Subgroups — #11776835

Order 1152.

540 conjugacy classes of subgroups.

The subgroup lattice has 201 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#10592624	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
4	#10589324	C₂	1	2	yes	maximal, complemented
5	#10589323	C₂	1	2	yes	maximal, complemented
6	#69850	—	3	4	no	maximal
7	C₂×C₃²×M₁₆	C₄	1	1	yes	characteristic
8	C₂×C₃²×M₁₆	C₄	1	1	yes	characteristic
9	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
10	C₂²×C₃⋊(C₃⋊C₈)	—	2	1	no	—
11	C₃²×#97	C₄	1	2	yes	—
12	C₃²×C₄×C₈	C₂²	1	2	yes	—
13	#63657	—	2	1	no	—
14	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
15	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	—
16	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
17	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
18	C₄×C₃⋊(C₃⋊C₈)	—	2	1	no	—
19	C₃×C₄×M₁₆	D₃	1	4	yes	—
20	#4925	—	3	4	no	—
21	C₂×C₄⋊(C₃⋊C₈)	—	3	4	no	—
22	#4458	—	3	8	no	—
23	#4457	—	3	8	no	—
24	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
26	C₂²×C₃²×C₄	Q₈	1	1	yes	characteristic
27	C₃²×M₁₆	—	2	2	no	—
28	C₃²×M₁₆	C₈	1	4	yes	complemented
29	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
30	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
31	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
32	C₃²×C₄²	D₄	1	1	yes	characteristic
33	C₃²×C₄²	Q₈	1	1	yes	characteristic
34	C₃²×C₄²	C₂³	1	1	yes	characteristic
35	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
36	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
37	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
38	C₂×C₃⋊(C₃⋊C₈)	—	4	2	no	—
39	#1565	—	9	1	no	Sylow
40	C₂×C₃×M₁₆	Dic₃	1	4	yes	—
41	C₂×C₃×M₁₆	Dic₃	1	4	yes	—
42	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
43	C₂²×C₃⋊C₈	—	6	4	no	—
44	C₃×#97	Dic₃	1	8	yes	—
45	C₃×C₄×C₈	C₂×D₃	1	8	yes	—
46	#830	—	6	4	no	—
47	C₄⋊(C₃⋊C₈)	—	3	8	no	—
48	C₄⋊(C₃⋊C₈)	—	3	4	no	—
49	C₄⋊(C₃⋊C₈)	—	3	4	no	—
50	C₄⋊(C₃⋊C₈)	—	6	4	no	—
51	C₄×C₃⋊C₈	—	6	4	no	—
52	C₂³×C₃²	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₃²×C₄	—	2	1	no	—
54	C₂×C₃²×C₄	—	2	1	no	—
55	C₂×C₃²×C₄	—	4	1	no	—
56	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
60	C₂×C₃²×C₄	M₁₆	1	2	yes	—
61	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
64	C₃²×C₈	—	2	2	no	—
65	C₃²×C₈	C₂×C₈	1	4	yes	complemented
66	C₃⋊(C₃⋊C₈)	—	4	1	no	—
67	C₃⋊(C₃⋊C₈)	—	4	2	no	—
68	C₃⋊(C₃⋊C₈)	—	4	1	no	—
69	#431	—	9	1	no	—
70	C₂×C₄⋊C₈	—	9	1	no	—
71	C₄×M₁₆	C₃⋊D₃	1	1	yes	characteristic
72	#335	—	9	2	no	—
73	#334	—	9	2	no	—
74	C₂²×C₃×C₄	Dic₆	1	4	yes	—
75	C₂²×C₃×C₄	D₁₂	1	4	yes	—
76	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
77	C₃×M₁₆	—	2	8	no	—
78	C₃×M₁₆	C₃⋊C₈	1	16	yes	complemented
79	C₂×C₃×C₈	C₂×Dic₃	1	8	yes	—
80	C₂×C₃×C₈	C₂×Dic₃	1	8	yes	—
81	C₃×C₄²	Dic₆	1	4	yes	—
82	C₃×C₄²	D₁₂	1	4	yes	—
83	C₃×C₄²	C₂×Dic₃	1	4	yes	—
84	C₃×C₄²	C₂²×D₃	1	4	yes	—
85	C₂×C₃⋊C₈	—	6	4	no	—
86	C₂×C₃⋊C₈	—	6	4	no	—
87	C₂×C₃⋊C₈	—	6	8	no	—
88	C₂×C₃⋊C₈	—	12	8	no	—
89	C₂²×C₃²	—	2	1	no	—
90	C₂²×C₃²	—	2	1	no	—
91	C₂²×C₃²	C₄⋊C₈	1	2	yes	—
92	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₃²×C₄	—	2	1	no	—
94	C₃²×C₄	—	2	1	no	—
95	C₃²×C₄	—	2	1	no	—
96	C₃²×C₄	C₄⋊C₈	1	2	yes	—
97	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
98	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
99	C₃²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
100	C₃²×C₄	#137	1	1	yes	characteristic
101	C₂×M₁₆	C₃⋊Dic₃	1	1	yes	characteristic
102	C₂×M₁₆	C₃⋊Dic₃	1	1	yes	characteristic
103	C₂²×C₈	—	18	1	no	—
104	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
105	C₄⋊C₈	—	9	1	no	—
106	C₄⋊C₈	—	9	2	no	—
107	C₄⋊C₈	—	9	1	no	—
108	C₄⋊C₈	—	18	1	no	—
