Subgroups — #11786279

Order 1152.

768 conjugacy classes of subgroups.

The subgroup lattice has 189 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#97	C₂	1	1	yes	maximal, characteristic
2	C₂×#63645	C₂	1	2	yes	maximal
3	C₂×#63644	C₂	1	2	yes	maximal
4	#10592635	C₂	1	1	yes	maximal, characteristic
5	#10592634	C₂	1	1	yes	maximal, characteristic
6	#69816	—	3	4	no	maximal
7	C₂²×C₃²×C₈	C₂²	1	2	yes	—
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
10	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
13	C₃²×#97	C₄	1	4	yes	complemented
14	#63658	—	2	2	no	—
15	#63645	—	2	4	no	—
16	#63644	—	2	4	no	—
17	C₂×C₃×#97	D₃	1	4	yes	—
18	C₂×#818	—	3	8	no	—
19	C₂×#817	—	3	8	no	—
20	#4936	—	3	4	no	—
21	#4935	—	3	4	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	1	no	—
26	C₂²×C₃⋊Dic₃	—	2	1	no	—
27	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
28	C₂×C₃²×C₈	D₄	1	4	yes	—
29	C₂×C₃²×C₈	Q₈	1	4	yes	—
30	C₃²×C₄²	C₂×C₄	1	2	yes	—
31	C₃²×C₄²	Q₈	1	2	yes	—
32	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
33	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
34	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
35	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
36	C₄⋊(C₃⋊Dic₃)	—	4	4	no	—
37	#1847	—	9	1	no	Sylow
38	C₂²×C₃×C₈	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	—	3	4	no	—
42	C₂×C₄⋊Dic₃	—	3	4	no	—
43	C₂×C₄⋊Dic₃	—	6	8	no	—
44	C₃×#97	Dic₃	1	16	yes	complemented
45	C₃⋊#95	—	6	8	no	—
46	#818	—	6	16	no	—
47	#817	—	6	16	no	—
48	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
49	C₂×C₃²×C₄	—	2	2	no	—
50	C₂×C₃²×C₄	—	2	2	no	—
51	C₂×C₃²×C₄	C₄⋊C₄	1	4	yes	—
52	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
54	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
55	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₃⋊Dic₃	—	4	2	no	—
57	C₂×C₃⋊Dic₃	—	4	1	no	—
58	C₂×C₃⋊Dic₃	—	4	1	no	—
59	C₂×C₃⋊Dic₃	—	4	4	no	—
60	C₂×C₃⋊Dic₃	—	4	2	no	—
61	C₂×C₃⋊Dic₃	—	4	2	no	—
62	C₃²×C₈	C₄⋊C₄	1	8	yes	complemented
63	C₂×#107	—	9	2	no	—
64	C₂×#106	—	9	2	no	—
65	C₂×#97	C₃⋊D₃	1	1	yes	characteristic
66	#384	—	9	1	no	—
67	#382	—	9	1	no	—
68	C₂²×C₃×C₄	D₁₂	1	8	yes	—
69	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
70	C₂²×Dic₃	—	6	4	no	—
71	C₂²×Dic₃	—	6	4	no	—
72	C₂²×Dic₃	—	6	8	no	—
73	C₂×C₃×C₈	Dic₆	1	16	yes	—
74	C₂×C₃×C₈	D₁₂	1	16	yes	—
75	C₂×C₃×C₈	C₂×Dic₃	1	16	yes	—
76	C₃×C₄²	Dic₆	1	8	yes	—
77	C₃×C₄²	C₂×Dic₃	1	8	yes	—
78	C₄⋊Dic₃	—	6	8	no	—
79	C₄⋊Dic₃	—	6	8	no	—
80	C₄⋊Dic₃	—	6	8	no	—
81	C₄⋊Dic₃	—	6	8	no	—
82	C₄⋊Dic₃	—	12	16	no	—
83	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₃²	#127	1	1	yes	characteristic
85	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
86	C₂²×C₃²	AGL(1,ℤ/8)	1	2	yes	—
87	C₂²×C₃²	#137	1	2	yes	—
88	C₃²×C₄	—	2	4	no	—
89	C₃²×C₄	C₂×C₄⋊C₄	1	2	yes	—
90	C₃²×C₄	C₄⋊Q₈	1	2	yes	—
91	C₃⋊Dic₃	—	8	2	no	—
92	C₃⋊Dic₃	—	8	1	no	—
93	C₃⋊Dic₃	—	8	1	no	—
94	C₂²×C₈	C₂×C₃⋊D₃	1	2	yes	—
95	C₂×C₄⋊C₄	—	9	1	no	—
96	C₂×C₄⋊C₄	—	9	1	no	—
97	C₂×C₄⋊C₄	—	9	2	no	—
98	C₂×C₄⋊C₄	—	18	2	no	—
99	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
100	#107	—	18	4	no	—
101	#106	—	18	4	no	—
102	#97	C₃⋊Dic₃	1	4	yes	complemented
