Subgroups — C₃×#67808 #11772384

Order 1152.

305 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#819	C₂	1	2	yes	maximal
3	C₃×#1381	—	3	1	no	maximal
4	#67808	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₃×#208	—	2	2	no	—
7	C₂×C₃²×C₁₆	—	2	2	no	—
8	C₂×C₃²×C₁₆	C₄	1	2	yes	—
9	C₃×#819	—	2	2	no	—
10	C₃×#819	C₄	1	4	yes	—
11	C₂²×C₃×C₁₆	—	2	1	no	—
12	C₂²×C₃×C₁₆	D₃	1	1	yes	characteristic
13	C₂²×C₃×C₁₆	C₂×C₃	1	1	yes	characteristic
14	C₂×C₃×#108	—	3	2	no	—
15	C₂×#819	C₂×C₃	1	2	yes	—
16	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
17	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
19	C₂×C₃²×C₈	D₄	1	2	yes	—
20	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
21	C₃×#208	—	2	4	no	—
22	C₃×#208	—	4	2	no	—
23	C₂×C₃×C₃⋊C₈	—	4	2	no	—
24	C₃²×C₁₆	—	2	2	no	—
25	#1381	—	3	1	no	Sylow
26	C₂×C₃×M₁₆	—	6	2	no	—
27	C₂²×C₃×C₈	—	2	1	no	—
28	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
29	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
30	C₂×#208	—	2	2	no	—
31	C₂×C₃×C₁₆	—	2	4	no	—
32	C₂×C₃×C₁₆	—	2	2	no	—
33	C₂×C₃×C₁₆	—	2	2	no	—
34	C₂×C₃×C₁₆	—	2	2	no	—
35	C₂×C₃×C₁₆	Dic₃	1	2	yes	—
36	C₂×C₃×C₁₆	C₃×C₄	1	2	yes	—
37	C₃×#108	—	3	4	no	—
38	C₃×#108	—	6	2	no	—
39	#819	—	2	2	no	—
40	#819	C₃×C₄	1	4	yes	—
41	C₂³×C₃²	Q₁₆	1	1	yes	characteristic
42	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
43	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
45	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
46	C₃²×C₈	C₄²	1	1	yes	characteristic
47	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
48	C₃²×C₈	C₄⋊C₄	1	2	yes	—
49	C₃×C₃⋊C₈	—	4	4	no	—
50	C₂²×C₁₆	C₃×D₃	1	1	yes	characteristic
51	C₂×#108	—	3	2	no	—
52	C₂²×C₃×C₄	—	2	1	no	—
53	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
54	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
55	C₃×M₁₆	—	6	4	no	—
56	C₃×M₁₆	—	12	2	no	—
57	C₂×C₃×C₈	—	2	1	no	—
58	C₂×C₃×C₈	—	2	2	no	—
59	C₂×C₃×C₈	—	2	1	no	—
60	C₂×C₃×C₈	—	2	2	no	—
61	C₂×C₃×C₈	—	12	2	no	—
62	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
63	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
64	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
65	C₂×C₃×C₈	#67	1	2	yes	—
66	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
67	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
68	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
69	C₂×C₃×C₈	C₃×Q₈	1	1	yes	characteristic
70	#208	—	2	4	no	—
71	#208	—	4	2	no	—
72	C₂×C₃⋊C₈	—	4	2	no	—
73	C₃×C₁₆	—	2	4	no	—
74	C₃×C₁₆	—	2	2	no	—
75	C₃×C₁₆	—	2	2	no	—
76	C₂²×C₃²	—	2	2	no	—
77	C₂²×C₃²	#103	1	2	yes	—
78	C₂²×C₃²	#107	1	1	yes	characteristic
79	C₃²×C₄	#95	1	1	yes	characteristic
80	C₃²×C₄	#102	1	2	yes	—
81	C₃²×C₄	#106	1	1	yes	characteristic
82	C₂×M₁₆	—	6	2	no	—
83	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
84	C₂×C₁₆	—	2	2	no	—
85	C₂×C₁₆	C₃×Dic₃	1	2	yes	—
86	#108	—	3	4	no	—
87	#108	—	6	2	no	—
88	C₂³×C₃	—	2	1	no	—
89	C₂³×C₃	Dic₁₂	1	1	yes	characteristic
90	C₂³×C₃	C₃×Q₁₆	1	1	yes	characteristic
91	C₂×C₃×C₄	—	2	2	no	—
92	C₂×C₃×C₄	—	2	1	no	—
93	C₂×C₃×C₄	—	2	1	no	—
94	C₂×C₃×C₄	—	2	2	no	—
95	C₂×C₃×C₄	#204	1	2	yes	—
96	C₂×C₃×C₄	D₂₄	1	1	yes	characteristic
97	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
98	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
