Subgroups — C₃×#11681 #10630288

Order 768.

324 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3170	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₃×#1566	C₂	1	1	yes	maximal, characteristic
4	C₃×#1334	C₂	1	4	yes	maximal, complemented
5	#11681	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₃×#533	—	2	1	no	—
7	C₂×C₃×M₃₂	—	2	1	no	—
8	C₃×#500	C₄	1	1	yes	characteristic
9	C₃×#494	C₄	1	1	yes	characteristic
10	C₃×#487	—	2	1	no	—
11	C₃×#467	—	2	1	no	—
12	C₃×#467	C₄	1	2	yes	—
13	C₃×C₄×Q₁₆	—	2	1	no	—
14	C₃×C₄×Q₁₆	C₂²	1	2	yes	—
15	C₃×#423	—	2	1	no	—
16	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₃×#346	C₂²	1	2	yes	—
18	C₃×C₄×C₁₆	—	2	1	no	—
19	C₃×#334	—	2	1	no	—
20	C₃×#334	C₂²	1	2	yes	—
21	#3170	C₂×C₃	1	1	yes	characteristic
22	C₄×M₃₂	C₂×C₃	1	1	yes	characteristic
23	#1566	C₂×C₃	1	1	yes	characteristic
24	#1334	C₂×C₃	1	4	yes	complemented
25	C₂×C₃×Q₁₆	—	2	1	no	—
26	C₂×C₃×Q₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₃×M₁₆	—	4	1	no	—
28	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
30	C₃×C₄⋊Q₈	—	2	1	no	—
31	C₃×C₄⋊Q₈	—	2	1	no	—
32	C₃×#125	—	4	1	no	—
33	C₃×C₂²⋊Q₈	—	2	2	no	—
34	C₃×C₂²⋊Q₈	—	4	1	no	—
35	C₃×C₄×Q₈	—	2	2	no	—
36	C₃×C₄×Q₈	—	4	1	no	—
37	C₃×#117	—	4	1	no	—
38	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
39	C₃×M₃₂	—	2	4	no	—
40	C₂×C₃×C₁₆	—	2	2	no	—
41	C₃×#107	—	2	1	no	—
42	C₃×#107	C₂×C₄	1	2	yes	—
43	C₃×C₄⋊C₈	—	2	2	no	—
44	C₃×C₄⋊C₈	—	4	1	no	—
45	C₃×#103	—	2	2	no	—
46	C₃×#103	—	2	2	no	—
47	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₃×C₄×C₈	D₄	1	1	yes	characteristic
49	C₃×C₄×C₈	D₄	1	1	yes	characteristic
50	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
51	#533	—	2	1	no	—
52	C₂×M₃₂	—	2	1	no	—
53	#500	C₃×C₄	1	1	yes	characteristic
54	#494	C₃×C₄	1	1	yes	characteristic
55	#487	—	2	1	no	—
56	#467	—	2	1	no	—
57	#467	C₃×C₄	1	2	yes	—
58	C₄×Q₁₆	—	2	1	no	—
59	C₄×Q₁₆	C₂²×C₃	1	2	yes	—
60	#423	—	2	1	no	—
61	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
62	#346	C₂²×C₃	1	2	yes	—
63	C₄×C₁₆	—	2	1	no	—
64	#334	—	2	1	no	—
65	#334	C₂²×C₃	1	2	yes	—
66	C₂×C₃×Q₈	—	2	2	no	—
67	C₂×C₃×Q₈	—	4	1	no	—
68	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
70	C₂²×C₃×C₄	SD₁₆	1	1	yes	characteristic
71	C₃×Q₁₆	—	2	4	no	—
72	C₃×Q₁₆	—	4	2	no	—
73	C₃×M₁₆	—	8	2	no	—
74	C₂×C₃×C₈	—	2	2	no	—
75	C₂×C₃×C₈	—	2	2	no	—
76	C₂×C₃×C₈	—	4	2	no	—
77	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
81	C₃×C₄⋊C₄	—	2	2	no	—
82	C₃×C₄⋊C₄	—	4	1	no	—
83	C₃×C₄⋊C₄	—	4	1	no	—
84	C₃×C₄⋊C₄	—	4	2	no	—
85	C₃×C₄⋊C₄	—	4	1	no	—
86	C₃×C₄⋊C₄	—	4	2	no	—
87	C₃×C₂²⋊C₄	—	4	2	no	—
88	C₃×C₄²	—	4	2	no	—
89	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
90	C₃×C₄²	D₈	1	1	yes	characteristic
91	C₃×C₄²	SD₁₆	1	1	yes	characteristic
92	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
93	C₃×C₁₆	—	2	4	no	—
94	C₂×Q₁₆	—	2	1	no	—
95	C₂×Q₁₆	C₂×C₃×C₄	1	2	yes	—
96	C₂×M₁₆	—	4	1	no	—
97	C₂²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
98	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
99	C₄⋊Q₈	—	2	1	no	—
100	C₄⋊Q₈	—	2	1	no	—
101	#125	—	4	1	no	—
102	C₂²⋊Q₈	—	2	2	no	—
103	C₂²⋊Q₈	—	4	1	no	—
104	C₄×Q₈	—	2	2	no	—
105	C₄×Q₈	—	4	1	no	—
106	#117	—	4	1	no	—
107	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
108	M₃₂	—	2	4	no	—
109	C₂×C₁₆	—	2	2	no	—
110	#107	—	2	1	no	—
111	#107	C₂×C₃×C₄	1	2	yes	—
112	C₄⋊C₈	—	2	2	no	—
113	C₄⋊C₈	—	4	1	no	—
114	#103	—	2	2	no	—
115	#103	—	2	2	no	—
