Subgroups — C₂×C₃×#536 #87331

Order 384.

1512 conjugacy classes of subgroups.

The subgroup lattice has 212 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×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#536	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃×C₄⋊Q₈	C₂	1	2	yes	maximal, complemented
5	C₂×C₃×#124	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×#123	C₂	1	4	yes	maximal, complemented
7	C₂×C₃×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
8	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#536	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₃×#41	—	2	8	no	—
11	C₂×C₃×#41	C₂²	1	8	yes	—
12	C₂²×C₃×Q₈	—	2	8	no	—
13	C₂²×C₃×Q₈	C₂²	1	2	yes	—
14	C₂²×C₃×Q₈	C₂²	1	2	yes	—
15	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
16	C₂²×C₃×Q₈	C₂²	1	8	yes	complemented
17	C₂²×C₃×D₄	C₂²	1	2	yes	—
18	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
19	C₂³×C₃×C₄	C₂²	1	2	yes	—
20	C₂³×C₃×C₄	C₂²	1	2	yes	—
21	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
22	C₃×C₄⋊Q₈	C₂²	1	16	yes	complemented
23	C₃×#124	C₂²	1	16	yes	complemented
24	C₃×#123	C₂²	1	32	yes	complemented
25	C₃×C₂²⋊Q₈	C₂²	1	32	yes	complemented
26	C₃×#117	C₂²	1	8	yes	complemented
27	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
28	C₂×C₃×C₄⋊C₄	C₂²	1	8	yes	—
29	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
30	C₂×C₃×C₂²⋊C₄	C₂²	1	8	yes	—
31	C₂×C₃×C₄²	C₂²	1	2	yes	—
32	C₂²×#41	C₂×C₃	1	1	yes	characteristic
33	C₂³×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
34	#536	C₂×C₃	1	16	yes	direct product factor
35	C₂×C₄⋊Q₈	C₂×C₃	1	2	yes	complemented
36	C₂×#124	C₂×C₃	1	2	yes	complemented
37	C₂×#123	C₂×C₃	1	4	yes	complemented
38	C₂×C₂²⋊Q₈	C₂×C₃	1	4	yes	complemented
39	C₂×#117	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂⁴×C₃	C₂³	1	2	yes	—
41	C₂⁴×C₃	C₂³	1	1	yes	characteristic
42	C₃×#41	—	2	32	no	—
43	C₂×C₃×Q₈	—	2	8	no	—
44	C₂×C₃×Q₈	—	2	8	no	—
45	C₂×C₃×Q₈	—	2	32	no	—
46	C₂×C₃×Q₈	—	2	4	no	—
47	C₂×C₃×Q₈	C₂³	1	8	yes	—
48	C₂×C₃×Q₈	C₂³	1	8	yes	—
49	C₂×C₃×Q₈	C₂³	1	4	yes	—
50	C₂×C₃×D₄	—	2	8	no	—
51	C₂×C₃×D₄	—	2	4	no	—
52	C₂×C₃×D₄	C₂³	1	8	yes	—
53	C₂×C₃×D₄	C₂³	1	4	yes	—
54	C₂²×C₃×C₄	—	2	8	no	—
55	C₂²×C₃×C₄	—	2	8	no	—
56	C₂²×C₃×C₄	D₄	1	8	yes	—
57	C₂²×C₃×C₄	C₂³	1	8	yes	—
58	C₂²×C₃×C₄	C₂³	1	8	yes	—
59	C₂²×C₃×C₄	C₂³	1	4	yes	—
60	C₂²×C₃×C₄	C₂³	1	4	yes	—
61	C₂²×C₃×C₄	C₂³	1	2	yes	—
62	C₂²×C₃×C₄	C₂³	1	4	yes	—
63	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
64	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
65	C₃×C₄⋊C₄	C₂³	1	8	yes	—
66	C₃×C₄⋊C₄	C₂³	1	32	yes	—
67	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
68	C₃×C₂²⋊C₄	C₂³	1	32	yes	—
69	C₃×C₄²	C₂³	1	8	yes	—
70	C₂×#41	—	2	8	no	—