109	C₂²⋊C₈	—	18	1	no	—
110	#97	C₃⋊Dic₃	1	2	yes	—
111	C₄×C₈	—	18	1	no	—
112	C₄×C₈	C₂×C₃⋊D₃	1	2	yes	—
113	C₂³×C₃	C₄⋊Dic₃	1	4	yes	—
114	C₂×C₃×C₄	—	2	4	no	—
115	C₂×C₃×C₄	—	2	4	no	—
116	C₂×C₃×C₄	—	4	4	no	—
117	C₂×C₃×C₄	C₂×C₃⋊C₈	1	8	yes	—
118	C₂×C₃×C₄	#208	1	8	yes	—
119	C₂×C₃×C₄	C₄⋊Dic₃	1	4	yes	—
120	C₂×C₃×C₄	C₄⋊Dic₃	1	4	yes	—
121	C₂×C₃×C₄	C₄⋊Dic₃	1	4	yes	—
122	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
123	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
124	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
125	C₃×C₈	—	2	8	no	—
126	C₃×C₈	C₂×C₃⋊C₈	1	16	yes	complemented
127	C₃⋊C₈	—	12	4	no	—
128	C₃⋊C₈	—	12	8	no	—
129	C₃⋊C₈	—	12	4	no	—
130	C₂×C₃²	—	2	2	no	—
131	C₂×C₃²	C₂×C₄⋊C₈	1	1	yes	characteristic
132	C₂×C₃²	#428	1	1	yes	characteristic
133	C₂×C₃²	#430	1	1	yes	characteristic
134	C₂²×C₄	#634	1	1	yes	characteristic
135	C₂²×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
137	M₁₆	—	2	2	no	—
138	M₁₆	C₃⋊(C₃⋊C₈)	1	4	yes	complemented
139	C₂×C₈	—	18	1	no	—
140	C₂×C₈	—	18	1	no	—
141	C₂×C₈	—	18	2	no	—
142	C₂×C₈	—	36	2	no	—
143	C₂×C₈	C₂×C₃⋊Dic₃	1	2	yes	—
144	C₂×C₈	C₂×C₃⋊Dic₃	1	2	yes	—
145	C₄²	#634	1	1	yes	characteristic
146	C₄²	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
147	C₄²	C₂×C₃⋊Dic₃	1	1	yes	characteristic
148	C₄²	C₂²×C₃⋊D₃	1	1	yes	characteristic
149	C₂²×C₃	—	2	4	no	—
150	C₂²×C₃	—	2	4	no	—
151	C₂²×C₃	C₄⋊(C₃⋊C₈)	1	8	yes	—
152	C₂²×C₃	C₂×C₄⋊Dic₃	1	4	yes	—
153	C₃×C₄	—	2	4	no	—
154	C₃×C₄	—	2	4	no	—
155	C₃×C₄	—	2	4	no	—
156	C₃×C₄	C₄⋊(C₃⋊C₈)	1	8	yes	—
157	C₃×C₄	#908	1	4	yes	—
158	C₃×C₄	#909	1	4	yes	—
159	C₃×C₄	C₂²×C₃⋊C₈	1	4	yes	—
160	C₃×C₄	C₂×#208	1	4	yes	—
161	C₃²	#1565	1	1	yes	characteristic, complemented, Sylow
162	C₂³	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	2	yes	—
167	C₂×C₄	#3748	1	2	yes	—
168	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
169	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
170	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
171	C₂×C₄	C₂×#634	1	1	yes	characteristic
172	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
173	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
174	C₈	—	2	2	no	—
175	C₈	—	36	2	no	—
176	C₈	—	36	1	no	—
177	C₈	—	36	1	no	—
178	C₈	C₂×C₃⋊(C₃⋊C₈)	1	4	yes	complemented
179	C₂×C₃	—	2	8	no	—
180	C₂×C₃	C₂×C₄⋊(C₃⋊C₈)	1	4	yes	—
181	C₂×C₃	#5121	1	4	yes	—
182	C₂×C₃	#5125	1	4	yes	—
183	C₂²	—	2	1	no	—
184	C₂²	—	2	1	no	—
185	C₂²	C₄⋊(C₃⋊(C₃⋊C₈))	1	2	yes	—
186	C₂²	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	C₄⋊(C₃⋊(C₃⋊C₈))	1	2	yes	—
191	C₄	#64117	1	1	yes	characteristic
192	C₄	#64118	1	1	yes	characteristic
193	C₄	C₂²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
194	C₄	C₂×#3748	1	1	yes	characteristic
195	C₃	#69850	1	4	yes	complemented
196	C₂	—	2	2	no	—
197	C₂	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	1	1	yes	characteristic
198	C₂	#10592820	1	1	yes	characteristic
199	C₂	#10592824	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)
commutator	C₃²×C₄	C₂²×C₈
Frattini	C₄²	C₂²×C₃⋊D₃
Fitting	C₃²×C₄×M₁₆	C₂
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂³×C₃²	C₄⋊C₄
cosocle	C₃²×C₄²	C₂³
hypercenter	C₄×M₁₆	C₃⋊D₃
hypocenter	C₃²	#1565
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. #10589324	C₂	semidirect
5. #10589323	C₂	semidirect
28. C₃²×M₁₆	C₈	semidirect
65. C₃²×C₈	C₂×C₈	semidirect
78. C₃×M₁₆	C₃⋊C₈	semidirect
126. C₃×C₈	C₂×C₃⋊C₈	semidirect
138. M₁₆	C₃⋊(C₃⋊C₈)	semidirect
161. C₃²	#1565	semidirect
178. C₈	C₂×C₃⋊(C₃⋊C₈)	semidirect
195. C₃	#69850	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×M₁₆	2
#10592624	2
C₂×C₄⋊(C₃⋊(C₃⋊C₈))	2
#10589324	2
#10589323	2
#69850	3