Subgroups — #11795879

Order 1152.

552 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×#63639	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	#10589327	C₂	1	2	yes	maximal, complemented
5	C₄×C₃⋊(C₃⋊C₁₆)	C₂	1	2	yes	maximal, complemented
6	#69881	—	3	4	no	maximal
7	C₂×C₃²×M₁₆	C₄	1	1	yes	characteristic
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₃²×#117	C₄	1	1	yes	characteristic
10	C₃²×#97	C₄	1	2	yes	—
11	C₃²×C₄×C₈	C₂²	1	2	yes	—
12	#63639	C₄	1	4	yes	complemented
13	C₂×C₃⋊(C₃⋊C₁₆)	C₄	1	4	yes	complemented
14	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
16	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
17	C₃×#405	D₃	1	4	yes	—
18	C₂×#812	—	3	4	no	—
19	C₂²×C₃⋊C₁₆	—	3	4	no	—
20	#4461	—	3	8	no	—
21	C₄×C₃⋊C₁₆	—	3	8	no	—
22	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
23	C₃²×M₁₆	C₈	1	4	yes	—
24	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
26	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
27	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
29	C₃²×C₄⋊C₄	C₈	1	2	yes	—
30	C₃²×C₂²⋊C₄	C₈	1	2	yes	—
31	C₃²×C₄²	C₂×C₄	1	2	yes	—
32	C₃⋊(C₃⋊C₁₆)	C₂×C₄	1	2	yes	—
33	C₃⋊(C₃⋊C₁₆)	C₂×C₄	1	2	yes	—
34	C₃⋊(C₃⋊C₁₆)	C₂×C₄	1	4	yes	—
35	#2108	—	9	1	no	Sylow
36	C₂×C₃×M₁₆	Dic₃	1	4	yes	—
37	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
38	C₃×#117	Dic₃	1	4	yes	—
39	C₃×#97	Dic₃	1	8	yes	—
40	C₃×C₄×C₈	C₂×D₃	1	8	yes	—
41	#812	—	3	16	no	—
42	C₂×C₃⋊C₁₆	—	3	16	no	—
43	C₂×C₃⋊C₁₆	—	3	4	no	—
44	C₂×C₃⋊C₁₆	—	3	8	no	—
45	C₂×C₃⋊C₁₆	—	3	4	no	—
46	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
47	C₂×C₃²×C₄	C₄²	1	2	yes	—
48	C₂×C₃²×C₄	C₂×C₈	1	4	yes	—
49	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
50	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
51	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
52	C₃²×C₈	C₄²	1	2	yes	—
53	C₃²×C₈	C₂×C₈	1	4	yes	—
54	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
55	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
56	C₂×M₃₂	—	9	1	no	—
57	C₂²×C₁₆	—	9	1	no	—
58	#405	C₃⋊D₃	1	1	yes	characteristic
59	#346	—	9	2	no	—
60	C₄×C₁₆	—	9	2	no	—
61	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
62	C₃×M₁₆	C₃⋊C₈	1	16	yes	—
63	C₂×C₃×C₈	C₄×D₃	1	8	yes	—
64	C₂×C₃×C₈	C₄×D₃	1	8	yes	—
65	C₂×C₃×C₈	C₂×Dic₃	1	8	yes	—
66	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
67	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
68	C₃×C₄⋊C₄	C₃⋊C₈	1	8	yes	—
69	C₃×C₂²⋊C₄	C₃⋊C₈	1	8	yes	—
70	C₃×C₄²	C₂×Dic₃	1	8	yes	—
71	C₃⋊C₁₆	—	3	8	no	—
72	C₃⋊C₁₆	—	3	8	no	—
73	C₃⋊C₁₆	—	3	16	no	—
74	C₂²×C₃²	—	2	2	no	—
75	C₂²×C₃²	C₄×C₈	1	2	yes	—
76	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
77	C₃²×C₄	—	2	4	no	—
78	C₃²×C₄	C₄×C₈	1	2	yes	—
79	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
80	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
81	C₂×M₁₆	C₃⋊Dic₃	1	1	yes	characteristic
82	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
83	#117	C₃⋊Dic₃	1	1	yes	characteristic