103	#95	—	18	2	no	—
104	C₂³×C₃	C₂×D₁₂	1	4	yes	—
105	C₂×C₃×C₄	—	2	8	no	—
106	C₂×C₃×C₄	—	2	8	no	—
107	C₂×C₃×C₄	C₄⋊Dic₃	1	16	yes	—
108	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
109	C₂×C₃×C₄	C₂×Dic₆	1	8	yes	—
110	C₂×C₃×C₄	C₂×D₁₂	1	8	yes	—
111	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
112	C₂×Dic₃	—	12	8	no	—
113	C₂×Dic₃	—	12	4	no	—
114	C₂×Dic₃	—	12	4	no	—
115	C₂×Dic₃	—	12	16	no	—
116	C₂×Dic₃	—	12	8	no	—
117	C₂×Dic₃	—	12	8	no	—
118	C₃×C₈	C₄⋊Dic₃	1	32	yes	complemented
119	C₂×C₃²	#384	1	1	yes	characteristic
120	C₂×C₃²	#428	1	2	yes	—
121	C₂×C₃²	#496	1	1	yes	characteristic
122	C₂×C₃²	#497	1	1	yes	characteristic
123	C₂×C₃²	#501	1	2	yes	—
124	C₂²×C₄	—	18	1	no	—
125	C₂²×C₄	—	18	1	no	—
126	C₂²×C₄	—	18	2	no	—
127	C₂²×C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
128	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
129	C₂×C₈	#634	1	4	yes	—
130	C₂×C₈	C₄⋊(C₃⋊D₃)	1	4	yes	—
131	C₂×C₈	C₂×C₃⋊Dic₃	1	4	yes	—
132	C₄⋊C₄	—	18	2	no	—
133	C₄⋊C₄	—	18	2	no	—
134	C₄⋊C₄	—	18	2	no	—
135	C₄⋊C₄	—	18	2	no	—
136	C₄⋊C₄	—	36	4	no	—
137	C₄²	#634	1	2	yes	—
138	C₄²	C₂×C₃⋊Dic₃	1	2	yes	—
139	C₂²×C₃	#869	1	4	yes	—
140	C₂²×C₃	#874	1	4	yes	—
141	C₂²×C₃	#908	1	8	yes	—
142	C₂²×C₃	#909	1	8	yes	—
143	C₂²×C₃	C₂×C₄⋊Dic₃	1	4	yes	—
144	C₃×C₄	—	2	16	no	—
145	C₃×C₄	#869	1	8	yes	—
146	C₃×C₄	C₂×C₄⋊Dic₃	1	8	yes	—
147	Dic₃	—	24	8	no	—
148	Dic₃	—	24	4	no	—
149	Dic₃	—	24	4	no	—
150	C₃²	#1847	1	1	yes	characteristic, complemented, Sylow
151	C₂³	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	36	2	no	—
155	C₂×C₄	—	36	1	no	—
156	C₂×C₄	—	36	1	no	—
157	C₂×C₄	—	36	4	no	—
158	C₂×C₄	—	36	2	no	—
159	C₂×C₄	—	36	2	no	—
160	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	4	yes	—
161	C₂×C₄	C₂×#634	1	1	yes	characteristic
162	C₂×C₄	C₂×#634	1	2	yes	—
163	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	2	yes	—
164	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
165	C₈	C₄⋊(C₃⋊Dic₃)	1	8	yes	complemented
166	C₂×C₃	#4702	1	8	yes	—
167	C₂×C₃	#4712	1	4	yes	—
168	C₂×C₃	#4715	1	4	yes	—
169	C₂×C₃	#4935	1	4	yes	—
170	C₂×C₃	#5121	1	8	yes	—
171	C₂²	#64078	1	1	yes	characteristic
172	C₂²	#64083	1	1	yes	characteristic
173	C₂²	#64117	1	2	yes	—
174	C₂²	#64118	1	2	yes	—
175	C₂²	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
176	C₄	—	2	4	no	—
177	C₄	—	72	2	no	—
178	C₄	—	72	1	no	—
179	C₄	—	72	1	no	—
180	C₄	#64078	1	2	yes	—
181	C₄	C₂×C₄⋊(C₃⋊Dic₃)	1	2	yes	—
182	C₃	#69816	1	4	yes	complemented
183	C₂	#10592401	1	2	yes	—
184	C₂	#10592411	1	1	yes	characteristic
185	C₂	#10592414	1	1	yes	characteristic
186	C₂	#10592634	1	1	yes	characteristic
187	C₂	#10592820	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×#97	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×#97	C₃⋊D₃
hypocenter	C₃²	#1847
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
13. C₃²×#97	C₄	semidirect
44. C₃×#97	Dic₃	semidirect
62. C₃²×C₈	C₄⋊C₄	semidirect
102. #97	C₃⋊Dic₃	semidirect
118. C₃×C₈	C₄⋊Dic₃	semidirect
150. C₃²	#1847	semidirect
165. C₈	C₄⋊(C₃⋊Dic₃)	semidirect
182. C₃	#69816	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#97	2
C₂×#63645	2
C₂×#63644	2
#10592635	2
#10592634	2
#69816	3