Subgroups — C₃×#21372 #10639979

Order 768.

714 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2921	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#433	C₂	1	1	yes	maximal, characteristic
3	C₃×#2873	C₂	1	1	yes	maximal, characteristic
4	C₃×#1950	C₂	1	4	yes	maximal, complemented
5	C₃×#1855	C₂	1	4	yes	maximal
6	C₃×#1778	C₂	1	2	yes	maximal, complemented
7	C₃×C₄×#108	C₂	1	2	yes	maximal, complemented
8	#21372	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂²×C₃×M₁₆	C₂²	1	2	yes	—
10	C₂×C₃×#117	C₂²	1	1	yes	characteristic
11	C₃×#433	—	2	4	no	—
12	C₃×#433	C₄	1	8	yes	—
13	C₂×C₃×#108	C₂²	1	4	yes	—
14	C₂×C₃×#108	C₂²	1	4	yes	—
15	C₃×#423	C₂²	1	2	yes	—
16	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
17	C₃×#407	C₂²	1	2	yes	—
18	C₂×C₃×C₂²⋊C₈	C₂²	1	2	yes	—
19	C₃×#405	—	2	4	no	—
20	C₃×C₄×M₁₆	C₂²	1	2	yes	—
21	C₃×C₄×M₁₆	C₂²	1	2	yes	—
22	C₃×C₄×M₁₆	C₂²	1	2	yes	—
23	C₂×C₃×#97	C₂²	1	1	yes	characteristic
24	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
25	C₃×#341	C₂²	1	4	yes	—
26	C₃×#341	C₂²	1	4	yes	—
27	#2921	C₂×C₃	1	1	yes	characteristic
28	C₂×#433	C₂×C₃	1	1	yes	characteristic
29	#2873	C₂×C₃	1	1	yes	characteristic
30	#1950	C₂×C₃	1	4	yes	complemented
31	#1855	C₂×C₃	1	4	yes	—
32	#1778	C₂×C₃	1	2	yes	complemented
33	C₄×#108	C₂×C₃	1	2	yes	complemented
34	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
35	C₂×C₃×M₁₆	—	2	2	no	—
36	C₂×C₃×M₁₆	—	2	8	no	—
37	C₂×C₃×M₁₆	C₂³	1	4	yes	—
38	C₂×C₃×M₁₆	C₂³	1	4	yes	—
39	C₂²×C₃×C₈	—	2	2	no	—
40	C₂²×C₃×C₈	C₂³	1	2	yes	—
41	C₂²×C₃×C₈	C₂³	1	2	yes	—
42	C₃×#117	C₂×C₄	1	4	yes	—
43	C₃×#117	D₄	1	4	yes	—
44	C₂×C₃×C₄⋊C₄	Q₈	1	2	yes	—
45	C₂×C₃×C₂²⋊C₄	Q₈	1	2	yes	—
46	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
47	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
48	C₃×#108	—	2	8	no	—
49	C₃×#108	—	2	8	no	—
50	C₃×C₄⋊C₈	—	2	4	no	—
51	C₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
52	C₃×C₂²⋊C₈	—	2	4	no	—
53	C₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
54	C₃×#97	—	2	2	no	—
55	C₃×#97	—	2	2	no	—
56	C₃×#97	—	2	2	no	—
57	C₃×#97	C₂×C₄	1	4	yes	—
58	C₃×C₄×C₈	—	2	2	no	—
59	C₃×C₄×C₈	—	2	2	no	—
60	C₃×C₄×C₈	—	2	2	no	—
61	C₃×C₄×C₈	C₂×C₄	1	4	yes	—
62	C₂²×M₁₆	C₂²×C₃	1	2	yes	—
63	C₂×#117	C₂²×C₃	1	1	yes	characteristic
64	#433	—	2	4	no	—
65	#433	C₃×C₄	1	8	yes	—
66	C₂×#108	C₂²×C₃	1	4	yes	—
67	C₂×#108	C₂²×C₃	1	4	yes	—
68	#423	C₂²×C₃	1	2	yes	—
69	C₂×C₄⋊C₈	C₂²×C₃	1	2	yes	—
70	#407	C₂²×C₃	1	2	yes	—
71	C₂×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
72	#405	—	2	4	no	—
73	C₄×M₁₆	C₂²×C₃	1	2	yes	—
74	C₄×M₁₆	C₂²×C₃	1	2	yes	—
75	C₄×M₁₆	C₂²×C₃	1	2	yes	—