84	M₃₂	—	9	4	no	—
85	C₂×C₁₆	—	9	4	no	—
86	C₂×C₁₆	—	9	1	no	—
87	C₂×C₁₆	—	9	2	no	—
88	C₂×C₁₆	—	9	1	no	—
89	#97	C₃⋊Dic₃	1	2	yes	—
90	C₄×C₈	C₂×C₃⋊D₃	1	2	yes	—
91	C₂³×C₃	C₂×C₃⋊C₈	1	4	yes	—
92	C₂×C₃×C₄	C₂×C₃⋊C₈	1	16	yes	—
93	C₂×C₃×C₄	C₂×C₃⋊C₈	1	4	yes	—
94	C₂×C₃×C₄	C₂×C₃⋊C₈	1	8	yes	—
95	C₂×C₃×C₄	C₄×Dic₃	1	8	yes	—
96	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
97	C₃×C₈	C₂×C₃⋊C₈	1	16	yes	—
98	C₃×C₈	C₄×Dic₃	1	8	yes	—
99	C₃×C₈	C₂×C₄×D₃	1	4	yes	—
100	C₃×C₈	C₂×C₄×D₃	1	4	yes	—
101	C₂×C₃²	—	2	2	no	—
102	C₂×C₃²	C₂×C₄×C₈	1	1	yes	characteristic
103	C₂×C₃²	#504	1	2	yes	—
104	C₂²×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
105	M₁₆	C₃⋊(C₃⋊C₈)	1	4	yes	—
106	C₂×C₈	C₄×C₃⋊D₃	1	2	yes	—
107	C₂×C₈	C₄×C₃⋊D₃	1	2	yes	—
108	C₂×C₈	C₂×C₃⋊Dic₃	1	2	yes	—
109	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
110	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
111	C₄⋊C₄	C₃⋊(C₃⋊C₈)	1	2	yes	—
112	C₂²⋊C₄	C₃⋊(C₃⋊C₈)	1	2	yes	—
113	C₄²	C₂×C₃⋊Dic₃	1	2	yes	—
114	C₁₆	—	9	2	no	—
115	C₁₆	—	9	2	no	—
116	C₁₆	—	9	4	no	—
117	C₂²×C₃	—	2	8	no	—
118	C₂²×C₃	C₄×C₃⋊C₈	1	8	yes	—
119	C₂²×C₃	C₂²×C₃⋊C₈	1	4	yes	—
120	C₃×C₄	—	2	16	no	—
121	C₃×C₄	C₄×C₃⋊C₈	1	8	yes	—
122	C₃×C₄	C₂²×C₃⋊C₈	1	4	yes	—
123	C₃×C₄	C₂×C₄×Dic₃	1	4	yes	—
124	C₃²	#2108	1	1	yes	characteristic, complemented, Sylow
125	C₂³	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
126	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	4	yes	—
127	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	2	yes	—
128	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
129	C₂×C₄	C₄×C₃⋊Dic₃	1	2	yes	—
130	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
131	C₈	C₂×C₃⋊(C₃⋊C₈)	1	4	yes	—
132	C₈	C₄×C₃⋊Dic₃	1	2	yes	—
133	C₈	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
134	C₈	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
135	C₂×C₃	—	2	8	no	—
136	C₂×C₃	C₂×C₄×C₃⋊C₈	1	4	yes	—
137	C₂×C₃	#5140	1	8	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	C₄×C₃⋊(C₃⋊C₈)	1	2	yes	—
140	C₂²	C₂²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	C₄×C₃⋊(C₃⋊C₈)	1	2	yes	—
143	C₄	C₂²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
144	C₄	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
145	C₃	#69881	1	4	yes	complemented
146	C₂	—	2	2	no	—
147	C₂	C₂×C₄×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
148	C₂	#10592839	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×C₃⋊D₃
commutator	C₂×C₃²	C₂×C₄×C₈
Fitting	C₃²×#405	C₂
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂³×C₃²	C₂×C₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	#405	C₃⋊D₃
hypocenter	C₃²	#2108
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
4. #10589327	C₂	semidirect
5. C₄×C₃⋊(C₃⋊C₁₆)	C₂	semidirect
12. #63639	C₄	semidirect
13. C₂×C₃⋊(C₃⋊C₁₆)	C₄	semidirect
124. C₃²	#2108	semidirect
145. C₃	#69881	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#405	2
C₂×#63639	2
C₂²×C₃⋊(C₃⋊C₁₆)	2
#10589327	2
C₄×C₃⋊(C₃⋊C₁₆)	2
#69881	3