Subgroups — C₂×C₃×#417 #84759

Order 384.

816 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	C₂²×C₃×#41	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#417	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×#417	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	—	2	1	no	—
17	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
18	C₂×C₃×M₁₆	—	2	4	no	—
19	C₂×C₃×M₁₆	C₂²	1	4	yes	complemented
20	C₂²×C₃×C₈	C₂²	1	2	yes	—
21	C₂×C₃×C₄⋊C₄	—	2	2	no	—
22	C₂×C₃×C₄⋊C₄	—	2	1	no	—
23	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
24	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	complemented
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	C₂²×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
30	#417	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	complemented
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	complemented
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	4	no	—
52	C₂²×C₃×C₄	—	2	2	no	—
53	C₂²×C₃×C₄	—	4	4	no	—
54	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
55	C₂²×C₃×C₄	D₄	1	2	yes	—
56	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
57	C₂²×C₃×C₄	D₄	1	4	yes	—
58	C₂²×C₃×C₄	C₂³	1	2	yes	—
59	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
60	C₃×M₁₆	—	2	8	no	—
61	C₂×C₃×C₈	—	2	4	no	—
62	C₂×C₃×C₈	C₂³	1	4	yes	—
63	C₃×C₄⋊C₄	—	2	2	no	—
64	C₃×C₄⋊C₄	—	2	4	no	—
65	C₃×C₄⋊C₄	C₂³	1	4	yes	—
66	C₂×#41	—	2	4	no	—
67	C₂×#41	—	2	2	no	—
68	C₂×#41	C₃×C₄	1	8	yes	complemented
69	C₂×#41	C₂²×C₃	1	4	yes	—
70	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
71	C₂²×D₄	—	2	1	no	—
72	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
73	C₂³×C₄	—	2	1	no	—
74	C₂³×C₄	—	2	1	no	—
75	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
76	C₂×M₁₆	—	2	4	no	—
77	C₂×M₁₆	C₂²×C₃	1	4	yes	complemented
78	C₂²×C₈	C₂²×C₃	1	2	yes	—
79	C₂×C₄⋊C₄	—	2	2	no	—
80	C₂×C₄⋊C₄	—	2	1	no	—
81	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
82	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	complemented
83	#103	C₂²×C₃	1	8	yes	complemented
84	#102	C₂²×C₃	1	8	yes	complemented
85	C₂³×C₃	—	2	2	no	—
86	C₂³×C₃	—	2	2	no	—
87	C₂³×C₃	—	2	4	no	—
88	C₂³×C₃	—	2	2	no	—
89	C₂³×C₃	—	4	2	no	—
90	C₂³×C₃	—	4	2	no	—
91	C₂³×C₃	C₂²⋊C₄	1	4	yes	—
92	C₂³×C₃	C₂×D₄	1	2	yes	—
93	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
94	C₃×Q₈	—	2	4	no	—
95	C₃×Q₈	—	2	2	no	—
96	C₃×Q₈	C₂²×C₄	1	4	yes	complemented
97	C₃×D₄	—	2	16	no	—
98	C₃×D₄	—	2	4	no	—
99	C₃×D₄	—	2	2	no	—
100	C₃×D₄	C₂²×C₄	1	4	yes	complemented
101	C₂×C₃×C₄	—	2	4	no	—
102	C₂×C₃×C₄	—	2	2	no	—
103	C₂×C₃×C₄	—	2	16	no	—
104	C₂×C₃×C₄	—	2	4	no	—
105	C₂×C₃×C₄	—	4	8	no	—
106	C₂×C₃×C₄	—	4	4	no	—
107	C₂×C₃×C₄	C₂²⋊C₄	1	4	yes	—
108	C₂×C₃×C₄	C₂²⋊C₄	1	8	yes	—
109	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
110	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
111	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
112	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
113	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
114	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
115	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
116	C₃×C₈	—	2	4	no	—
117	C₂⁴	—	2	1	no	—
118	C₂⁴	C₃×D₄	1	1	yes	characteristic
119	#41	—	2	16	no	—
120	#41	—	2	8	no	—
121	#41	C₂×C₃×C₄	1	16	yes	complemented
122	C₂×Q₈	—	2	2	no	—
123	C₂×Q₈	—	2	1	no	—
124	C₂×Q₈	C₂×C₃×C₄	1	4	yes	complemented
125	C₂×Q₈	C₂³×C₃	1	2	yes	—
126	C₂×D₄	—	2	8	no	—
127	C₂×D₄	—	2	4	no	—
128	C₂×D₄	—	2	1	no	—
129	C₂×D₄	—	2	2	no	—
130	C₂×D₄	C₂×C₃×C₄	1	4	yes	complemented
131	C₂×D₄	C₂³×C₃	1	2	yes	—
132	C₂²×C₄	—	2	1	no	—
133	C₂²×C₄	—	2	8	no	—
134	C₂²×C₄	—	2	4	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	4	4	no	—
138	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
139	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
140	C₂²×C₄	C₃×D₄	1	2	yes	—
141	C₂²×C₄	C₃×D₄	1	4	yes	—
142	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
143	C₂²×C₄	C₂³×C₃	1	2	yes	—
144	M₁₆	—	2	8	no	—
145	C₂×C₈	—	2	4	no	—
