Subgroups — #11785486

Order 1152.

780 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	C₂×C₃²×#131	C₂	1	1	yes	maximal, characteristic
2	#10592939	C₂	1	2	yes	maximal
3	#10592805	C₂	1	2	yes	maximal, complemented
4	#10589423	C₂	1	2	yes	maximal, complemented
5	#72012	—	3	4	no	maximal
6	C₂×C₃²×#41	C₂²	1	1	yes	characteristic
7	C₃²×#131	—	2	2	no	—
8	C₃²×#131	—	2	2	no	—
9	C₂×C₃²×M₁₆	C₄	1	2	yes	complemented
10	C₂×C₃²×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₃²×C₈	C₄	1	2	yes	complemented
12	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
13	#64135	C₂²	1	4	yes	—
14	#63661	—	2	4	no	—
15	#63645	—	2	2	no	—
16	#63644	—	2	2	no	—
17	C₂×C₃×#131	D₃	1	4	yes	—
18	#5240	—	3	8	no	—
19	#5106	—	3	8	no	—
20	#4557	—	3	8	no	—
21	C₃²×#41	—	2	2	no	—
22	C₃²×#41	—	2	2	no	—
23	C₂×C₃²×Q₈	Q₈	1	1	yes	characteristic
24	C₂×C₃²×D₄	D₄	1	2	yes	—
25	C₂×C₃²×D₄	Q₈	1	1	yes	characteristic
26	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
27	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
28	C₃²×M₁₆	—	2	2	no	—
29	C₃²×M₁₆	—	2	4	no	—
30	C₂×C₃²×C₈	—	2	2	no	—
31	C₂×C₃²×C₈	—	2	2	no	—
32	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
33	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃²×C₈	D₄	1	2	yes	—
36	C₂×C₃²×C₈	D₄	1	2	yes	—
37	#3757	—	2	4	no	—
38	C₄⋊(C₃⋊Dic₃)	—	2	4	no	—
39	C₄×C₃⋊Dic₃	—	2	4	no	—
40	#1812	—	9	1	no	Sylow
41	C₂×C₃×#41	C₂×D₃	1	4	yes	—
42	C₃×#131	—	2	8	no	—
43	C₃×#131	—	2	8	no	—
44	C₂×C₃×M₁₆	Dic₃	1	8	yes	complemented
45	C₂×C₃×M₁₆	C₂×D₃	1	4	yes	—
46	C₂²×C₃×C₈	Dic₃	1	8	yes	complemented
47	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
48	#926	—	3	16	no	—
49	#834	—	6	16	no	—
50	#818	—	6	8	no	—
51	#817	—	6	8	no	—
52	C₂³×C₃²	C₄⋊C₄	1	2	yes	—
53	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
54	C₃²×Q₈	—	2	2	no	—
55	C₃²×D₄	—	2	4	no	—
56	C₃²×D₄	—	2	2	no	—
57	C₂×C₃²×C₄	—	2	2	no	—
58	C₂×C₃²×C₄	—	2	2	no	—
59	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
60	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
62	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₃²×C₄	#41	1	2	yes	—
64	C₂×C₃⋊Dic₃	—	4	4	no	—
65	C₃²×C₈	—	2	2	no	—
66	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
68	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
69	C₂×#131	C₃⋊D₃	1	1	yes	characteristic
70	#427	—	9	2	no	—
71	#410	—	9	2	no	—
72	H(3,ℤ/4)	—	9	2	no	—
73	C₃×#41	—	2	8	no	—
74	C₃×#41	—	2	8	no	—
75	C₂×C₃×Q₈	Dic₆	1	4	yes	—
76	C₂×C₃×D₄	Dic₆	1	4	yes	—
77	C₂×C₃×D₄	D₁₂	1	8	yes	—
78	C₂²×C₃×C₄	C₂×Dic₃	1	8	yes	—
79	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
80	C₃×M₁₆	—	2	8	no	—
81	C₃×M₁₆	—	2	16	no	—
82	C₂×C₃×C₈	—	2	8	no	—
83	C₂×C₃×C₈	—	2	8	no	—
84	C₂×C₃×C₈	C₂×Dic₃	1	8	yes	—
85	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
86	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
87	C₂×C₃×C₈	#67	1	8	yes	—
88	C₂×C₃×C₈	#67	1	8	yes	—
89	#217	—	6	16	no	—
90	C₄⋊Dic₃	—	6	16	no	—
91	C₄×Dic₃	—	6	16	no	—
92	C₂²×C₃²	—	2	2	no	—
93	C₂²×C₃²	—	4	1	no	—
94	C₂²×C₃²	—	4	2	no	—
95	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂²×C₃²	#121	1	2	yes	—
97	C₃²×C₄	—	2	2	no	—
98	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₃²×C₄	#117	1	1	yes	characteristic
100	C₃²×C₄	C₂²⋊Q₈	1	2	yes	—
101	C₃⋊Dic₃	—	8	4	no	—
102	C₂×#41	C₂×C₃⋊D₃	1	1	yes	characteristic
103	#131	—	2	2	no	—
104	#131	—	2	2	no	—
105	C₂×M₁₆	C₃⋊Dic₃	1	2	yes	complemented
106	C₂×M₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
107	C₂²×C₈	C₃⋊Dic₃	1	2	yes	complemented
