Subgroups — C₃×C₄²×D₈ #10639104

Order 768.

1350 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×D₈	C₂	1	3	yes	maximal
2	C₃×C₄²×D₄	C₂	1	2	yes	maximal, complemented
3	C₃×C₄×#107	C₂	1	3	yes	maximal, complemented
4	C₃×C₄×#102	C₂	1	6	yes	maximal, complemented
5	C₃×C₄²×C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄²×D₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂²×C₃×D₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄×D₄	C₂²	1	6	yes	—
9	C₂²×C₃×C₄²	—	2	2	no	—
10	C₃×C₄×D₈	C₄	1	24	yes	direct product factor
11	C₂×C₃×#107	C₂²	1	3	yes	—
12	C₂×C₃×#102	C₂²	1	6	yes	—
13	C₂×C₃×C₄×C₈	C₂²	1	3	yes	—
14	C₂×C₃×C₄×C₈	C₂²	1	3	yes	—
15	C₃×C₄×C₄⋊C₄	C₂²	1	6	yes	—
16	C₃×C₄×C₂²⋊C₄	—	2	6	no	—
17	C₃×C₄³	C₂²	1	1	yes	characteristic
18	C₃×#340	C₂²	1	6	yes	—
19	C₂×C₄×D₈	C₂×C₃	1	3	yes	—
20	C₄²×D₄	C₂×C₃	1	2	yes	complemented
21	C₄×#107	C₂×C₃	1	3	yes	complemented
22	C₄×#102	C₂×C₃	1	6	yes	complemented
23	C₄²×C₈	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂²×C₃×D₄	C₂³	1	2	yes	—
25	C₂³×C₃×C₄	—	2	6	no	—
26	C₂×C₃×D₈	C₂×C₄	1	12	yes	—
27	C₂²×C₃×C₈	C₂³	1	3	yes	—
28	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
29	C₃×C₄×D₄	—	2	12	no	—
30	C₃×C₄×D₄	C₂×C₄	1	24	yes	complemented
31	C₂×C₃×C₄⋊C₄	C₂³	1	6	yes	—
32	C₂×C₃×C₂²⋊C₄	—	2	6	no	—
33	C₂×C₃×C₄²	—	2	6	no	—
34	C₂×C₃×C₄²	—	4	8	no	—
35	C₂×C₃×C₄²	D₄	1	3	yes	—
36	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
37	C₂×C₃×C₄²	C₂³	1	3	yes	—
38	C₃×#107	C₂×C₄	1	12	yes	complemented
39	C₃×#102	C₂×C₄	1	24	yes	complemented
40	C₃×C₄×C₈	—	2	6	no	—
41	C₃×C₄×C₈	C₂×C₄	1	12	yes	complemented
42	C₃×#95	—	2	6	no	—
43	C₂²×D₈	C₂²×C₃	1	1	yes	characteristic
44	C₂×C₄×D₄	C₂²×C₃	1	6	yes	—
45	C₂²×C₄²	—	2	2	no	—
46	C₄×D₈	C₃×C₄	1	24	yes	direct product factor
47	C₂×#107	C₂²×C₃	1	3	yes	—
48	C₂×#102	C₂²×C₃	1	6	yes	—
49	C₂×C₄×C₈	C₂²×C₃	1	3	yes	—
50	C₂×C₄×C₈	C₂²×C₃	1	3	yes	—
51	C₄×C₄⋊C₄	C₂²×C₃	1	6	yes	—
52	C₄×C₂²⋊C₄	—	2	6	no	—
53	C₄³	C₂²×C₃	1	1	yes	characteristic
54	#340	C₂²×C₃	1	6	yes	—
55	C₂⁴×C₃	—	2	2	no	—
56	C₂×C₃×D₄	—	2	6	no	—
57	C₂×C₃×D₄	C₂²×C₄	1	12	yes	—
58	C₂²×C₃×C₄	—	2	6	no	—
59	C₂²×C₃×C₄	—	2	24	no	—
60	C₂²×C₃×C₄	—	4	12	no	—
61	C₂²×C₃×C₄	—	4	12	no	—
62	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
63	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
64	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
65	C₃×D₈	C₄²	1	16	yes	direct product factor
66	C₂×C₃×C₈	—	2	6	no	—
67	C₂×C₃×C₈	C₂²×C₄	1	6	yes	—
68	C₂×C₃×C₈	C₂²×C₄	1	6	yes	—
69	C₃×C₄⋊C₄	—	2	6	no	—
70	C₃×C₄⋊C₄	C₂²×C₄	1	12	yes	—
71	C₃×C₂²⋊C₄	—	2	24	no	—
72	C₃×C₄²	—	2	6	no	—
73	C₃×C₄²	—	4	12	no	—
74	C₃×C₄²	D₈	1	4	yes	direct product factor
75	C₃×C₄²	C₂²×C₄	1	6	yes	—
76	C₃×C₄²	#41	1	6	yes	—
77	C₂²×D₄	C₂³×C₃	1	2	yes	—
78	C₂³×C₄	—	2	6	no	—
79	C₂×D₈	C₂×C₃×C₄	1	12	yes	—
80	C₂²×C₈	C₂³×C₃	1	3	yes	—
81	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
82	C₄×D₄	—	2	12	no	—
83	C₄×D₄	C₂×C₃×C₄	1	24	yes	complemented
84	C₂×C₄⋊C₄	C₂³×C₃	1	6	yes	—
85	C₂×C₂²⋊C₄	—	2	6	no	—
86	C₂×C₄²	—	2	6	no	—
87	C₂×C₄²	—	4	8	no	—