99	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
100	C₂×C₃×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₃×C₄	C₃×D₈	1	1	yes	characteristic
102	C₂×C₃×C₄	C₃×SD₁₆	1	2	yes	—
103	C₃×C₈	—	2	1	no	—
104	C₃×C₈	—	2	2	no	—
105	C₃×C₈	—	2	1	no	—
106	C₃×C₈	—	12	4	no	—
107	C₃×C₈	C₄×Dic₃	1	1	yes	characteristic
108	C₃×C₈	Dic₃⋊C₄	1	2	yes	—
109	C₃×C₈	#217	1	1	yes	characteristic
110	C₃×C₈	C₃×C₄²	1	1	yes	characteristic
111	C₃×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
112	C₃×C₈	C₃×C₄⋊C₄	1	2	yes	—
113	C₃⋊C₈	—	4	4	no	—
114	C₂×C₃²	—	2	2	no	—
115	C₂×C₃²	#340	1	1	yes	characteristic
116	C₂×C₃²	#368	1	2	yes	—
117	C₂²×C₄	C₃×D₁₂	1	1	yes	characteristic
118	M₁₆	—	6	4	no	—
119	M₁₆	—	12	2	no	—
120	C₂×C₈	—	12	2	no	—
121	C₂×C₈	C₃×Dic₆	1	1	yes	characteristic
122	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
123	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
124	C₂×C₈	C₃×#67	1	2	yes	—
125	C₁₆	—	2	2	no	—
126	C₂²×C₃	—	2	4	no	—
127	C₂²×C₃	—	2	2	no	—
128	C₂²×C₃	—	2	2	no	—
129	C₂²×C₃	—	2	2	no	—
130	C₂²×C₃	—	2	1	no	—
131	C₂²×C₃	#816	1	2	yes	—
132	C₂²×C₃	#818	1	1	yes	characteristic
133	C₂²×C₃	C₃×#103	1	2	yes	—
134	C₂²×C₃	C₃×#107	1	1	yes	characteristic
135	C₃×C₄	—	2	1	no	—
136	C₃×C₄	—	2	2	no	—
137	C₃×C₄	—	2	1	no	—
138	C₃×C₄	#817	1	1	yes	characteristic
139	C₃×C₄	#821	1	2	yes	—
140	C₃×C₄	C₃⋊#95	1	1	yes	characteristic
141	C₃×C₄	C₃×#95	1	1	yes	characteristic
142	C₃×C₄	C₃×#102	1	2	yes	—
143	C₃×C₄	C₃×#106	1	1	yes	characteristic
144	C₃²	#1381	1	1	yes	characteristic, complemented, Sylow
145	C₂³	C₃×Dic₁₂	1	1	yes	characteristic
146	C₂×C₄	C₃×#204	1	2	yes	—
147	C₂×C₄	C₃×D₂₄	1	1	yes	characteristic
148	C₂×C₄	C₃×C₄⋊Dic₃	1	1	yes	characteristic
149	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
150	C₈	—	12	4	no	—
151	C₈	C₃×C₄×Dic₃	1	1	yes	characteristic
152	C₈	C₃×Dic₃⋊C₄	1	2	yes	—
153	C₈	C₃×#217	1	1	yes	characteristic
154	C₂×C₃	—	2	4	no	—
155	C₂×C₃	—	2	2	no	—
156	C₂×C₃	—	2	2	no	—
157	C₂×C₃	—	2	2	no	—
158	C₂×C₃	—	2	1	no	—
159	C₂×C₃	#4506	1	2	yes	—
160	C₂×C₃	#4551	1	1	yes	characteristic
161	C₂×C₃	C₃×#340	1	1	yes	characteristic
162	C₂×C₃	C₃×#368	1	2	yes	—
163	C₂²	—	2	2	no	—
164	C₂²	C₃×#816	1	2	yes	—
165	C₂²	C₃×#818	1	1	yes	characteristic
166	C₄	C₃×#817	1	1	yes	characteristic
167	C₄	C₃×#821	1	2	yes	—
168	C₄	C₃×C₃⋊#95	1	1	yes	characteristic
169	C₃	—	2	1	no	—
170	C₃	#67808	1	1	yes	characteristic, direct product factor
171	C₃	C₃×#1381	1	1	yes	characteristic, complemented
172	C₂	—	2	2	no	—
173	C₂	C₃×#4506	1	2	yes	—
174	C₂	C₃×#4551	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	D₂₄
commutator	C₃×C₈	C₃×C₄²
Frattini	C₂²×C₈	C₂×C₃×D₃
Fitting	C₂²×C₃²×C₁₆	C₂
socle	C₂²×C₃²	#107
prime radical	C₂³×C₃²	Q₁₆
cosocle	C₂²×C₃×C₈	C₂²×C₃
hypercenter	C₂²×C₃×C₁₆	D₃
hypocenter	C₃	C₃×#1381
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈ ◁ C₂²×C₃×C₁₆

Complements

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

N	H ≅ G/N	type
4. #67808	C₃	direct (N × H)
144. C₃²	#1381	semidirect
170. C₃	#67808	direct (N × H)
171. C₃	C₃×#1381	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃²×C₁₆	2
C₂×C₃×#819	2
C₃×#1381	3
#67808	3

Subgroups — C3×#67808 #11772384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#67808 #11772384