Subgroups — C₃×#2499 #84363

Order 384.

484 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic
3	C₃×#397	C₂	1	4	yes	maximal
4	C₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#387	C₂	1	4	yes	maximal, complemented
7	C₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×C₄×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
10	#2499	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
12	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
13	C₃×C₂²⋊Q₈	—	2	4	no	—
14	C₃×#117	—	2	4	no	—
15	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
18	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
24	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
25	C₂×C₃×C₄²	C₂²	1	2	yes	—
26	C₃×#95	C₂²	1	2	yes	—
27	C₃×#95	C₂²	1	4	yes	—
28	C₃×#95	C₂²	1	4	yes	—
29	C₃×#95	C₂²	1	4	yes	—
30	C₂×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic
31	C₂×#117	C₂×C₃	1	1	yes	characteristic
32	#397	C₂×C₃	1	4	yes	—
33	#391	C₂×C₃	1	1	yes	characteristic, complemented
34	#391	C₂×C₃	1	1	yes	characteristic, complemented
35	#387	C₂×C₃	1	4	yes	complemented
36	#384	C₂×C₃	1	1	yes	characteristic, complemented
37	#382	C₂×C₃	1	1	yes	characteristic, complemented
38	C₄×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
39	C₂⁴×C₃	C₂³	1	1	yes	characteristic
40	C₂×C₃×Q₈	—	2	2	no	—
41	C₂×C₃×Q₈	—	2	2	no	—
42	C₂×C₃×Q₈	—	2	2	no	—
43	C₂²×C₃×C₄	—	2	2	no	—
44	C₂²×C₃×C₄	—	2	2	no	—
45	C₂²×C₃×C₄	D₄	1	4	yes	—
46	C₂²×C₃×C₄	C₂³	1	2	yes	—
47	C₂²×C₃×C₄	C₂³	1	2	yes	—
48	C₂²×C₃×C₄	C₂³	1	2	yes	—
49	C₂²×C₃×C₄	C₂³	1	2	yes	—
50	C₂²×C₃×C₄	C₂³	1	2	yes	—
51	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
52	C₂²×C₃×C₄	C₂³	1	2	yes	—
53	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
54	C₃×C₄⋊C₄	—	2	2	no	—
55	C₃×C₄⋊C₄	—	2	2	no	—
56	C₃×C₄⋊C₄	—	2	2	no	—
57	C₃×C₄⋊C₄	—	2	4	no	—
58	C₃×C₄⋊C₄	—	2	8	no	—
59	C₃×C₂²⋊C₄	—	2	4	no	—
60	C₃×C₂²⋊C₄	—	2	4	no	—
61	C₃×C₂²⋊C₄	—	2	4	no	—
62	C₃×C₄²	—	2	4	no	—
63	C₃×C₄²	—	2	2	no	—
64	C₃×C₄²	—	2	2	no	—
65	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
66	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
67	C₂²⋊Q₈	—	2	4	no	—
68	#117	—	2	4	no	—
69	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
72	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
73	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
74	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
75	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
76	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
77	C₂×C₄²	C₂²×C₃	1	2	yes	—
78	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
80	#95	C₂²×C₃	1	2	yes	—
81	#95	C₂²×C₃	1	4	yes	—
82	#95	C₂²×C₃	1	4	yes	—
83	#95	C₂²×C₃	1	4	yes	—
84	C₂³×C₃	—	2	2	no	—
85	C₂³×C₃	—	2	1	no	—
86	C₂³×C₃	—	2	1	no	—
87	C₂³×C₃	—	2	2	no	—
88	C₂³×C₃	C₂×D₄	1	2	yes	—
89	C₂³×C₃	C₂⁴	1	1	yes	characteristic
90	C₃×Q₈	—	4	4	no	—
91	C₂×C₃×C₄	—	2	2	no	—
92	C₂×C₃×C₄	—	2	1	no	—
93	C₂×C₃×C₄	—	2	1	no	—
94	C₂×C₃×C₄	—	2	2	no	—
95	C₂×C₃×C₄	—	2	8	no	—
96	C₂×C₃×C₄	—	2	2	no	—
97	C₂×C₃×C₄	—	2	2	no	—
98	C₂×C₃×C₄	—	2	2	no	—
99	C₂×C₃×C₄	—	2	2	no	—
100	C₂×C₃×C₄	—	2	2	no	—
101	C₂×C₃×C₄	—	2	2	no	—
102	C₂×C₃×C₄	—	2	4	no	—
103	C₂×C₃×C₄	—	2	4	no	—
104	C₂×C₃×C₄	—	2	2	no	—
