Subgroups — C₃×#71421 #11797022

Order 1152.

365 conjugacy classes of subgroups.

The subgroup lattice has 216 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	C₃×#5101	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₃×Dic₃⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	C₃×#2151	—	3	1	no	maximal
6	#71421	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
8	C₃×#926	C₄	1	1	yes	characteristic
9	C₂×C₃×#208	C₄	1	1	yes	characteristic
10	C₃²×M₃₂	—	2	2	no	—
11	C₂×C₃²×C₁₆	C₂²	1	2	yes	—
12	C₃×#815	C₄	1	2	yes	—
13	C₃×C₈×Dic₃	C₂²	1	2	yes	—
14	C₂×C₃×C₃⋊C₁₆	—	2	2	no	—
15	C₂×C₃×C₃⋊C₁₆	C₂²	1	2	yes	—
16	C₂×C₃×M₃₂	—	2	1	no	—
17	C₂×C₃×M₃₂	D₃	1	1	yes	characteristic
18	C₂×C₃×M₃₂	C₂×C₃	1	1	yes	characteristic
19	C₂²×C₃×C₁₆	—	3	1	no	—
20	C₃×#405	—	3	1	no	—
21	#5101	C₂×C₃	1	1	yes	characteristic
22	C₂²×C₃⋊C₁₆	C₂×C₃	1	1	yes	characteristic
23	C₃×C₄⋊C₁₆	—	3	4	no	—
24	Dic₃⋊C₁₆	C₂×C₃	1	4	yes	complemented
25	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃²×C₈	D₄	1	2	yes	—
28	C₂×C₃²×C₈	Q₈	1	2	yes	—
29	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
30	C₃×#217	C₈	1	2	yes	—
31	C₃×C₄⋊Dic₃	C₈	1	2	yes	—
32	C₃×C₄×Dic₃	C₂×C₄	1	2	yes	—
33	C₃×#208	—	2	2	no	—
34	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
35	C₃²×C₁₆	—	2	2	no	—
36	C₃×C₃⋊C₁₆	—	2	2	no	—
37	#2151	—	3	1	no	Sylow
38	C₂×C₃×M₁₆	—	3	1	no	—
39	C₂²×C₃×C₈	—	2	1	no	—
40	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
42	C₃×#117	—	3	1	no	—
43	#926	C₃×C₄	1	1	yes	characteristic
44	C₂×#208	C₃×C₄	1	1	yes	characteristic
45	C₃×M₃₂	—	2	4	no	—
46	C₃×M₃₂	—	2	2	no	—
47	C₃×M₃₂	—	2	2	no	—
48	C₂×C₃×C₁₆	—	2	2	no	—
49	C₂×C₃×C₁₆	—	3	2	no	—
50	C₂×C₃×C₁₆	—	6	2	no	—
51	C₂×C₃×C₁₆	C₂×D₃	1	2	yes	—
52	C₂×C₃×C₁₆	C₂²×C₃	1	2	yes	—
53	C₃×#97	—	3	2	no	—
54	C₃×C₄×C₈	—	3	2	no	—
55	#815	C₃×C₄	1	2	yes	—
56	C₈×Dic₃	C₂²×C₃	1	2	yes	—
57	C₂×C₃⋊C₁₆	—	2	2	no	—
58	C₂×C₃⋊C₁₆	C₂²×C₃	1	2	yes	—
59	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
60	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
61	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
62	C₂×C₃²×C₄	M₁₆	1	2	yes	—
63	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₃×Dic₃	C₂×C₈	1	4	yes	—
65	C₃²×C₈	C₄⋊C₄	1	2	yes	—
66	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
67	C₃²×C₈	C₂×Q₈	1	1	yes	characteristic
68	C₃×C₃⋊C₈	—	2	2	no	—
69	C₂×M₃₂	C₃×D₃	1	1	yes	characteristic
70	C₂²×C₁₆	—	3	1	no	—
71	#405	—	3	1	no	—
72	C₄⋊C₁₆	—	3	4	no	—
73	C₂²×C₃×C₄	—	2	1	no	—
74	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
75	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
76	C₃×M₁₆	—	6	2	no	—
77	C₂×C₃×C₈	—	2	2	no	—
78	C₂×C₃×C₈	—	2	1	no	—
79	C₂×C₃×C₈	—	2	2	no	—
