Subgroups — #11776945

Order 1152.

498 conjugacy classes of subgroups.

The subgroup lattice has 192 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	#10592625	C₂	1	1	yes	maximal, characteristic
3	#10592623	C₂	1	1	yes	maximal, characteristic
4	#10589324	C₂	1	2	yes	maximal, complemented
5	#10589323	C₂	1	2	yes	maximal, complemented
6	#69851	—	3	4	no	maximal
7	C₂×C₃²×M₁₆	C₄	1	2	yes	—
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₂×#3748	—	2	1	no	—
10	C₃²×#97	C₄	1	1	yes	characteristic
11	C₃²×#97	C₄	1	1	yes	characteristic
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	#63630	—	2	1	no	—
19	C₃×C₄×M₁₆	D₃	1	4	yes	—
20	#4926	—	3	4	no	—
21	#4924	—	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	4	no	—
28	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
29	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
30	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
31	C₃²×C₄²	D₄	1	1	yes	characteristic
32	C₃²×C₄²	Q₈	1	1	yes	characteristic
33	C₃²×C₄²	C₂³	1	1	yes	characteristic
34	#3748	—	4	2	no	—
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	#1569	—	9	1	no	Sylow
39	C₂×C₃×M₁₆	Dic₃	1	8	yes	—
40	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
41	C₂×#208	—	6	4	no	—
42	C₃×#97	Dic₃	1	4	yes	—
43	C₃×#97	Dic₃	1	4	yes	—
44	C₃×C₄×C₈	C₂×D₃	1	8	yes	—
45	#830	—	6	4	no	—
46	C₄⋊(C₃⋊C₈)	—	3	8	no	—
47	C₄⋊(C₃⋊C₈)	—	3	4	no	—
48	C₄⋊(C₃⋊C₈)	—	3	4	no	—
49	C₄⋊(C₃⋊C₈)	—	6	4	no	—
50	#803	—	6	4	no	—
51	C₂³×C₃²	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₃²×C₄	—	2	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
61	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
62	C₃²×C₈	—	2	2	no	—
63	C₃²×C₈	M₁₆	1	4	yes	complemented
64	C₃⋊(C₃⋊C₈)	—	4	1	no	—
65	C₃⋊(C₃⋊C₈)	—	4	2	no	—
66	C₃⋊(C₃⋊C₈)	—	4	1	no	—
67	#432	—	9	1	no	—
68	#423	—	9	1	no	—
69	C₄×M₁₆	C₃⋊D₃	1	1	yes	characteristic
70	#335	—	9	2	no	—
71	#334	—	9	2	no	—
72	C₂²×C₃×C₄	Dic₆	1	4	yes	—
73	C₂²×C₃×C₄	D₁₂	1	4	yes	—
74	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
75	C₃×M₁₆	—	2	16	no	—
76	C₂×C₃×C₈	C₂×Dic₃	1	8	yes	—
77	C₂×C₃×C₈	C₂×Dic₃	1	8	yes	—
78	C₃×C₄²	Dic₆	1	4	yes	—
79	C₃×C₄²	D₁₂	1	4	yes	—
80	C₃×C₄²	C₂×Dic₃	1	4	yes	—
81	C₃×C₄²	C₂²×D₃	1	4	yes	—
82	#208	—	12	8	no	—
83	C₂×C₃⋊C₈	—	6	4	no	—
84	C₂×C₃⋊C₈	—	6	4	no	—
85	C₂×C₃⋊C₈	—	6	8	no	—
86	C₂²×C₃²	—	2	1	no	—
87	C₂²×C₃²	—	2	1	no	—
88	C₂²×C₃²	—	2	1	no	—
89	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₃²×C₄	—	2	1	no	—
91	C₃²×C₄	—	2	1	no	—
92	C₃²×C₄	—	2	1	no	—
93	C₃²×C₄	—	2	1	no	—
94	C₃²×C₄	C₂×M₁₆	1	2	yes	—
95	C₃²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
96	C₃²×C₄	#137	1	1	yes	characteristic
97	C₂×M₁₆	—	18	1	no	—
98	C₂×M₁₆	C₃⋊Dic₃	1	2	yes	—
99	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
100	C₄⋊C₈	—	9	1	no	—
101	C₄⋊C₈	—	9	2	no	—
102	C₄⋊C₈	—	9	1	no	—
103	C₄⋊C₈	—	18	1	no	—
104	C₂²⋊C₈	—	18	1	no	—
105	#97	—	18	1	no	—
106	#97	C₃⋊Dic₃	1	1	yes	characteristic
107	#97	C₃⋊Dic₃	1	1	yes	characteristic
