₂³×C₃	C₃×SD₁₆	1	8	yes	—
114	C₂³×C₃	C₂×D₁₂	1	3	yes	—
115	C₂³×C₃	C₂×C₃×D₄	1	3	yes	—
116	C₂×C₃×C₄	—	2	1	no	—
117	C₂×C₃×C₄	—	2	12	no	—
118	C₂×C₃×C₄	—	2	6	no	—
119	C₂×C₃×C₄	—	2	3	no	—
120	C₂×C₃×C₄	—	2	6	no	—
121	C₂×C₃×C₄	—	6	8	no	—
122	C₂×C₃×C₄	—	12	24	no	—
123	C₂×C₃×C₄	C₄⋊Dic₃	1	12	yes	—
124	C₂×C₃×C₄	C₃×C₄⋊C₄	1	12	yes	—
125	C₂×C₃×C₄	C₂×Dic₆	1	6	yes	—
126	C₂×C₃×C₄	C₂×D₁₂	1	3	yes	—
127	C₂×C₃×C₄	C₂²×Dic₃	1	6	yes	—
128	C₂×C₃×C₄	C₂²×C₃×C₄	1	6	yes	—
129	C₂×C₃×C₄	C₂×C₃×D₄	1	3	yes	—
130	C₂×C₃×C₄	C₂×C₃×Q₈	1	6	yes	—
131	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
133	C₂×Dic₃	—	2	8	no	—
134	C₂×Dic₃	—	4	24	no	—
135	C₃×C₈	—	2	8	no	—
136	C₃×C₈	C₂²×Dic₃	1	8	yes	complemented
137	C₃×C₈	C₂²×C₃×C₄	1	8	yes	complemented
138	C₂×C₃²	C₂×#106	1	12	yes	complemented
139	C₂×C₃²	C₂²×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₃²	C₂²×SD₁₆	1	2	yes	—
141	C₂⁴	C₃×D₁₂	1	1	yes	characteristic
142	C₂²×C₄	—	6	12	no	—
143	C₂²×C₄	—	12	8	no	—
144	C₂²×C₄	C₃×Dic₆	1	4	yes	—
145	C₂²×C₄	C₃×D₁₂	1	3	yes	—
146	C₂²×C₄	C₂×C₃×Dic₃	1	4	yes	—
147	C₂²×C₄	C₂²×C₃×D₃	1	3	yes	—
148	C₂×C₈	C₂×C₃×Dic₃	1	24	yes	complemented
149	C₂×C₈	C₂²×C₃×D₃	1	4	yes	—
150	C₄⋊C₄	—	3	8	no	—
151	C₄⋊C₄	—	6	12	no	—
152	C₂²×C₃	—	2	12	no	—
153	C₂²×C₃	—	2	6	no	—
154	C₂²×C₃	—	2	1	no	—
155	C₂²×C₃	—	2	16	no	—
156	C₂²×C₃	#817	1	16	yes	direct product factor
157	C₂²×C₃	C₃×#106	1	16	yes	complemented
158	C₂²×C₃	C₂×#204	1	12	yes	—
159	C₂²×C₃	C₂×C₄⋊Dic₃	1	6	yes	—
160	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	6	yes	—
161	C₂²×C₃	C₂×C₃×SD₁₆	1	12	yes	—
162	C₂²×C₃	C₂²×D₁₂	1	1	yes	characteristic
163	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
164	C₃×C₄	—	2	6	no	—
165	C₃×C₄	—	2	1	no	—
166	C₃×C₄	—	2	1	no	—
167	C₃×C₄	—	12	8	no	—
168	C₃×C₄	C₂×C₄⋊Dic₃	1	6	yes	—
169	C₃×C₄	C₂×C₃×C₄⋊C₄	1	6	yes	—
170	C₃×C₄	C₂²×Dic₆	1	1	yes	characteristic
171	C₃×C₄	C₂³×Dic₃	1	1	yes	characteristic
172	C₃×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
173	C₃×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
174	Dic₃	—	4	8	no	—
175	C₃²	C₂²×#106	1	1	yes	characteristic, complemented, Sylow
176	C₂³	C₃×#204	1	8	yes	—
177	C₂³	C₃×C₄⋊Dic₃	1	4	yes	—
178	C₂³	C₂×C₃×D₁₂	1	3	yes	—
179	C₂×C₄	—	6	8	no	—
180	C₂×C₄	—	12	24	no	—
181	C₂×C₄	C₃×C₄⋊Dic₃	1	12	yes	—
182	C₂×C₄	C₂×C₃×Dic₆	1	6	yes	—
183	C₂×C₄	C₂×C₃×D₁₂	1	3	yes	—
184	C₂×C₄	C₂²×C₃×Dic₃	1	6	yes	—
185	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
