Subgroups — C₃×#67821 #11772510

Order 1152.

293 conjugacy classes of subgroups.

The subgroup lattice has 253 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	C₂×C₃×#819	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#819	C₂	1	1	yes	maximal, characteristic
4	C₃×#1388	—	3	1	no	maximal
5	#67821	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₃×#208	—	2	1	no	—
8	C₂×C₃×#208	—	2	1	no	—
9	C₃²×M₃₂	—	2	1	no	—
10	C₃²×M₃₂	C₄	1	2	yes	—
11	C₂×C₃²×C₁₆	—	2	1	no	—
12	C₃×#819	—	2	1	no	—
13	C₃×#819	—	2	1	no	—
14	C₃×#819	C₄	1	2	yes	—
15	C₃×#819	C₄	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	C₂×C₃×#108	—	3	1	no	—
20	C₂×C₃×#108	—	3	1	no	—
21	C₂×#819	C₂×C₃	1	1	yes	characteristic
22	C₂×#819	C₂×C₃	1	1	yes	characteristic
23	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
24	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
28	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
29	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
30	C₃×#208	—	2	2	no	—
31	C₃×#208	—	2	2	no	—
32	C₃×#208	—	4	1	no	—
33	C₃×#208	—	4	1	no	—
34	C₂×C₃×C₃⋊C₈	—	4	1	no	—
35	C₂×C₃×C₃⋊C₈	—	4	1	no	—
36	C₃²×C₁₆	—	2	2	no	—
37	#1388	—	3	1	no	Sylow
38	C₂×C₃×M₁₆	—	6	1	no	—
39	C₂×C₃×M₁₆	—	6	1	no	—
40	C₂²×C₃×C₈	—	2	1	no	—
41	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
42	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
43	C₂×#208	—	2	1	no	—
44	C₂×#208	—	2	1	no	—
45	C₃×M₃₂	—	2	2	no	—
46	C₃×M₃₂	—	2	1	no	—
47	C₃×M₃₂	—	2	1	no	—
48	C₃×M₃₂	—	2	2	no	—
49	C₃×M₃₂	Dic₃	1	2	yes	—
50	C₃×M₃₂	C₃×C₄	1	2	yes	—
51	C₂×C₃×C₁₆	—	2	2	no	—
52	C₂×C₃×C₁₆	—	2	1	no	—
53	C₂×C₃×C₁₆	—	2	1	no	—
54	C₃×#108	—	3	2	no	—
55	C₃×#108	—	3	2	no	—
56	C₃×#108	—	6	1	no	—
57	C₃×#108	—	6	1	no	—
58	#819	—	2	1	no	—
59	#819	—	2	1	no	—
60	#819	C₃×C₄	1	2	yes	—
61	#819	C₃×C₄	1	2	yes	—
62	C₂³×C₃²	Q₁₆	1	1	yes	characteristic
63	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
67	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
68	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
69	C₃²×C₈	C₄²	1	1	yes	characteristic
70	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
71	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
72	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
73	C₃×C₃⋊C₈	—	4	2	no	—
74	C₃×C₃⋊C₈	—	4	2	no	—
75	C₂×M₃₂	C₃×D₃	1	1	yes	characteristic
76	C₂×#108	—	3	1	no	—
77	C₂×#108	—	3	1	no	—
78	C₂²×C₃×C₄	—	2	1	no	—
79	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
80	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
81	C₃×M₁₆	—	6	2	no	—
82	C₃×M₁₆	—	6	2	no	—
83	C₃×M₁₆	—	12	1	no	—
84	C₃×M₁₆	—	12	1	no	—
85	C₂×C₃×C₈	—	2	1	no	—
86	C₂×C₃×C₈	—	2	1	no	—
87	C₂×C₃×C₈	—	2	1	no	—
88	C₂×C₃×C₈	—	2	1	no	—
89	C₂×C₃×C₈	—	2	1	no	—
90	C₂×C₃×C₈	—	2	1	no	—
91	C₂×C₃×C₈	—	12	1	no	—
92	C₂×C₃×C₈	—	12	1	no	—
93	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
94	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
95	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
