Subgroups — #11786027

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	C₃²×#384	C₂	1	1	yes	maximal, characteristic
2	#10592634	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#10589395	C₂	1	4	yes	maximal
5	#69574	—	3	4	no	maximal
6	C₂×C₃²×C₄⋊C₄	—	2	1	no	—
7	C₂×C₃²×C₄⋊C₄	—	2	1	no	—
8	C₂×C₃²×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
11	C₂×C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
12	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
13	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
14	C₄×C₃⋊(C₃⋊C₈)	C₄	1	4	yes	complemented
15	C₃×#384	D₃	1	4	yes	—
16	#4935	—	3	4	no	—
17	C₂×C₄×C₃⋊C₈	—	3	4	no	—
18	#4529	—	3	16	no	—
19	C₂²×C₃²×C₄	—	2	2	no	—
20	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
21	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
22	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
23	C₂²×C₃⋊Dic₃	—	2	2	no	—
24	C₃²×C₄⋊C₄	—	2	4	no	—
25	C₃²×C₄⋊C₄	—	4	2	no	—
26	C₃²×C₄⋊C₄	—	4	2	no	—
27	C₃²×C₄²	C₂×C₄	1	2	yes	—
28	C₃²×C₄²	Q₈	1	2	yes	—
29	C₄⋊(C₃⋊Dic₃)	—	2	4	no	—
30	C₄⋊(C₃⋊Dic₃)	—	4	2	no	—
31	C₄⋊(C₃⋊Dic₃)	—	4	2	no	—
32	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
33	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
34	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
35	#1834	—	9	1	no	Sylow
36	C₂×C₃×C₄⋊C₄	—	2	4	no	—
37	C₂×C₃×C₄⋊C₄	—	2	4	no	—
38	C₂×C₃×C₄⋊C₄	C₂×D₃	1	8	yes	—
39	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
40	C₂×C₄⋊Dic₃	—	3	8	no	—
41	C₂×C₄⋊Dic₃	—	6	4	no	—
42	C₂×C₄⋊Dic₃	—	6	4	no	—
43	C₂²×C₃⋊C₈	—	3	8	no	—
44	C₄×C₃⋊C₈	—	3	16	no	—
45	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
46	C₂×C₃²×C₄	—	4	2	no	—
47	C₂×C₃²×C₄	—	4	2	no	—
48	C₂×C₃²×C₄	—	4	2	no	—
49	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
50	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
51	C₂×C₃²×C₄	D₈	1	2	yes	—
52	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
53	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
54	C₂×C₃²×C₄	Q₁₆	1	2	yes	—
55	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₃²×C₄	#41	1	2	yes	—
58	C₂×C₃²×C₄	#41	1	2	yes	—
59	C₂×C₃⋊Dic₃	—	4	2	no	—
60	C₂×C₃⋊Dic₃	—	4	2	no	—
61	C₂×C₃⋊Dic₃	—	4	2	no	—
62	C₃⋊(C₃⋊C₈)	—	2	4	no	—
63	C₂×C₄×C₈	—	9	1	no	—
64	#384	—	9	1	no	—
65	#384	C₃⋊D₃	1	1	yes	characteristic
66	#340	—	9	4	no	—
67	C₂²×C₃×C₄	—	2	8	no	—
68	C₂²×C₃×C₄	#67	1	4	yes	—
69	C₂²×C₃×C₄	#67	1	4	yes	—
70	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
71	C₂²×Dic₃	—	6	8	no	—
72	C₃×C₄⋊C₄	—	2	16	no	—
73	C₃×C₄⋊C₄	—	4	8	no	—
74	C₃×C₄⋊C₄	—	4	8	no	—
75	C₃×C₄²	Dic₆	1	8	yes	—
76	C₃×C₄²	C₄×D₃	1	8	yes	—
77	C₄⋊Dic₃	—	6	16	no	—
78	C₄⋊Dic₃	—	12	8	no	—
79	C₄⋊Dic₃	—	12	8	no	—
80	C₂×C₃⋊C₈	—	3	16	no	—
81	C₂×C₃⋊C₈	—	6	8	no	—
82	C₂×C₃⋊C₈	—	6	8	no	—
83	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₃²	#124	1	1	yes	characteristic
85	C₂²×C₃²	#124	1	1	yes	characteristic
86	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
87	C₂²×C₃²	C₂×SD₁₆	1	1	yes	characteristic
88	C₂²×C₃²	C₂×SD₁₆	1	1	yes	characteristic
89	C₂²×C₃²	C₂×Q₁₆	1	1	yes	characteristic
90	C₃²×C₄	—	8	2	no	—
91	C₃²×C₄	#106	1	4	yes	—
92	C₃²×C₄	#107	1	4	yes	—
93	C₃²×C₄	#117	1	2	yes	—
94	C₃²×C₄	#125	1	2	yes	—
95	C₃⋊Dic₃	—	8	2	no	—
96	C₂²×C₈	—	9	2	no	—
97	C₂×C₄⋊C₄	—	2	1	no	—
98	C₂×C₄⋊C₄	—	2	1	no	—
99	C₂×C₄⋊C₄	—	9	2	no	—
100	C₂×C₄⋊C₄	—	18	1	no	—
101	C₂×C₄⋊C₄	—	18	1	no	—
102	C₂×C₄⋊C₄	C₂×C₃⋊D₃	1	2	yes	—
103	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
104	C₄×C₈	—	9	4	no	—
105	C₂³×C₃	C₂×#67	1	4	yes	—
106	C₂×C₃×C₄	—	4	8	no	—
107	C₂×C₃×C₄	—	4	8	no	—
108	C₂×C₃×C₄	—	4	8	no	—