186	C₈	C₂²×C₃×Dic₃	1	8	yes	complemented
187	C₂×C₃	—	2	1	no	—
188	C₂×C₃	—	2	12	no	—
189	C₂×C₃	—	2	2	no	—
190	C₂×C₃	C₂×#817	1	12	yes	direct product factor
191	C₂×C₃	C₂×C₃×#106	1	12	yes	complemented
192	C₂×C₃	C₂²×#204	1	2	yes	—
193	C₂×C₃	C₂²×C₄⋊Dic₃	1	1	yes	characteristic
194	C₂×C₃	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
195	C₂×C₃	C₂²×C₃×SD₁₆	1	2	yes	—
196	C₂²	C₃×#817	1	16	yes	direct product factor
197	C₂²	C₂×C₃×#204	1	12	yes	—
198	C₂²	C₂×C₃×C₄⋊Dic₃	1	6	yes	—
199	C₂²	C₂²×C₃×D₁₂	1	1	yes	characteristic
200	C₄	—	12	8	no	—
201	C₄	C₂×C₃×C₄⋊Dic₃	1	6	yes	—
202	C₄	C₂²×C₃×Dic₆	1	1	yes	characteristic
203	C₄	C₂³×C₃×Dic₃	1	1	yes	characteristic
204	C₃	—	2	1	no	—
205	C₃	C₂²×#817	1	1	yes	characteristic, direct product factor
206	C₃	C₂²×C₃×#106	1	1	yes	characteristic, complemented
207	C₂	C₂×C₃×#817	1	12	yes	direct product factor
208	C₂	C₂²×C₃×#204	1	2	yes	—
209	C₂	C₂²×C₃×C₄⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₃	D₁₂
commutator	C₃×C₄	C₂³×C₃×C₄
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₂³×C₃²×C₈	C₂
socle, prime radical	C₂⁴×C₃²	D₄
cosocle	C₂×C₃×C₄	C₂⁴×C₃
hypercenter	C₂³×C₃×C₈	D₃
hypocenter	C₃	C₂²×C₃×#106
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴×C₃ ◁ C₂³×C₃×C₄ ◁ C₂³×C₃×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×#817	C₂	direct (N × H)
5. C₂²×#817	C₃	direct (N × H)
8. C₂²×C₃²×C₈	C₄	semidirect
12. C₃×#817	C₂²	direct (N × H)
19. C₂×#817	C₂×C₃	direct (N × H)
27. C₂×C₃²×C₈	C₂×C₄	semidirect
39. C₂²×C₃×C₈	Dic₃	semidirect
40. C₂²×C₃×C₈	C₃×C₄	semidirect
48. #817	C₂²×C₃	direct (N × H)
59. C₃²×C₈	C₂²×C₄	semidirect
84. C₂×C₃×C₈	C₂×Dic₃	semidirect
85. C₂×C₃×C₈	C₂×C₃×C₄	semidirect
92. C₂²×C₃²	#106	semidirect
102. C₂²×C₈	C₃×Dic₃	semidirect
136. C₃×C₈	C₂²×Dic₃	semidirect
137. C₃×C₈	C₂²×C₃×C₄	semidirect
138. C₂×C₃²	C₂×#106	semidirect
148. C₂×C₈	C₂×C₃×Dic₃	semidirect
156. C₂²×C₃	#817	direct (N × H)
157. C₂²×C₃	C₃×#106	semidirect
175. C₃²	C₂²×#106	semidirect
186. C₈	C₂²×C₃×Dic₃	semidirect
190. C₂×C₃	C₂×#817	direct (N × H)
191. C₂×C₃	C₂×C₃×#106	semidirect
196. C₂²	C₃×#817	direct (N × H)
205. C₃	C₂²×#817	direct (N × H)
206. C₃	C₂²×C₃×#106	semidirect
207. C₂	C₂×C₃×#817	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×C₃²×C₈	2
C₂²×C₃×C₄⋊Dic₃	2
C₂×C₃×#817	2
C₂²×C₃×#106	3
C₂²×#817	3

Subgroups — C22×C3×#817 #11858378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#817 #11858378