80	C₂×C₃×C₈	—	2	1	no	—
81	C₂×C₃×C₈	—	3	2	no	—
82	C₂×C₃×C₈	Dic₆	1	2	yes	—
83	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
84	C₂×C₃×C₈	#67	1	2	yes	—
85	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
86	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
87	C₂×C₃×C₈	C₃×Q₈	1	2	yes	—
88	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
89	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
90	C₃×C₄⋊C₄	—	3	2	no	—
91	C₃×C₂²⋊C₄	—	3	2	no	—
92	C₃×C₄²	—	3	2	no	—
93	#217	C₃×C₈	1	2	yes	—
94	C₄⋊Dic₃	C₃×C₈	1	2	yes	—
95	C₄×Dic₃	C₂×C₃×C₄	1	2	yes	—
96	#208	—	2	2	no	—
97	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
98	C₃×C₁₆	—	2	4	no	—
99	C₃×C₁₆	—	2	2	no	—
100	C₃×C₁₆	—	2	2	no	—
101	C₃×C₁₆	—	6	2	no	—
102	C₃⋊C₁₆	—	2	2	no	—
103	C₂²×C₃²	—	2	2	no	—
104	C₂²×C₃²	C₄⋊C₈	1	2	yes	—
105	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
106	C₃²×C₄	C₄⋊C₈	1	2	yes	—
107	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
109	C₃×Dic₃	—	2	4	no	—
110	C₂×M₁₆	—	3	1	no	—
111	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
112	#117	—	3	1	no	—
113	M₃₂	—	2	2	no	—
114	C₂×C₁₆	—	3	2	no	—
115	C₂×C₁₆	—	6	2	no	—
116	C₂×C₁₆	C₂×C₃×D₃	1	2	yes	—
117	#97	—	3	2	no	—
118	C₄×C₈	—	3	2	no	—
119	C₂³×C₃	—	2	1	no	—
120	C₂³×C₃	D₃×C₈	1	1	yes	characteristic
121	C₂³×C₃	C₂×C₃×C₈	1	1	yes	characteristic
122	C₂×C₃×C₄	—	2	1	no	—
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	—	2	2	no	—
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	—	3	4	no	—
127	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
128	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
129	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
130	C₂×C₃×C₄	C₃×C₄⋊C₄	1	2	yes	—
131	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
132	C₂×C₃×C₄	C₃×M₁₆	1	2	yes	—
133	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
135	C₂×Dic₃	C₂×C₃×C₈	1	4	yes	—
136	C₃×C₈	—	2	2	no	—
137	C₃×C₈	—	2	1	no	—
138	C₃×C₈	—	2	1	no	—
139	C₃×C₈	—	6	2	no	—
140	C₃×C₈	Dic₃⋊C₄	1	2	yes	—
141	C₃×C₈	C₃×C₄⋊C₄	1	2	yes	—
142	C₃×C₈	C₂×Dic₆	1	1	yes	characteristic
143	C₃×C₈	C₂×#67	1	1	yes	characteristic
144	C₃×C₈	C₂×C₃×D₄	1	1	yes	characteristic
145	C₃×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
146	C₃⋊C₈	—	2	2	no	—
147	C₂×C₃²	—	2	2	no	—
148	C₂×C₃²	C₂×C₄⋊C₈	1	1	yes	characteristic
149	C₂×C₃²	#504	1	2	yes	—
150	C₂²×C₄	C₃×C₄×D₃	1	1	yes	characteristic
151	M₁₆	—	6	2	no	—
152	C₂×C₈	—	3	2	no	—
153	C₂×C₈	C₃×Dic₆	1	2	yes	—
154	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
155	C₂×C₈	C₃×#67	1	2	yes	—
156	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
157	C₄⋊C₄	—	3	2	no	—
158	C₂²⋊C₄	—	3	2	no	—
