Subgroups — C₃×#13279 #10631886

Order 768.

366 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×C₃×#500	C₂	1	1	yes	maximal, characteristic
2	C₃×#2150	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×#108	C₂	1	1	yes	maximal, characteristic
4	C₃×#1387	C₂	1	4	yes	maximal
5	#13279	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂×C₃×C₄⋊Q₈	—	2	1	no	—
7	C₂×C₃×M₃₂	C₂²	1	2	yes	—
8	C₃×#500	—	2	1	no	—
9	C₃×#500	—	2	1	no	—
10	C₃×#500	C₄	1	2	yes	—
11	C₃×#500	C₄	1	2	yes	—
12	C₂×C₃×#108	C₂²	1	2	yes	—
13	C₂×C₃×#107	—	2	1	no	—
14	C₂×C₃×#107	C₂²	1	2	yes	—
15	C₃×C₄×M₁₆	—	2	1	no	—
16	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄⋊C₁₆	—	2	1	no	—
18	C₃×C₄⋊C₁₆	—	2	1	no	—
19	C₃×#341	—	2	1	no	—
20	C₂×#500	C₂×C₃	1	1	yes	characteristic
21	#2150	C₂×C₃	1	1	yes	characteristic
22	C₄×#108	C₂×C₃	1	1	yes	characteristic
23	#1387	C₂×C₃	1	4	yes	—
24	C₂²×C₃×Q₈	—	4	1	no	—
25	C₂×C₃×M₁₆	—	2	2	no	—
26	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
28	C₃×C₄⋊Q₈	—	2	2	no	—
29	C₃×C₄⋊Q₈	—	2	2	no	—
30	C₃×C₄⋊Q₈	—	4	1	no	—
31	C₃×C₄⋊Q₈	—	4	1	no	—
32	C₂×C₃×C₄⋊C₄	—	2	2	no	—
33	C₂×C₃×C₄⋊C₄	—	4	1	no	—
34	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
35	C₃×M₃₂	—	2	4	no	—
36	C₂×C₃×C₁₆	—	2	2	no	—
37	C₃×#108	—	2	2	no	—
38	C₃×#108	Q₈	1	4	yes	complemented
39	C₃×#107	—	2	2	no	—
40	C₃×#107	—	2	2	no	—
41	C₃×#107	—	2	2	no	—
42	C₃×#107	C₂×C₄	1	4	yes	—
43	C₃×#97	—	4	1	no	—
44	C₃×C₄×C₈	—	4	1	no	—
45	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
47	C₃×C₄×C₈	D₄	1	1	yes	characteristic
48	C₃×C₄×C₈	D₄	1	1	yes	characteristic
49	C₂×C₄⋊Q₈	—	2	1	no	—
50	C₂×M₃₂	C₂²×C₃	1	2	yes	—
51	#500	—	2	1	no	—
52	#500	—	2	1	no	—
53	#500	C₃×C₄	1	2	yes	—
54	#500	C₃×C₄	1	2	yes	—
55	C₂×#108	C₂²×C₃	1	2	yes	—
56	C₂×#107	—	2	1	no	—
57	C₂×#107	C₂²×C₃	1	2	yes	—
58	C₄×M₁₆	—	2	1	no	—
59	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
60	C₄⋊C₁₆	—	2	1	no	—
61	C₄⋊C₁₆	—	2	1	no	—
62	#341	—	2	1	no	—
63	C₂×C₃×Q₈	—	4	2	no	—
64	C₂×C₃×Q₈	—	4	2	no	—
65	C₂×C₃×Q₈	—	8	4	no	—
66	C₂²×C₃×C₄	—	4	2	no	—
67	C₂²×C₃×C₄	SD₁₆	1	1	yes	characteristic
68	C₂²×C₃×C₄	SD₁₆	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
70	C₃×M₁₆	—	2	4	no	—
71	C₃×M₁₆	—	4	2	no	—
72	C₂×C₃×C₈	—	2	2	no	—
73	C₂×C₃×C₈	—	4	2	no	—
74	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
79	C₂×C₃×C₈	#41	1	2	yes	—
80	C₃×C₄⋊C₄	—	2	4	no	—
81	C₃×C₄⋊C₄	—	4	2	no	—
82	C₃×C₄⋊C₄	—	4	2	no	—
83	C₃×C₄⋊C₄	—	4	2	no	—
84	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
85	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
86	C₃×C₄²	D₈	1	1	yes	characteristic
87	C₃×C₄²	D₈	1	1	yes	characteristic
88	C₃×C₁₆	—	4	2	no	—
89	C₂²×Q₈	—	4	1	no	—
90	C₂×M₁₆	—	2	2	no	—
91	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
92	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
93	C₄⋊Q₈	—	2	2	no	—
94	C₄⋊Q₈	—	2	2	no	—
95	C₄⋊Q₈	—	4	1	no	—
96	C₄⋊Q₈	—	4	1	no	—
97	C₂×C₄⋊C₄	—	2	2	no	—
98	C₂×C₄⋊C₄	—	4	1	no	—
99	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
100	M₃₂	—	2	4	no	—
101	C₂×C₁₆	—	2	2	no	—
102	#108	—	2	2	no	—
103	#108	C₃×Q₈	1	4	yes	complemented
104	#107	—	2	2	no	—
105	#107	—	2	2	no	—
106	#107	—	2	2	no	—
107	#107	C₂×C₃×C₄	1	4	yes	—
108	#97	—	4	1	no	—
109	C₄×C₈	—	4	1	no	—
110	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
111	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
112	C₄×C₈	C₃×D₄	1	1	yes	characteristic
113	C₄×C₈	C₃×D₄	1	1	yes	characteristic
