Subgroups — C₃×#29820 #10648427

Order 768.

780 conjugacy classes of subgroups.

The subgroup lattice has 258 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	C₃×#3039	C₂	1	2	yes	maximal
3	C₃×#2922	C₂	1	1	yes	maximal, characteristic
4	C₃×#2047	C₂	1	4	yes	maximal, complemented
5	C₃×#2020	C₂	1	4	yes	maximal
6	C₃×#1966	C₂	1	2	yes	maximal, complemented
7	C₃×#1837	C₂	1	1	yes	maximal, characteristic, complemented
8	#29820	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂×C₃×#137	C₂²	1	4	yes	—
10	C₂²×C₃×Q₁₆	C₂²	1	4	yes	—
11	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₃×C₄×#41	C₂²	1	1	yes	characteristic
13	C₃×#494	—	2	2	no	—
14	C₃×#494	—	2	2	no	—
15	C₃×#462	—	2	4	no	—
16	C₃×#461	—	2	4	no	—
17	C₃×#433	—	2	2	no	—
18	C₃×#431	—	2	2	no	—
19	C₃×#424	C₂²	1	2	yes	—
20	C₃×#423	C₂²	1	2	yes	—
21	C₃×#419	C₂²	1	4	yes	—
22	C₂×C₃×#101	C₂²	1	8	yes	—
23	C₃×#408	C₂²	1	2	yes	—
24	C₃×C₄×M₁₆	C₂²	1	1	yes	characteristic
25	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
26	C₃×#341	C₂²	1	4	yes	—
27	C₂×#494	C₂×C₃	1	1	yes	characteristic, complemented
28	#3039	C₂×C₃	1	2	yes	—
29	#2922	C₂×C₃	1	1	yes	characteristic
30	#2047	C₂×C₃	1	4	yes	complemented
31	#2020	C₂×C₃	1	4	yes	—
32	#1966	C₂×C₃	1	2	yes	complemented
33	#1837	C₂×C₃	1	1	yes	characteristic, complemented
34	C₂×C₃×#41	C₂³	1	1	yes	characteristic
35	C₂²×C₃×Q₈	C₂³	1	4	yes	—
36	C₃×#137	—	4	8	no	—
37	C₂×C₃×Q₁₆	—	2	4	no	—
38	C₂×C₃×Q₁₆	—	2	8	no	—
39	C₂×C₃×Q₁₆	—	4	8	no	—
40	C₂×C₃×SD₁₆	—	2	4	no	—
41	C₂×C₃×M₁₆	C₂³	1	4	yes	—
42	C₂×C₃×M₁₆	C₂³	1	2	yes	—
43	C₂²×C₃×C₈	C₂³	1	2	yes	—
44	C₃×C₄⋊Q₈	—	2	4	no	—
45	C₃×C₄⋊Q₈	—	2	2	no	—
46	C₃×C₄⋊Q₈	—	2	2	no	—
47	C₃×C₄×Q₈	D₄	1	2	yes	—
48	C₃×C₄×D₄	—	2	2	no	—
49	C₃×C₄×D₄	D₄	1	2	yes	—
50	C₃×#117	—	2	1	no	—
51	C₃×#117	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄⋊C₄	—	2	4	no	—
53	C₂×C₃×C₄²	—	2	1	no	—
54	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
55	C₃×C₄⋊C₈	—	2	4	no	—
56	C₃×C₄⋊C₈	—	2	4	no	—
57	C₃×#103	—	2	8	no	—
58	C₃×#101	—	4	8	no	—
59	C₃×C₂²⋊C₈	—	2	4	no	—
60	C₃×#97	—	2	1	no	—
61	C₃×C₄×C₈	—	2	1	no	—
62	C₃×C₄×C₈	D₄	1	2	yes	—
63	C₃×C₄×C₈	D₄	1	2	yes	—
64	C₂×#137	C₂²×C₃	1	4	yes	—
65	C₂²×Q₁₆	C₂²×C₃	1	4	yes	—
66	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
67	C₄×#41	C₂²×C₃	1	1	yes	characteristic
68	#494	—	2	2	no	—
69	#494	—	2	2	no	—
70	#462	—	2	4	no	—
71	#461	—	2	4	no	—
72	#433	—	2	2	no	—
73	#431	—	2	2	no	—
74	#424	C₂²×C₃	1	2	yes	—
75	#423	C₂²×C₃	1	2	yes	—
76	#419	C₂²×C₃	1	4	yes	—
77	C₂×#101	C₂²×C₃	1	8	yes	—
78	#408	C₂²×C₃	1	2	yes	—
79	C₄×M₁₆	C₂²×C₃	1	1	yes	characteristic
80	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
81	#341	C₂²×C₃	1	4	yes	—
