Subgroups — C₂×C₃×#416 #84756

Order 384.

792 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×#41	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#416	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#103	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×#102	C₂	1	2	yes	maximal, complemented
7	C₂×#416	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂×C₃×#41	—	2	4	no	—
9	C₂×C₃×#41	—	2	2	no	—
10	C₂×C₃×#41	C₄	1	8	yes	complemented
11	C₂×C₃×#41	C₂²	1	4	yes	—
12	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃×D₄	—	2	1	no	—
14	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
15	C₂³×C₃×C₄	—	2	1	no	—
16	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
17	C₂²×C₃×C₈	—	2	4	no	—
18	C₂²×C₃×C₈	C₂²	1	2	yes	—
19	C₂²×C₃×C₈	C₂²	1	4	yes	complemented
20	C₃×#117	—	2	2	no	—
21	C₃×#117	C₂²	1	4	yes	complemented
22	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
24	C₂×C₃×C₄²	—	2	1	no	—
25	C₃×#103	C₂²	1	8	yes	complemented
26	C₃×#102	C₂²	1	8	yes	complemented
27	C₂²×#41	C₂×C₃	1	1	yes	characteristic
28	C₂³×C₈	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂×#117	C₂×C₃	1	1	yes	characteristic, complemented
30	#416	C₂×C₃	1	8	yes	direct product factor
31	C₂×#103	C₂×C₃	1	2	yes	complemented
32	C₂×#102	C₂×C₃	1	2	yes	complemented
33	C₂⁴×C₃	—	2	1	no	—
34	C₂⁴×C₃	D₄	1	1	yes	characteristic
35	C₃×#41	—	2	16	no	—
36	C₃×#41	—	2	8	no	—
37	C₃×#41	C₂×C₄	1	16	yes	complemented
38	C₂×C₃×Q₈	—	2	2	no	—
39	C₂×C₃×Q₈	—	2	1	no	—
40	C₂×C₃×Q₈	C₂×C₄	1	4	yes	—
41	C₂×C₃×Q₈	C₂³	1	2	yes	—
42	C₂×C₃×D₄	—	2	8	no	—
43	C₂×C₃×D₄	—	2	4	no	—
44	C₂×C₃×D₄	—	2	1	no	—
45	C₂×C₃×D₄	—	2	2	no	—
46	C₂×C₃×D₄	C₂×C₄	1	4	yes	—
47	C₂×C₃×D₄	C₂³	1	2	yes	—
48	C₂²×C₃×C₄	—	2	1	no	—
49	C₂²×C₃×C₄	—	2	8	no	—
50	C₂²×C₃×C₄	—	2	4	no	—
51	C₂²×C₃×C₄	—	2	2	no	—
52	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
53	C₂²×C₃×C₄	D₄	1	2	yes	—
54	C₂²×C₃×C₄	D₄	1	4	yes	—
55	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
56	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
57	C₂²×C₃×C₄	C₂³	1	2	yes	—
58	C₂×C₃×C₈	—	2	4	no	—
59	C₂×C₃×C₈	—	2	8	no	—
60	C₂×C₃×C₈	C₂³	1	4	yes	—
61	C₃×C₄⋊C₄	—	2	2	no	—
62	C₃×C₄⋊C₄	C₂³	1	4	yes	—
63	C₃×C₂²⋊C₄	—	2	4	no	—
64	C₃×C₄²	—	2	4	no	—
65	C₂×#41	—	2	4	no	—
66	C₂×#41	—	2	2	no	—
67	C₂×#41	C₃×C₄	1	8	yes	complemented
68	C₂×#41	C₂²×C₃	1	4	yes	—
69	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
70	C₂²×D₄	—	2	1	no	—
71	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
72	C₂³×C₄	—	2	1	no	—
73	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
74	C₂²×C₈	—	2	4	no	—
75	C₂²×C₈	C₂²×C₃	1	2	yes	—
76	C₂²×C₈	C₂²×C₃	1	4	yes	complemented
77	#117	—	2	2	no	—
78	#117	C₂²×C₃	1	4	yes	complemented
79	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
80	C₂×C₂²⋊C₄	—	2	1	no	—
