Subgroups — C₃×#9842 #10628449

Order 768.

626 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#492	C₂	1	1	yes	maximal, characteristic
2	C₃×#1990	C₂	1	2	yes	maximal, complemented
3	C₃×#1957	C₂	1	2	yes	maximal
4	C₃×#1845	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	#9842	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂²×C₃×SD₁₆	—	2	2	no	—
8	C₂×C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×#127	C₂²	1	1	yes	characteristic
10	C₃×#492	—	2	2	no	—
11	C₃×#492	C₄	1	4	yes	complemented
12	C₂×C₃×#107	—	2	2	no	—
13	C₂×C₃×#102	—	2	4	no	—
14	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
15	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₃×#384	C₂²	1	2	yes	—
17	C₃×#340	—	2	4	no	—
18	C₃×#322	—	2	2	no	—
19	C₂×#492	C₂×C₃	1	1	yes	characteristic
20	#1990	C₂×C₃	1	2	yes	complemented
21	#1957	C₂×C₃	1	2	yes	—
22	#1845	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂×#322	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂²×C₃×Q₈	—	2	1	no	—
25	C₂²×C₃×D₄	—	2	1	no	—
26	C₂²×C₃×D₄	—	2	2	no	—
27	C₂×C₃×SD₁₆	—	2	8	no	—
28	C₂×C₃×SD₁₆	—	4	4	no	—
29	C₂×C₃×SD₁₆	—	4	4	no	—
30	C₂²×C₃×C₈	D₄	1	4	yes	—
31	C₂²×C₃×C₈	D₄	1	2	yes	—
32	C₃×C₄⋊Q₈	—	2	1	no	—
33	C₃×C₄⋊Q₈	—	2	2	no	—
34	C₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
35	C₃×#127	—	2	1	no	—
36	C₃×#127	—	2	2	no	—
37	C₃×#127	C₂×C₄	1	2	yes	—
38	C₂×C₃×C₄⋊C₄	—	2	2	no	—
39	C₂×C₃×C₄⋊C₄	—	2	2	no	—
40	C₂×C₃×C₄⋊C₄	—	2	2	no	—
41	C₂×C₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
43	C₃×#107	—	4	4	no	—
44	C₃×#102	—	4	8	no	—
45	C₃×C₄×C₈	—	2	2	no	—
46	C₃×C₄×C₈	—	2	2	no	—
47	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
48	C₃×C₄×C₈	D₄	1	2	yes	—
49	C₂²×SD₁₆	—	2	2	no	—
50	C₂×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
51	C₂×#127	C₂²×C₃	1	1	yes	characteristic
52	#492	—	2	2	no	—
53	#492	C₃×C₄	1	4	yes	complemented
54	C₂×#107	—	2	2	no	—
55	C₂×#102	—	2	4	no	—
56	C₂×C₄×C₈	C₂²×C₃	1	2	yes	—
57	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
58	#384	C₂²×C₃	1	2	yes	—
59	#340	—	2	4	no	—
60	#322	—	2	2	no	—
61	C₂⁴×C₃	—	4	1	no	—
62	C₂×C₃×Q₈	—	2	2	no	—
63	C₂×C₃×Q₈	—	4	1	no	—
64	C₂×C₃×Q₈	—	4	2	no	—
65	C₂×C₃×Q₈	—	4	2	no	—
66	C₂×C₃×D₄	—	2	2	no	—
67	C₂×C₃×D₄	—	2	4	no	—
68	C₂×C₃×D₄	—	4	2	no	—
69	C₂×C₃×D₄	—	4	1	no	—
70	C₂×C₃×D₄	—	4	2	no	—
71	C₂×C₃×D₄	—	4	2	no	—
72	C₂×C₃×D₄	—	4	4	no	—
73	C₂×C₃×D₄	—	4	4	no	—
74	C₂²×C₃×C₄	—	4	1	no	—
75	C₂²×C₃×C₄	—	4	2	no	—
76	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
78	C₃×SD₁₆	—	4	16	no	—
79	C₂×C₃×C₈	—	2	4	no	—
80	C₂×C₃×C₈	—	2	4	no	—
81	C₂×C₃×C₈	—	2	4	no	—
82	C₂×C₃×C₈	C₂²⋊C₄	1	4	yes	—
83	C₂×C₃×C₈	D₈	1	4	yes	—
