Subgroups — #11785346

Order 1152.

1008 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃²×C₈	C₂	1	1	yes	maximal, characteristic
2	#10592911	C₂	1	2	yes	maximal
3	C₂×#63644	C₂	1	2	yes	maximal, complemented
4	#10589417	C₂	1	2	yes	maximal, complemented
5	#71970	—	3	4	no	maximal
6	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
7	C₂²×C₃²×C₈	—	2	4	no	—
8	C₂²×C₃²×C₈	C₄	1	4	yes	complemented
9	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₂×#3757	—	2	2	no	—
12	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	4	yes	—
13	#63658	—	2	2	no	—
14	#63644	—	2	4	no	—
15	C₂³×C₃×C₈	D₃	1	4	yes	—
16	#5212	—	3	8	no	—
17	C₂×#817	—	3	8	no	—
18	#4551	—	3	8	no	—
19	C₂⁴×C₃²	D₄	1	1	yes	characteristic
20	C₂²×C₃²×C₄	—	2	2	no	—
21	C₂²×C₃²×C₄	—	2	2	no	—
22	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
23	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
24	C₂²×C₃²×C₄	Q₈	1	2	yes	—
25	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
26	C₂²×C₃⋊Dic₃	—	2	4	no	—
27	C₂×C₃²×C₈	—	2	2	no	—
28	C₂×C₃²×C₈	—	2	8	no	—
29	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
30	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
31	C₂×C₃²×C₈	D₄	1	4	yes	—
32	#3757	—	4	4	no	—
33	C₄⋊(C₃⋊Dic₃)	—	2	4	no	—
34	C₄⋊(C₃⋊Dic₃)	—	2	4	no	—
35	#1807	—	9	1	no	Sylow
36	C₂³×C₃×C₄	C₂×D₃	1	4	yes	—
37	C₂²×C₃×C₈	—	2	16	no	—
38	C₂²×C₃×C₈	Dic₃	1	16	yes	complemented
39	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
40	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
41	C₂×#217	—	6	8	no	—
42	C₂×C₄⋊Dic₃	—	3	16	no	—
43	C₃⋊#95	—	6	8	no	—
44	#817	—	6	16	no	—
45	C₂³×C₃²	—	2	2	no	—
46	C₂³×C₃²	—	2	2	no	—
47	C₂³×C₃²	C₄⋊C₄	1	2	yes	—
48	C₂³×C₃²	SD₁₆	1	4	yes	—
49	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
50	C₂×C₃²×C₄	—	2	2	no	—
51	C₂×C₃²×C₄	—	2	2	no	—
52	C₂×C₃²×C₄	—	2	2	no	—
53	C₂×C₃²×C₄	—	2	4	no	—
54	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
55	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
57	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₃²×C₄	#41	1	2	yes	—
59	C₂×C₃⋊Dic₃	—	4	4	no	—
60	C₂×C₃⋊Dic₃	—	4	4	no	—
61	C₂×C₃⋊Dic₃	—	4	4	no	—
62	C₃²×C₈	—	2	2	no	—
63	C₃²×C₈	C₂²⋊C₄	1	4	yes	complemented
64	C₂³×C₈	C₃⋊D₃	1	1	yes	characteristic
65	C₂×#106	—	9	2	no	—
66	#380	—	9	2	no	—
67	#340	—	9	2	no	—
68	C₂⁴×C₃	D₁₂	1	4	yes	—
69	C₂²×C₃×C₄	—	2	8	no	—
70	C₂²×C₃×C₄	—	2	8	no	—
71	C₂²×C₃×C₄	Dic₆	1	8	yes	—
72	C₂²×C₃×C₄	D₁₂	1	4	yes	—
73	C₂²×C₃×C₄	C₂×Dic₃	1	8	yes	—
74	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
75	C₂²×Dic₃	—	6	16	no	—
76	C₂×C₃×C₈	—	2	8	no	—
77	C₂×C₃×C₈	—	2	32	no	—
78	C₂×C₃×C₈	C₂×Dic₃	1	8	yes	—
79	C₂×C₃×C₈	C₂×Dic₃	1	8	yes	—
80	C₂×C₃×C₈	#67	1	16	yes	—
81	#217	—	12	16	no	—
82	C₄⋊Dic₃	—	6	16	no	—
83	C₄⋊Dic₃	—	6	16	no	—
84	C₂²×C₃²	—	2	2	no	—
85	C₂²×C₃²	—	2	2	no	—
86	C₂²×C₃²	—	2	8	no	—
87	C₂²×C₃²	#106	1	4	yes	complemented
88	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₃²	#121	1	2	yes	—
90	C₂²×C₃²	C₂×SD₁₆	1	2	yes	—
91	C₂²×C₃²	#135	1	2	yes	—
92	C₃²×C₄	—	2	2	no	—
93	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₃²×C₄	#117	1	1	yes	characteristic
95	C₃²×C₄	C₂²⋊Q₈	1	2	yes	—
96	C₃⋊Dic₃	—	8	4	no	—
97	C₂³×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
98	C₂²×C₈	—	2	4	no	—
99	C₂²×C₈	C₃⋊Dic₃	1	4	yes	complemented
100	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
101	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
102	C₂×C₄⋊C₄	—	9	4	no	—
