Subgroups — C₂×#83780 #11701656

Order 768.

2074 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×#131	C₂	1	1	yes	maximal, characteristic
2	#83780	C₂	1	16	yes	maximal, direct product factor
3	C₂×#5140	C₂	1	8	yes	maximal, complemented
4	C₂²×#812	C₂	1	6	yes	maximal, complemented
5	C₂×#3407	—	3	1	no	maximal, Sylow
6	C₂²×C₃×#41	C₄	1	1	yes	characteristic
7	C₂×C₃×#131	C₄	1	4	yes	—
8	C₂×C₃×#131	C₄	1	8	yes	—
9	C₂×C₃×#131	C₂²	1	4	yes	—
10	C₂×C₃×#131	C₂²	1	8	yes	—
11	C₂²×C₃×M₁₆	C₄	1	3	yes	—
12	C₂³×C₃×C₈	C₂²	1	3	yes	—
13	#5140	C₂²	1	64	yes	complemented
14	C₂×#812	C₂²	1	4	yes	—
15	C₂×#812	C₂²	1	12	yes	complemented
16	C₂×#812	C₂²	1	48	yes	complemented
17	C₂²×C₃⋊C₁₆	C₂²	1	12	yes	complemented
18	C₂²×#131	D₃	1	1	yes	characteristic
19	#3407	—	3	16	no	—
20	C₂×#504	—	3	8	no	—
21	C₂²×M₃₂	—	3	6	no	—
22	C₂×C₃×#41	C₈	1	4	yes	—
23	C₂×C₃×#41	C₈	1	8	yes	—
24	C₂×C₃×#41	C₂×C₄	1	4	yes	—
25	C₂×C₃×#41	C₂×C₄	1	8	yes	—
26	C₂²×C₃×Q₈	C₈	1	1	yes	characteristic
27	C₂²×C₃×D₄	C₈	1	3	yes	—
28	C₂³×C₃×C₄	C₂×C₄	1	3	yes	—
29	C₃×#131	C₂×C₄	1	16	yes	—
30	C₃×#131	C₂×C₄	1	32	yes	—
31	C₃×#131	C₂³	1	16	yes	—
32	C₂×C₃×M₁₆	C₂×C₄	1	12	yes	—
33	C₂×C₃×M₁₆	C₂×C₄	1	24	yes	—
34	C₂²×C₃×C₈	C₂×C₄	1	3	yes	—
35	C₂²×C₃×C₈	C₂×C₄	1	6	yes	—
36	C₂²×C₃×C₈	C₂×C₄	1	12	yes	—
37	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
38	C₂²×C₃×C₈	C₂³	1	6	yes	—
39	C₂²×C₃×C₈	C₂³	1	12	yes	—
40	#812	C₂³	1	16	yes	—
41	#812	C₂³	1	48	yes	complemented
42	C₂×C₃⋊C₁₆	C₂³	1	8	yes	—
43	C₂×C₃⋊C₁₆	C₂³	1	48	yes	complemented
44	C₂²×#41	Dic₃	1	1	yes	characteristic
45	C₂×#131	Dic₃	1	4	yes	—
46	C₂×#131	Dic₃	1	8	yes	—
47	C₂×#131	C₂×D₃	1	4	yes	—
48	C₂×#131	C₂×D₃	1	8	yes	—
49	C₂²×M₁₆	Dic₃	1	3	yes	—
50	C₂³×C₈	C₂×D₃	1	3	yes	—
51	#504	—	3	64	no	—
52	C₂×M₃₂	—	3	4	no	—
53	C₂×M₃₂	—	3	12	no	—
54	C₂×M₃₂	—	3	48	no	—
55	C₂²×C₁₆	—	3	12	no	—
56	C₂⁴×C₃	C₂×C₈	1	3	yes	—
57	C₃×#41	C₂×C₈	1	16	yes	—
58	C₃×#41	C₂×C₈	1	32	yes	—
59	C₃×#41	C₂²×C₄	1	16	yes	—
60	C₂×C₃×Q₈	C₂×C₈	1	8	yes	—
61	C₂×C₃×Q₈	C₂×C₈	1	4	yes	—
62	C₂×C₃×D₄	C₂×C₈	1	12	yes	—
63	C₂×C₃×D₄	C₂×C₈	1	24	yes	—
64	C₂²×C₃×C₄	C₂×C₈	1	3	yes	—
65	C₂²×C₃×C₄	C₂×C₈	1	6	yes	—
66	C₂²×C₃×C₄	C₂×C₈	1	12	yes	—
67	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₃×C₄	C₂²×C₄	1	6	yes	—
69	C₂²×C₃×C₄	C₂²×C₄	1	12	yes	—
70	C₃×M₁₆	C₂²×C₄	1	48	yes	—
71	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
73	C₂×C₃×C₈	C₂²×C₄	1	6	yes	—
74	C₂×C₃×C₈	C₂²×C₄	1	12	yes	—
