Subgroups — #11784254

Order 1152.

888 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	C₂×C₃²×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₃⋊Dic₃	C₂	1	2	yes	maximal
3	C₄×C₄⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	#10589416	C₂	1	2	yes	maximal
6	#70465	—	3	4	no	maximal
7	C₂²×C₃²×C₈	C₂²	1	2	yes	—
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₃⋊Dic₃)	C₄	1	2	yes	—
10	C₂×C₄×C₃⋊Dic₃	C₂²	1	2	yes	—
11	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₃²×C₄⋊C₈	C₄	1	4	yes	complemented
14	#63658	C₄	1	2	yes	—
15	C₈×C₃⋊Dic₃	—	2	4	no	—
16	C₄×C₃⋊(C₃⋊C₈)	C₄	1	4	yes	complemented
17	C₂×C₃×C₄⋊C₈	D₃	1	4	yes	—
18	C₂×C₈×Dic₃	—	3	8	no	—
19	C₄×C₄⋊Dic₃	—	3	4	no	—
20	C₂×C₄×C₃⋊C₈	—	3	4	no	—
21	#4550	—	3	8	no	—
22	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
25	C₂²×C₃⋊Dic₃	C₂×C₄	1	4	yes	—
26	C₂×C₃²×C₈	—	2	2	no	—
27	C₂×C₃²×C₈	—	2	2	no	—
28	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
29	C₃²×C₄²	C₂×C₄	1	2	yes	—
30	C₃²×C₄²	C₂×C₄	1	2	yes	—
31	C₄⋊(C₃⋊Dic₃)	C₈	1	8	yes	complemented
32	C₄×C₃⋊Dic₃	—	2	2	no	—
33	C₄×C₃⋊Dic₃	—	2	2	no	—
34	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
35	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
36	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
37	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
38	C₂×C₃⋊(C₃⋊C₈)	D₄	1	2	yes	—
39	C₂×C₃⋊(C₃⋊C₈)	Q₈	1	2	yes	—
40	C₈×C₄⋊C₄	—	9	1	no	Sylow
41	C₂²×C₃×C₈	C₂×D₃	1	8	yes	—
42	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
43	C₂×C₄⋊Dic₃	—	3	8	no	—
44	C₂×C₄×Dic₃	—	3	8	no	—
45	C₂²×C₃⋊C₈	—	3	4	no	—
46	C₂²×C₃⋊C₈	—	3	4	no	—
47	C₃×C₄⋊C₈	Dic₃	1	16	yes	complemented
48	C₃⋊#95	—	3	8	no	—
49	C₈×Dic₃	—	6	16	no	—
50	C₄×C₃⋊C₈	—	3	16	no	—
51	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₃²×C₄	—	2	2	no	—
53	C₂×C₃²×C₄	C₄²	1	2	yes	—
54	C₂×C₃²×C₄	C₄²	1	2	yes	—
55	C₂×C₃²×C₄	C₂×C₈	1	4	yes	—
56	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₃²×C₄	#41	1	1	yes	characteristic
61	C₂×C₃²×C₄	#41	1	1	yes	characteristic
62	C₂×C₃⋊Dic₃	—	2	4	no	—
63	C₂×C₃⋊Dic₃	—	2	4	no	—
64	C₂×C₃⋊Dic₃	C₂×C₈	1	8	yes	—
65	C₃²×C₈	—	2	4	no	—
66	C₃⋊(C₃⋊C₈)	—	2	2	no	—
67	C₃⋊(C₃⋊C₈)	C₄⋊C₄	1	4	yes	complemented
68	C₂×C₄⋊C₈	C₃⋊D₃	1	1	yes	characteristic
69	C₂×C₄×C₈	—	9	1	no	—
70	C₂×C₄×C₈	—	9	2	no	—
71	C₄×C₄⋊C₄	—	9	1	no	—
72	#336	—	9	2	no	—
73	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