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	—	2	4	no	—
107	C₂×C₃×C₄	—	2	2	no	—
108	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
109	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
110	C₂×C₃×C₄	#41	1	4	yes	—
111	C₂×C₃×C₄	#41	1	2	yes	—
112	C₂×C₃×C₄	#41	1	2	yes	—
113	C₂⁴	C₂³×C₃	1	1	yes	characteristic
114	C₂×Q₈	—	2	2	no	—
115	C₂×Q₈	—	2	2	no	—
116	C₂×Q₈	—	2	2	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	2	2	no	—
119	C₂²×C₄	C₃×D₄	1	4	yes	—
120	C₂²×C₄	C₂³×C₃	1	2	yes	—
121	C₂²×C₄	C₂³×C₃	1	2	yes	—
122	C₂²×C₄	C₂³×C₃	1	2	yes	—
123	C₂²×C₄	C₂³×C₃	1	2	yes	—
124	C₂²×C₄	C₂³×C₃	1	2	yes	—
125	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
126	C₂²×C₄	C₂³×C₃	1	2	yes	—
127	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	2	4	no	—
132	C₄⋊C₄	—	2	8	no	—
133	C₂²⋊C₄	—	2	4	no	—
134	C₂²⋊C₄	—	2	4	no	—
135	C₂²⋊C₄	—	2	4	no	—
136	C₄²	—	2	4	no	—
137	C₄²	—	2	2	no	—
138	C₄²	—	2	2	no	—
139	C₂²×C₃	—	2	2	no	—
140	C₂²×C₃	—	2	2	no	—
141	C₂²×C₃	—	2	2	no	—
142	C₂²×C₃	—	4	2	no	—
143	C₂²×C₃	—	4	2	no	—
144	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
145	C₂²×C₃	C₂×#41	1	1	yes	characteristic
146	C₂²×C₃	C₂×#41	1	1	yes	characteristic
147	C₂²×C₃	C₂×#41	1	2	yes	—
148	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
149	C₃×C₄	—	2	2	no	—
150	C₃×C₄	—	2	4	no	—
151	C₃×C₄	—	4	2	no	—
152	C₃×C₄	—	4	2	no	—
153	C₃×C₄	—	4	2	no	—
154	C₃×C₄	—	4	2	no	—
155	C₃×C₄	—	4	2	no	—
156	C₃×C₄	#123	1	4	yes	—
157	C₂³	—	2	2	no	—
158	C₂³	—	2	1	no	—
159	C₂³	—	2	1	no	—
160	C₂³	—	2	2	no	—
161	C₂³	C₂×C₃×D₄	1	2	yes	—
162	C₂³	C₂⁴×C₃	1	1	yes	characteristic
163	Q₈	—	4	4	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
182	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
183	C₂×C₄	C₃×#41	1	4	yes	—
184	C₂×C₄	C₃×#41	1	2	yes	—
185	C₂×C₄	C₃×#41	1	2	yes	—
186	C₂×C₃	—	4	2	no	—
187	C₂×C₃	C₂×#123	1	1	yes	characteristic
188	C₂×C₃	#532	1	1	yes	characteristic
189	C₂×C₃	#536	1	1	yes	characteristic
190	C₂×C₃	#552	1	2	yes	—
191	C₂×C₃	#556	1	2	yes	—
192	C₂²	—	2	2	no	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
198	C₂²	C₂×C₃×#41	1	1	yes	characteristic
199	C₂²	C₂×C₃×#41	1	1	yes	characteristic
200	C₂²	C₂×C₃×#41	1	2	yes	—
201	C₂²	C₃×2₋¹⁺⁴	1	2	yes	—
202	C₄	—	2	2	no	—
203	C₄	—	2	4	no	—
204	C₄	—	4	2	no	—
205	C₄	—	4	2	no	—
206	C₄	—	4	2	no	—
207	C₄	—	4	2	no	—
208	C₄	—	4	2	no	—
209	C₄	C₃×#123	1	4	yes	—
210	C₃	#2499	1	1	yes	characteristic, direct product factor, Sylow
211	C₂	—	4	2	no	—
212	C₂	C₂×C₃×#123	1	1	yes	characteristic
213	C₂	C₃×#532	1	1	yes	characteristic
214	C₂	C₃×#536	1	1	yes	characteristic
215	C₂	C₃×#552	1	2	yes	—
216	C₂	C₃×#556	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₃×#391	C₂	semidirect
5. C₃×#391	C₂	semidirect
6. C₃×#387	C₂	semidirect
7. C₃×#384	C₂	semidirect
8. C₃×#382	C₂	semidirect
10. #2499	C₃	direct (N × H)
33. #391	C₂×C₃	semidirect
34. #391	C₂×C₃	semidirect
35. #387	C₂×C₃	semidirect
36. #384	C₂×C₃	semidirect
37. #382	C₂×C₃	semidirect
210. C₃	#2499	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×C₂²⋊Q₈	2
C₂×C₃×#117	2
C₃×#397	2
C₃×#391	2
C₃×#387	2
C₃×#384	2
C₃×#382	2
C₃×C₄×C₂²⋊C₄	2
#2499	3

Subgroups — C3×#2499 #84363

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2499 #84363