116	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
117	C₄×C₈	C₃×D₄	1	1	yes	characteristic
118	C₄×C₈	C₃×D₄	1	1	yes	characteristic
119	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
120	C₂³×C₃	#102	1	1	yes	characteristic
121	C₃×Q₈	—	4	2	no	—
122	C₃×Q₈	—	4	2	no	—
123	C₃×Q₈	—	8	2	no	—
124	C₂×C₃×C₄	—	2	2	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	4	2	no	—
129	C₂×C₃×C₄	—	4	2	no	—
130	C₂×C₃×C₄	#102	1	1	yes	characteristic
131	C₂×C₃×C₄	#102	1	1	yes	characteristic
132	C₂×C₃×C₄	#102	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
136	C₃×C₈	—	2	1	no	—
137	C₃×C₈	—	2	1	no	—
138	C₃×C₈	—	8	2	no	—
139	C₃×C₈	#104	1	2	yes	—
140	C₃×C₈	#131	1	2	yes	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	—	4	1	no	—
143	C₂²×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
145	C₂²×C₄	C₃×SD₁₆	1	1	yes	characteristic
146	Q₁₆	—	2	4	no	—
147	Q₁₆	—	4	2	no	—
148	M₁₆	—	8	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
153	C₂×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
155	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	4	1	no	—
158	C₄⋊C₄	—	4	1	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	1	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₂²⋊C₄	—	4	2	no	—
163	C₄²	—	4	2	no	—
164	C₄²	C₃×C₂²⋊C₄	1	1	yes	characteristic
165	C₄²	C₃×D₈	1	1	yes	characteristic
166	C₄²	C₃×SD₁₆	1	1	yes	characteristic
167	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
168	C₁₆	—	2	4	no	—
169	C₂²×C₃	—	2	1	no	—
170	C₂²×C₃	—	2	1	no	—
171	C₂²×C₃	—	2	1	no	—
172	C₂²×C₃	C₂×#102	1	1	yes	characteristic
173	C₃×C₄	—	2	1	no	—
174	C₃×C₄	—	2	1	no	—
175	C₃×C₄	—	2	1	no	—
176	C₃×C₄	—	8	2	no	—
177	C₃×C₄	—	8	2	no	—
178	C₃×C₄	#360	1	2	yes	—
179	C₃×C₄	#362	1	2	yes	—
180	C₃×C₄	#408	1	1	yes	characteristic
181	C₃×C₄	C₂×#104	1	1	yes	characteristic
182	C₂³	C₃×#102	1	1	yes	characteristic
183	Q₈	—	4	2	no	—
184	Q₈	—	4	2	no	—
185	Q₈	—	8	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	C₃×#102	1	1	yes	characteristic
193	C₂×C₄	C₃×#102	1	1	yes	characteristic
194	C₂×C₄	C₃×#102	1	1	yes	characteristic
195	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
196	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
197	C₂×C₄	C₂×C₃×SD₁₆	1	1	yes	characteristic
198	C₈	—	2	1	no	—
199	C₈	—	2	1	no	—
200	C₈	—	8	2	no	—
201	C₈	C₃×#104	1	2	yes	—
202	C₈	C₃×#131	1	2	yes	—
203	C₂×C₃	—	4	1	no	—
204	C₂×C₃	#1492	1	1	yes	characteristic
205	C₂×C₃	C₂×#360	1	1	yes	characteristic
206	C₂×C₃	C₂×#362	1	1	yes	characteristic
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	C₂×C₃×#102	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	8	2	no	—
215	C₄	—	8	2	no	—
216	C₄	C₃×#360	1	2	yes	—
217	C₄	C₃×#362	1	2	yes	—
218	C₄	C₃×#408	1	1	yes	characteristic
219	C₄	C₂×C₃×#104	1	1	yes	characteristic
220	C₃	#11681	1	1	yes	characteristic, direct product factor, Sylow
221	C₂	—	4	1	no	—
222	C₂	C₃×#1492	1	1	yes	characteristic
223	C₂	C₂×C₃×#360	1	1	yes	characteristic
224	C₂	C₂×C₃×#362	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#102
commutator	C₂×C₈	C₂²×C₃×C₄
Frattini, cosocle	C₄×C₈	C₂³×C₃
prime radical	C₂³×C₃	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃×#1334	C₂	semidirect
5. #11681	C₃	direct (N × H)
24. #1334	C₂×C₃	semidirect
220. C₃	#11681	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#3170	2
C₃×C₄×M₃₂	2
C₃×#1566	2
C₃×#1334	2
#11681	3

Subgroups — C3×#11681 #10630288

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#11681 #10630288