109	C₂×C₃×C₄	Dic₃⋊C₄	1	8	yes	—
110	C₂×C₃×C₄	Dic₃⋊C₄	1	8	yes	—
111	C₂×C₃×C₄	#213	1	8	yes	—
112	C₂×C₃×C₄	#214	1	8	yes	—
113	C₂×C₃×C₄	#215	1	8	yes	—
114	C₂×C₃×C₄	#216	1	8	yes	—
115	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
116	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
117	C₂×C₃×C₄	#237	1	8	yes	—
118	C₂×C₃×C₄	#239	1	8	yes	—
119	C₂×Dic₃	—	12	8	no	—
120	C₂×Dic₃	—	12	8	no	—
121	C₂×Dic₃	—	12	8	no	—
122	C₃⋊C₈	—	6	16	no	—
123	C₂×C₃²	#382	1	1	yes	characteristic
124	C₂×C₃²	C₂×#106	1	1	yes	characteristic
125	C₂×C₃²	C₂×#107	1	1	yes	characteristic
126	C₂×C₃²	#486	1	1	yes	characteristic
127	C₂×C₃²	#486	1	1	yes	characteristic
128	C₂×C₃²	#487	1	1	yes	characteristic
129	C₂×C₃²	#487	1	1	yes	characteristic
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	18	2	no	—
132	C₂²×C₄	#638	1	1	yes	characteristic
133	C₂²×C₄	#638	1	1	yes	characteristic
134	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
135	C₂×C₈	—	9	4	no	—
136	C₂×C₈	—	18	2	no	—
137	C₂×C₈	—	18	2	no	—
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄⋊C₄	—	4	2	no	—
141	C₄⋊C₄	—	18	4	no	—
142	C₄⋊C₄	—	36	2	no	—
143	C₄⋊C₄	—	36	2	no	—
144	C₄²	#634	1	2	yes	—
145	C₄²	C₄×C₃⋊D₃	1	2	yes	—
146	C₂²×C₃	C₂×Dic₃⋊C₄	1	4	yes	—
147	C₂²×C₃	C₂×#213	1	4	yes	—
148	C₂²×C₃	C₂×#214	1	4	yes	—
149	C₂²×C₃	#936	1	4	yes	—
150	C₂²×C₃	C₂×#215	1	4	yes	—
151	C₂²×C₃	C₂×#216	1	4	yes	—
152	C₂²×C₃	#947	1	4	yes	—
153	C₃×C₄	—	8	8	no	—
154	C₃×C₄	#807	1	16	yes	—
155	C₃×C₄	#808	1	16	yes	—
156	C₃×C₄	#890	1	8	yes	—
157	C₃×C₄	#892	1	8	yes	—
158	Dic₃	—	24	8	no	—
159	C₃²	#1834	1	1	yes	characteristic, complemented, Sylow
160	C₂³	C₂×#638	1	1	yes	characteristic
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	36	2	no	—
165	C₂×C₄	—	36	2	no	—
166	C₂×C₄	—	36	2	no	—
167	C₂×C₄	#3750	1	2	yes	—
168	C₂×C₄	#3750	1	2	yes	—
169	C₂×C₄	#3753	1	2	yes	—
170	C₂×C₄	#3754	1	2	yes	—
171	C₂×C₄	#3755	1	2	yes	—
172	C₂×C₄	#3756	1	2	yes	—
173	C₂×C₄	C₂×#634	1	1	yes	characteristic
174	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
175	C₂×C₄	#3830	1	2	yes	—
176	C₂×C₄	#3832	1	2	yes	—
177	C₈	—	18	4	no	—
178	C₂×C₃	C₂×#807	1	4	yes	—
179	C₂×C₃	C₂×#808	1	4	yes	—
180	C₂×C₃	#4977	1	4	yes	—
181	C₂×C₃	#5070	1	4	yes	—
182	C₂×C₃	#5079	1	4	yes	—
183	C₂×C₃	#5088	1	4	yes	—
184	C₂×C₃	#5093	1	4	yes	—
185	C₂²	C₂×#3750	1	1	yes	characteristic
186	C₂²	C₂×#3753	1	1	yes	characteristic
187	C₂²	C₂×#3754	1	1	yes	characteristic
188	C₂²	#64145	1	1	yes	characteristic
189	C₂²	C₂×#3755	1	1	yes	characteristic
190	C₂²	C₂×#3756	1	1	yes	characteristic
191	C₂²	#64156	1	1	yes	characteristic
192	C₄	—	8	2	no	—
193	C₄	—	72	2	no	—
194	C₄	#63634	1	4	yes	—
195	C₄	#63635	1	4	yes	—
196	C₄	#64099	1	2	yes	—
197	C₄	#64101	1	2	yes	—
198	C₃	#69574	1	4	yes	complemented
199	C₂	C₂×#63634	1	1	yes	characteristic
200	C₂	C₂×#63635	1	1	yes	characteristic
201	C₂	#10592676	1	1	yes	characteristic
202	C₂	#10592769	1	1	yes	characteristic
203	C₂	#10592778	1	1	yes	characteristic
204	C₂	#10592787	1	1	yes	characteristic
205	C₂	#10592792	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#638
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×C₃⋊D₃
Fitting	C₃²×#384	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	#384	C₃⋊D₃
hypocenter	C₃²	#1834
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

Complements

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

N	H ≅ G/N	type
14. C₄×C₃⋊(C₃⋊C₈)	C₄	semidirect
159. C₃²	#1834	semidirect
198. C₃	#69574	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#384	2
#10592634	2
C₂×C₄×C₃⋊(C₃⋊C₈)	2
#10589395	2
#69574	3