88	C₂×C₄²	C₃×D₄	1	3	yes	—
89	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
90	C₂×C₄²	C₂³×C₃	1	3	yes	—
91	#107	C₂×C₃×C₄	1	12	yes	complemented
92	#102	C₂×C₃×C₄	1	24	yes	complemented
93	C₄×C₈	—	2	6	no	—
94	C₄×C₈	C₂×C₃×C₄	1	12	yes	complemented
95	#95	—	2	6	no	—
96	C₂³×C₃	—	2	12	no	—
97	C₂³×C₃	—	4	2	no	—
98	C₂³×C₃	—	4	6	no	—
99	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
100	C₃×D₄	—	2	12	no	—
101	C₃×D₄	C₂×C₄²	1	8	yes	complemented
102	C₂×C₃×C₄	—	2	6	no	—
103	C₂×C₃×C₄	—	2	12	no	—
104	C₂×C₃×C₄	—	4	6	no	—
105	C₂×C₃×C₄	—	4	6	no	—
106	C₂×C₃×C₄	—	4	48	no	—
107	C₂×C₃×C₄	C₄×D₄	1	6	yes	—
108	C₂×C₃×C₄	C₄×D₄	1	6	yes	—
109	C₂×C₃×C₄	C₂×D₈	1	6	yes	—
110	C₂×C₃×C₄	#135	1	6	yes	—
111	C₂×C₃×C₄	C₂³×C₄	1	3	yes	—
112	C₂×C₃×C₄	C₂×#41	1	3	yes	—
113	C₃×C₈	—	2	6	no	—
114	C₃×C₈	C₂×C₄²	1	4	yes	complemented
115	C₂⁴	—	2	2	no	—
116	C₂×D₄	—	2	6	no	—
117	C₂×D₄	C₂²×C₃×C₄	1	12	yes	—
118	C₂²×C₄	—	2	6	no	—
119	C₂²×C₄	—	2	24	no	—
120	C₂²×C₄	—	4	12	no	—
121	C₂²×C₄	—	4	12	no	—
122	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
123	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
124	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
125	D₈	C₃×C₄²	1	16	yes	direct product factor
126	C₂×C₈	—	2	6	no	—
127	C₂×C₈	C₂²×C₃×C₄	1	6	yes	—
128	C₂×C₈	C₂²×C₃×C₄	1	6	yes	—
129	C₄⋊C₄	—	2	6	no	—
130	C₄⋊C₄	C₂²×C₃×C₄	1	12	yes	—
131	C₂²⋊C₄	—	2	24	no	—
132	C₄²	—	2	6	no	—
133	C₄²	—	4	12	no	—
134	C₄²	C₃×D₈	1	4	yes	direct product factor
135	C₄²	C₂²×C₃×C₄	1	6	yes	—
136	C₄²	C₃×#41	1	6	yes	—
137	C₂²×C₃	—	2	8	no	—
138	C₂²×C₃	—	4	12	no	—
139	C₂²×C₃	—	4	12	no	—
140	C₂²×C₃	C₂×C₄×D₄	1	3	yes	—
141	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
142	C₂²×C₃	C₂×#135	1	3	yes	—
143	C₃×C₄	—	2	6	no	—
144	C₃×C₄	—	4	12	no	—
145	C₃×C₄	C₄×D₈	1	12	yes	direct product factor
146	C₃×C₄	C₂²×C₄²	1	1	yes	characteristic
147	C₃×C₄	C₄×#41	1	3	yes	—
148	C₂³	—	2	12	no	—
149	C₂³	—	4	2	no	—
150	C₂³	—	4	6	no	—
151	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
152	D₄	—	2	12	no	—
153	D₄	C₂×C₃×C₄²	1	8	yes	complemented
154	C₂×C₄	—	2	6	no	—
155	C₂×C₄	—	2	12	no	—
156	C₂×C₄	—	4	6	no	—
157	C₂×C₄	—	4	6	no	—
158	C₂×C₄	—	4	48	no	—
159	C₂×C₄	C₃×C₄×D₄	1	6	yes	—
160	C₂×C₄	C₃×C₄×D₄	1	6	yes	—
161	C₂×C₄	C₂×C₃×D₈	1	6	yes	—
162	C₂×C₄	C₃×#135	1	6	yes	—
163	C₂×C₄	C₂³×C₃×C₄	1	3	yes	—
164	C₂×C₄	C₂×C₃×#41	1	3	yes	—
165	C₈	—	2	6	no	—
166	C₈	C₂×C₃×C₄²	1	4	yes	complemented
167	C₂×C₃	—	4	8	no	—
168	C₂×C₃	C₄²×D₄	1	1	yes	characteristic
169	C₂×C₃	C₂×C₄×D₈	1	3	yes	—
170	C₂×C₃	C₄×#135	1	3	yes	—
171	C₂²	—	2	8	no	—
172	C₂²	—	4	12	no	—
173	C₂²	—	4	12	no	—
174	C₂²	C₂×C₃×C₄×D₄	1	3	yes	—
175	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
176	C₂²	C₂×C₃×#135	1	3	yes	—
177	C₄	—	2	6	no	—
178	C₄	—	4	12	no	—
179	C₄	C₃×C₄×D₈	1	12	yes	direct product factor
180	C₄	C₂²×C₃×C₄²	1	1	yes	characteristic
181	C₄	C₃×C₄×#41	1	3	yes	—
182	C₃	C₄²×D₈	1	1	yes	characteristic, direct product factor, Sylow
183	C₂	—	4	8	no	—
184	C₂	C₃×C₄²×D₄	1	1	yes	characteristic
185	C₂	C₂×C₃×C₄×D₈	1	3	yes	—
186	C₂	C₃×C₄×#135	1	3	yes	—