76	C₂×#97	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
78	#341	C₂²×C₃	1	4	yes	—
79	#341	C₂²×C₃	1	4	yes	—
80	C₂⁴×C₃	C₂×Q₈	1	1	yes	characteristic
81	C₂²×C₃×C₄	—	2	4	no	—
82	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
83	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
84	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₂×Q₈	1	2	yes	—
86	C₂²×C₃×C₄	C₂×Q₈	1	2	yes	—
87	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
88	C₃×M₁₆	—	2	8	no	—
89	C₃×M₁₆	—	2	4	no	—
90	C₃×M₁₆	—	2	4	no	—
91	C₃×M₁₆	—	2	4	no	—
92	C₃×M₁₆	#41	1	8	yes	—
93	C₂×C₃×C₈	—	2	8	no	—
94	C₂×C₃×C₈	—	2	4	no	—
95	C₂×C₃×C₈	—	2	4	no	—
96	C₂×C₃×C₈	—	2	4	no	—
97	C₂×C₃×C₈	—	2	4	no	—
98	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
99	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
100	C₃×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
101	C₃×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
102	C₃×C₄²	C₂²×C₄	1	2	yes	—
103	C₃×C₄²	C₂²×C₄	1	2	yes	—
104	C₃×C₄²	C₂×D₄	1	2	yes	—
105	C₃×C₄²	C₂×D₄	1	2	yes	—
106	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
107	C₂×M₁₆	—	2	2	no	—
108	C₂×M₁₆	—	2	8	no	—
109	C₂×M₁₆	C₂³×C₃	1	4	yes	—
110	C₂×M₁₆	C₂³×C₃	1	4	yes	—
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	C₂³×C₃	1	2	yes	—
113	C₂²×C₈	C₂³×C₃	1	2	yes	—
114	#117	C₂×C₃×C₄	1	4	yes	—
115	#117	C₃×D₄	1	4	yes	—
116	C₂×C₄⋊C₄	C₃×Q₈	1	2	yes	—
117	C₂×C₂²⋊C₄	C₃×Q₈	1	2	yes	—
118	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
119	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
120	#108	—	2	8	no	—
121	#108	—	2	8	no	—
122	C₄⋊C₈	—	2	4	no	—
123	C₄⋊C₈	C₂×C₃×C₄	1	8	yes	—
124	C₂²⋊C₈	—	2	4	no	—
125	C₂²⋊C₈	C₂×C₃×C₄	1	8	yes	—
126	#97	—	2	2	no	—
127	#97	—	2	2	no	—
128	#97	—	2	2	no	—
129	#97	C₂×C₃×C₄	1	4	yes	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	C₂×C₃×C₄	1	4	yes	—
134	C₂³×C₃	—	2	2	no	—
135	C₂³×C₃	—	4	2	no	—
136	C₂³×C₃	C₂×C₄⋊C₄	1	2	yes	—
137	C₂³×C₃	C₂²×Q₈	1	1	yes	characteristic
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	—	2	4	no	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	2	2	no	—
142	C₂×C₃×C₄	—	4	2	no	—
143	C₂×C₃×C₄	—	4	2	no	—
144	C₂×C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
145	C₂×C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
146	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
147	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂×#41	1	4	yes	—
150	C₃×C₈	—	2	4	no	—
151	C₃×C₈	—	2	4	no	—
152	C₃×C₈	—	2	8	no	—
153	C₃×C₈	—	4	4	no	—
154	C₂⁴	C₂×C₃×Q₈	1	1	yes	characteristic
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
157	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