159	C₄²	—	3	2	no	—
160	C₁₆	—	2	2	no	—
161	C₁₆	—	6	2	no	—
162	C₂²×C₃	—	2	2	no	—
163	C₂²×C₃	—	2	2	no	—
164	C₂²×C₃	—	2	1	no	—
165	C₂²×C₃	—	2	2	no	—
166	C₂²×C₃	—	4	2	no	—
167	C₂²×C₃	Dic₃⋊C₈	1	2	yes	—
168	C₂²×C₃	C₃×C₄⋊C₈	1	2	yes	—
169	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
170	C₂²×C₃	C₂²×C₃×C₈	1	1	yes	characteristic
171	C₃×C₄	—	2	1	no	—
172	C₃×C₄	—	2	1	no	—
173	C₃×C₄	—	2	2	no	—
174	C₃×C₄	—	6	4	no	—
175	C₃×C₄	Dic₃⋊C₈	1	2	yes	—
176	C₃×C₄	C₃×C₄⋊C₈	1	2	yes	—
177	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
178	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
179	C₃×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
180	C₃×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
181	Dic₃	—	2	4	no	—
182	C₃²	#2151	1	1	yes	characteristic, complemented, Sylow
183	C₂³	C₃×D₃×C₈	1	1	yes	characteristic
184	C₂×C₄	—	3	4	no	—
185	C₂×C₄	C₃×D₃×C₈	1	1	yes	characteristic
186	C₂×C₄	C₃×ΓL(1,𝔽₂₅)	1	2	yes	—
187	C₂×C₄	C₃×Dic₃⋊C₄	1	2	yes	—
188	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
189	C₈	—	6	2	no	—
190	C₈	C₃×Dic₃⋊C₄	1	2	yes	—
191	C₈	C₂×C₃×Dic₆	1	1	yes	characteristic
192	C₈	C₂×C₃×#67	1	1	yes	characteristic
193	C₂×C₃	—	2	2	no	—
194	C₂×C₃	—	2	2	no	—
195	C₂×C₃	—	2	2	no	—
196	C₂×C₃	—	2	1	no	—
197	C₂×C₃	—	4	2	no	—
198	C₂×C₃	#4907	1	2	yes	—
199	C₂×C₃	C₂×Dic₃⋊C₈	1	1	yes	characteristic
200	C₂×C₃	C₂×C₃×C₄⋊C₈	1	1	yes	characteristic
201	C₂×C₃	C₃×#504	1	2	yes	—
202	C₂²	—	2	2	no	—
203	C₂²	C₃×Dic₃⋊C₈	1	2	yes	—
204	C₂²	C₂×C₃×D₃×C₈	1	1	yes	characteristic
205	C₄	—	6	4	no	—
206	C₄	C₃×Dic₃⋊C₈	1	2	yes	—
207	C₄	C₂×C₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
208	C₄	C₂×C₃×Dic₃⋊C₄	1	1	yes	characteristic
209	C₃	—	2	1	no	—
210	C₃	#71421	1	1	yes	characteristic, direct product factor
211	C₃	C₃×#2151	1	1	yes	characteristic, complemented
212	C₂	—	2	2	no	—
213	C₂	C₃×#4907	1	2	yes	—
214	C₂	C₂×C₃×Dic₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₃×C₈
Frattini	C₂×C₈	C₂²×C₃×D₃
Fitting	C₂×C₃²×M₃₂	C₂
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂³×C₃²	C₂×C₈
cosocle	C₂×C₃×C₈	C₂³×C₃
hypercenter	C₂×C₃×M₃₂	D₃
hypocenter	C₃	C₃×#2151
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₂×C₃×M₃₂

Complements

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

N	H ≅ G/N	type
4. C₃×Dic₃⋊C₁₆	C₂	semidirect
6. #71421	C₃	direct (N × H)
24. Dic₃⋊C₁₆	C₂×C₃	semidirect
182. C₃²	#2151	semidirect
210. C₃	#71421	direct (N × H)
211. C₃	C₃×#2151	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
C₃×#5101	2
C₂²×C₃×C₃⋊C₁₆	2
C₃×Dic₃⋊C₁₆	2
C₃×#2151	3
#71421	3

Subgroups — C3×#71421 #11797022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#71421 #11797022