84	C₂×C₃×C₈	SD₁₆	1	4	yes	—
85	C₃×C₄⋊C₄	—	2	4	no	—
86	C₃×C₄⋊C₄	—	4	2	no	—
87	C₃×C₄⋊C₄	—	4	4	no	—
88	C₃×C₄⋊C₄	—	4	4	no	—
89	C₃×C₄⋊C₄	—	4	4	no	—
90	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
91	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
92	C₃×C₄²	#41	1	2	yes	—
93	C₂²×Q₈	—	2	1	no	—
94	C₂²×D₄	—	2	1	no	—
95	C₂²×D₄	—	2	2	no	—
96	C₂×SD₁₆	—	2	8	no	—
97	C₂×SD₁₆	—	4	4	no	—
98	C₂×SD₁₆	—	4	4	no	—
99	C₂²×C₈	C₃×D₄	1	4	yes	—
100	C₂²×C₈	C₃×D₄	1	2	yes	—
101	C₄⋊Q₈	—	2	1	no	—
102	C₄⋊Q₈	—	2	2	no	—
103	C₄⋊Q₈	C₂×C₃×C₄	1	2	yes	—
104	#127	—	2	1	no	—
105	#127	—	2	2	no	—
106	#127	C₂×C₃×C₄	1	2	yes	—
107	C₂×C₄⋊C₄	—	2	2	no	—
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄⋊C₄	—	2	2	no	—
111	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
112	#107	—	4	4	no	—
113	#102	—	4	8	no	—
114	C₄×C₈	—	2	2	no	—
115	C₄×C₈	—	2	2	no	—
116	C₄×C₈	C₂×C₃×C₄	1	2	yes	—
117	C₄×C₈	C₃×D₄	1	2	yes	—
118	C₂³×C₃	—	4	2	no	—
119	C₂³×C₃	—	8	1	no	—
120	C₂³×C₃	—	8	1	no	—
121	C₂³×C₃	—	8	2	no	—
122	C₂³×C₃	—	8	2	no	—
123	C₂³×C₃	#127	1	1	yes	characteristic
124	C₃×Q₈	—	4	4	no	—
125	C₃×Q₈	—	8	2	no	—
126	C₃×D₄	—	4	4	no	—
127	C₃×D₄	—	4	8	no	—
128	C₃×D₄	—	8	4	no	—
129	C₃×D₄	—	8	2	no	—
130	C₂×C₃×C₄	—	4	2	no	—
131	C₂×C₃×C₄	—	8	1	no	—
132	C₂×C₃×C₄	—	8	1	no	—
133	C₂×C₃×C₄	—	8	2	no	—
134	C₂×C₃×C₄	—	8	2	no	—
135	C₂×C₃×C₄	—	8	2	no	—
136	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
138	C₂×C₃×C₄	#121	1	2	yes	—
139	C₂×C₃×C₄	#124	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
141	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
142	C₂×C₃×C₄	AGL(1,ℤ/8)	1	4	yes	—
143	C₂×C₃×C₄	#137	1	4	yes	—
144	C₃×C₈	—	4	4	no	—
145	C₃×C₈	#102	1	8	yes	complemented
146	C₂⁴	—	4	1	no	—
147	C₂×Q₈	—	2	2	no	—
148	C₂×Q₈	—	4	1	no	—
149	C₂×Q₈	—	4	2	no	—
150	C₂×Q₈	—	4	2	no	—
151	C₂×D₄	—	2	2	no	—
152	C₂×D₄	—	2	4	no	—
153	C₂×D₄	—	4	2	no	—
154	C₂×D₄	—	4	1	no	—
155	C₂×D₄	—	4	2	no	—
156	C₂×D₄	—	4	2	no	—
157	C₂×D₄	—	4	4	no	—
158	C₂×D₄	—	4	4	no	—
159	C₂²×C₄	—	4	1	no	—
160	C₂²×C₄	—	4	2	no	—
161	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
162	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
163	SD₁₆	—	4	16	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	C₃×C₂²⋊C₄	1	4	yes	—
168	C₂×C₈	C₃×D₈	1	4	yes	—
169	C₂×C₈	C₃×SD₁₆	1	4	yes	—
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	4	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄²	C₂²×C₃×C₄	1	1	yes	characteristic
176	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
177	C₄²	C₃×#41	1	2	yes	—
178	C₂²×C₃	—	8	2	no	—
179	C₂²×C₃	—	8	4	no	—
180	C₂²×C₃	—	16	2	no	—
181	C₂²×C₃	—	16	1	no	—
182	C₂²×C₃	—	16	1	no	—
183	C₂²×C₃	#390	1	1	yes	characteristic