114	C₂³×C₃	C₂×SD₁₆	1	1	yes	characteristic
115	C₃×Q₈	—	8	4	no	—
116	C₃×Q₈	—	8	4	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	4	4	no	—
120	C₂×C₃×C₄	—	8	4	no	—
121	C₂×C₃×C₄	#102	1	1	yes	characteristic
122	C₂×C₃×C₄	#102	1	1	yes	characteristic
123	C₂×C₃×C₄	#102	1	1	yes	characteristic
124	C₂×C₃×C₄	#102	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
128	C₂×C₃×C₄	#137	1	2	yes	—
129	C₃×C₈	—	2	1	no	—
130	C₃×C₈	—	2	1	no	—
131	C₃×C₈	—	4	4	no	—
132	C₃×C₈	C₄×Q₈	1	2	yes	—
133	C₃×C₈	#124	1	1	yes	characteristic
134	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
135	C₂×Q₈	—	4	2	no	—
136	C₂×Q₈	—	4	2	no	—
137	C₂×Q₈	—	8	4	no	—
138	C₂²×C₄	—	4	2	no	—
139	C₂²×C₄	C₃×SD₁₆	1	1	yes	characteristic
140	C₂²×C₄	C₃×SD₁₆	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
142	M₁₆	—	2	4	no	—
143	M₁₆	—	4	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
149	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
150	C₂×C₈	C₂×C₃×Q₈	1	2	yes	—
151	C₂×C₈	C₃×#41	1	2	yes	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄²	C₃×C₂²⋊C₄	1	1	yes	characteristic
157	C₄²	C₃×C₂²⋊C₄	1	1	yes	characteristic
158	C₄²	C₃×D₈	1	1	yes	characteristic
159	C₄²	C₃×D₈	1	1	yes	characteristic
160	C₁₆	—	4	2	no	—
161	C₂²×C₃	—	2	1	no	—
162	C₂²×C₃	—	2	1	no	—
163	C₂²×C₃	C₂×#102	1	1	yes	characteristic
164	C₂²×C₃	#475	1	2	yes	—
165	C₃×C₄	—	2	1	no	—
166	C₃×C₄	—	2	1	no	—
167	C₃×C₄	—	8	4	no	—
168	C₃×C₄	#361	1	2	yes	—
169	C₃×C₄	#362	1	2	yes	—
170	C₃×C₄	#391	1	1	yes	characteristic
171	C₃×C₄	#419	1	1	yes	characteristic
172	C₃×C₄	#474	1	2	yes	—
173	C₂³	C₂×C₃×SD₁₆	1	1	yes	characteristic
174	Q₈	—	8	4	no	—
175	Q₈	—	8	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	8	4	no	—
180	C₂×C₄	C₃×#102	1	1	yes	characteristic
181	C₂×C₄	C₃×#102	1	1	yes	characteristic
182	C₂×C₄	C₃×#102	1	1	yes	characteristic
183	C₂×C₄	C₃×#102	1	1	yes	characteristic
184	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
185	C₂×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
186	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
187	C₂×C₄	C₃×#137	1	2	yes	—
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	4	4	no	—
191	C₈	C₃×C₄×Q₈	1	2	yes	—
192	C₈	C₃×#124	1	1	yes	characteristic
193	C₈	C₃×C₄⋊Q₈	1	1	yes	characteristic
194	C₂×C₃	—	2	2	no	—
195	C₂×C₃	#1988	1	1	yes	characteristic
196	C₂×C₃	C₂×#361	1	1	yes	characteristic
197	C₂×C₃	C₂×#362	1	1	yes	characteristic
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	C₂×C₃×#102	1	1	yes	characteristic
201	C₂²	C₃×#475	1	2	yes	—
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	8	4	no	—
205	C₄	C₃×#361	1	2	yes	—
206	C₄	C₃×#362	1	2	yes	—
207	C₄	C₃×#391	1	1	yes	characteristic
208	C₄	C₃×#419	1	1	yes	characteristic
209	C₄	C₃×#474	1	2	yes	—
210	C₃	#13279	1	1	yes	characteristic, direct product factor, Sylow
211	C₂	—	2	2	no	—
212	C₂	C₃×#1988	1	1	yes	characteristic
213	C₂	C₂×C₃×#361	1	1	yes	characteristic
214	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	C₂²×C₈	C₂³×C₃
prime radical	C₂³×C₃	C₂×SD₁₆
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
5. #13279	C₃	direct (N × H)
38. C₃×#108	Q₈	semidirect
103. #108	C₃×Q₈	semidirect
210. C₃	#13279	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#500	2
C₃×#2150	2
C₃×C₄×#108	2
C₃×#1387	2
#13279	3

Subgroups — C3×#13279 #10631886

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#13279 #10631886