Subgroups — C₃×#67664 #11772310

Order 1152.

353 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#97	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#812	C₂	1	2	yes	maximal
3	C₃×#1377	—	3	1	no	maximal
4	#67664	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₂²×C₃²×C₈	C₄	1	1	yes	characteristic
6	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₃²×C₄²	C₄	1	1	yes	characteristic
8	C₃²×#97	—	2	2	no	—
9	C₃×#812	—	2	2	no	—
10	C₃×#812	C₄	1	4	yes	complemented
11	C₂×C₃×C₃⋊C₁₆	—	2	2	no	—
12	C₂×C₃×M₃₂	—	3	2	no	—
13	C₂×C₃×#97	—	2	1	no	—
14	C₂×C₃×#97	D₃	1	1	yes	characteristic
15	C₂×C₃×#97	C₂×C₃	1	1	yes	characteristic
16	C₂×#812	C₂×C₃	1	2	yes	—
17	C₂²×C₃²×C₄	C₈	1	2	yes	—
18	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃²×C₈	—	2	1	no	—
20	C₂×C₃²×C₈	—	2	1	no	—
21	C₂×C₃²×C₈	C₈	1	2	yes	—
22	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
23	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
24	C₂×C₃²×C₈	D₄	1	2	yes	—
25	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
26	C₃²×C₄²	—	2	1	no	—
27	C₃²×C₄²	—	2	1	no	—
28	C₃×C₃⋊C₁₆	—	2	4	no	—
29	#1377	—	3	1	no	Sylow
30	C₂²×C₃×C₈	—	2	1	no	—
31	C₂²×C₃×C₈	—	2	1	no	—
32	C₂²×C₃×C₈	Dic₃	1	1	yes	characteristic
33	C₂²×C₃×C₈	C₃×C₄	1	1	yes	characteristic
34	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
35	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
36	C₂×C₃×C₄²	—	2	1	no	—
37	C₂×C₃×C₄²	Dic₃	1	1	yes	characteristic
38	C₂×C₃×C₄²	C₃×C₄	1	1	yes	characteristic
39	C₃×M₃₂	—	3	4	no	—
40	C₃×M₃₂	—	6	2	no	—
41	C₂×C₃×C₁₆	—	6	2	no	—
42	C₃×#97	—	2	4	no	—
43	C₃×#97	—	2	2	no	—
44	C₃×#97	—	2	2	no	—
45	#812	—	2	2	no	—
46	#812	C₃×C₄	1	4	yes	complemented
47	C₂×C₃⋊C₁₆	—	2	2	no	—
48	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
49	C₂×C₃²×C₄	—	2	2	no	—
50	C₂×C₃²×C₄	—	4	2	no	—
51	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
52	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
55	C₂×C₃²×C₄	M₁₆	1	1	yes	characteristic
56	C₂×C₃²×C₄	M₁₆	1	1	yes	characteristic
57	C₃²×C₈	—	2	2	no	—
58	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
59	C₃²×C₈	C₄⋊C₄	1	2	yes	—
60	C₂×M₃₂	—	3	2	no	—
61	C₂×#97	C₃×D₃	1	1	yes	characteristic
62	C₂²×C₃×C₄	—	2	2	no	—
63	C₂²×C₃×C₄	—	2	1	no	—
64	C₂²×C₃×C₄	C₃⋊C₈	1	2	yes	—
65	C₂²×C₃×C₄	C₃×C₈	1	2	yes	—
66	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
67	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
68	C₂×C₃×C₈	—	2	2	no	—
69	C₂×C₃×C₈	—	2	2	no	—
70	C₂×C₃×C₈	—	2	1	no	—
71	C₂×C₃×C₈	—	2	1	no	—
72	C₂×C₃×C₈	—	2	1	no	—
73	C₂×C₃×C₈	—	2	1	no	—
74	C₂×C₃×C₈	—	2	2	no	—
75	C₂×C₃×C₈	—	2	1	no	—
76	C₂×C₃×C₈	—	2	2	no	—
77	C₂×C₃×C₈	—	2	1	no	—
78	C₂×C₃×C₈	—	2	2	no	—
79	C₂×C₃×C₈	C₃⋊C₈	1	2	yes	—
80	C₂×C₃×C₈	C₃×C₈	1	2	yes	—
81	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
82	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
83	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
