Subgroups — #11773535

Order 1152.

672 conjugacy classes of subgroups.

The subgroup lattice has 177 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#431	C₂	1	1	yes	maximal, characteristic
2	C₄×#3748	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#10589320	C₂	1	2	yes	maximal, complemented
5	C₈×C₃⋊(C₃⋊C₈)	C₂	1	2	yes	maximal, complemented
6	#70739	—	3	4	no	maximal
7	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₂×#3748	C₄	1	2	yes	—
9	C₂²×C₃⋊(C₃⋊C₈)	C₄	1	2	yes	—
10	C₃²×C₄⋊C₈	C₄	1	2	yes	—
11	C₃²×C₂²⋊C₈	C₄	1	2	yes	—
12	C₃²×C₄×C₈	C₂²	1	2	yes	—
13	#63630	C₄	1	2	yes	—
14	C₄×C₃⋊(C₃⋊C₈)	C₄	1	2	yes	—
15	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
16	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
17	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
18	C₃×#431	D₃	1	4	yes	—
19	C₄×#208	—	3	4	no	—
20	C₂×C₄×C₃⋊C₈	—	3	4	no	—
21	#4454	—	3	8	no	—
22	C₈×C₃⋊C₈	—	3	8	no	—
23	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
27	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
28	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
29	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
30	C₃²×C₄²	C₂³	1	1	yes	characteristic
31	#3748	C₈	1	8	yes	complemented
32	C₂×C₃⋊(C₃⋊C₈)	—	2	4	no	—
33	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
34	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
35	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
36	C₈×M₁₆	—	9	1	no	Sylow
37	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
38	C₂×#208	—	3	8	no	—
39	C₂²×C₃⋊C₈	—	3	8	no	—
40	C₃×C₄⋊C₈	Dic₃	1	8	yes	—
41	C₃×C₂²⋊C₈	Dic₃	1	8	yes	—
42	C₃×C₄×C₈	C₂×D₃	1	8	yes	—
43	#803	—	3	8	no	—
44	C₄×C₃⋊C₈	—	3	8	no	—
45	C₄×C₃⋊C₈	—	3	4	no	—
46	C₄×C₃⋊C₈	—	3	4	no	—
47	C₄×C₃⋊C₈	—	3	8	no	—
48	C₂³×C₃²	C₄²	1	1	yes	characteristic
49	C₂×C₃²×C₄	—	2	2	no	—
50	C₂×C₃²×C₄	—	2	2	no	—
51	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
52	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
53	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
54	C₂×C₃²×C₄	C₂×C₈	1	4	yes	—
55	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
58	C₃²×C₈	—	2	4	no	—
59	C₃⋊(C₃⋊C₈)	—	2	2	no	—
60	C₃⋊(C₃⋊C₈)	C₂×C₈	1	8	yes	complemented
61	C₃⋊(C₃⋊C₈)	M₁₆	1	4	yes	complemented
62	#431	C₃⋊D₃	1	1	yes	characteristic
63	C₄×M₁₆	—	9	1	no	—
64	C₂×C₄×C₈	—	9	1	no	—
65	#322	—	9	2	no	—
66	C₈²	—	9	2	no	—
67	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
68	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
69	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
70	C₂×C₃×C₈	C₂×Dic₃	1	16	yes	—
71	C₃×C₄²	C₄×D₃	1	4	yes	—
72	C₃×C₄²	C₄×D₃	1	4	yes	—
73	C₃×C₄²	C₂×Dic₃	1	4	yes	—
74	C₃×C₄²	C₂²×D₃	1	4	yes	—
75	#208	—	3	32	no	—
76	C₂×C₃⋊C₈	—	3	8	no	—
77	C₂×C₃⋊C₈	—	3	16	no	—
78	C₂×C₃⋊C₈	—	3	8	no	—
79	C₂×C₃⋊C₈	—	6	16	no	—
80	C₂²×C₃²	—	2	1	no	—
81	C₂²×C₃²	—	2	1	no	—
82	C₂²×C₃²	C₄×C₈	1	2	yes	—
83	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
84	C₃²×C₄	—	2	1	no	—
85	C₃²×C₄	—	2	1	no	—
86	C₃²×C₄	C₄×C₈	1	2	yes	—
87	C₃²×C₄	C₂²×C₈	1	2	yes	—
88	C₃²×C₄	C₂×M₁₆	1	2	yes	—
89	C₃²×C₄	#131	1	2	yes	—
90	C₂×M₁₆	—	9	2	no	—
91	C₂²×C₈	—	9	2	no	—
92	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
93	C₄⋊C₈	C₃⋊Dic₃	1	2	yes	—
