Subgroups — C₃×#32684 #10651291

Order 768.

648 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2567	C₂	1	2	yes	maximal
2	C₃×#2087	C₂	1	2	yes	maximal
3	C₃×#2057	C₂	1	4	yes	maximal
4	C₃×#1984	C₂	1	1	yes	maximal, characteristic
5	C₃×#1942	C₂	1	2	yes	maximal
6	C₃×#1921	C₂	1	4	yes	maximal
7	#32684	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×#107	C₂²	1	2	yes	—
11	C₂×C₃×#107	C₂²	1	2	yes	—
12	C₂×C₃×#107	C₂²	1	2	yes	—
13	C₂×C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₃×#395	—	2	2	no	—
16	C₃×#395	—	2	2	no	—
17	C₃×#391	—	2	2	no	—
18	C₃×#389	C₂²	1	4	yes	—
19	C₃×#389	C₂²	1	4	yes	—
20	C₃×#384	—	2	2	no	—
21	C₃×C₄×C₄⋊C₄	C₂²	1	2	yes	—
22	C₃×#340	C₂²	1	4	yes	—
23	C₃×#340	C₂²	1	4	yes	—
24	C₃×#340	C₂²	1	2	yes	—
25	C₃×#336	C₂²	1	4	yes	—
26	#2567	C₂×C₃	1	2	yes	—
27	#2087	C₂×C₃	1	2	yes	—
28	#2057	C₂×C₃	1	4	yes	—
29	#1984	C₂×C₃	1	1	yes	characteristic
30	#1942	C₂×C₃	1	2	yes	—
31	#1921	C₂×C₃	1	4	yes	—
32	C₂²×C₃×Q₈	—	2	2	no	—
33	C₂²×C₃×Q₈	D₄	1	1	yes	characteristic
34	C₂²×C₃×C₈	C₂³	1	2	yes	—
35	C₂²×C₃×C₈	C₂³	1	2	yes	—
36	C₃×C₄⋊Q₈	—	4	2	no	—
37	C₃×C₄⋊Q₈	—	4	2	no	—
38	C₃×C₄×Q₈	—	4	1	no	—
39	C₃×C₄×Q₈	—	4	1	no	—
40	C₂×C₃×C₄⋊C₄	—	2	2	no	—
41	C₂×C₃×C₄⋊C₄	—	2	4	no	—
42	C₂×C₃×C₄⋊C₄	—	2	4	no	—
43	C₂×C₃×C₄⋊C₄	D₄	1	2	yes	—
44	C₂×C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
45	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
46	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
47	C₂×C₃×C₄²	—	2	2	no	—
48	C₂×C₃×C₄²	C₂³	1	2	yes	—
49	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
50	C₃×#107	—	4	2	no	—
51	C₃×#107	—	4	2	no	—
52	C₃×#107	Q₈	1	8	yes	complemented
53	C₃×C₄⋊C₈	—	4	1	no	—
54	C₃×C₄×C₈	—	4	1	no	—
55	C₃×#95	—	2	2	no	—
56	C₃×#95	—	2	4	no	—
57	C₃×#95	—	2	4	no	—
58	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
59	C₂×C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
60	C₂×#107	C₂²×C₃	1	2	yes	—
61	C₂×#107	C₂²×C₃	1	2	yes	—
62	C₂×#107	C₂²×C₃	1	2	yes	—
63	C₂×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
65	#395	—	2	2	no	—
66	#395	—	2	2	no	—
67	#391	—	2	2	no	—
68	#389	C₂²×C₃	1	4	yes	—
69	#389	C₂²×C₃	1	4	yes	—
70	#384	—	2	2	no	—
71	C₄×C₄⋊C₄	C₂²×C₃	1	2	yes	—
72	#340	C₂²×C₃	1	4	yes	—
73	#340	C₂²×C₃	1	4	yes	—
74	#340	C₂²×C₃	1	2	yes	—
75	#336	C₂²×C₃	1	4	yes	—
76	C₂×C₃×Q₈	—	4	1	no	—
77	C₂×C₃×Q₈	—	4	1	no	—
78	C₂×C₃×Q₈	—	4	4	no	—
79	C₂×C₃×Q₈	—	8	4	no	—
80	C₂×C₃×Q₈	Q₁₆	1	4	yes	—