146	C₂×C₈	C₂³×C₃	1	4	yes	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	C₂³×C₃	1	4	yes	—
150	C₂²×C₃	—	2	2	no	—
151	C₂²×C₃	—	2	2	no	—
152	C₂²×C₃	—	2	8	no	—
153	C₂²×C₃	—	2	4	no	—
154	C₂²×C₃	—	4	2	no	—
155	C₂²×C₃	—	4	2	no	—
156	C₂²×C₃	—	4	8	no	—
157	C₂²×C₃	C₂×C₂²⋊C₄	1	4	yes	—
158	C₂²×C₃	C₂×C₂²⋊C₄	1	2	yes	—
159	C₂²×C₃	AGL(1,ℤ/8)	1	2	yes	—
160	C₂²×C₃	#137	1	2	yes	—
161	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
162	C₃×C₄	—	2	4	no	—
163	C₃×C₄	—	4	4	no	—
164	C₃×C₄	C₂×C₂²⋊C₄	1	4	yes	—
165	C₃×C₄	C₂×C₂²⋊C₄	1	2	yes	—
166	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
167	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
168	C₂³	—	2	2	no	—
169	C₂³	—	2	2	no	—
170	C₂³	—	2	4	no	—
171	C₂³	—	2	2	no	—
172	C₂³	—	4	2	no	—
173	C₂³	—	4	2	no	—
174	C₂³	C₃×C₂²⋊C₄	1	4	yes	—
175	C₂³	C₂×C₃×D₄	1	2	yes	—
176	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
177	Q₈	—	2	4	no	—
178	Q₈	—	2	2	no	—
179	Q₈	C₂²×C₃×C₄	1	4	yes	complemented
180	D₄	—	2	16	no	—
181	D₄	—	2	4	no	—
182	D₄	—	2	2	no	—
183	D₄	C₂²×C₃×C₄	1	4	yes	complemented
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	16	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	4	8	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	C₃×C₂²⋊C₄	1	4	yes	—
191	C₂×C₄	C₃×C₂²⋊C₄	1	8	yes	—
192	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
193	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
194	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
195	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
196	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
197	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
198	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
199	C₈	—	2	4	no	—
200	C₂×C₃	—	2	4	no	—
201	C₂×C₃	—	4	4	no	—
202	C₂×C₃	#417	1	4	yes	direct product factor
203	C₂×C₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
204	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
205	C₂×C₃	C₂×#137	1	1	yes	characteristic
206	C₂²	—	2	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	8	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	4	2	no	—
212	C₂²	—	4	8	no	—
213	C₂²	C₂×C₃×C₂²⋊C₄	1	4	yes	—
214	C₂²	C₂×C₃×C₂²⋊C₄	1	2	yes	—
215	C₂²	C₃×AGL(1,ℤ/8)	1	2	yes	—
216	C₂²	C₃×#137	1	2	yes	—
217	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
218	C₄	—	2	4	no	—
219	C₄	—	4	4	no	—
220	C₄	C₂×C₃×C₂²⋊C₄	1	4	yes	—
221	C₄	C₂×C₃×C₂²⋊C₄	1	2	yes	—
222	C₄	C₂³×C₃×C₄	1	1	yes	characteristic
223	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
224	C₃	C₂×#417	1	1	yes	characteristic, direct product factor, Sylow
225	C₂	—	2	4	no	—
226	C₂	—	4	4	no	—
227	C₂	C₃×#417	1	4	yes	direct product factor
228	C₂	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
229	C₂	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
230	C₂	C₂×C₃×#137	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×M₁₆	C₂	semidirect
3. C₂²×C₃×C₄⋊C₄	C₂	semidirect
4. C₃×#417	C₂	direct (N × H)
5. C₂×C₃×#103	C₂	semidirect
6. C₂×C₃×#102	C₂	semidirect
7. C₂×#417	C₃	direct (N × H)
10. C₂×C₃×#41	C₄	semidirect
19. C₂×C₃×M₁₆	C₂²	semidirect
24. C₂×C₃×C₄⋊C₄	C₂²	semidirect
25. C₃×#103	C₂²	semidirect
26. C₃×#102	C₂²	semidirect
28. C₂²×M₁₆	C₂×C₃	semidirect
29. C₂²×C₄⋊C₄	C₂×C₃	semidirect
30. #417	C₂×C₃	direct (N × H)
31. C₂×#103	C₂×C₃	semidirect
32. C₂×#102	C₂×C₃	semidirect
37. C₃×#41	C₂×C₄	semidirect
40. C₂×C₃×Q₈	C₂×C₄	semidirect
46. C₂×C₃×D₄	C₂×C₄	semidirect
68. C₂×#41	C₃×C₄	semidirect
77. C₂×M₁₆	C₂²×C₃	semidirect
82. C₂×C₄⋊C₄	C₂²×C₃	semidirect
83. #103	C₂²×C₃	semidirect
84. #102	C₂²×C₃	semidirect
96. C₃×Q₈	C₂²×C₄	semidirect
100. C₃×D₄	C₂²×C₄	semidirect
121. #41	C₂×C₃×C₄	semidirect
124. C₂×Q₈	C₂×C₃×C₄	semidirect
130. C₂×D₄	C₂×C₃×C₄	semidirect
179. Q₈	C₂²×C₃×C₄	semidirect
183. D₄	C₂²×C₃×C₄	semidirect
202. C₂×C₃	#417	direct (N × H)
224. C₃	C₂×#417	direct (N × H)
227. C₂	C₃×#417	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
C₂²×C₃×M₁₆	2
C₂²×C₃×C₄⋊C₄	2
C₃×#417	2
C₂×C₃×#103	2
C₂×C₃×#102	2
C₂×#417	3

Subgroups — C2×C3×#417 #84759

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#417 #84759