Subgroups — #11790096

Order 1152.

744 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×#63644	C₂	1	1	yes	maximal, characteristic
3	C₂×#63644	C₂	1	1	yes	maximal, characteristic
4	#10592627	C₂	1	2	yes	maximal
5	#10589417	C₂	1	2	yes	maximal
6	#69769	—	3	4	no	maximal
7	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
12	C₂×#3750	—	2	2	no	—
13	C₃²×C₄×C₈	—	2	2	no	—
14	#63658	—	2	2	no	—
15	#63644	—	2	2	no	—
16	#63644	C₄	1	4	yes	complemented
17	C₂×C₃×C₄×C₈	D₃	1	4	yes	—
18	C₂×#817	—	3	4	no	—
19	C₂×#817	—	3	4	no	—
20	#4928	—	3	8	no	—
21	#4551	—	3	8	no	—
22	C₂²×C₃²×C₄	D₄	1	2	yes	—
23	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
24	C₂²×C₃⋊Dic₃	—	2	2	no	—
25	C₂²×C₃⋊Dic₃	—	2	2	no	—
26	C₂×C₃²×C₈	—	2	1	no	—
27	C₂×C₃²×C₈	—	2	1	no	—
28	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
29	C₂×C₃²×C₈	D₄	1	2	yes	—
30	C₂×C₃²×C₈	Q₈	1	2	yes	—
31	C₂×C₃²×C₈	Q₈	1	2	yes	—
32	C₃²×C₄²	—	2	1	no	—
33	C₃²×C₄²	—	2	1	no	—
34	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
35	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
36	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
37	C₄⋊(C₃⋊Dic₃)	C₂×C₄	1	4	yes	—
38	#3750	—	4	4	no	—
39	#1941	—	9	1	no	Sylow
40	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
41	C₂²×C₃×C₈	C₂×D₃	1	4	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	C₂×Dic₃⋊C₄	—	6	8	no	—
46	C₃×C₄×C₈	—	2	8	no	—
47	C₃⋊#95	—	6	8	no	—
48	#817	—	3	16	no	—
49	#817	—	6	8	no	—
50	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
51	C₂×C₃²×C₄	—	2	2	no	—
52	C₂×C₃²×C₄	—	2	2	no	—
53	C₂×C₃²×C₄	SD₁₆	1	4	yes	—
54	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
56	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₃²×C₄	#41	1	1	yes	characteristic
59	C₂×C₃²×C₄	#41	1	1	yes	characteristic
60	C₂×C₃⋊Dic₃	—	2	4	no	—
61	C₂×C₃⋊Dic₃	—	4	2	no	—
62	C₂×C₃⋊Dic₃	—	4	2	no	—
63	C₂×C₃⋊Dic₃	—	4	2	no	—
64	C₂×C₃⋊Dic₃	—	4	2	no	—
65	C₂×C₃⋊Dic₃	—	4	2	no	—
66	C₃²×C₈	—	2	2	no	—
67	C₃²×C₈	C₄⋊C₄	1	4	yes	complemented
68	C₂×#106	—	9	1	no	—
69	C₂×#106	—	9	1	no	—
70	C₂×C₄×C₈	C₃⋊D₃	1	1	yes	characteristic
71	#389	—	9	2	no	—
72	#340	—	9	2	no	—
73	C₂²×C₃×C₄	D₁₂	1	8	yes	—
74	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
75	C₂²×Dic₃	—	6	8	no	—
76	C₂²×Dic₃	—	6	8	no	—
77	C₂×C₃×C₈	—	2	4	no	—
78	C₂×C₃×C₈	—	2	4	no	—
79	C₂×C₃×C₈	Dic₆	1	8	yes	—
80	C₂×C₃×C₈	Dic₆	1	8	yes	—
81	C₂×C₃×C₈	C₄×D₃	1	8	yes	—
82	C₂×C₃×C₈	#67	1	8	yes	—
83	C₃×C₄²	—	2	4	no	—
84	C₃×C₄²	—	2	4	no	—
85	C₄⋊Dic₃	—	3	16	no	—
86	C₄⋊Dic₃	—	6	8	no	—
87	C₄⋊Dic₃	—	6	8	no	—
88	C₄⋊Dic₃	—	6	8	no	—
89	Dic₃⋊C₄	—	12	16	no	—
90	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
91	C₂²×C₃²	#121	1	1	yes	characteristic
92	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
93	C₂²×C₃²	C₂×SD₁₆	1	2	yes	—
94	C₂²×C₃²	#135	1	2	yes	—
95	C₃²×C₄	—	2	4	no	—
96	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
98	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
99	C₃²×C₄	#125	1	1	yes	characteristic
100	C₃⋊Dic₃	—	4	4	no	—
101	C₃⋊Dic₃	—	8	2	no	—
102	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
103	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
104	C₂×C₄⋊C₄	—	9	2	no	—
105	C₂×C₄⋊C₄	—	9	2	no	—
106	C₂×C₄⋊C₄	—	18	2	no	—
107	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
108	#106	—	9	4	no	—
109	#106	—	18	2	no	—
110	C₄×C₈	—	2	2	no	—
111	#95	—	18	2	no	—
112	C₂³×C₃	C₂×D₁₂	1	4	yes	—