81	C₂²×C₃×C₄	—	2	4	no	—
82	C₂²×C₃×C₄	—	2	4	no	—
83	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
84	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
87	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
88	C₂×C₃×C₈	—	2	2	no	—
89	C₂×C₃×C₈	—	2	2	no	—
90	C₂×C₃×C₈	—	2	2	no	—
91	C₂×C₃×C₈	—	2	4	no	—
92	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
93	C₃×C₄⋊C₄	—	2	4	no	—
94	C₃×C₄⋊C₄	—	2	4	no	—
95	C₃×C₄⋊C₄	—	2	4	no	—
96	C₃×C₄⋊C₄	—	4	2	no	—
97	C₃×C₄⋊C₄	—	4	1	no	—
98	C₃×C₄⋊C₄	—	4	4	no	—
99	C₃×C₄⋊C₄	—	8	4	no	—
100	C₃×C₄⋊C₄	—	8	4	no	—
101	C₃×C₄⋊C₄	C₂×Q₈	1	8	yes	—
102	C₃×C₄²	—	2	1	no	—
103	C₃×C₄²	—	2	1	no	—
104	C₃×C₄²	—	2	2	no	—
105	C₃×C₄²	—	2	2	no	—
106	C₃×C₄²	—	4	4	no	—
107	C₂²×Q₈	—	2	2	no	—
108	C₂²×Q₈	C₃×D₄	1	1	yes	characteristic
109	C₂²×C₈	C₂³×C₃	1	2	yes	—
110	C₂²×C₈	C₂³×C₃	1	2	yes	—
111	C₄⋊Q₈	—	4	2	no	—
112	C₄⋊Q₈	—	4	2	no	—
113	C₄×Q₈	—	4	1	no	—
114	C₄×Q₈	—	4	1	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	2	4	no	—
117	C₂×C₄⋊C₄	—	2	4	no	—
118	C₂×C₄⋊C₄	C₃×D₄	1	2	yes	—
119	C₂×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
120	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
121	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
122	C₂×C₄²	—	2	2	no	—
123	C₂×C₄²	C₂³×C₃	1	2	yes	—
124	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
125	#107	—	4	2	no	—
126	#107	—	4	2	no	—
127	#107	C₃×Q₈	1	8	yes	complemented
128	C₄⋊C₈	—	4	1	no	—
129	C₄×C₈	—	4	1	no	—
130	#95	—	2	2	no	—
131	#95	—	2	4	no	—
132	#95	—	2	4	no	—
133	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
134	C₃×Q₈	—	4	4	no	—
135	C₃×Q₈	—	8	8	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	2	2	no	—
142	C₂×C₃×C₄	—	2	8	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	2	4	no	—
145	C₂×C₃×C₄	—	4	4	no	—
146	C₂×C₃×C₄	—	4	4	no	—
147	C₂×C₃×C₄	—	4	4	no	—
148	C₂×C₃×C₄	—	4	4	no	—
149	C₂×C₃×C₄	—	4	4	no	—
150	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
151	C₂×C₃×C₄	C₂×Q₁₆	1	4	yes	—
152	C₂×C₃×C₄	C₂²×Q₈	1	2	yes	—
153	C₂×C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
154	C₂×C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂×C₃×C₄	2₋¹⁺⁴	1	2	yes	—
156	C₃×C₈	—	2	4	no	—
157	C₃×C₈	—	4	2	no	—
158	C₂×Q₈	—	4	1	no	—
159	C₂×Q₈	—	4	1	no	—
160	C₂×Q₈	—	4	4	no	—
161	C₂×Q₈	—	8	4	no	—
162	C₂×Q₈	C₃×Q₁₆	1	4	yes	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
166	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