	subgroup	G/H
center	C₂×C₃×C₄²	D₄
commutator	C₄	C₂²×C₃×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
socle	C₂³×C₃	C₂²×D₄
prime radical	C₂²×C₃×D₈	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄² ◁ C₃×C₄³ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₄²×D₄	C₂	semidirect
3. C₃×C₄×#107	C₂	semidirect
4. C₃×C₄×#102	C₂	semidirect
5. C₃×C₄²×C₈	C₂	semidirect
6. C₄²×D₈	C₃	direct (N × H)
10. C₃×C₄×D₈	C₄	direct (N × H)
20. C₄²×D₄	C₂×C₃	semidirect
21. C₄×#107	C₂×C₃	semidirect
22. C₄×#102	C₂×C₃	semidirect
23. C₄²×C₈	C₂×C₃	semidirect
30. C₃×C₄×D₄	C₂×C₄	semidirect
38. C₃×#107	C₂×C₄	semidirect
39. C₃×#102	C₂×C₄	semidirect
41. C₃×C₄×C₈	C₂×C₄	semidirect
46. C₄×D₈	C₃×C₄	direct (N × H)
65. C₃×D₈	C₄²	direct (N × H)
74. C₃×C₄²	D₈	direct (N × H)
83. C₄×D₄	C₂×C₃×C₄	semidirect
91. #107	C₂×C₃×C₄	semidirect
92. #102	C₂×C₃×C₄	semidirect
94. C₄×C₈	C₂×C₃×C₄	semidirect
101. C₃×D₄	C₂×C₄²	semidirect
114. C₃×C₈	C₂×C₄²	semidirect
125. D₈	C₃×C₄²	direct (N × H)
134. C₄²	C₃×D₈	direct (N × H)
145. C₃×C₄	C₄×D₈	direct (N × H)
153. D₄	C₂×C₃×C₄²	semidirect
166. C₈	C₂×C₃×C₄²	semidirect
179. C₄	C₃×C₄×D₈	direct (N × H)
182. C₃	C₄²×D₈	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×C₄×D₈	2
C₃×C₄²×D₄	2
C₃×C₄×#107	2
C₃×C₄×#102	2
C₃×C₄²×C₈	2
C₄²×D₈	3

Subgroups — C3×C42×D8 #10639104

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄²×D₈ #10639104