113	C₂×C₃×C₄	—	2	8	no	—
114	C₂×C₃×C₄	—	2	8	no	—
115	C₂×C₃×C₄	#204	1	16	yes	—
116	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
117	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
118	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
119	C₂×C₃×C₄	#235	1	4	yes	—
120	C₂×C₃×C₄	#235	1	4	yes	—
121	C₂×C₃×C₄	C₂×#67	1	4	yes	—
122	C₂×Dic₃	—	6	16	no	—
123	C₂×Dic₃	—	12	8	no	—
124	C₂×Dic₃	—	12	8	no	—
125	C₂×Dic₃	—	12	8	no	—
126	C₂×Dic₃	—	12	8	no	—
127	C₂×Dic₃	—	12	8	no	—
128	C₃×C₈	—	2	8	no	—
129	C₃×C₈	Dic₃⋊C₄	1	16	yes	complemented
130	C₂×C₃²	#389	1	1	yes	characteristic
131	C₂×C₃²	C₄×SD₁₆	1	2	yes	—
132	C₂×C₃²	#465	1	2	yes	—
133	C₂×C₃²	#498	1	1	yes	characteristic
134	C₂×C₃²	#499	1	1	yes	characteristic
135	C₂²×C₄	—	18	2	no	—
136	C₂²×C₄	—	18	2	no	—
137	C₂²×C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
138	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	#634	1	2	yes	—
142	C₂×C₈	#634	1	2	yes	—
143	C₂×C₈	C₄×C₃⋊D₃	1	2	yes	—
144	C₂×C₈	#638	1	2	yes	—
145	C₄⋊C₄	—	9	4	no	—
146	C₄⋊C₄	—	18	2	no	—
147	C₄⋊C₄	—	18	2	no	—
148	C₄⋊C₄	—	18	2	no	—
149	C₄⋊C₄	—	36	4	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	2	1	no	—
152	C₂²×C₃	#869	1	4	yes	—
153	C₂²×C₃	C₄×D₁₂	1	4	yes	—
154	C₂²×C₃	C₂×#204	1	8	yes	—
155	C₂²×C₃	#904	1	8	yes	—
156	C₂²×C₃	#930	1	4	yes	—
157	C₃×C₄	—	2	16	no	—
158	C₃×C₄	C₄×Dic₆	1	4	yes	—
159	C₃×C₄	#870	1	4	yes	—
160	C₃×C₄	C₂×Dic₃⋊C₄	1	4	yes	—
161	C₃×C₄	#924	1	4	yes	—
162	Dic₃	—	12	16	no	—
163	Dic₃	—	24	8	no	—
164	C₃²	#1941	1	1	yes	characteristic, complemented, Sylow
165	C₂³	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	18	4	no	—
169	C₂×C₄	—	36	2	no	—
170	C₂×C₄	—	36	2	no	—
171	C₂×C₄	—	36	2	no	—
172	C₂×C₄	—	36	2	no	—
173	C₂×C₄	—	36	2	no	—
174	C₂×C₄	#3744	1	4	yes	—
175	C₂×C₄	C₂×#634	1	1	yes	characteristic
176	C₂×C₄	C₂×#634	1	1	yes	characteristic
177	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
178	C₂×C₄	#3828	1	1	yes	characteristic
179	C₂×C₄	#3828	1	1	yes	characteristic
180	C₂×C₄	C₂×#638	1	1	yes	characteristic
181	C₈	—	2	2	no	—
182	C₈	#3750	1	4	yes	complemented
183	C₂×C₃	#4680	1	4	yes	—
184	C₂×C₃	#4683	1	4	yes	—
185	C₂×C₃	C₄×#204	1	8	yes	—
186	C₂×C₃	#4928	1	4	yes	—
187	C₂×C₃	#5114	1	8	yes	—
188	C₂²	#64078	1	1	yes	characteristic
189	C₂²	C₄×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
190	C₂²	C₂×#3744	1	2	yes	—
191	C₂²	#64113	1	2	yes	—
192	C₂²	#64139	1	1	yes	characteristic
193	C₄	—	2	4	no	—
194	C₄	—	36	4	no	—
195	C₄	—	72	2	no	—
196	C₄	C₄×#634	1	1	yes	characteristic
197	C₄	#64079	1	1	yes	characteristic
198	C₄	C₂×#3750	1	1	yes	characteristic
199	C₄	#64133	1	1	yes	characteristic
200	C₃	#69769	1	4	yes	complemented
201	C₂	#10592379	1	1	yes	characteristic
202	C₂	#10592382	1	1	yes	characteristic
203	C₂	C₄×#3744	1	2	yes	—
204	C₂	#10592627	1	1	yes	characteristic
205	C₂	#10592813	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₄⋊(C₃⋊D₃)
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×C₄×C₈	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×C₄×C₈	C₃⋊D₃
hypocenter	C₃²	#1941
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
16. #63644	C₄	semidirect
67. C₃²×C₈	C₄⋊C₄	semidirect
129. C₃×C₈	Dic₃⋊C₄	semidirect
164. C₃²	#1941	semidirect
182. C₈	#3750	semidirect
200. C₃	#69769	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×#63644	2
#10592627	2
#10589417	2
#69769	3