169	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	C₂×C₃×Q₈	1	4	yes	—
175	C₄⋊C₄	—	2	4	no	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	1	no	—
180	C₄⋊C₄	—	4	4	no	—
181	C₄⋊C₄	—	8	4	no	—
182	C₄⋊C₄	—	8	4	no	—
183	C₄⋊C₄	C₂×C₃×Q₈	1	8	yes	—
184	C₄²	—	2	1	no	—
185	C₄²	—	2	1	no	—
186	C₄²	—	2	2	no	—
187	C₄²	—	2	2	no	—
188	C₄²	—	4	4	no	—
189	C₂²×C₃	#534	1	1	yes	characteristic
190	C₂²×C₃	D₄×Q₈	1	2	yes	—
191	C₂²×C₃	C₂²×Q₁₆	1	1	yes	characteristic
192	C₂²×C₃	#575	1	1	yes	characteristic
193	C₂²×C₃	#576	1	2	yes	—
194	C₃×C₄	—	2	4	no	—
195	C₃×C₄	—	2	2	no	—
196	C₃×C₄	—	4	8	no	—
197	C₃×C₄	—	4	1	no	—
198	C₃×C₄	—	4	2	no	—
199	C₃×C₄	—	8	4	no	—
200	C₃×C₄	#544	1	2	yes	—
201	C₃×C₄	Q₈²	1	1	yes	characteristic
202	C₃×C₄	#563	1	1	yes	characteristic
203	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
204	Q₈	—	4	4	no	—
205	Q₈	—	8	8	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	C₂×C₃×Q₁₆	1	2	yes	—
221	C₂×C₄	C₂×C₃×Q₁₆	1	4	yes	—
222	C₂×C₄	C₂²×C₃×Q₈	1	2	yes	—
223	C₂×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
224	C₂×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
225	C₂×C₄	C₃×2₋¹⁺⁴	1	2	yes	—
226	C₈	—	2	4	no	—
227	C₈	—	4	2	no	—
228	C₂×C₃	#2734	1	1	yes	characteristic
229	C₂×C₃	#3189	1	1	yes	characteristic
230	C₂×C₃	#3199	1	1	yes	characteristic
231	C₂×C₃	Q₈×Q₁₆	1	2	yes	—
232	C₂×C₃	#3389	1	2	yes	—
233	C₂²	C₃×#534	1	1	yes	characteristic
234	C₂²	C₃×D₄×Q₈	1	2	yes	—
235	C₂²	C₂²×C₃×Q₁₆	1	1	yes	characteristic
236	C₂²	C₃×#575	1	1	yes	characteristic
237	C₂²	C₃×#576	1	2	yes	—
238	C₄	—	2	4	no	—
239	C₄	—	2	2	no	—
240	C₄	—	4	8	no	—
241	C₄	—	4	2	no	—
242	C₄	—	4	1	no	—
243	C₄	—	8	4	no	—
244	C₄	C₃×#544	1	2	yes	—
245	C₄	C₃×Q₈²	1	1	yes	characteristic
246	C₄	C₃×#563	1	1	yes	characteristic
247	C₃	#32684	1	1	yes	characteristic, direct product factor, Sylow
248	C₂	C₃×#2734	1	1	yes	characteristic
249	C₂	C₃×#3189	1	1	yes	characteristic
250	C₂	C₃×#3199	1	1	yes	characteristic
251	C₂	C₃×Q₈×Q₁₆	1	2	yes	—
252	C₂	C₃×#3389	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
7. #32684	C₃	direct (N × H)
52. C₃×#107	Q₈	semidirect
127. #107	C₃×Q₈	semidirect
247. C₃	#32684	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×#2567	2
C₃×#2087	2
C₃×#2057	2
C₃×#1984	2
C₃×#1942	2
C₃×#1921	2
#32684	3

Subgroups — C3×#32684 #10651291

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#32684 #10651291