81	C₂×C₄²	—	2	1	no	—
82	#103	C₂²×C₃	1	8	yes	complemented
83	#102	C₂²×C₃	1	8	yes	complemented
84	C₂³×C₃	—	2	4	no	—
85	C₂³×C₃	—	2	4	no	—
86	C₂³×C₃	—	2	2	no	—
87	C₂³×C₃	—	4	4	no	—
88	C₂³×C₃	C₂²⋊C₄	1	4	yes	—
89	C₂³×C₃	C₂×D₄	1	2	yes	—
90	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
91	C₃×Q₈	—	2	4	no	—
92	C₃×Q₈	—	2	2	no	—
93	C₃×Q₈	C₂²×C₄	1	4	yes	—
94	C₃×D₄	—	2	16	no	—
95	C₃×D₄	—	2	4	no	—
96	C₃×D₄	—	2	2	no	—
97	C₃×D₄	C₂²×C₄	1	4	yes	—
98	C₂×C₃×C₄	—	2	4	no	—
99	C₂×C₃×C₄	—	2	2	no	—
100	C₂×C₃×C₄	—	2	16	no	—
101	C₂×C₃×C₄	—	2	4	no	—
102	C₂×C₃×C₄	—	4	4	no	—
103	C₂×C₃×C₄	C₂²⋊C₄	1	4	yes	—
104	C₂×C₃×C₄	C₂²⋊C₄	1	8	yes	—
105	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
106	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
107	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
108	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
109	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
110	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
111	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
112	C₃×C₈	—	2	4	no	—
113	C₂⁴	—	2	1	no	—
114	C₂⁴	C₃×D₄	1	1	yes	characteristic
115	#41	—	2	16	no	—
116	#41	—	2	8	no	—
117	#41	C₂×C₃×C₄	1	16	yes	complemented
118	C₂×Q₈	—	2	2	no	—
119	C₂×Q₈	—	2	1	no	—
120	C₂×Q₈	C₂×C₃×C₄	1	4	yes	—
121	C₂×Q₈	C₂³×C₃	1	2	yes	—
122	C₂×D₄	—	2	8	no	—
123	C₂×D₄	—	2	4	no	—
124	C₂×D₄	—	2	1	no	—
125	C₂×D₄	—	2	2	no	—
126	C₂×D₄	C₂×C₃×C₄	1	4	yes	—
127	C₂×D₄	C₂³×C₃	1	2	yes	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	2	8	no	—
130	C₂²×C₄	—	2	4	no	—
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
133	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
134	C₂²×C₄	C₃×D₄	1	2	yes	—
135	C₂²×C₄	C₃×D₄	1	4	yes	—
136	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
137	C₂²×C₄	C₂³×C₃	1	2	yes	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	2	8	no	—
140	C₂×C₈	C₂³×C₃	1	4	yes	—
141	C₄⋊C₄	—	2	2	no	—
142	C₄⋊C₄	C₂³×C₃	1	4	yes	—
143	C₂²⋊C₄	—	2	4	no	—
144	C₄²	—	2	4	no	—
145	C₂²×C₃	—	2	4	no	—
146	C₂²×C₃	—	2	8	no	—
147	C₂²×C₃	—	2	4	no	—
148	C₂²×C₃	—	4	4	no	—
149	C₂²×C₃	—	4	8	no	—
150	C₂²×C₃	C₂×C₂²⋊C₄	1	4	yes	—
151	C₂²×C₃	C₂×C₂²⋊C₄	1	2	yes	—
152	C₂²×C₃	#135	1	4	yes	—
153	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
154	C₃×C₄	—	2	4	no	—
155	C₃×C₄	—	4	4	no	—
156	C₃×C₄	C₂×C₂²⋊C₄	1	4	yes	—
157	C₃×C₄	C₂×C₂²⋊C₄	1	2	yes	—
158	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
159	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
160	C₂³	—	2	4	no	—
161	C₂³	—	2	4	no	—
162	C₂³	—	2	2	no	—
163	C₂³	—	4	4	no	—
164	C₂³	C₃×C₂²⋊C₄	1	4	yes	—
165	C₂³	C₂×C₃×D₄	1	2	yes	—
166	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
167	Q₈	—	2	4	no	—
168	Q₈	—	2	2	no	—
169	Q₈	C₂²×C₃×C₄	1	4	yes	—
170	D₄	—	2	16	no	—
