Subgroups — #11790099

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×#106	C₂	1	1	yes	maximal, characteristic
2	C₂×#63644	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
4	#10592683	C₂	1	2	yes	maximal
5	#10589395	C₂	1	2	yes	maximal
6	#70560	—	3	4	no	maximal
7	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃²×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
10	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
11	C₂×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
12	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₃²×#106	C₄	1	4	yes	complemented
14	#63658	—	2	2	no	—
15	#63644	—	2	2	no	—
16	C₈×C₃⋊Dic₃	—	2	2	no	—
17	C₂×C₃×#106	D₃	1	4	yes	—
18	C₂×#817	—	3	4	no	—
19	C₂×C₈×Dic₃	—	3	4	no	—
20	#4984	—	3	8	no	—
21	#4529	—	3	8	no	—
22	C₂²×C₃²×C₄	—	2	2	no	—
23	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
24	C₂²×C₃⋊Dic₃	—	2	2	no	—
25	C₂²×C₃⋊Dic₃	D₄	1	2	yes	—
26	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
27	C₂×C₃²×C₈	D₄	1	2	yes	—
28	C₂×C₃²×C₈	Q₈	1	2	yes	—
29	C₃²×C₄⋊C₄	—	2	2	no	—
30	C₃²×C₄⋊C₄	C₂×C₄	1	4	yes	—
31	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
32	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
33	C₄⋊(C₃⋊Dic₃)	—	4	4	no	—
34	C₄×C₃⋊Dic₃	—	2	1	no	—
35	C₄×C₃⋊Dic₃	—	2	1	no	—
36	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
37	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
38	C₂×C₃⋊(C₃⋊C₈)	Q₈	1	2	yes	—
39	#1941	—	9	1	no	Sylow
40	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
41	C₂×C₃×C₄⋊C₄	C₂×D₃	1	8	yes	—
42	C₂×C₄⋊Dic₃	—	3	8	no	—
43	C₂×C₄⋊Dic₃	—	6	8	no	—
44	C₂×C₄×Dic₃	—	3	4	no	—
45	C₂²×C₃⋊C₈	—	3	4	no	—
46	C₃×#106	Dic₃	1	16	yes	complemented
47	C₃⋊#95	—	6	8	no	—
48	#817	—	6	8	no	—
49	C₈×Dic₃	—	6	8	no	—
50	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
51	C₂×C₃²×C₄	—	2	4	no	—
52	C₂×C₃²×C₄	—	4	2	no	—
53	C₂×C₃²×C₄	—	4	2	no	—
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	2	no	—
61	C₂×C₃⋊Dic₃	—	2	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	SD₁₆	1	4	yes	—
66	C₃²×C₈	C₄⋊C₄	1	4	yes	complemented
67	C₃⋊(C₃⋊C₈)	—	2	2	no	—
68	C₂×#106	—	9	1	no	—
69	C₂×#106	C₃⋊D₃	1	1	yes	characteristic
70	C₂×C₄×C₈	—	9	1	no	—
71	#389	—	9	2	no	—
72	#340	—	9	2	no	—
73	C₂²×C₃×C₄	—	2	8	no	—
74	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
75	C₂²×Dic₃	—	3	8	no	—
76	C₂²×Dic₃	—	6	8	no	—
77	C₂×C₃×C₈	Dic₆	1	8	yes	—
78	C₂×C₃×C₈	D₁₂	1	8	yes	—
79	C₂×C₃×C₈	C₂×Dic₃	1	8	yes	—
80	C₃×C₄⋊C₄	—	2	8	no	—
81	C₃×C₄⋊C₄	C₂×Dic₃	1	16	yes	—
82	C₄⋊Dic₃	—	6	8	no	—
83	C₄⋊Dic₃	—	6	8	no	—
84	C₄⋊Dic₃	—	12	16	no	—
85	C₄×Dic₃	—	6	4	no	—
86	C₄×Dic₃	—	6	4	no	—
87	C₂×C₃⋊C₈	—	3	8	no	—
88	C₂×C₃⋊C₈	—	6	4	no	—
89	C₂×C₃⋊C₈	—	6	4	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	—	4	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	—	2	4	no	—
101	C₃⋊Dic₃	—	8	2	no	—
102	C₂²×C₈	—	9	1	no	—
103	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
104	C₂×C₄⋊C₄	—	9	2	no	—
105	C₂×C₄⋊C₄	—	18	2	no	—
106	C₂×C₄⋊C₄	C₂×C₃⋊D₃	1	2	yes	—
107	C₂×C₄²	—	9	1	no	—
108	#106	—	18	2	no	—
109	#106	C₃⋊Dic₃	1	4	yes	complemented
110	C₄×C₈	—	18	2	no	—
111	#95	—	18	2	no	—
112	C₂³×C₃	D₃×D₄	1	4	yes	—
113	C₂×C₃×C₄	—	2	16	no	—
114	C₂×C₃×C₄	—	4	8	no	—
115	C₂×C₃×C₄	—	4	8	no	—
116	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