82	C₃×#41	—	2	4	no	—
83	C₂×C₃×Q₈	—	2	4	no	—
84	C₂×C₃×Q₈	—	2	4	no	—
85	C₂×C₃×Q₈	—	2	8	no	—
86	C₂×C₃×Q₈	—	2	8	no	—
87	C₂×C₃×Q₈	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×D₄	—	2	1	no	—
89	C₂×C₃×D₄	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₃×C₄	—	2	2	no	—
91	C₂²×C₃×C₄	—	2	1	no	—
92	C₂²×C₃×C₄	—	2	4	no	—
93	C₂²×C₃×C₄	#41	1	2	yes	—
94	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
95	C₃×Q₁₆	—	4	16	no	—
96	C₃×Q₁₆	—	4	8	no	—
97	C₃×SD₁₆	—	4	8	no	—
98	C₃×M₁₆	—	2	4	no	—
99	C₃×M₁₆	—	2	8	no	—
100	C₂×C₃×C₈	—	2	4	no	—
101	C₂×C₃×C₈	—	2	2	no	—
102	C₂×C₃×C₈	—	2	4	no	—
103	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
104	C₃×C₄⋊C₄	—	2	2	no	—
105	C₃×C₄⋊C₄	—	4	4	no	—
106	C₃×C₄⋊C₄	—	4	4	no	—
107	C₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
108	C₃×C₂²⋊C₄	—	2	1	no	—
109	C₃×C₂²⋊C₄	—	2	2	no	—
110	C₃×C₄²	—	2	2	no	—
111	C₃×C₄²	—	2	1	no	—
112	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
113	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
114	C₃×C₄²	C₂×D₄	1	2	yes	—
115	C₂×#41	C₂³×C₃	1	1	yes	characteristic
116	C₂²×Q₈	C₂³×C₃	1	4	yes	—
117	#137	—	4	8	no	—
118	C₂×Q₁₆	—	2	4	no	—
119	C₂×Q₁₆	—	2	8	no	—
120	C₂×Q₁₆	—	4	8	no	—
121	C₂×SD₁₆	—	2	4	no	—
122	C₂×M₁₆	C₂³×C₃	1	4	yes	—
123	C₂×M₁₆	C₂³×C₃	1	2	yes	—
124	C₂²×C₈	C₂³×C₃	1	2	yes	—
125	C₄⋊Q₈	—	2	4	no	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	—	2	2	no	—
128	C₄×Q₈	C₃×D₄	1	2	yes	—
129	C₄×D₄	—	2	2	no	—
130	C₄×D₄	C₃×D₄	1	2	yes	—
131	#117	—	2	1	no	—
132	#117	C₂³×C₃	1	1	yes	characteristic
133	C₂×C₄⋊C₄	—	2	4	no	—
134	C₂×C₄²	—	2	1	no	—
135	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
136	C₄⋊C₈	—	2	4	no	—
137	C₄⋊C₈	—	2	4	no	—
138	#103	—	2	8	no	—
139	#101	—	4	8	no	—
140	C₂²⋊C₈	—	2	4	no	—
141	#97	—	2	1	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	C₃×D₄	1	2	yes	—
144	C₄×C₈	C₃×D₄	1	2	yes	—
145	C₂³×C₃	—	2	1	no	—
146	C₂³×C₃	C₂×#41	1	1	yes	characteristic
147	C₃×Q₈	—	2	2	no	—
148	C₃×Q₈	—	4	4	no	—
149	C₃×Q₈	—	4	8	no	—
150	C₃×Q₈	—	4	4	no	—
151	C₃×D₄	—	2	2	no	—
152	C₃×D₄	—	4	2	no	—
153	C₂×C₃×C₄	—	2	4	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	4	2	no	—
157	C₂×C₃×C₄	—	4	4	no	—
158	C₂×C₃×C₄	—	4	4	no	—
159	C₂×C₃×C₄	—	4	8	no	—
160	C₂×C₃×C₄	#121	1	2	yes	—
161	C₂×C₃×C₄	#121	1	2	yes	—
162	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₃×C₄	H(5,𝔽₂)	1	4	yes	—
165	C₃×C₈	—	2	4	no	—
166	C₃×C₈	—	4	2	no	—
167	C₃×C₈	—	4	4	no	—
168	#41	—	2	4	no	—
169	C₂×Q₈	—	2	4	no	—
170	C₂×Q₈	—	2	4	no	—
171	C₂×Q₈	—	2	8	no	—
172	C₂×Q₈	—	2	8	no	—
173	C₂×Q₈	C₂×C₃×D₄	1	1	yes	characteristic
174	C₂×D₄	—	2	1	no	—
175	C₂×D₄	C₂×C₃×D₄	1	1	yes	characteristic
