Subgroups — #11772350

Order 1152.

408 conjugacy classes of subgroups.

The subgroup lattice has 171 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×M₃₂	C₂	1	1	yes	maximal, characteristic
2	#10592800	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	#67815	—	3	4	no	maximal
5	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
6	#64135	C₄	1	1	yes	characteristic
7	C₂×#3748	C₄	1	1	yes	characteristic
8	C₃²×M₃₂	—	2	1	no	—
9	C₃²×M₃₂	C₄	1	2	yes	complemented
10	C₂×C₃²×C₁₆	—	2	1	no	—
11	#63642	—	2	1	no	—
12	C₈×C₃⋊Dic₃	—	2	1	no	—
13	C₂×C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
14	C₂×C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
15	C₂×C₃⋊(C₃⋊C₁₆)	C₄	1	2	yes	complemented
16	C₂×C₃×M₃₂	D₃	1	4	yes	—
17	#5101	—	3	4	no	—
18	C₂²×C₃⋊C₁₆	—	3	4	no	—
19	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
23	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
24	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
25	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
26	#3757	—	2	1	no	—
27	C₄⋊(C₃⋊Dic₃)	C₈	1	2	yes	—
28	C₄×C₃⋊Dic₃	—	2	1	no	—
29	#3748	—	2	1	no	—
30	#3748	C₈	1	2	yes	—
31	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
32	C₃²×C₁₆	—	2	2	no	—
33	C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
34	#1379	—	9	1	no	Sylow
35	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
36	#926	—	3	4	no	—
37	C₂×#208	—	3	4	no	—
38	C₃×M₃₂	—	2	4	no	—
39	C₃×M₃₂	Dic₃	1	8	yes	complemented
40	C₂×C₃×C₁₆	—	2	4	no	—
41	#815	—	6	4	no	—
42	C₈×Dic₃	—	6	4	no	—
43	C₂×C₃⋊C₁₆	—	3	8	no	—
44	C₂×C₃⋊C₁₆	—	6	4	no	—
45	C₂×C₃⋊C₁₆	—	6	4	no	—
46	C₂³×C₃²	M₁₆	1	1	yes	characteristic
47	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
48	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
51	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
52	C₂×C₃²×C₄	M₁₆	1	1	yes	characteristic
53	C₂×C₃⋊Dic₃	—	2	2	no	—
54	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
55	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
56	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
57	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
58	C₃⋊(C₃⋊C₈)	—	2	2	no	—
59	C₂×M₃₂	C₃⋊D₃	1	1	yes	characteristic
60	C₂²×C₁₆	—	9	1	no	—
61	#405	—	9	1	no	—
62	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
63	C₂×C₃×C₈	Dic₆	1	4	yes	—
64	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
65	C₂×C₃×C₈	D₁₂	1	4	yes	—
66	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
67	C₂×C₃×C₈	#67	1	4	yes	—
68	C₂×C₃×C₈	#67	1	4	yes	—
69	#217	—	6	4	no	—
70	C₄⋊Dic₃	—	3	8	no	—
71	C₄×Dic₃	—	6	4	no	—
72	#208	—	3	8	no	—
73	#208	—	6	4	no	—
74	C₂×C₃⋊C₈	—	6	4	no	—
75	C₃×C₁₆	—	2	8	no	—
76	C₃⋊C₁₆	—	6	8	no	—
77	C₂²×C₃²	—	2	2	no	—
78	C₂²×C₃²	#97	1	1	yes	characteristic
79	C₂²×C₃²	C₂²⋊C₈	1	1	yes	characteristic
80	C₂²×C₃²	C₄⋊C₈	1	1	yes	characteristic
81	C₃²×C₄	#95	1	1	yes	characteristic
82	C₃²×C₄	C₄×C₈	1	1	yes	characteristic
83	C₃²×C₄	C₂²⋊C₈	1	1	yes	characteristic
84	C₃²×C₄	C₄⋊C₈	1	1	yes	characteristic
85	C₃⋊Dic₃	—	4	2	no	—
86	C₂×M₁₆	—	9	1	no	—
87	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
88	#117	—	9	1	no	—
89	M₃₂	—	2	1	no	—
90	M₃₂	C₃⋊Dic₃	1	2	yes	complemented