75	C₂×C₃×C₈	C₂²×C₄	1	12	yes	—
76	C₂×C₃×C₈	C₂²×C₄	1	24	yes	—
77	C₂×C₃×C₈	C₂⁴	1	12	yes	—
78	C₂×C₃×C₈	C₂⁴	1	2	yes	—
79	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
80	C₃⋊C₁₆	C₂⁴	1	16	yes	—
81	C₂×#41	C₃⋊C₈	1	4	yes	—
82	C₂×#41	C₃⋊C₈	1	8	yes	—
83	C₂×#41	C₂×Dic₃	1	4	yes	—
84	C₂×#41	C₂×Dic₃	1	8	yes	—
85	C₂²×Q₈	C₃⋊C₈	1	1	yes	characteristic
86	C₂²×D₄	C₃⋊C₈	1	3	yes	—
87	C₂³×C₄	C₂×Dic₃	1	3	yes	—
88	#131	C₂×Dic₃	1	16	yes	—
89	#131	C₂×Dic₃	1	32	yes	—
90	#131	C₂²×D₃	1	16	yes	—
91	C₂×M₁₆	C₂×Dic₃	1	12	yes	—
92	C₂×M₁₆	C₂×Dic₃	1	24	yes	—
93	C₂²×C₈	C₂×Dic₃	1	3	yes	—
94	C₂²×C₈	C₂×Dic₃	1	6	yes	—
95	C₂²×C₈	C₂×Dic₃	1	12	yes	—
96	C₂²×C₈	C₂²×D₃	1	12	yes	—
97	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
98	C₂²×C₈	C₂²×D₃	1	6	yes	—
99	M₃₂	—	3	16	no	—
100	M₃₂	—	3	48	no	—
101	C₂×C₁₆	—	3	48	no	—
102	C₂×C₁₆	—	3	8	no	—
103	C₂³×C₃	—	4	6	no	—
104	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
105	C₂³×C₃	C₂²×C₈	1	6	yes	—
106	C₂³×C₃	C₂²×C₈	1	12	yes	—
107	C₃×Q₈	C₂²×C₈	1	16	yes	—
108	C₃×D₄	C₂²×C₈	1	48	yes	—
109	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
110	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₃×C₄	C₂²×C₈	1	12	yes	—
112	C₂×C₃×C₄	C₂²×C₈	1	6	yes	—
113	C₂×C₃×C₄	C₂²×C₈	1	24	yes	—
114	C₂×C₃×C₄	C₂²×C₈	1	12	yes	—
115	C₂×C₃×C₄	C₂³×C₄	1	2	yes	—
116	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₃×C₄	C₂³×C₄	1	12	yes	—
118	C₃×C₈	C₂³×C₄	1	2	yes	—
119	C₃×C₈	C₂³×C₄	1	1	yes	characteristic
120	C₃×C₈	C₂³×C₄	1	12	yes	—
121	C₃×C₈	C₂⁵	1	1	yes	characteristic
122	C₂⁴	C₂×C₃⋊C₈	1	3	yes	—
123	#41	C₂×C₃⋊C₈	1	16	yes	—
124	#41	C₂×C₃⋊C₈	1	32	yes	—
125	#41	C₂²×Dic₃	1	16	yes	—
126	C₂×Q₈	C₂×C₃⋊C₈	1	8	yes	—
127	C₂×Q₈	C₂×C₃⋊C₈	1	4	yes	—
128	C₂×D₄	C₂×C₃⋊C₈	1	12	yes	—
129	C₂×D₄	C₂×C₃⋊C₈	1	24	yes	—
130	C₂²×C₄	C₂×C₃⋊C₈	1	3	yes	—
131	C₂²×C₄	C₂×C₃⋊C₈	1	6	yes	—
132	C₂²×C₄	C₂×C₃⋊C₈	1	12	yes	—
133	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
134	C₂²×C₄	C₂²×Dic₃	1	12	yes	—
135	C₂²×C₄	C₂²×Dic₃	1	6	yes	—
136	M₁₆	C₂²×Dic₃	1	48	yes	—
137	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
138	C₂×C₈	C₂²×Dic₃	1	2	yes	—
139	C₂×C₈	C₂²×Dic₃	1	12	yes	—
140	C₂×C₈	C₂²×Dic₃	1	6	yes	—
141	C₂×C₈	C₂²×Dic₃	1	24	yes	—
142	C₂×C₈	C₂²×Dic₃	1	12	yes	—
143	C₂×C₈	C₂³×D₃	1	12	yes	—
144	C₂×C₈	C₂³×D₃	1	2	yes	—
145	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
146	C₁₆	—	3	16	no	—
147	C₂²×C₃	—	2	2	no	—