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	C₃×#41	1	2	yes	—
180	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
181	Q₁₆	—	4	16	no	—
182	Q₁₆	—	4	8	no	—
183	SD₁₆	—	4	8	no	—
184	M₁₆	—	2	4	no	—
185	M₁₆	—	2	8	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	C₂×C₃×D₄	1	4	yes	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	4	4	no	—
192	C₄⋊C₄	—	4	4	no	—
193	C₄⋊C₄	C₂×C₃×D₄	1	2	yes	—
194	C₂²⋊C₄	—	2	1	no	—
195	C₂²⋊C₄	—	2	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	1	no	—
198	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
199	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
200	C₄²	C₂×C₃×D₄	1	2	yes	—
201	C₂²×C₃	—	2	2	no	—
202	C₂²×C₃	—	4	1	no	—
203	C₂²×C₃	—	4	2	no	—
204	C₂²×C₃	C₂×#121	1	1	yes	characteristic
205	C₂²×C₃	#540	1	2	yes	—
206	C₃×C₄	—	2	2	no	—
207	C₃×C₄	—	4	1	no	—
208	C₃×C₄	—	4	2	no	—
209	C₃×C₄	—	8	4	no	—
210	C₃×C₄	#473	1	4	yes	—
211	C₃×C₄	#535	1	2	yes	—
212	C₃×C₄	#535	1	2	yes	—
213	C₂³	—	2	1	no	—
214	C₂³	C₂×C₃×#41	1	1	yes	characteristic
215	Q₈	—	2	2	no	—
216	Q₈	—	4	4	no	—
217	Q₈	—	4	8	no	—
218	Q₈	—	4	4	no	—
219	D₄	—	2	2	no	—
220	D₄	—	4	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	8	no	—
228	C₂×C₄	C₃×#121	1	2	yes	—
229	C₂×C₄	C₃×#121	1	2	yes	—
230	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
231	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
232	C₂×C₄	C₃×H(5,𝔽₂)	1	4	yes	—
233	C₈	—	2	4	no	—
234	C₈	—	4	4	no	—
235	C₈	—	4	2	no	—
236	C₂×C₃	—	2	2	no	—
237	C₂×C₃	—	8	1	no	—
238	C₂×C₃	#2619	1	1	yes	characteristic
239	C₂×C₃	C₂×#473	1	2	yes	—
240	C₂²	—	2	2	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	2	no	—
243	C₂²	C₂×C₃×#121	1	1	yes	characteristic
244	C₂²	C₃×#540	1	2	yes	—
245	C₄	—	2	2	no	—
246	C₄	—	4	1	no	—
247	C₄	—	4	2	no	—
248	C₄	—	8	4	no	—
249	C₄	C₃×#473	1	4	yes	—
250	C₄	C₃×#535	1	2	yes	—
251	C₄	C₃×#535	1	2	yes	—
252	C₃	#29820	1	1	yes	characteristic, direct product factor, Sylow
253	C₂	—	2	2	no	—
254	C₂	—	8	1	no	—
255	C₂	C₃×#2619	1	1	yes	characteristic
256	C₂	C₂×C₃×#473	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
prime radical	C₂×C₃×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ 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×#2047	C₂	semidirect
6. C₃×#1966	C₂	semidirect
7. C₃×#1837	C₂	semidirect
8. #29820	C₃	direct (N × H)
27. C₂×#494	C₂×C₃	semidirect
30. #2047	C₂×C₃	semidirect
32. #1966	C₂×C₃	semidirect
33. #1837	C₂×C₃	semidirect
252. C₃	#29820	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
C₃×#3039	2
C₃×#2922	2
C₃×#2047	2
C₃×#2020	2
C₃×#1966	2
C₃×#1837	2
#29820	3

Subgroups — C3×#29820 #10648427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#29820 #10648427