Subgroups — #10804051

Order 768.

678 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	C₃×#3160	C₂	1	1	yes	maximal, characteristic
2	#83328	C₂	1	2	yes	maximal
3	#71839	C₂	1	2	yes	maximal
4	#70663	C₂	1	4	yes	maximal
5	#70660	C₂	1	4	yes	maximal
6	#69792	C₂	1	1	yes	maximal, characteristic
7	C₄×#815	C₂	1	1	yes	maximal, characteristic
8	#21734	—	3	1	no	maximal, Sylow
9	C₃×#533	C₂²	1	2	yes	—
10	C₃×#500	C₄	1	2	yes	complemented
11	C₃×#499	C₄	1	4	yes	complemented
12	C₃×#498	C₄	1	2	yes	complemented
13	C₃×#427	C₂²	1	4	yes	—
14	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×#819	C₂²	1	4	yes	—
16	C₂×#815	C₂²	1	2	yes	—
17	C₂×#815	C₂²	1	2	yes	—
18	#5024	—	2	4	no	—
19	Q₈×C₃⋊C₈	—	2	4	no	—
20	#4999	C₂²	1	4	yes	—
21	#4997	C₂²	1	4	yes	—
22	C₄²×Dic₃	C₂²	1	1	yes	characteristic
23	#4924	C₂²	1	2	yes	—
24	C₂×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
25	#4532	C₂²	1	8	yes	—
26	#3160	D₃	1	1	yes	characteristic
27	#2960	—	3	2	no	—
28	#1994	—	3	2	no	—
29	#1946	—	3	4	no	—
30	#1848	—	3	1	no	—
31	#1773	—	3	4	no	—
32	C₄×#97	—	3	1	no	—
33	C₂²×C₃×C₈	C₂³	1	2	yes	—
34	C₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
35	C₃×#125	C₂×C₄	1	4	yes	—
36	C₃×C₂²⋊Q₈	—	2	4	no	—
37	C₃×C₄×Q₈	—	2	4	no	—
38	C₃×#117	C₂³	1	4	yes	—
39	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
40	C₂×C₄×Dic₃	—	2	2	no	—
41	C₂×C₄×Dic₃	C₂³	1	2	yes	—
42	C₂×#208	C₂³	1	4	yes	—
43	C₂²×C₃⋊C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₃⋊C₈	C₂³	1	1	yes	characteristic
45	C₃×#107	C₂×C₄	1	8	yes	—
46	C₃×#106	C₂×C₄	1	8	yes	—
47	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
49	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
50	#819	—	2	8	no	—
51	#815	—	2	4	no	—
52	#815	Q₈	1	8	yes	complemented
53	C₄⋊(C₃⋊C₈)	—	2	4	no	—
54	C₄⋊(C₃⋊C₈)	—	2	8	no	—
55	#803	—	2	4	no	—
56	C₄×C₃⋊C₈	—	2	4	no	—
57	C₄×C₃⋊C₈	D₄	1	4	yes	—
58	#533	C₂×D₃	1	2	yes	—
59	#500	Dic₃	1	2	yes	complemented
60	#499	Dic₃	1	4	yes	complemented
61	#498	Dic₃	1	2	yes	complemented
62	#446	—	6	4	no	—
63	C₈×Q₈	—	6	4	no	—
64	C₂×#108	—	3	4	no	—
65	#427	C₂×D₃	1	4	yes	—
66	#424	—	3	4	no	—
67	#423	—	3	2	no	—
68	#405	—	3	4	no	—
69	C₂×#97	—	3	2	no	—
70	C₂×#97	—	3	2	no	—
71	C₂×C₄×C₈	—	3	1	no	—
72	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
73	C₄³	—	3	1	no	—
74	#339	—	3	8	no	—
75	C₂×C₃×Q₈	—	2	4	no	—
76	C₂²×C₃×C₄	#41	1	2	yes	—