103	C₂×C₂²⋊C₄	—	18	2	no	—
104	#106	—	18	4	no	—
105	#95	—	18	2	no	—
106	C₂³×C₃	—	2	8	no	—
107	C₂³×C₃	—	2	8	no	—
108	C₂³×C₃	#204	1	16	yes	—
109	C₂³×C₃	C₄⋊Dic₃	1	8	yes	—
110	C₂³×C₃	C₂×D₁₂	1	4	yes	—
111	C₂×C₃×C₄	—	2	8	no	—
112	C₂×C₃×C₄	—	2	16	no	—
113	C₂×C₃×C₄	—	2	8	no	—
114	C₂×C₃×C₄	—	2	8	no	—
115	C₂×C₃×C₄	C₄⋊Dic₃	1	8	yes	—
116	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
117	C₂×C₃×C₄	#235	1	8	yes	—
118	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
119	C₂×C₃×C₄	C₂×#67	1	8	yes	—
120	C₂×Dic₃	—	12	16	no	—
121	C₂×Dic₃	—	12	16	no	—
122	C₂×Dic₃	—	12	16	no	—
123	C₃×C₈	—	2	8	no	—
124	C₃×C₈	#217	1	16	yes	complemented
125	C₂×C₃²	—	2	4	no	—
126	C₂×C₃²	#380	1	1	yes	characteristic
127	C₂×C₃²	C₂×#106	1	1	yes	characteristic
128	C₂×C₃²	#427	1	1	yes	characteristic
129	C₂×C₃²	#465	1	4	yes	—
130	C₂⁴	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	2	2	no	—
133	C₂²×C₄	—	18	4	no	—
134	C₂²×C₄	#634	1	2	yes	—
135	C₂²×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
137	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	8	no	—
140	C₂×C₈	C₂×C₃⋊Dic₃	1	2	yes	—
141	C₂×C₈	C₂×C₃⋊Dic₃	1	2	yes	—
142	C₂×C₈	#638	1	4	yes	—
143	C₄⋊C₄	—	18	4	no	—
144	C₄⋊C₄	—	18	4	no	—
145	C₂²⋊C₄	—	36	4	no	—
146	C₂²×C₃	—	2	8	no	—
147	C₂²×C₃	—	2	8	no	—
148	C₂²×C₃	—	2	32	no	—
149	C₂²×C₃	#817	1	16	yes	complemented
150	C₂²×C₃	C₂×#204	1	8	yes	—
151	C₂²×C₃	#904	1	8	yes	—
152	C₂²×C₃	C₂×C₄⋊Dic₃	1	4	yes	—
153	C₂²×C₃	#930	1	8	yes	—
154	C₃×C₄	—	2	8	no	—
155	C₃×C₄	#924	1	8	yes	—
156	C₃×C₄	#926	1	4	yes	—
157	C₃×C₄	C₂×#217	1	4	yes	—
158	Dic₃	—	24	16	no	—
159	C₃²	#1807	1	1	yes	characteristic, complemented, Sylow
160	C₂³	—	2	2	no	—
161	C₂³	—	2	2	no	—
162	C₂³	#3744	1	4	yes	—
163	C₂³	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
164	C₂³	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	36	4	no	—
170	C₂×C₄	—	36	4	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	4	yes	complemented
179	C₂×C₃	—	2	16	no	—
180	C₂×C₃	C₂×#817	1	4	yes	—
181	C₂×C₃	#5106	1	4	yes	—
182	C₂×C₃	#5114	1	16	yes	—
183	C₂×C₃	#5212	1	4	yes	—
184	C₂²	—	2	2	no	—
185	C₂²	—	2	8	no	—
186	C₂²	—	2	2	no	—
187	C₂²	#63644	1	4	yes	complemented
188	C₂²	C₂×#3744	1	2	yes	—
189	C₂²	#64113	1	2	yes	—
190	C₂²	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
191	C₂²	#64139	1	2	yes	—
192	C₄	—	2	2	no	—
193	C₄	—	72	4	no	—
194	C₄	#64133	1	2	yes	—
195	C₄	#64135	1	1	yes	characteristic
196	C₄	C₂×#3757	1	1	yes	characteristic
197	C₃	#71970	1	4	yes	complemented
198	C₂	—	2	4	no	—
199	C₂	C₂×#63644	1	1	yes	characteristic
200	C₂	#10592805	1	1	yes	characteristic
201	C₂	#10592813	1	4	yes	—
202	C₂	#10592911	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊(C₃⋊D₃)
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×C₃⋊D₃
Fitting	C₂³×C₃²×C₈	C₂
socle	C₂³×C₃²	C₂×D₄
prime radical	C₂⁴×C₃²	D₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂³×C₈	C₃⋊D₃
hypocenter	C₃²	#1807
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#63644	C₂	semidirect
4. #10589417	C₂	semidirect
8. C₂²×C₃²×C₈	C₄	semidirect
38. C₂²×C₃×C₈	Dic₃	semidirect
63. C₃²×C₈	C₂²⋊C₄	semidirect
87. C₂²×C₃²	#106	semidirect
99. C₂²×C₈	C₃⋊Dic₃	semidirect
124. C₃×C₈	#217	semidirect
149. C₂²×C₃	#817	semidirect
159. C₃²	#1807	semidirect
178. C₈	#3757	semidirect
187. C₂²	#63644	semidirect
197. C₃	#71970	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₃²×C₈	2
#10592911	2
C₂×#63644	2
#10589417	2
#71970	3