84	C₂×C₃×C₈	#67	1	2	yes	—
85	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
86	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
87	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
88	C₂×C₃×C₈	C₃×Q₈	1	1	yes	characteristic
89	C₃×C₄²	—	2	2	no	—
90	C₃×C₄²	—	2	2	no	—
91	C₃×C₄²	—	2	1	no	—
92	C₃×C₄²	—	2	1	no	—
93	C₃×C₄²	—	2	1	no	—
94	C₃×C₄²	—	2	1	no	—
95	C₃×C₁₆	—	6	4	no	—
96	C₃⋊C₁₆	—	2	4	no	—
97	C₂²×C₃²	—	2	2	no	—
98	C₂²×C₃²	C₄×C₈	1	1	yes	characteristic
99	C₂²×C₃²	C₂²⋊C₈	1	1	yes	characteristic
100	C₂²×C₃²	C₄⋊C₈	1	1	yes	characteristic
101	C₃²×C₄	—	4	2	no	—
102	C₃²×C₄	#95	1	1	yes	characteristic
103	C₃²×C₄	#97	1	1	yes	characteristic
104	C₃²×C₄	C₂²⋊C₈	1	1	yes	characteristic
105	C₃²×C₄	C₄⋊C₈	1	1	yes	characteristic
106	C₂²×C₈	C₃×Dic₃	1	1	yes	characteristic
107	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
108	C₂×C₄²	C₃×Dic₃	1	1	yes	characteristic
109	M₃₂	—	3	4	no	—
110	M₃₂	—	6	2	no	—
111	C₂×C₁₆	—	6	2	no	—
112	#97	—	2	2	no	—
113	C₂³×C₃	—	2	1	no	—
114	C₂³×C₃	C₂×C₃⋊C₈	1	1	yes	characteristic
115	C₂³×C₃	C₂×C₃×C₈	1	1	yes	characteristic
116	C₂×C₃×C₄	—	2	4	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	2	1	no	—
120	C₂×C₃×C₄	—	2	1	no	—
121	C₂×C₃×C₄	—	2	1	no	—
122	C₂×C₃×C₄	—	2	1	no	—
123	C₂×C₃×C₄	—	2	1	no	—
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	—	4	4	no	—
126	C₂×C₃×C₄	—	4	2	no	—
127	C₂×C₃×C₄	—	4	2	no	—
128	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
129	C₂×C₃×C₄	#208	1	1	yes	characteristic
130	C₂×C₃×C₄	#208	1	1	yes	characteristic
131	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
132	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
133	C₂×C₃×C₄	#217	1	1	yes	characteristic
134	C₂×C₃×C₄	C₃×C₄²	1	1	yes	characteristic
135	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₃×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
138	C₂×C₃×C₄	C₃×M₁₆	1	1	yes	characteristic
139	C₂×C₃×C₄	C₃×M₁₆	1	1	yes	characteristic
140	C₃×C₈	—	2	4	no	—
141	C₃×C₈	—	2	2	no	—
142	C₃×C₈	—	2	2	no	—
143	C₃×C₈	—	2	2	no	—
144	C₃×C₈	—	2	2	no	—
145	C₃×C₈	Dic₃⋊C₄	1	2	yes	—
146	C₃×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
147	C₃×C₈	C₃×C₂²⋊C₄	1	2	yes	—
148	C₃×C₈	C₃×C₄⋊C₄	1	2	yes	—
149	C₂×C₃²	—	2	1	no	—
150	C₂×C₃²	—	2	1	no	—
151	C₂×C₃²	#336	1	1	yes	characteristic
152	C₂×C₃²	#349	1	2	yes	—
153	C₂²×C₄	C₃×C₃⋊C₈	1	2	yes	—
154	C₂²×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	C₃×C₃⋊C₈	1	2	yes	—
158	C₂×C₈	C₃×Dic₆	1	1	yes	characteristic
159	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
160	C₂×C₈	C₃×D₁₂	1	1	yes	characteristic
161	C₂×C₈	C₃×#67	1	2	yes	—
162	C₄²	—	2	1	no	—
163	C₄²	—	2	1	no	—
164	C₁₆	—	6	4	no	—
165	C₂²×C₃	—	2	4	no	—
166	C₂²×C₃	—	2	2	no	—
167	C₂²×C₃	—	2	2	no	—
168	C₂²×C₃	—	2	1	no	—
169	C₂²×C₃	—	2	1	no	—
170	C₂²×C₃	—	2	1	no	—
171	C₂²×C₃	C₄×C₃⋊C₈	1	1	yes	characteristic