77	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
78	C₂²×Dic₃	—	2	2	no	—
79	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
80	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
81	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
82	C₂×C₃×C₈	#41	1	4	yes	—
83	C₃×C₄⋊C₄	—	2	8	no	—
84	C₃×C₄⋊C₄	—	2	4	no	—
85	C₃×C₄⋊C₄	C₂²×C₄	1	8	yes	—
86	C₃×C₂²⋊C₄	—	2	4	no	—
87	C₃×C₄²	—	2	4	no	—
88	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
89	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
90	C₃×C₄²	C₂×D₄	1	2	yes	—
91	C₄×Dic₃	—	2	2	no	—
92	C₄×Dic₃	—	4	4	no	—
93	C₄×Dic₃	C₂×Q₈	1	4	yes	—
94	#208	—	2	8	no	—
95	C₂×C₃⋊C₈	—	2	2	no	—
96	C₂×C₃⋊C₈	—	2	4	no	—
97	C₂×C₃⋊C₈	C₂×D₄	1	2	yes	—
98	C₂×C₃⋊C₈	C₂×D₄	1	2	yes	—
99	C₂×C₃⋊C₈	C₂×Q₈	1	4	yes	—
100	C₂×M₁₆	—	3	4	no	—
101	C₂²×C₈	—	3	1	no	—
102	C₂²×C₈	—	3	1	no	—
103	C₂²×C₈	C₂²×D₃	1	2	yes	—
104	C₄⋊Q₈	C₂×Dic₃	1	4	yes	—
105	#125	C₂×Dic₃	1	4	yes	—
106	C₂²⋊Q₈	—	2	4	no	—
107	C₄×Q₈	—	2	4	no	—
108	#117	C₂²×D₃	1	4	yes	—
109	C₂×C₄²	—	3	2	no	—
110	C₂×C₄²	—	6	2	no	—
111	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
112	#108	—	6	8	no	—
113	#107	C₂×Dic₃	1	8	yes	—
114	#106	C₂×Dic₃	1	8	yes	—
115	C₄⋊C₈	—	6	8	no	—
116	C₄⋊C₈	—	6	4	no	—
117	#97	—	3	8	no	—
118	#97	—	6	4	no	—
119	#97	—	6	4	no	—
120	C₄×C₈	—	3	4	no	—
121	C₄×C₈	—	6	4	no	—
122	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
123	C₄×C₈	C₂×Dic₃	1	2	yes	—
124	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
125	C₂³×C₃	C₂×#41	1	1	yes	characteristic
126	C₃×Q₈	—	4	8	no	—
127	C₂×C₃×C₄	—	2	4	no	—
128	C₂×C₃×C₄	—	2	8	no	—
129	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
130	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
131	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂²×Q₈	1	2	yes	—
134	C₂×C₃×C₄	C₂×#41	1	2	yes	—
135	C₂×Dic₃	—	2	4	no	—
136	C₂×Dic₃	—	4	4	no	—
137	C₃×C₈	C₄×Q₈	1	4	yes	—
138	C₃×C₈	C₄×Q₈	1	4	yes	—
139	C₃⋊C₈	—	2	2	no	—
140	C₃⋊C₈	—	4	4	no	—
141	C₃⋊C₈	C₄⋊Q₈	1	4	yes	—
142	C₂×Q₈	—	2	4	no	—
143	C₂²×C₄	—	6	2	no	—
144	C₂²×C₄	#237	1	2	yes	—
145	C₂²×C₄	C₂³×D₃	1	1	yes	characteristic
146	M₁₆	—	6	8	no	—
147	C₂×C₈	—	3	2	no	—
148	C₂×C₈	—	3	4	no	—
149	C₂×C₈	—	3	2	no	—
150	C₂×C₈	—	6	2	no	—
151	C₂×C₈	—	6	4	no	—
152	C₂×C₈	D₃×Q₈	1	4	yes	—
153	C₂×C₈	#239	1	4	yes	—
154	C₂×C₈	C₂²×Dic₃	1	2	yes	—
155	C₂×C₈	C₂²×Dic₃	1	2	yes	—