94	C₂²⋊C₈	C₃⋊Dic₃	1	2	yes	—
95	#97	—	9	2	no	—
96	C₄×C₈	—	9	2	no	—
97	C₄×C₈	—	9	1	no	—
98	C₄×C₈	—	9	2	no	—
99	C₄×C₈	—	9	1	no	—
100	C₄×C₈	C₂×C₃⋊D₃	1	2	yes	—
101	C₂³×C₃	C₄×Dic₃	1	4	yes	—
102	C₂×C₃×C₄	—	2	8	no	—
103	C₂×C₃×C₄	—	2	8	no	—
104	C₂×C₃×C₄	D₃×C₈	1	16	yes	—
105	C₂×C₃×C₄	C₄×Dic₃	1	4	yes	—
106	C₂×C₃×C₄	C₄×Dic₃	1	4	yes	—
107	C₂×C₃×C₄	C₄×Dic₃	1	4	yes	—
108	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
109	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
110	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
111	C₃×C₈	—	2	16	no	—
112	C₃⋊C₈	—	3	32	no	—
113	C₃⋊C₈	—	3	16	no	—
114	C₃⋊C₈	—	6	8	no	—
115	C₂×C₃²	—	2	2	no	—
116	C₂×C₃²	C₂×C₄×C₈	1	1	yes	characteristic
117	C₂×C₃²	C₄×M₁₆	1	1	yes	characteristic
118	C₂×C₃²	#405	1	1	yes	characteristic
119	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
120	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
121	C₂²×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
122	M₁₆	—	9	8	no	—
123	C₂×C₈	—	9	2	no	—
124	C₂×C₈	—	9	4	no	—
125	C₂×C₈	—	9	2	no	—
126	C₂×C₈	—	18	4	no	—
127	C₂×C₈	C₂×C₃⋊Dic₃	1	4	yes	—
128	C₄²	C₄×C₃⋊D₃	1	1	yes	characteristic
129	C₄²	C₄×C₃⋊D₃	1	1	yes	characteristic
130	C₄²	C₂×C₃⋊Dic₃	1	1	yes	characteristic
131	C₄²	C₂²×C₃⋊D₃	1	1	yes	characteristic
132	C₂²×C₃	—	2	4	no	—
133	C₂²×C₃	—	2	4	no	—
134	C₂²×C₃	C₈×Dic₃	1	8	yes	—
135	C₂²×C₃	C₂×C₄×Dic₃	1	4	yes	—
136	C₃×C₄	—	2	4	no	—
137	C₃×C₄	—	2	4	no	—
138	C₃×C₄	C₈×Dic₃	1	8	yes	—
139	C₃×C₄	C₂×D₃×C₈	1	8	yes	—
140	C₃×C₄	D₃×M₁₆	1	8	yes	—
141	C₃×C₄	#948	1	8	yes	—
142	C₃²	C₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
143	C₂³	C₄×C₃⋊Dic₃	1	1	yes	characteristic
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	C₈×C₃⋊D₃	1	4	yes	—
147	C₂×C₄	C₄×C₃⋊Dic₃	1	1	yes	characteristic
148	C₂×C₄	C₄×C₃⋊Dic₃	1	1	yes	characteristic
149	C₂×C₄	C₄×C₃⋊Dic₃	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
152	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
153	C₈	—	2	4	no	—
154	C₈	—	9	8	no	—
155	C₈	—	9	4	no	—
156	C₈	—	18	2	no	—
157	C₂×C₃	—	2	8	no	—
158	C₂×C₃	#4997	1	4	yes	—
159	C₂×C₃	C₂×C₈×Dic₃	1	4	yes	—
160	C₂×C₃	Dic₃×M₁₆	1	4	yes	—
161	C₂²	—	2	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	C₈×C₃⋊Dic₃	1	2	yes	—
164	C₂²	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	C₈×C₃⋊Dic₃	1	2	yes	—
168	C₄	C₂×C₈×C₃⋊D₃	1	2	yes	—
169	C₄	M₁₆×C₃⋊D₃	1	2	yes	—
170	C₄	#64157	1	2	yes	—
171	C₃	#70739	1	4	yes	complemented
172	C₂	—	2	2	no	—
173	C₂	#10592696	1	1	yes	characteristic
174	C₂	C₂×C₈×C₃⋊Dic₃	1	1	yes	characteristic
175	C₂	M₁₆×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×C₃⋊D₃
commutator	C₂×C₃²	C₂×C₄×C₈
Fitting	C₃²×#431	C₂
socle	C₂²×C₃²	C₂×C₄²
prime radical	C₂³×C₃²	C₄²
cosocle	C₃²×C₄²	C₂³
hypercenter	#431	C₃⋊D₃
hypocenter	C₃²	C₈×M₁₆
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄² ◁ #431

Complements

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

N	H ≅ G/N	type
4. #10589320	C₂	semidirect
5. C₈×C₃⋊(C₃⋊C₈)	C₂	semidirect
31. #3748	C₈	semidirect
60. C₃⋊(C₃⋊C₈)	C₂×C₈	semidirect
61. C₃⋊(C₃⋊C₈)	M₁₆	semidirect
142. C₃²	C₈×M₁₆	semidirect
171. C₃	#70739	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#431	2
C₄×#3748	2
C₂×C₄×C₃⋊(C₃⋊C₈)	2
#10589320	2
C₈×C₃⋊(C₃⋊C₈)	2
#70739	3