158	C₂²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₃×Q₈	1	2	yes	—
160	C₂²×C₄	C₂×C₃×Q₈	1	2	yes	—
161	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
162	M₁₆	—	2	8	no	—
163	M₁₆	—	2	4	no	—
164	M₁₆	—	2	4	no	—
165	M₁₆	—	2	4	no	—
166	M₁₆	C₃×#41	1	8	yes	—
167	C₂×C₈	—	2	8	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	2	4	no	—
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	C₂²×C₃×C₄	1	4	yes	—
173	C₂×C₈	C₂²×C₃×C₄	1	4	yes	—
174	C₄⋊C₄	C₃×C₄⋊C₄	1	8	yes	—
175	C₂²⋊C₄	C₃×C₄⋊C₄	1	8	yes	—
176	C₄²	C₂²×C₃×C₄	1	2	yes	—
177	C₄²	C₂²×C₃×C₄	1	2	yes	—
178	C₄²	C₂×C₃×D₄	1	2	yes	—
179	C₄²	C₂×C₃×D₄	1	2	yes	—
180	C₂²×C₃	—	2	2	no	—
181	C₂²×C₃	—	2	4	no	—
182	C₂²×C₃	—	2	2	no	—
183	C₂²×C₃	—	8	2	no	—
184	C₂²×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
185	C₂²×C₃	#533	1	2	yes	—
186	C₃×C₄	—	2	4	no	—
187	C₃×C₄	—	2	4	no	—
188	C₃×C₄	—	2	2	no	—
189	C₃×C₄	C₄×#41	1	2	yes	—
190	C₃×C₄	#525	1	2	yes	—
191	C₂³	—	2	2	no	—
192	C₂³	—	4	2	no	—
193	C₂³	C₂×C₃×C₄⋊C₄	1	2	yes	—
194	C₂³	C₂²×C₃×Q₈	1	1	yes	characteristic
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	C₂×C₃×C₄⋊C₄	1	4	yes	—
202	C₂×C₄	C₂×C₃×C₄⋊C₄	1	4	yes	—
203	C₂×C₄	C₂×C₃×C₄⋊C₄	1	2	yes	—
204	C₂×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
205	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
206	C₂×C₄	C₂×C₃×#41	1	4	yes	—
207	C₈	—	2	4	no	—
208	C₈	—	2	4	no	—
209	C₈	—	2	8	no	—
210	C₈	—	4	4	no	—
211	C₂×C₃	—	2	2	no	—
212	C₂×C₃	—	4	2	no	—
213	C₂×C₃	#2296	1	1	yes	characteristic
214	C₂×C₃	#2916	1	2	yes	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	4	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	8	2	no	—
219	C₂²	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
220	C₂²	C₃×#533	1	2	yes	—
221	C₄	—	2	4	no	—
222	C₄	—	2	4	no	—
223	C₄	—	2	2	no	—
224	C₄	C₃×C₄×#41	1	2	yes	—
225	C₄	C₃×#525	1	2	yes	—
226	C₃	#21372	1	1	yes	characteristic, direct product factor, Sylow
227	C₂	—	2	2	no	—
228	C₂	—	4	2	no	—
229	C₂	C₃×#2296	1	1	yes	characteristic
230	C₂	C₃×#2916	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₃×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₃	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃×#1950	C₂	semidirect
6. C₃×#1778	C₂	semidirect
7. C₃×C₄×#108	C₂	semidirect
8. #21372	C₃	direct (N × H)
30. #1950	C₂×C₃	semidirect
32. #1778	C₂×C₃	semidirect
33. C₄×#108	C₂×C₃	semidirect
226. C₃	#21372	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×#2921	2
C₂×C₃×#433	2
C₃×#2873	2
C₃×#1950	2
C₃×#1855	2
C₃×#1778	2
C₃×C₄×#108	2
#21372	3

Subgroups — C3×#21372 #10639979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#21372 #10639979