172	C₂²×C₃	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
173	C₂²×C₃	#830	1	1	yes	characteristic
174	C₂²×C₃	C₃×C₄×C₈	1	1	yes	characteristic
175	C₂²×C₃	C₃×C₂²⋊C₈	1	1	yes	characteristic
176	C₂²×C₃	C₃×C₄⋊C₈	1	1	yes	characteristic
177	C₃×C₄	—	2	1	no	—
178	C₃×C₄	—	2	1	no	—
179	C₃×C₄	—	2	1	no	—
180	C₃×C₄	—	2	1	no	—
181	C₃×C₄	—	4	4	no	—
182	C₃×C₄	—	4	2	no	—
183	C₃×C₄	—	4	2	no	—
184	C₃×C₄	#803	1	1	yes	characteristic
185	C₃×C₄	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
186	C₃×C₄	#830	1	1	yes	characteristic
187	C₃×C₄	C₃⋊#95	1	1	yes	characteristic
188	C₃×C₄	C₃×#95	1	1	yes	characteristic
189	C₃×C₄	C₃×#97	1	1	yes	characteristic
190	C₃×C₄	C₃×C₂²⋊C₈	1	1	yes	characteristic
191	C₃×C₄	C₃×C₄⋊C₈	1	1	yes	characteristic
192	C₃²	#1377	1	1	yes	characteristic, complemented, Sylow
193	C₂³	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
197	C₂×C₄	C₃×#208	1	1	yes	characteristic
198	C₂×C₄	C₃×#208	1	1	yes	characteristic
199	C₂×C₄	C₃×C₄×Dic₃	1	1	yes	characteristic
200	C₂×C₄	C₃×C₄⋊Dic₃	1	1	yes	characteristic
201	C₂×C₄	C₃×#217	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	C₃×Dic₃⋊C₄	1	2	yes	—
204	C₈	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
205	C₂×C₃	—	2	2	no	—
206	C₂×C₃	—	2	2	no	—
207	C₂×C₃	—	2	1	no	—
208	C₂×C₃	—	2	1	no	—
209	C₂×C₃	—	2	1	no	—
210	C₂×C₃	—	2	1	no	—
211	C₂×C₃	—	2	1	no	—
212	C₂×C₃	—	2	2	no	—
213	C₂×C₃	#4524	1	1	yes	characteristic
214	C₂×C₃	#4553	1	2	yes	—
215	C₂×C₃	C₃×#336	1	1	yes	characteristic
216	C₂×C₃	C₃×#349	1	2	yes	—
217	C₂²	—	2	2	no	—
218	C₂²	C₃×C₄×C₃⋊C₈	1	1	yes	characteristic
219	C₂²	C₃×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
220	C₂²	C₃×#830	1	1	yes	characteristic
221	C₄	—	4	2	no	—
222	C₄	C₃×#803	1	1	yes	characteristic
223	C₄	C₃×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
224	C₄	C₃×#830	1	1	yes	characteristic
225	C₄	C₃×C₃⋊#95	1	1	yes	characteristic
226	C₃	—	2	1	no	—
227	C₃	#67664	1	1	yes	characteristic, direct product factor
228	C₃	C₃×#1377	1	1	yes	characteristic, complemented
229	C₂	—	2	1	no	—
230	C₂	—	2	1	no	—
231	C₂	C₃×#4524	1	1	yes	characteristic
232	C₂	C₃×#4553	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	#217
commutator	C₂²×C₃	C₃×C₄×C₈
Frattini	C₂²×C₈	C₂×C₃×D₃
Fitting	C₂×C₃²×#97	C₂
socle	C₂²×C₃²	C₂²⋊C₈
prime radical	C₂³×C₃²	C₂×C₈
cosocle	C₂²×C₃×C₈	C₂²×C₃
hypercenter	C₂×C₃×#97	D₃
hypocenter	C₃	C₃×#1377
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₈ ◁ C₂×C₃×#97

Complements

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

N	H ≅ G/N	type
4. #67664	C₃	direct (N × H)
10. C₃×#812	C₄	semidirect
46. #812	C₃×C₄	semidirect
192. C₃²	#1377	semidirect
227. C₃	#67664	direct (N × H)
228. C₃	C₃×#1377	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#97	2
C₂×C₃×#812	2
C₃×#1377	3
#67664	3

Subgroups — C3×#67664 #11772310

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#67664 #11772310