Subgroups — C₃×#29812 #10648419

Order 768.

824 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#433	C₂	1	1	yes	maximal, characteristic
3	C₃×#2621	C₂	1	2	yes	maximal
4	C₃×#2056	C₂	1	4	yes	maximal, complemented
5	C₃×#2001	C₂	1	4	yes	maximal
6	C₃×#1960	C₂	1	2	yes	maximal, complemented
7	C₃×#1835	C₂	1	1	yes	maximal, characteristic, complemented
8	#29812	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂²×C₃×Q₁₆	C₂²	1	4	yes	—
10	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₃×C₂²⋊Q₈	C₂²	1	4	yes	—
12	C₂×C₃×#117	C₂²	1	1	yes	characteristic
13	C₃×#494	—	2	4	no	—
14	C₃×#433	—	2	4	no	—
15	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃×#103	C₂²	1	8	yes	—
17	C₂×C₃×C₂²⋊C₈	C₂²	1	2	yes	—
18	C₂×C₃×#97	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
20	C₃×#398	—	2	4	no	—
21	C₃×#397	—	2	4	no	—
22	C₃×#391	C₂²	1	4	yes	—
23	C₃×#382	C₂²	1	2	yes	—
24	C₃×#340	C₂²	1	4	yes	—
25	C₂×#494	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂×#433	C₂×C₃	1	1	yes	characteristic
27	#2621	C₂×C₃	1	2	yes	—
28	#2056	C₂×C₃	1	4	yes	complemented
29	#2001	C₂×C₃	1	4	yes	—
30	#1960	C₂×C₃	1	2	yes	complemented
31	#1835	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂²×C₃×Q₈	C₂³	1	4	yes	—
33	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
34	C₂×C₃×Q₁₆	—	2	8	no	—
35	C₂×C₃×Q₁₆	—	4	8	no	—
36	C₂²×C₃×C₈	C₂³	1	2	yes	—
37	C₂²×C₃×C₈	C₂³	1	2	yes	—
38	C₃×C₄⋊Q₈	—	2	4	no	—
39	C₃×C₄⋊Q₈	—	2	4	no	—
40	C₃×C₂²⋊Q₈	—	4	8	no	—
41	C₃×#117	—	2	2	no	—
42	C₃×#117	—	2	2	no	—
43	C₂×C₃×C₄⋊C₄	—	2	4	no	—
44	C₂×C₃×C₄⋊C₄	—	2	4	no	—
45	C₂×C₃×C₄⋊C₄	D₄	1	2	yes	—
46	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
47	C₂×C₃×C₂²⋊C₄	—	2	4	no	—
48	C₂×C₃×C₂²⋊C₄	D₄	1	2	yes	—
49	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
51	C₃×C₄⋊C₈	—	2	4	no	—
52	C₃×#103	—	4	8	no	—
53	C₃×C₂²⋊C₈	—	2	4	no	—
54	C₃×#97	—	2	2	no	—
55	C₃×C₄×C₈	D₄	1	4	yes	—
56	C₃×#95	—	2	4	no	—
57	C₃×#95	—	2	8	no	—
58	C₂²×Q₁₆	C₂²×C₃	1	4	yes	—
59	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
60	C₂×C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
61	C₂×#117	C₂²×C₃	1	1	yes	characteristic
62	#494	—	2	4	no	—
63	#433	—	2	4	no	—
64	C₂×C₄⋊C₈	C₂²×C₃	1	2	yes	—
65	C₂×#103	C₂²×C₃	1	8	yes	—
66	C₂×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
67	C₂×#97	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
69	#398	—	2	4	no	—
70	#397	—	2	4	no	—
71	#391	C₂²×C₃	1	4	yes	—
72	#382	C₂²×C₃	1	2	yes	—
73	#340	C₂²×C₃	1	4	yes	—
74	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₃×Q₈	—	2	4	no	—
76	C₂×C₃×Q₈	—	2	4	no	—
77	C₂×C₃×Q₈	—	2	8	no	—
78	C₂×C₃×Q₈	—	2	8	no	—
79	C₂²×C₃×C₄	—	2	1	no	—
80	C₂²×C₃×C₄	—	2	1	no	—
81	C₂²×C₃×C₄	—	2	4	no	—
82	C₂²×C₃×C₄	—	2	4	no	—
83	C₂²×C₃×C₄	—	2	4	no	—
84	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
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	C₃×Q₁₆	—	4	16	no	—
89	C₂×C₃×C₈	—	2	4	no	—
90	C₂×C₃×C₈	—	2	2	no	—
91	C₂×C₃×C₈	—	2	4	no	—
92	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
93	C₃×C₄⋊C₄	—	2	4	no	—
94	C₃×C₄⋊C₄	—	2	8	no	—
95	C₃×C₄⋊C₄	—	4	8	no	—
96	C₃×C₄⋊C₄	—	4	8	no	—
97	C₃×C₂²⋊C₄	—	2	4	no	—
98	C₃×C₂²⋊C₄	—	4	8	no	—
99	C₃×C₄²	—	2	1	no	—
100	C₃×C₄²	—	2	1	no	—
101	C₃×C₄²	C₂×D₄	1	2	yes	—
102	C₃×C₄²	#41	1	2	yes	—
103	C₂²×Q₈	C₂³×C₃	1	4	yes	—
104	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
105	C₂×Q₁₆	—	2	8	no	—
106	C₂×Q₁₆	—	4	8	no	—
107	C₂²×C₈	C₂³×C₃	1	2	yes	—
108	C₂²×C₈	C₂³×C₃	1	2	yes	—
109	C₄⋊Q₈	—	2	4	no	—
110	C₄⋊Q₈	—	2	4	no	—
111	C₂²⋊Q₈	—	4	8	no	—
112	#117	—	2	2	no	—
113	#117	—	2	2	no	—
114	C₂×C₄⋊C₄	—	2	4	no	—
115	C₂×C₄⋊C₄	—	2	4	no	—
116	C₂×C₄⋊C₄	C₃×D₄	1	2	yes	—