71	C₂×#41	C₂²×C₃	1	8	yes	—
72	C₂²×Q₈	—	2	8	no	—
73	C₂²×Q₈	C₂²×C₃	1	2	yes	—
74	C₂²×Q₈	C₂²×C₃	1	2	yes	—
75	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
76	C₂²×Q₈	C₂²×C₃	1	8	yes	complemented
77	C₂²×D₄	C₂²×C₃	1	2	yes	—
78	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
79	C₂³×C₄	C₂²×C₃	1	2	yes	—
80	C₂³×C₄	C₂²×C₃	1	2	yes	—
81	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
82	C₄⋊Q₈	C₂²×C₃	1	16	yes	complemented
83	#124	C₂²×C₃	1	16	yes	complemented
84	#123	C₂²×C₃	1	32	yes	complemented
85	C₂²⋊Q₈	C₂²×C₃	1	32	yes	complemented
86	#117	C₂²×C₃	1	8	yes	complemented
87	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
88	C₂×C₄⋊C₄	C₂²×C₃	1	8	yes	—
89	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
90	C₂×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
91	C₂×C₄²	C₂²×C₃	1	2	yes	—
92	C₂³×C₃	—	2	4	no	—
93	C₂³×C₃	—	2	4	no	—
94	C₂³×C₃	—	2	8	no	—
95	C₂³×C₃	C₂×D₄	1	4	yes	—
96	C₂³×C₃	C₂⁴	1	4	yes	—
97	C₂³×C₃	C₂⁴	1	2	yes	—
98	C₂³×C₃	C₂⁴	1	1	yes	characteristic
99	C₃×Q₈	—	2	16	no	—
100	C₃×Q₈	—	2	16	no	—
101	C₃×Q₈	—	2	8	no	—
102	C₃×D₄	—	2	16	no	—
103	C₃×D₄	—	2	8	no	—
104	C₂×C₃×C₄	—	2	4	no	—
105	C₂×C₃×C₄	—	2	16	no	—
106	C₂×C₃×C₄	—	2	8	no	—
107	C₂×C₃×C₄	—	2	8	no	—
108	C₂×C₃×C₄	—	2	16	no	—
109	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
110	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
111	C₂×C₃×C₄	C₂×D₄	1	16	yes	—
112	C₂×C₃×C₄	C₂⁴	1	8	yes	—
113	C₂×C₃×C₄	C₂⁴	1	8	yes	—
114	C₂×C₃×C₄	C₂⁴	1	4	yes	—
115	C₂×C₃×C₄	C₂⁴	1	2	yes	—
116	C₂×C₃×C₄	C₂⁴	1	2	yes	—
117	C₂⁴	C₂³×C₃	1	2	yes	—
118	C₂⁴	C₂³×C₃	1	1	yes	characteristic
119	#41	—	2	32	no	—
120	C₂×Q₈	—	2	8	no	—
121	C₂×Q₈	—	2	8	no	—
122	C₂×Q₈	—	2	32	no	—
123	C₂×Q₈	—	2	4	no	—
124	C₂×Q₈	C₂³×C₃	1	8	yes	—
125	C₂×Q₈	C₂³×C₃	1	8	yes	—
126	C₂×Q₈	C₂³×C₃	1	4	yes	—
127	C₂×D₄	—	2	8	no	—
128	C₂×D₄	—	2	4	no	—
129	C₂×D₄	C₂³×C₃	1	8	yes	—
130	C₂×D₄	C₂³×C₃	1	4	yes	—
131	C₂²×C₄	—	2	8	no	—
132	C₂²×C₄	—	2	8	no	—
133	C₂²×C₄	C₃×D₄	1	8	yes	—
134	C₂²×C₄	C₂³×C₃	1	4	yes	—
135	C₂²×C₄	C₂³×C₃	1	4	yes	—
136	C₂²×C₄	C₂³×C₃	1	2	yes	—
137	C₂²×C₄	C₂³×C₃	1	8	yes	—
138	C₂²×C₄	C₂³×C₃	1	8	yes	—
139	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
140	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
141	C₂²×C₄	C₂³×C₃	1	4	yes	—
142	C₄⋊C₄	C₂³×C₃	1	8	yes	—
143	C₄⋊C₄	C₂³×C₃	1	32	yes	—
144	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
145	C₂²⋊C₄	C₂³×C₃	1	32	yes	—
146	C₄²	C₂³×C₃	1	8	yes	—
147	C₂²×C₃	—	2	8	no	—
148	C₂²×C₃	—	2	4	no	—
149	C₂²×C₃	—	2	4	no	—
150	C₂²×C₃	—	2	8	no	—
151	C₂²×C₃	—	4	8	no	—
152	C₂²×C₃	C₂²×D₄	1	4	yes	—
153	C₂²×C₃	C₂²×D₄	1	2	yes	—
154	C₂²×C₃	2₋¹⁺⁴	1	4	yes	—
155	C₂²×C₃	C₂⁵	1	1	yes	characteristic
156	C₃×C₄	—	2	4	no	—
157	C₃×C₄	—	2	8	no	—
158	C₃×C₄	—	2	8	no	—