108	C₄×C₈	C₂×C₃⋊D₃	1	2	yes	—
109	C₂³×C₃	C₄⋊Dic₃	1	4	yes	—
110	C₂×C₃×C₄	—	2	4	no	—
111	C₂×C₃×C₄	—	2	4	no	—
112	C₂×C₃×C₄	—	2	8	no	—
113	C₂×C₃×C₄	—	4	4	no	—
114	C₂×C₃×C₄	C₄⋊Dic₃	1	4	yes	—
115	C₂×C₃×C₄	C₄⋊Dic₃	1	4	yes	—
116	C₂×C₃×C₄	C₄⋊Dic₃	1	4	yes	—
117	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
118	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
119	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
120	C₃×C₈	—	2	8	no	—
121	C₃×C₈	#208	1	16	yes	complemented
122	C₃⋊C₈	—	12	4	no	—
123	C₃⋊C₈	—	12	8	no	—
124	C₃⋊C₈	—	12	4	no	—
125	C₂×C₃²	—	4	1	no	—
126	C₂×C₃²	#423	1	1	yes	characteristic
127	C₂×C₃²	#428	1	1	yes	characteristic
128	C₂×C₃²	#430	1	1	yes	characteristic
129	C₂²×C₄	#634	1	1	yes	characteristic
130	C₂²×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
132	M₁₆	—	2	4	no	—
133	M₁₆	—	36	2	no	—
134	C₂×C₈	—	18	1	no	—
135	C₂×C₈	—	18	1	no	—
136	C₂×C₈	—	18	2	no	—
137	C₂×C₈	C₂×C₃⋊Dic₃	1	2	yes	—
138	C₂×C₈	C₂×C₃⋊Dic₃	1	2	yes	—
139	C₄²	#634	1	1	yes	characteristic
140	C₄²	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
141	C₄²	C₂×C₃⋊Dic₃	1	1	yes	characteristic
142	C₄²	C₂²×C₃⋊D₃	1	1	yes	characteristic
143	C₂²×C₃	—	2	4	no	—
144	C₂²×C₃	—	2	4	no	—
145	C₂²×C₃	—	2	4	no	—
146	C₂²×C₃	C₂×C₄⋊Dic₃	1	4	yes	—
147	C₃×C₄	—	2	4	no	—
148	C₃×C₄	—	2	4	no	—
149	C₃×C₄	—	2	4	no	—
150	C₃×C₄	—	2	4	no	—
151	C₃×C₄	#908	1	4	yes	—
152	C₃×C₄	#909	1	4	yes	—
153	C₃×C₄	C₂×#208	1	8	yes	—
154	C₃²	#1569	1	1	yes	characteristic, complemented, Sylow
155	C₂³	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
161	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
162	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
163	C₂×C₄	C₂×#634	1	1	yes	characteristic
164	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
165	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
166	C₈	—	2	2	no	—
167	C₈	—	36	2	no	—
168	C₈	—	36	1	no	—
169	C₈	—	36	1	no	—
170	C₈	#3748	1	4	yes	complemented
171	C₂×C₃	—	4	4	no	—
172	C₂×C₃	#4924	1	4	yes	—
173	C₂×C₃	#5121	1	4	yes	—
174	C₂×C₃	#5125	1	4	yes	—
175	C₂²	—	2	1	no	—
176	C₂²	—	2	1	no	—
177	C₂²	—	2	1	no	—
178	C₂²	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
179	C₄	—	2	1	no	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	#64117	1	1	yes	characteristic
184	C₄	#64118	1	1	yes	characteristic
185	C₄	C₂×#3748	1	2	yes	—
186	C₃	#69851	1	4	yes	complemented
187	C₂	—	4	1	no	—
188	C₂	#10592623	1	1	yes	characteristic
189	C₂	#10592820	1	1	yes	characteristic
190	C₂	#10592824	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)
commutator	C₂×C₃²×C₄	C₂²×C₄
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	#1569
LCS	G ▷ C₂×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
63. C₃²×C₈	M₁₆	semidirect
121. C₃×C₈	#208	semidirect
154. C₃²	#1569	semidirect
170. C₈	#3748	semidirect
186. C₃	#69851	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
#10592625	2
#10592623	2
#10589324	2
#10589323	2
#69851	3