91	C₂×C₁₆	—	2	1	no	—
92	C₂×C₁₆	—	9	2	no	—
93	C₂×C₁₆	—	18	1	no	—
94	C₂×C₁₆	—	18	1	no	—
95	#97	—	18	1	no	—
96	C₄×C₈	—	18	1	no	—
97	C₂³×C₃	ΓL(1,𝔽₂₅)	1	4	yes	—
98	C₂×C₃×C₄	D₃×C₈	1	4	yes	—
99	C₂×C₃×C₄	D₃×C₈	1	4	yes	—
100	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	4	yes	—
101	C₂×C₃×C₄	C₄×Dic₃	1	4	yes	—
102	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
103	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
104	C₂×Dic₃	—	6	8	no	—
105	C₃×C₈	Dic₃⋊C₄	1	4	yes	—
106	C₃×C₈	C₄⋊Dic₃	1	4	yes	—
107	C₃×C₈	(C₂×D₃)⋊C₄	1	4	yes	—
108	C₃×C₈	#217	1	4	yes	—
109	C₃⋊C₈	—	6	8	no	—
110	C₂×C₃²	—	2	1	no	—
111	C₂×C₃²	—	2	1	no	—
112	C₂×C₃²	#336	1	1	yes	characteristic
113	C₂×C₃²	#350	1	2	yes	—
114	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
115	M₁₆	—	9	2	no	—
116	M₁₆	—	18	1	no	—
117	C₂×C₈	—	18	1	no	—
118	C₂×C₈	#634	1	1	yes	characteristic
119	C₂×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
120	C₂×C₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
121	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
122	C₂×C₈	#638	1	1	yes	characteristic
123	C₂×C₈	#638	1	1	yes	characteristic
124	C₄⋊C₄	—	9	2	no	—
125	C₂²⋊C₄	—	18	1	no	—
126	C₄²	—	18	1	no	—
127	C₁₆	—	2	2	no	—
128	C₁₆	—	18	2	no	—
129	C₂²×C₃	—	2	8	no	—
130	C₂²×C₃	Dic₃⋊C₈	1	4	yes	—
131	C₂²×C₃	#815	1	4	yes	—
132	C₂²×C₃	(C₂×D₃)⋊C₈	1	4	yes	—
133	C₃×C₄	C₈×Dic₃	1	4	yes	—
134	C₃×C₄	Dic₃⋊C₈	1	4	yes	—
135	C₃×C₄	(C₂×D₃)⋊C₈	1	4	yes	—
136	C₃×C₄	C₃⋊#95	1	4	yes	—
137	Dic₃	—	12	8	no	—
138	C₃²	#1379	1	1	yes	characteristic, complemented, Sylow
139	C₂³	#3743	1	1	yes	characteristic
140	C₂×C₄	—	18	2	no	—
141	C₂×C₄	C₈×C₃⋊D₃	1	1	yes	characteristic
142	C₂×C₄	C₈×C₃⋊D₃	1	1	yes	characteristic
143	C₂×C₄	#3743	1	1	yes	characteristic
144	C₂×C₄	C₄×C₃⋊Dic₃	1	1	yes	characteristic
145	C₂×C₄	#3750	1	1	yes	characteristic
146	C₂×C₄	#3752	1	1	yes	characteristic
147	C₈	—	18	2	no	—
148	C₈	#3750	1	1	yes	characteristic
149	C₈	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
150	C₈	#3752	1	1	yes	characteristic
151	C₈	#3757	1	1	yes	characteristic
152	C₂×C₃	—	2	4	no	—
153	C₂×C₃	—	2	4	no	—
154	C₂×C₃	#4515	1	8	yes	—
155	C₂×C₃	#4550	1	4	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	#63641	1	1	yes	characteristic
158	C₂²	#63642	1	1	yes	characteristic
159	C₂²	#63647	1	1	yes	characteristic
160	C₄	—	36	2	no	—
161	C₄	C₈×C₃⋊Dic₃	1	1	yes	characteristic
162	C₄	#63641	1	1	yes	characteristic
163	C₄	#63647	1	1	yes	characteristic
164	C₄	#63658	1	1	yes	characteristic
165	C₃	#67815	1	4	yes	complemented
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	#10589381	1	2	yes	—
169	C₂	#10589416	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#638
commutator	C₃²×C₄	C₄×C₈
Frattini	C₂²×C₈	C₂×C₃⋊D₃
Fitting	C₂×C₃²×M₃₂	C₂
socle	C₂²×C₃²	C₂²⋊C₈
prime radical	C₂³×C₃²	M₁₆
cosocle	C₂²×C₃²×C₈	C₂²
hypercenter	C₂×M₃₂	C₃⋊D₃
hypocenter	C₃²	#1379
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
9. C₃²×M₃₂	C₄	semidirect
15. C₂×C₃⋊(C₃⋊C₁₆)	C₄	semidirect
39. C₃×M₃₂	Dic₃	semidirect
90. M₃₂	C₃⋊Dic₃	semidirect
138. C₃²	#1379	semidirect
165. C₃	#67815	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×M₃₂	2
#10592800	2
C₂²×C₃⋊(C₃⋊C₁₆)	2
#67815	3