159	C₃×C₄	C₂²×D₄	1	4	yes	—
160	C₃×C₄	C₂²×D₄	1	4	yes	—
161	C₂³	—	2	4	no	—
162	C₂³	—	2	4	no	—
163	C₂³	—	2	8	no	—
164	C₂³	C₂×C₃×D₄	1	4	yes	—
165	C₂³	C₂⁴×C₃	1	4	yes	—
166	C₂³	C₂⁴×C₃	1	2	yes	—
167	C₂³	C₂⁴×C₃	1	1	yes	characteristic
168	Q₈	—	2	16	no	—
169	Q₈	—	2	16	no	—
170	Q₈	—	2	8	no	—
171	D₄	—	2	16	no	—
172	D₄	—	2	8	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	16	no	—
175	C₂×C₄	—	2	8	no	—
176	C₂×C₄	—	2	8	no	—
177	C₂×C₄	—	2	16	no	—
178	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
179	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
180	C₂×C₄	C₂×C₃×D₄	1	16	yes	—
181	C₂×C₄	C₂⁴×C₃	1	8	yes	—
182	C₂×C₄	C₂⁴×C₃	1	8	yes	—
183	C₂×C₄	C₂⁴×C₃	1	4	yes	—
184	C₂×C₄	C₂⁴×C₃	1	2	yes	—
185	C₂×C₄	C₂⁴×C₃	1	2	yes	—
186	C₂×C₃	—	2	4	no	—
187	C₂×C₃	—	4	4	no	—
188	C₂×C₃	#536	1	4	yes	direct product factor
189	C₂×C₃	C₂³×D₄	1	1	yes	characteristic
190	C₂×C₃	C₂×2₋¹⁺⁴	1	2	yes	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	8	no	—
193	C₂²	—	2	4	no	—
194	C₂²	—	2	8	no	—
195	C₂²	—	4	8	no	—
196	C₂²	C₂²×C₃×D₄	1	4	yes	—
197	C₂²	C₂²×C₃×D₄	1	2	yes	—
198	C₂²	C₃×2₋¹⁺⁴	1	4	yes	—
199	C₂²	C₂⁵×C₃	1	1	yes	characteristic
200	C₄	—	2	4	no	—
201	C₄	—	2	8	no	—
202	C₄	—	2	8	no	—
203	C₄	C₂²×C₃×D₄	1	4	yes	—
204	C₄	C₂²×C₃×D₄	1	4	yes	—
205	C₃	C₂×#536	1	1	yes	characteristic, direct product factor, Sylow
206	C₂	—	2	4	no	—
207	C₂	—	4	4	no	—
208	C₂	C₃×#536	1	4	yes	direct product factor
209	C₂	C₂³×C₃×D₄	1	1	yes	characteristic
210	C₂	C₂×C₃×2₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃
prime radical	C₂²×C₃×#41	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₃×Q₈	C₂	semidirect
3. C₃×#536	C₂	direct (N × H)
4. C₂×C₃×C₄⋊Q₈	C₂	semidirect
5. C₂×C₃×#124	C₂	semidirect
6. C₂×C₃×#123	C₂	semidirect
7. C₂×C₃×C₂²⋊Q₈	C₂	semidirect
8. C₂×C₃×#117	C₂	semidirect
9. C₂×#536	C₃	direct (N × H)
16. C₂²×C₃×Q₈	C₂²	semidirect
22. C₃×C₄⋊Q₈	C₂²	semidirect
23. C₃×#124	C₂²	semidirect
24. C₃×#123	C₂²	semidirect
25. C₃×C₂²⋊Q₈	C₂²	semidirect
26. C₃×#117	C₂²	semidirect
33. C₂³×Q₈	C₂×C₃	semidirect
34. #536	C₂×C₃	direct (N × H)
35. C₂×C₄⋊Q₈	C₂×C₃	semidirect
36. C₂×#124	C₂×C₃	semidirect
37. C₂×#123	C₂×C₃	semidirect
38. C₂×C₂²⋊Q₈	C₂×C₃	semidirect
39. C₂×#117	C₂×C₃	semidirect
76. C₂²×Q₈	C₂²×C₃	semidirect
82. C₄⋊Q₈	C₂²×C₃	semidirect
83. #124	C₂²×C₃	semidirect
84. #123	C₂²×C₃	semidirect
85. C₂²⋊Q₈	C₂²×C₃	semidirect
86. #117	C₂²×C₃	semidirect
188. C₂×C₃	#536	direct (N × H)
205. C₃	C₂×#536	direct (N × H)
208. C₂	C₃×#536	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×Q₈	2
C₃×#536	2
C₂×C₃×C₄⋊Q₈	2
C₂×C₃×#124	2
C₂×C₃×#123	2
C₂×C₃×C₂²⋊Q₈	2
C₂×C₃×#117	2
C₂×#536	3

Subgroups — C2×C3×#536 #87331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#536 #87331