148	C₂²×C₃	—	2	12	no	—
149	C₂²×C₃	—	4	12	no	—
150	C₂²×C₃	C₂³×C₈	1	2	yes	—
151	C₂²×C₃	C₂³×C₈	1	12	yes	—
152	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
153	C₃×C₄	C₂³×C₈	1	2	yes	—
154	C₃×C₄	C₂³×C₈	1	1	yes	characteristic
155	C₃×C₄	C₂³×C₈	1	12	yes	—
156	C₃×C₄	C₂⁴×C₄	1	1	yes	characteristic
157	C₂³	—	4	6	no	—
158	C₂³	C₂²×C₃⋊C₈	1	1	yes	characteristic
159	C₂³	C₂²×C₃⋊C₈	1	6	yes	—
160	C₂³	C₂²×C₃⋊C₈	1	12	yes	—
161	Q₈	C₂²×C₃⋊C₈	1	16	yes	—
162	D₄	C₂²×C₃⋊C₈	1	48	yes	—
163	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
164	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
165	C₂×C₄	C₂²×C₃⋊C₈	1	12	yes	—
166	C₂×C₄	C₂²×C₃⋊C₈	1	6	yes	—
167	C₂×C₄	C₂²×C₃⋊C₈	1	24	yes	—
168	C₂×C₄	C₂²×C₃⋊C₈	1	12	yes	—
169	C₂×C₄	C₂³×Dic₃	1	2	yes	—
170	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
171	C₂×C₄	C₂³×Dic₃	1	12	yes	—
172	C₈	C₂³×Dic₃	1	2	yes	—
173	C₈	C₂³×Dic₃	1	1	yes	characteristic
174	C₈	C₂³×Dic₃	1	12	yes	—
175	C₈	C₂⁴×D₃	1	1	yes	characteristic
176	C₂×C₃	—	2	2	no	—
177	C₂×C₃	—	2	12	no	—
178	C₂×C₃	#3407	1	2	yes	complemented
179	C₂×C₃	C₂⁴×C₈	1	1	yes	characteristic
180	C₂²	—	2	2	no	—
181	C₂²	—	2	12	no	—
182	C₂²	—	4	12	no	—
183	C₂²	C₂³×C₃⋊C₈	1	2	yes	—
184	C₂²	C₂³×C₃⋊C₈	1	12	yes	—
185	C₂²	C₂³×C₃⋊C₈	1	1	yes	characteristic
186	C₄	C₂³×C₃⋊C₈	1	2	yes	—
187	C₄	C₂³×C₃⋊C₈	1	1	yes	characteristic
188	C₄	C₂³×C₃⋊C₈	1	12	yes	—
189	C₄	C₂⁴×Dic₃	1	1	yes	characteristic
190	C₃	C₂×#3407	1	1	yes	characteristic, complemented, Sylow
191	C₂	—	2	2	no	—
192	C₂	—	2	12	no	—
193	C₂	#83780	1	2	yes	direct product factor
194	C₂	C₂⁴×C₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂³×D₃
commutator	C₂×C₃	C₂⁴×C₈
Frattini	C₈	C₂⁴×D₃
Fitting	C₂²×C₃×#131	C₂
socle	C₂²×C₃	C₂³×C₈
prime radical	C₂²×C₃×#41	C₄
cosocle	C₃×C₈	C₂⁵
hypercenter	C₂²×#131	D₃
hypocenter	C₃	C₂×#3407
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
2. #83780	C₂	direct (N × H)
3. C₂×#5140	C₂	semidirect
4. C₂²×#812	C₂	semidirect
13. #5140	C₂²	semidirect
15. C₂×#812	C₂²	semidirect
16. C₂×#812	C₂²	semidirect
17. C₂²×C₃⋊C₁₆	C₂²	semidirect
41. #812	C₂³	semidirect
43. C₂×C₃⋊C₁₆	C₂³	semidirect
178. C₂×C₃	#3407	semidirect
190. C₃	C₂×#3407	semidirect
193. C₂	#83780	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×#131	2
#83780	2
C₂×#5140	2
C₂²×#812	2
C₂×#3407	3

Subgroups — C2×#83780 #11701656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#83780 #11701656