74	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
75	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
76	C₂²×Dic₃	—	3	16	no	—
77	C₂×C₃×C₈	—	2	8	no	—
78	C₂×C₃×C₈	—	2	8	no	—
79	C₂×C₃×C₈	C₂×Dic₃	1	16	yes	—
80	C₃×C₄²	C₄×D₃	1	8	yes	—
81	C₃×C₄²	C₂×Dic₃	1	8	yes	—
82	C₄⋊Dic₃	—	3	32	no	—
83	C₄×Dic₃	—	6	8	no	—
84	C₄×Dic₃	—	6	8	no	—
85	C₂×C₃⋊C₈	—	3	8	no	—
86	C₂×C₃⋊C₈	—	3	8	no	—
87	C₂×C₃⋊C₈	—	3	8	no	—
88	C₂×C₃⋊C₈	—	3	8	no	—
89	C₂×C₃⋊C₈	—	6	4	no	—
90	C₂×C₃⋊C₈	—	6	4	no	—
91	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
92	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
93	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
94	C₂²×C₃²	C₂²×C₈	1	2	yes	—
95	C₂²×C₃²	#131	1	2	yes	—
96	C₃²×C₄	—	2	2	no	—
97	C₃²×C₄	C₄×C₈	1	4	yes	complemented
98	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₃²×C₄	#117	1	1	yes	characteristic
100	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
101	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
102	C₃⋊Dic₃	—	2	8	no	—
103	C₂²×C₈	—	9	1	no	—
104	C₂²×C₈	—	9	1	no	—
105	C₂²×C₈	C₂×C₃⋊D₃	1	2	yes	—
106	C₂×C₄⋊C₄	—	9	2	no	—
107	C₂×C₄²	—	9	2	no	—
108	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
109	C₄⋊C₈	C₃⋊Dic₃	1	4	yes	complemented
110	C₄×C₈	—	9	4	no	—
111	C₄×C₈	—	18	4	no	—
112	#95	—	9	2	no	—
113	C₂³×C₃	C₂×C₄×D₃	1	4	yes	—
114	C₂×C₃×C₄	—	2	8	no	—
115	C₂×C₃×C₄	D₃×C₈	1	16	yes	—
116	C₂×C₃×C₄	C₄×Dic₃	1	8	yes	—
117	C₂×C₃×C₄	C₄×Dic₃	1	8	yes	—
118	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
119	C₂×C₃×C₄	D₃×D₄	1	4	yes	—
120	C₂×C₃×C₄	#237	1	4	yes	—
121	C₂×C₃×C₄	D₃×Q₈	1	4	yes	—
122	C₂×C₃×C₄	#239	1	4	yes	—
123	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
124	C₂×Dic₃	—	3	32	no	—
125	C₂×Dic₃	—	6	16	no	—
126	C₂×Dic₃	—	6	16	no	—
127	C₃×C₈	—	2	16	no	—
128	C₃⋊C₈	—	3	16	no	—
129	C₃⋊C₈	—	6	8	no	—
130	C₂×C₃²	C₄×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₃²	C₂×C₄×C₈	1	1	yes	characteristic
132	C₂×C₃²	#405	1	1	yes	characteristic
133	C₂×C₃²	C₈×D₄	1	2	yes	—
134	C₂×C₃²	C₈×Q₈	1	2	yes	—
135	C₂²×C₄	—	9	4	no	—
136	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
137	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
138	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	9	2	no	—
142	C₂×C₈	—	9	2	no	—
143	C₂×C₈	—	9	2	no	—
144	C₂×C₈	—	9	2	no	—
145	C₂×C₈	—	18	1	no	—
146	C₂×C₈	—	18	1	no	—
147	C₂×C₈	C₂×C₃⋊Dic₃	1	4	yes	—
148	C₄⋊C₄	—	9	8	no	—