117	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
118	C₂×C₂²⋊C₄	—	2	4	no	—
119	C₂×C₂²⋊C₄	C₃×D₄	1	2	yes	—
120	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
121	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
122	C₄⋊C₈	—	2	4	no	—
123	#103	—	4	8	no	—
124	C₂²⋊C₈	—	2	4	no	—
125	#97	—	2	2	no	—
126	C₄×C₈	C₃×D₄	1	4	yes	—
127	#95	—	2	4	no	—
128	#95	—	2	8	no	—
129	C₂³×C₃	—	2	1	no	—
130	C₂³×C₃	—	2	2	no	—
131	C₂³×C₃	—	2	4	no	—
132	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
133	C₃×Q₈	—	4	8	no	—
134	C₃×Q₈	—	4	8	no	—
135	C₂×C₃×C₄	—	2	4	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	4	no	—
140	C₂×C₃×C₄	—	4	2	no	—
141	C₂×C₃×C₄	—	4	2	no	—
142	C₂×C₃×C₄	—	4	4	no	—
143	C₂×C₃×C₄	—	4	4	no	—
144	C₂×C₃×C₄	—	4	4	no	—
145	C₂×C₃×C₄	—	4	4	no	—
146	C₂×C₃×C₄	—	4	8	no	—
147	C₂×C₃×C₄	#121	1	4	yes	—
148	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
150	C₂×C₃×C₄	H(5,𝔽₂)	1	4	yes	—
151	C₃×C₈	—	2	4	no	—
152	C₃×C₈	—	4	2	no	—
153	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
154	C₂×Q₈	—	2	4	no	—
155	C₂×Q₈	—	2	4	no	—
156	C₂×Q₈	—	2	8	no	—
157	C₂×Q₈	—	2	8	no	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
164	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
166	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
167	Q₁₆	—	4	16	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	C₂×C₃×D₄	1	4	yes	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	8	no	—
174	C₄⋊C₄	—	4	8	no	—
175	C₄⋊C₄	—	4	8	no	—
176	C₂²⋊C₄	—	2	4	no	—
177	C₂²⋊C₄	—	4	8	no	—
178	C₄²	—	2	1	no	—
179	C₄²	—	2	1	no	—
180	C₄²	C₂×C₃×D₄	1	2	yes	—
181	C₄²	C₃×#41	1	2	yes	—
182	C₂²×C₃	—	4	1	no	—
183	C₂²×C₃	—	4	4	no	—
184	C₂²×C₃	—	4	2	no	—
185	C₂²×C₃	C₂×#121	1	1	yes	characteristic
186	C₂²×C₃	#535	1	2	yes	—
187	C₂²×C₃	#578	1	4	yes	—
188	C₃×C₄	—	2	4	no	—
189	C₃×C₄	—	4	1	no	—
190	C₃×C₄	—	4	2	no	—
191	C₃×C₄	—	8	4	no	—
192	C₃×C₄	—	8	4	no	—
193	C₃×C₄	#535	1	2	yes	—
194	C₃×C₄	#540	1	2	yes	—
195	C₂³	—	2	4	no	—
196	C₂³	—	2	1	no	—
197	C₂³	—	2	2	no	—
198	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
199	Q₈	—	4	8	no	—
200	Q₈	—	4	8	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	8	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	C₃×#121	1	4	yes	—
214	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
215	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
216	C₂×C₄	C₃×H(5,𝔽₂)	1	4	yes	—
217	C₈	—	2	4	no	—
218	C₈	—	4	2	no	—
219	C₂×C₃	—	8	1	no	—
220	C₂×C₃	#2619	1	1	yes	characteristic
221	C₂×C₃	#3059	1	2	yes	—
222	C₂×C₃	#3223	1	4	yes	—
223	C₂²	—	4	1	no	—
224	C₂²	—	4	4	no	—
225	C₂²	—	4	2	no	—
226	C₂²	C₂×C₃×#121	1	1	yes	characteristic
227	C₂²	C₃×#535	1	2	yes	—
228	C₂²	C₃×#578	1	4	yes	—
229	C₄	—	2	4	no	—
230	C₄	—	4	1	no	—
231	C₄	—	4	2	no	—
232	C₄	—	8	4	no	—
233	C₄	—	8	4	no	—
234	C₄	C₃×#535	1	2	yes	—
235	C₄	C₃×#540	1	2	yes	—
236	C₃	#29812	1	1	yes	characteristic, direct product factor, Sylow
237	C₂	—	8	1	no	—
238	C₂	C₃×#2619	1	1	yes	characteristic
239	C₂	C₃×#3059	1	2	yes	—
240	C₂	C₃×#3223	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
prime radical	C₂⁴×C₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#494	C₂	semidirect
4. C₃×#2056	C₂	semidirect
6. C₃×#1960	C₂	semidirect
7. C₃×#1835	C₂	semidirect
8. #29812	C₃	direct (N × H)
25. C₂×#494	C₂×C₃	semidirect
28. #2056	C₂×C₃	semidirect
30. #1960	C₂×C₃	semidirect
31. #1835	C₂×C₃	semidirect
236. C₃	#29812	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×#494	2
C₂×C₃×#433	2
C₃×#2621	2
C₃×#2056	2
C₃×#2001	2
C₃×#1960	2
C₃×#1835	2
#29812	3

Subgroups — C3×#29812 #10648419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#29812 #10648419