156	C₄⋊C₄	—	2	8	no	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	C₂²×Dic₃	1	8	yes	—
159	C₂²⋊C₄	—	2	4	no	—
160	C₄²	—	2	4	no	—
161	C₄²	—	3	4	no	—
162	C₄²	—	6	2	no	—
163	C₄²	—	12	4	no	—
164	C₄²	D₃×D₄	1	2	yes	—
165	C₄²	C₂²×Dic₃	1	1	yes	characteristic
166	C₄²	C₂²×Dic₃	1	1	yes	characteristic
167	C₂²×C₃	—	2	2	no	—
168	C₂²×C₃	C₂×C₄×D₄	1	1	yes	characteristic
169	C₂²×C₃	#533	1	2	yes	—
170	C₃×C₄	—	2	2	no	—
171	C₃×C₄	—	4	8	no	—
172	C₃×C₄	#444	1	4	yes	—
173	C₃×C₄	C₂×C₄×Q₈	1	2	yes	—
174	C₃×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
175	C₃×C₄	#532	1	1	yes	characteristic
176	Dic₃	—	4	4	no	—
177	C₂³	C₂×#237	1	1	yes	characteristic
178	Q₈	—	4	8	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	—	6	4	no	—
182	C₂×C₄	—	12	4	no	—
183	C₂×C₄	D₄×Dic₃	1	2	yes	—
184	C₂×C₄	D₄×Dic₃	1	2	yes	—
185	C₂×C₄	C₂×D₃×D₄	1	1	yes	characteristic
186	C₂×C₄	C₂×D₃×Q₈	1	2	yes	—
187	C₂×C₄	C₂×#239	1	2	yes	—
188	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
189	C₈	—	3	4	no	—
190	C₈	—	6	2	no	—
191	C₈	—	12	4	no	—
192	C₈	Q₈×Dic₃	1	4	yes	—
193	C₈	Q₈×Dic₃	1	4	yes	—
194	C₂×C₃	—	2	2	no	—
195	C₂×C₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
196	C₂×C₃	C₂×#444	1	2	yes	—
197	C₂²	—	2	2	no	—
198	C₂²	#5728	1	2	yes	—
199	C₂²	C₂×D₄×Dic₃	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	4	8	no	—
202	C₄	—	12	4	no	—
203	C₄	#5197	1	4	yes	—
204	C₄	D₃×C₄⋊Q₈	1	1	yes	characteristic
205	C₄	#5725	1	1	yes	characteristic
206	C₄	C₂×Q₈×Dic₃	1	2	yes	—
207	C₃	#21734	1	1	yes	characteristic, complemented, Sylow
208	C₂	—	2	2	no	—
209	C₂	Dic₃×C₄⋊Q₈	1	1	yes	characteristic
210	C₂	C₂×#5197	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₄
commutator	C₂×C₃×C₄	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₃
Fitting	C₃×#3160	C₂
socle	C₂²×C₃	C₂×C₄×D₄
prime radical	C₂³×C₃	C₂×#41
cosocle	C₂²×C₃×C₄	C₂⁴
hypercenter	#3160	D₃
hypocenter	C₃	#21734
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

Complements

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

N	H ≅ G/N	type
10. C₃×#500	C₄	semidirect
11. C₃×#499	C₄	semidirect
12. C₃×#498	C₄	semidirect
52. #815	Q₈	semidirect
59. #500	Dic₃	semidirect
60. #499	Dic₃	semidirect
61. #498	Dic₃	semidirect
207. C₃	#21734	semidirect

Maximal subgroups

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

subgroup	index
C₃×#3160	2
#83328	2
#71839	2
#70663	2
#70660	2
#69792	2
C₄×#815	2
#21734	3