184	C₂²×C₃	#492	1	1	yes	characteristic
185	C₂²×C₃	#493	1	1	yes	characteristic
186	C₂²×C₃	#496	1	2	yes	—
187	C₂²×C₃	#497	1	2	yes	—
188	C₃×C₄	—	8	2	no	—
189	C₃×C₄	—	16	2	no	—
190	C₃×C₄	C₂×#102	1	2	yes	—
191	C₃×C₄	#440	1	4	yes	—
192	C₃×C₄	#468	1	4	yes	—
193	C₃×C₄	#486	1	2	yes	—
194	C₂³	—	4	2	no	—
195	C₂³	—	8	1	no	—
196	C₂³	—	8	1	no	—
197	C₂³	—	8	2	no	—
198	C₂³	—	8	2	no	—
199	C₂³	C₃×#127	1	1	yes	characteristic
200	Q₈	—	4	4	no	—
201	Q₈	—	8	2	no	—
202	D₄	—	4	4	no	—
203	D₄	—	4	8	no	—
204	D₄	—	8	4	no	—
205	D₄	—	8	2	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	8	1	no	—
208	C₂×C₄	—	8	1	no	—
209	C₂×C₄	—	8	2	no	—
210	C₂×C₄	—	8	2	no	—
211	C₂×C₄	—	8	2	no	—
212	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
213	C₂×C₄	C₃×C₄×D₄	1	2	yes	—
214	C₂×C₄	C₃×#121	1	2	yes	—
215	C₂×C₄	C₃×#124	1	1	yes	characteristic
216	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
217	C₂×C₄	C₂×C₃×SD₁₆	1	2	yes	—
218	C₂×C₄	C₃×AGL(1,ℤ/8)	1	4	yes	—
219	C₂×C₄	C₃×#137	1	4	yes	—
220	C₈	—	4	4	no	—
221	C₈	C₃×#102	1	8	yes	complemented
222	C₂×C₃	—	16	2	no	—
223	C₂×C₃	#1714	1	2	yes	—
224	C₂×C₃	#1721	1	1	yes	characteristic
225	C₂×C₃	#1722	1	1	yes	characteristic
226	C₂×C₃	#1957	1	1	yes	characteristic
227	C₂×C₃	#1973	1	2	yes	—
228	C₂²	—	8	2	no	—
229	C₂²	—	8	4	no	—
230	C₂²	—	16	2	no	—
231	C₂²	—	16	1	no	—
232	C₂²	—	16	1	no	—
233	C₂²	C₃×#390	1	1	yes	characteristic
234	C₂²	C₃×#492	1	1	yes	characteristic
235	C₂²	C₃×#493	1	1	yes	characteristic
236	C₂²	C₃×#496	1	2	yes	—
237	C₂²	C₃×#497	1	2	yes	—
238	C₄	—	8	2	no	—
239	C₄	—	16	2	no	—
240	C₄	C₂×C₃×#102	1	2	yes	—
241	C₄	C₃×#440	1	4	yes	—
242	C₄	C₃×#468	1	4	yes	—
243	C₄	C₃×#486	1	2	yes	—
244	C₃	#9842	1	1	yes	characteristic, direct product factor, Sylow
245	C₂	—	16	2	no	—
246	C₂	C₃×#1714	1	2	yes	—
247	C₂	C₃×#1721	1	1	yes	characteristic
248	C₂	C₃×#1722	1	1	yes	characteristic
249	C₂	C₃×#1957	1	1	yes	characteristic
250	C₂	C₃×#1973	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	#127
commutator	C₄²	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₂×C₃×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#1990	C₂	semidirect
4. C₃×#1845	C₂	semidirect
5. C₂×C₃×#322	C₂	semidirect
6. #9842	C₃	direct (N × H)
11. C₃×#492	C₄	semidirect
20. #1990	C₂×C₃	semidirect
22. #1845	C₂×C₃	semidirect
23. C₂×#322	C₂×C₃	semidirect
53. #492	C₃×C₄	semidirect
145. C₃×C₈	#102	semidirect
221. C₈	C₃×#102	semidirect
244. C₃	#9842	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×#492	2
C₃×#1990	2
C₃×#1957	2
C₃×#1845	2
C₂×C₃×#322	2
#9842	3

Subgroups — C3×#9842 #10628449

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#9842 #10628449