96	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
97	C₂×C₃×C₈	#67	1	1	yes	characteristic
98	C₂×C₃×C₈	#67	1	1	yes	characteristic
99	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
100	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
102	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
103	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
104	C₂×C₃×C₈	C₃×Q₈	1	1	yes	characteristic
105	#208	—	2	2	no	—
106	#208	—	2	2	no	—
107	#208	—	4	1	no	—
108	#208	—	4	1	no	—
109	C₂×C₃⋊C₈	—	4	1	no	—
110	C₂×C₃⋊C₈	—	4	1	no	—
111	C₃×C₁₆	—	2	4	no	—
112	C₃×C₁₆	—	2	2	no	—
113	C₃×C₁₆	—	2	2	no	—
114	C₂²×C₃²	—	4	1	no	—
115	C₂²×C₃²	#103	1	1	yes	characteristic
116	C₂²×C₃²	#103	1	1	yes	characteristic
117	C₂²×C₃²	#107	1	1	yes	characteristic
118	C₃²×C₄	#95	1	1	yes	characteristic
119	C₃²×C₄	#102	1	1	yes	characteristic
120	C₃²×C₄	#102	1	1	yes	characteristic
121	C₃²×C₄	#106	1	1	yes	characteristic
122	C₂×M₁₆	—	6	1	no	—
123	C₂×M₁₆	—	6	1	no	—
124	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
125	M₃₂	—	2	1	no	—
126	M₃₂	C₃×Dic₃	1	2	yes	—
127	C₂×C₁₆	—	2	1	no	—
128	#108	—	3	2	no	—
129	#108	—	3	2	no	—
130	#108	—	6	1	no	—
131	#108	—	6	1	no	—
132	C₂³×C₃	—	2	1	no	—
133	C₂³×C₃	Dic₁₂	1	1	yes	characteristic
134	C₂³×C₃	C₃×Q₁₆	1	1	yes	characteristic
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	#204	1	1	yes	characteristic
142	C₂×C₃×C₄	#204	1	1	yes	characteristic
143	C₂×C₃×C₄	D₂₄	1	1	yes	characteristic
144	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
145	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
146	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
147	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₃×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₃×C₄	C₃×D₈	1	1	yes	characteristic
151	C₂×C₃×C₄	C₃×SD₁₆	1	1	yes	characteristic
152	C₂×C₃×C₄	C₃×SD₁₆	1	1	yes	characteristic
153	C₃×C₈	—	2	1	no	—
154	C₃×C₈	—	2	1	no	—
155	C₃×C₈	—	2	1	no	—
156	C₃×C₈	—	2	1	no	—
157	C₃×C₈	—	12	2	no	—
158	C₃×C₈	—	12	2	no	—
159	C₃×C₈	C₄×Dic₃	1	1	yes	characteristic
160	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
161	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
162	C₃×C₈	#217	1	1	yes	characteristic
163	C₃×C₈	C₃×C₄²	1	1	yes	characteristic
164	C₃×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
165	C₃×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
166	C₃×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
167	C₃⋊C₈	—	4	2	no	—
168	C₃⋊C₈	—	4	2	no	—
169	C₂×C₃²	—	2	1	no	—
170	C₂×C₃²	—	2	1	no	—
171	C₂×C₃²	—	2	1	no	—
172	C₂×C₃²	#340	1	1	yes	characteristic
173	C₂²×C₄	C₃×D₁₂	1	1	yes	characteristic
174	M₁₆	—	6	2	no	—
175	M₁₆	—	6	2	no	—
176	M₁₆	—	12	1	no	—
177	M₁₆	—	12	1	no	—
178	C₂×C₈	—	12	1	no	—
179	C₂×C₈	—	12	1	no	—
180	C₂×C₈	C₃×Dic₆	1	1	yes	characteristic
181	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
182	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
183	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
184	C₂×C₈	C₃×#67	1	1	yes	characteristic