171	D₄	—	2	4	no	—
172	D₄	—	2	2	no	—
173	D₄	C₂²×C₃×C₄	1	4	yes	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	16	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	C₃×C₂²⋊C₄	1	4	yes	—
180	C₂×C₄	C₃×C₂²⋊C₄	1	8	yes	—
181	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
182	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
183	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
184	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
185	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
186	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
187	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
188	C₈	—	2	4	no	—
189	C₂×C₃	—	2	4	no	—
190	C₂×C₃	—	4	4	no	—
191	C₂×C₃	#416	1	4	yes	direct product factor
192	C₂×C₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
193	C₂×C₃	C₂×#135	1	2	yes	—
194	C₂²	—	2	4	no	—
195	C₂²	—	2	8	no	—
196	C₂²	—	2	4	no	—
197	C₂²	—	4	4	no	—
198	C₂²	—	4	8	no	—
199	C₂²	C₂×C₃×C₂²⋊C₄	1	4	yes	—
200	C₂²	C₂×C₃×C₂²⋊C₄	1	2	yes	—
201	C₂²	C₃×#135	1	4	yes	—
202	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
203	C₄	—	2	4	no	—
204	C₄	—	4	4	no	—
205	C₄	C₂×C₃×C₂²⋊C₄	1	4	yes	—
206	C₄	C₂×C₃×C₂²⋊C₄	1	2	yes	—
207	C₄	C₂³×C₃×C₄	1	1	yes	characteristic
208	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
209	C₃	C₂×#416	1	1	yes	characteristic, direct product factor, Sylow
210	C₂	—	2	4	no	—
211	C₂	—	4	4	no	—
212	C₂	C₃×#416	1	4	yes	direct product factor
213	C₂	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
214	C₂	C₂×C₃×#135	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	D₄
commutator	C₄	C₂³×C₃×C₄
Frattini, cosocle	C₂×C₄	C₂⁴×C₃
socle	C₂³×C₃	C₂×D₄
prime radical	C₂²×C₃×#41	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂³×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₃×C₈	C₂	semidirect
3. C₂×C₃×#117	C₂	semidirect
4. C₃×#416	C₂	direct (N × H)
5. C₂×C₃×#103	C₂	semidirect
6. C₂×C₃×#102	C₂	semidirect
7. C₂×#416	C₃	direct (N × H)
10. C₂×C₃×#41	C₄	semidirect
19. C₂²×C₃×C₈	C₂²	semidirect
21. C₃×#117	C₂²	semidirect
25. C₃×#103	C₂²	semidirect
26. C₃×#102	C₂²	semidirect
28. C₂³×C₈	C₂×C₃	semidirect
29. C₂×#117	C₂×C₃	semidirect
30. #416	C₂×C₃	direct (N × H)
31. C₂×#103	C₂×C₃	semidirect
32. C₂×#102	C₂×C₃	semidirect
37. C₃×#41	C₂×C₄	semidirect
67. C₂×#41	C₃×C₄	semidirect
76. C₂²×C₈	C₂²×C₃	semidirect
78. #117	C₂²×C₃	semidirect
82. #103	C₂²×C₃	semidirect
83. #102	C₂²×C₃	semidirect
117. #41	C₂×C₃×C₄	semidirect
191. C₂×C₃	#416	direct (N × H)
209. C₃	C₂×#416	direct (N × H)
212. C₂	C₃×#416	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×#41	2
C₂³×C₃×C₈	2
C₂×C₃×#117	2
C₃×#416	2
C₂×C₃×#103	2
C₂×C₃×#102	2
C₂×#416	3

Subgroups — C2×C3×#416 #84756

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#416 #84756