108	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
109	#117	—	9	4	no	—
110	#107	—	18	2	no	—
111	#106	—	18	2	no	—
112	#99	—	18	4	no	—
113	C₂³×C₃	C₄⋊Dic₃	1	8	yes	—
114	C₂³×C₃	C₂×D₁₂	1	4	yes	—
115	C₃×Q₈	—	2	8	no	—
116	C₃×D₄	—	2	16	no	—
117	C₃×D₄	—	2	8	no	—
118	C₂×C₃×C₄	—	2	8	no	—
119	C₂×C₃×C₄	—	2	8	no	—
120	C₂×C₃×C₄	C₄⋊Dic₃	1	8	yes	—
121	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
122	C₂×C₃×C₄	#235	1	8	yes	—
123	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
124	C₂×C₃×C₄	C₂×#67	1	8	yes	—
125	C₂×Dic₃	—	12	16	no	—
126	C₃×C₈	—	2	8	no	—
127	C₃×C₈	#217	1	4	yes	—
128	C₃×C₈	#217	1	4	yes	—
129	C₃×C₈	#217	1	8	yes	—
130	C₂×C₃²	—	2	2	no	—
131	C₂×C₃²	—	2	1	no	—
132	C₂×C₃²	—	4	2	no	—
133	C₂×C₃²	#380	1	1	yes	characteristic
134	#41	—	2	2	no	—
135	#41	—	2	2	no	—
136	C₂×Q₈	#634	1	1	yes	characteristic
137	C₂×D₄	#634	1	1	yes	characteristic
138	C₂×D₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
139	C₂²×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
140	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
141	M₁₆	—	2	2	no	—
142	M₁₆	—	2	4	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	C₂×C₃⋊Dic₃	1	2	yes	—
146	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
147	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
148	C₂×C₈	#638	1	2	yes	—
149	C₂×C₈	#638	1	2	yes	—
150	C₄⋊C₄	—	18	4	no	—
151	C₂²⋊C₄	—	18	4	no	—
152	C₄²	—	18	4	no	—
153	C₂²×C₃	—	2	8	no	—
154	C₂²×C₃	—	4	4	no	—
155	C₂²×C₃	—	4	8	no	—
156	C₂²×C₃	C₂×C₄⋊Dic₃	1	4	yes	—
157	C₂²×C₃	#930	1	8	yes	—
158	C₃×C₄	—	2	8	no	—
159	C₃×C₄	#924	1	8	yes	—
160	C₃×C₄	#926	1	4	yes	—
161	C₃×C₄	C₂×#217	1	4	yes	—
162	Dic₃	—	24	16	no	—
163	C₃²	#1812	1	1	yes	characteristic, complemented, Sylow
164	C₂³	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
165	C₂³	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
166	Q₈	—	2	2	no	—
167	D₄	—	2	4	no	—
168	D₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	36	4	no	—
172	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
173	C₂×C₄	C₂×#634	1	1	yes	characteristic
174	C₂×C₄	#3828	1	2	yes	—
175	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
176	C₂×C₄	C₂×#638	1	2	yes	—
177	C₈	—	2	2	no	—
178	C₈	#3757	1	1	yes	characteristic
179	C₈	#3757	1	1	yes	characteristic
180	C₈	#3757	1	2	yes	—
181	C₂×C₃	—	2	8	no	—
182	C₂×C₃	—	2	4	no	—
183	C₂×C₃	—	4	8	no	—
184	C₂×C₃	#5212	1	4	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	—	4	1	no	—
187	C₂²	—	4	2	no	—
188	C₂²	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
189	C₂²	#64139	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	72	4	no	—
192	C₄	#64133	1	2	yes	—
193	C₄	#64135	1	1	yes	characteristic
194	C₄	C₂×#3757	1	1	yes	characteristic
195	C₃	#72012	1	4	yes	complemented
196	C₂	—	2	2	no	—
197	C₂	—	2	1	no	—
198	C₂	—	4	2	no	—
199	C₂	#10592911	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#64135
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×#131	C₂
socle	C₂×C₃²	#380
prime radical	C₂×C₃²×#41	C₂²
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×#131	C₃⋊D₃
hypocenter	C₃²	#1812
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
3. #10592805	C₂	semidirect
4. #10589423	C₂	semidirect
9. C₂×C₃²×M₁₆	C₄	semidirect
11. C₂²×C₃²×C₈	C₄	semidirect
44. C₂×C₃×M₁₆	Dic₃	semidirect
46. C₂²×C₃×C₈	Dic₃	semidirect
105. C₂×M₁₆	C₃⋊Dic₃	semidirect
107. C₂²×C₈	C₃⋊Dic₃	semidirect
163. C₃²	#1812	semidirect
195. C₃	#72012	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#131	2
#10592939	2
#10592805	2
#10589423	2
#72012	3