185	C₂×C₈	C₃×#67	1	1	yes	characteristic
186	C₁₆	—	2	2	no	—
187	C₂²×C₃	—	2	1	no	—
188	C₂²×C₃	—	2	1	no	—
189	C₂²×C₃	—	2	1	no	—
190	C₂²×C₃	—	4	1	no	—
191	C₂²×C₃	—	4	1	no	—
192	C₂²×C₃	—	4	2	no	—
193	C₂²×C₃	#816	1	1	yes	characteristic
194	C₂²×C₃	#816	1	1	yes	characteristic
195	C₂²×C₃	#818	1	1	yes	characteristic
196	C₂²×C₃	C₃×#103	1	1	yes	characteristic
197	C₂²×C₃	C₃×#103	1	1	yes	characteristic
198	C₂²×C₃	C₃×#107	1	1	yes	characteristic
199	C₃×C₄	—	2	1	no	—
200	C₃×C₄	—	2	1	no	—
201	C₃×C₄	—	2	1	no	—
202	C₃×C₄	—	2	1	no	—
203	C₃×C₄	#817	1	1	yes	characteristic
204	C₃×C₄	#821	1	1	yes	characteristic
205	C₃×C₄	#821	1	1	yes	characteristic
206	C₃×C₄	C₃⋊#95	1	1	yes	characteristic
207	C₃×C₄	C₃×#95	1	1	yes	characteristic
208	C₃×C₄	C₃×#102	1	1	yes	characteristic
209	C₃×C₄	C₃×#102	1	1	yes	characteristic
210	C₃×C₄	C₃×#106	1	1	yes	characteristic
211	C₃²	#1388	1	1	yes	characteristic, complemented, Sylow
212	C₂³	C₃×Dic₁₂	1	1	yes	characteristic
213	C₂×C₄	C₃×#204	1	1	yes	characteristic
214	C₂×C₄	C₃×#204	1	1	yes	characteristic
215	C₂×C₄	C₃×D₂₄	1	1	yes	characteristic
216	C₂×C₄	C₃×C₄⋊Dic₃	1	1	yes	characteristic
217	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
218	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
219	C₈	—	12	2	no	—
220	C₈	—	12	2	no	—
221	C₈	C₃×C₄×Dic₃	1	1	yes	characteristic
222	C₈	C₃×Dic₃⋊C₄	1	1	yes	characteristic
223	C₈	C₃×Dic₃⋊C₄	1	1	yes	characteristic
224	C₈	C₃×#217	1	1	yes	characteristic
225	C₂×C₃	—	2	2	no	—
226	C₂×C₃	—	2	1	no	—
227	C₂×C₃	—	2	1	no	—
228	C₂×C₃	—	2	1	no	—
229	C₂×C₃	—	2	1	no	—
230	C₂×C₃	—	2	1	no	—
231	C₂×C₃	—	2	1	no	—
232	C₂×C₃	—	2	1	no	—
233	C₂×C₃	—	4	1	no	—
234	C₂×C₃	—	4	1	no	—
235	C₂×C₃	#4551	1	1	yes	characteristic
236	C₂×C₃	C₃×#340	1	1	yes	characteristic
237	C₂²	—	4	1	no	—
238	C₂²	C₃×#816	1	1	yes	characteristic
239	C₂²	C₃×#816	1	1	yes	characteristic
240	C₂²	C₃×#818	1	1	yes	characteristic
241	C₄	C₃×#817	1	1	yes	characteristic
242	C₄	C₃×#821	1	1	yes	characteristic
243	C₄	C₃×#821	1	1	yes	characteristic
244	C₄	C₃×C₃⋊#95	1	1	yes	characteristic
245	C₃	—	2	1	no	—
246	C₃	#67821	1	1	yes	characteristic, direct product factor
247	C₃	C₃×#1388	1	1	yes	characteristic, complemented
248	C₂	—	2	1	no	—
249	C₂	—	2	1	no	—
250	C₂	—	2	1	no	—
251	C₂	C₃×#4551	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	#821
commutator	C₃×C₈	C₃×C₄²
Frattini	C₂²×C₈	C₂×C₃×D₃
Fitting	C₂×C₃²×M₃₂	C₂
socle	C₂×C₃²	#340
prime radical	C₂³×C₃²	Q₁₆
cosocle	C₂²×C₃×C₈	C₂²×C₃
hypercenter	C₂×C₃×M₃₂	D₃
hypocenter	C₃	C₃×#1388
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈ ◁ C₂×C₃×M₃₂

Complements

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

N	H ≅ G/N	type
5. #67821	C₃	direct (N × H)
211. C₃²	#1388	semidirect
246. C₃	#67821	direct (N × H)
247. C₃	C₃×#1388	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
C₂×C₃×#819	2
C₃×#1388	3
#67821	3

Subgroups — C3×#67821 #11772510

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#67821 #11772510