149	C₄²	—	18	2	no	—
150	C₄²	—	18	2	no	—
151	C₄²	C₄×C₃⋊D₃	1	2	yes	—
152	C₄²	C₂×C₃⋊Dic₃	1	2	yes	—
153	C₂²×C₃	#887	1	4	yes	—
154	C₂²×C₃	#893	1	4	yes	—
155	C₂²×C₃	C₂×D₃×C₈	1	8	yes	—
156	C₂²×C₃	#907	1	8	yes	—
157	C₂²×C₃	C₂×C₄×Dic₃	1	4	yes	—
158	C₃×C₄	—	2	8	no	—
159	C₃×C₄	C₈×Dic₃	1	16	yes	complemented
160	C₃×C₄	D₃×C₄⋊C₄	1	4	yes	—
161	C₃×C₄	#892	1	4	yes	—
162	C₃×C₄	D₄×Dic₃	1	4	yes	—
163	C₃×C₄	Q₈×Dic₃	1	4	yes	—
164	Dic₃	—	6	32	no	—
165	C₃²	C₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
166	C₂³	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	9	8	no	—
169	C₂×C₄	—	18	4	no	—
170	C₂×C₄	—	18	4	no	—
171	C₂×C₄	C₈×C₃⋊D₃	1	4	yes	—
172	C₂×C₄	C₄×C₃⋊Dic₃	1	2	yes	—
173	C₂×C₄	C₄×C₃⋊Dic₃	1	2	yes	—
174	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
175	C₂×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
176	C₂×C₄	#3830	1	1	yes	characteristic
177	C₂×C₄	Q₈×C₃⋊D₃	1	1	yes	characteristic
178	C₂×C₄	#3832	1	1	yes	characteristic
179	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
180	C₈	—	2	4	no	—
181	C₈	—	9	4	no	—
182	C₈	—	18	2	no	—
183	C₂×C₃	#4830	1	8	yes	—
184	C₂×C₃	#4834	1	8	yes	—
185	C₂×C₃	Dic₃×C₄⋊C₄	1	4	yes	—
186	C₂×C₃	C₂×C₈×Dic₃	1	4	yes	—
187	C₂×C₃	#5122	1	4	yes	—
188	C₂²	#64096	1	1	yes	characteristic
189	C₂²	#64102	1	1	yes	characteristic
190	C₂²	C₂×C₈×C₃⋊D₃	1	2	yes	—
191	C₂²	#64116	1	2	yes	—
192	C₂²	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
193	C₄	—	2	2	no	—
194	C₄	—	18	8	no	—
195	C₄	C₈×C₃⋊Dic₃	1	4	yes	complemented
196	C₄	C₄⋊C₄×C₃⋊D₃	1	1	yes	characteristic
197	C₄	#64101	1	1	yes	characteristic
198	C₄	D₄×C₃⋊Dic₃	1	1	yes	characteristic
199	C₄	Q₈×C₃⋊Dic₃	1	1	yes	characteristic
200	C₃	#70465	1	4	yes	complemented
201	C₂	#10592529	1	2	yes	—
202	C₂	#10592533	1	2	yes	—
203	C₂	C₄⋊C₄×C₃⋊Dic₃	1	1	yes	characteristic
204	C₂	C₂×C₈×C₃⋊Dic₃	1	1	yes	characteristic
205	C₂	#10592821	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
13. C₃²×C₄⋊C₈	C₄	semidirect
16. C₄×C₃⋊(C₃⋊C₈)	C₄	semidirect
31. C₄⋊(C₃⋊Dic₃)	C₈	semidirect
47. C₃×C₄⋊C₈	Dic₃	semidirect
67. C₃⋊(C₃⋊C₈)	C₄⋊C₄	semidirect
97. C₃²×C₄	C₄×C₈	semidirect
109. C₄⋊C₈	C₃⋊Dic₃	semidirect
159. C₃×C₄	C₈×Dic₃	semidirect
165. C₃²	C₈×C₄⋊C₄	semidirect
195. C₄	C₈×C₃⋊Dic₃	semidirect
200. C₃	#70465	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃²×C₄⋊C₈	2
C₂×C₈×C₃⋊Dic₃	2
C₄×C₄⋊(C₃⋊Dic₃)	2
C₂×C₄×C₃⋊(C₃⋊C₈)	2
#10589416	2
#70465	3