117	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
118	C₂×C₃×C₄	#237	1	4	yes	—
119	C₂×C₃×C₄	D₃×Q₈	1	4	yes	—
120	C₂×C₃×C₄	#239	1	4	yes	—
121	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
122	C₂×Dic₃	—	3	16	no	—
123	C₂×Dic₃	—	6	8	no	—
124	C₂×Dic₃	—	6	8	no	—
125	C₂×Dic₃	—	12	8	no	—
126	C₂×Dic₃	—	12	8	no	—
127	C₂×Dic₃	—	12	8	no	—
128	C₃×C₈	C₄⋊Dic₃	1	16	yes	complemented
129	C₃⋊C₈	—	6	8	no	—
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	—	2	2	no	—
136	C₂²×C₄	—	9	2	no	—
137	C₂²×C₄	—	18	2	no	—
138	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
139	C₂×C₈	—	9	2	no	—
140	C₂×C₈	—	18	1	no	—
141	C₂×C₈	—	18	1	no	—
142	C₂×C₈	#634	1	2	yes	—
143	C₂×C₈	C₄⋊(C₃⋊D₃)	1	2	yes	—
144	C₂×C₈	C₂×C₃⋊Dic₃	1	2	yes	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	18	2	no	—
147	C₄⋊C₄	—	18	2	no	—
148	C₄⋊C₄	—	36	4	no	—
149	C₄⋊C₄	C₂×C₃⋊Dic₃	1	4	yes	—
150	C₄²	—	18	1	no	—
151	C₄²	—	18	1	no	—
152	C₂²×C₃	#888	1	4	yes	—
153	C₂²×C₃	#895	1	4	yes	—
154	C₂²×C₃	D₃×SD₁₆	1	8	yes	—
155	C₂²×C₃	#916	1	8	yes	—
156	C₂²×C₃	D₄×Dic₃	1	4	yes	—
157	C₃×C₄	—	4	16	no	—
158	C₃×C₄	#890	1	4	yes	—
159	C₃×C₄	#897	1	4	yes	—
160	C₃×C₄	C₂×C₄⋊Dic₃	1	4	yes	—
161	C₃×C₄	Q₈×Dic₃	1	4	yes	—
162	Dic₃	—	6	16	no	—
163	Dic₃	—	24	8	no	—
164	C₃²	#1941	1	1	yes	characteristic, complemented, Sylow
165	C₂³	D₄×C₃⋊D₃	1	1	yes	characteristic
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	9	4	no	—
170	C₂×C₄	—	18	2	no	—
171	C₂×C₄	—	18	2	no	—
172	C₂×C₄	—	36	2	no	—
173	C₂×C₄	—	36	2	no	—
174	C₂×C₄	—	36	2	no	—
175	C₂×C₄	C₂×#634	1	1	yes	characteristic
176	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
177	C₂×C₄	#3830	1	1	yes	characteristic
178	C₂×C₄	Q₈×C₃⋊D₃	1	1	yes	characteristic
179	C₂×C₄	#3832	1	1	yes	characteristic
180	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
181	C₈	—	18	2	no	—
182	C₈	C₄⋊(C₃⋊Dic₃)	1	4	yes	complemented
183	C₂×C₃	#4855	1	4	yes	—
184	C₂×C₃	#4858	1	4	yes	—
185	C₂×C₃	#4864	1	8	yes	—
186	C₂×C₃	#4984	1	4	yes	—
187	C₂×C₃	Dic₃×SD₁₆	1	8	yes	—
188	C₂²	#64097	1	1	yes	characteristic
189	C₂²	#64104	1	1	yes	characteristic
190	C₂²	SD₁₆×C₃⋊D₃	1	2	yes	—
191	C₂²	#64125	1	2	yes	—
192	C₂²	D₄×C₃⋊Dic₃	1	1	yes	characteristic
193	C₄	—	4	4	no	—
194	C₄	—	18	4	no	—
195	C₄	—	72	2	no	—
196	C₄	#64099	1	1	yes	characteristic
197	C₄	#64106	1	1	yes	characteristic
198	C₄	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
199	C₄	Q₈×C₃⋊Dic₃	1	1	yes	characteristic
200	C₃	#70560	1	4	yes	complemented
201	C₂	#10592554	1	1	yes	characteristic
202	C₂	#10592557	1	1	yes	characteristic
203	C₂	#10592563	1	2	yes	—
204	C₂	#10592683	1	1	yes	characteristic
205	C₂	SD₁₆×C₃⋊Dic₃	1	2	yes	—

	subgroup	G/H
center	C₂³	D₄×C₃⋊D₃
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×#106	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×#106	C₃⋊D₃
hypocenter	C₃²	#1941
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

Complements

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

N	H ≅ G/N	type
13. C₃²×#106	C₄	semidirect
46. C₃×#106	Dic₃	semidirect
66. C₃²×C₈	C₄⋊C₄	semidirect
109. #106	C₃⋊Dic₃	semidirect
128. C₃×C₈	C₄⋊Dic₃	semidirect
164. C₃²	#1941	semidirect
182. C₈	C₄⋊(C₃⋊Dic₃)	semidirect
200. C₃	#70560	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#106	2
C₂×#63644	2
C₂×C₈×C₃⋊Dic₃	2
#10592683	2
#10589395	2
#70560	3