Subgroups — #10713334

Order 768.

294 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#364	C₂	1	1	yes	maximal, characteristic
2	C₂×#4513	C₂	1	2	yes	maximal
3	#69779	C₂	1	1	yes	maximal, characteristic
4	#69766	C₂	1	1	yes	maximal, characteristic
5	#67819	C₂	1	2	yes	maximal
6	#13187	—	3	1	no	maximal, Sylow
7	C₂×C₃×M₃₂	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×#819	C₂²	1	1	yes	characteristic
10	#5106	C₂²	1	1	yes	characteristic
11	C₂×#818	—	2	1	no	—
12	C₂×#817	C₂²	1	2	yes	—
13	#5101	—	2	1	no	—
14	#4936	—	2	1	no	—
15	C₃×#364	C₄	1	4	yes	complemented
16	#4523	—	2	1	no	—
17	#4513	—	2	4	no	—
18	C₂×#365	—	3	2	no	—
19	C₂×#364	D₃	1	1	yes	characteristic
20	#1846	—	3	1	no	—
21	#1772	—	3	1	no	—
22	#1386	—	3	2	no	—
23	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
25	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
26	#926	—	2	1	no	—
27	C₂×C₄⋊Dic₃	—	2	2	no	—
28	C₂×#208	—	2	1	no	—
29	C₃×M₃₂	—	2	2	no	—
30	C₃×M₃₂	C₂×C₄	1	4	yes	—
31	C₂×C₃×C₁₆	—	2	2	no	—
32	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
33	C₃×C₄×C₈	Q₈	1	2	yes	—
34	C₃⋊#95	—	4	2	no	—
35	#819	—	2	1	no	—
36	#819	D₄	1	2	yes	—
37	#818	—	4	2	no	—
38	#818	Q₈	1	2	yes	—
39	#817	—	2	1	no	—
40	#817	—	2	2	no	—
41	#817	—	2	2	no	—
42	#815	—	4	1	no	—
43	C₈×Dic₃	—	4	1	no	—
44	C₂×M₃₂	C₂×D₃	1	2	yes	—
45	C₂×#108	—	3	1	no	—
46	#427	—	3	1	no	—
47	C₂×#107	—	6	1	no	—
48	C₂×#106	—	3	2	no	—
49	#405	—	6	1	no	—
50	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
51	#382	—	6	1	no	—
52	#365	—	6	4	no	—
53	#364	Dic₃	1	4	yes	complemented
54	#339	—	6	1	no	—
55	C₂²×C₃×C₄	—	2	1	no	—
56	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₃	—	4	2	no	—
58	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
59	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
60	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
61	C₂×C₃×C₈	SD₁₆	1	2	yes	—
62	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₃×C₈	#41	1	1	yes	characteristic
67	C₂×C₃×C₈	#41	1	1	yes	characteristic
68	C₃×C₄²	—	2	1	no	—
69	C₃×C₄²	C₄⋊C₄	1	2	yes	—
70	#217	—	4	1	no	—
71	C₄⋊Dic₃	—	2	2	no	—
72	C₄⋊Dic₃	—	4	2	no	—
73	C₄⋊Dic₃	—	4	2	no	—
74	C₄×Dic₃	—	4	1	no	—
75	#208	—	2	2	no	—
76	#208	—	4	1	no	—
77	C₂×C₃⋊C₈	—	4	1	no	—
78	C₃×C₁₆	—	2	4	no	—
79	C₂×M₁₆	—	6	1	no	—
80	C₂²×C₈	D₁₂	1	1	yes	characteristic
81	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
82	#117	—	6	1	no	—
83	C₂×C₄⋊C₄	—	6	2	no	—
84	C₂×C₄²	D₁₂	1	1	yes	characteristic
85	M₃₂	—	2	2	no	—
86	M₃₂	C₂×Dic₃	1	4	yes	—
87	C₂×C₁₆	—	2	2	no	—
88	#108	—	3	2	no	—
89	#108	—	6	1	no	—
90	#107	—	3	2	no	—
91	#107	—	12	2	no	—
92	#106	—	6	1	no	—
93	#106	—	6	2	no	—
94	#106	—	6	2	no	—
95	#97	—	12	1	no	—
96	C₄×C₈	—	12	1	no	—
97	C₄×C₈	Dic₆	1	2	yes	—
98	C₄×C₈	C₂×Dic₃	1	2	yes	—
99	#95	—	12	2	no	—
100	C₂³×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂×C₃×C₄	—	2	2	no	—
102	C₂×C₃×C₄	—	4	2	no	—
103	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₃×C₄	#121	1	1	yes	characteristic
105	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₃×C₄	#137	1	1	yes	characteristic
109	C₂×Dic₃	—	4	2	no	—
110	C₂×Dic₃	—	8	4	no	—
111	C₂×Dic₃	—	8	2	no	—
112	C₃×C₈	#106	1	2	yes	—
113	C₃×C₈	#106	1	2	yes	—
114	C₃×C₈	C₄×D₄	1	1	yes	characteristic
115	C₃×C₈	C₄×Q₈	1	1	yes	characteristic
116	C₃×C₈	#125	1	1	yes	characteristic
117	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
118	C₃⋊C₈	—	4	2	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	12	2	no	—
121	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
122	M₁₆	—	6	2	no	—
123	M₁₆	—	12	1	no	—
124	C₂×C₈	—	12	1	no	—
125	C₂×C₈	#204	1	1	yes	characteristic
126	C₂×C₈	#204	1	1	yes	characteristic
127	C₂×C₈	#204	1	2	yes	—
128	C₂×C₈	C₄⋊Dic₃	1	2	yes	—
129	C₂×C₈	C₂×Dic₆	1	1	yes	characteristic
130	C₂×C₈	D₃×D₄	1	1	yes	characteristic
131	C₂×C₈	#237	1	1	yes	characteristic
132	C₂×C₈	D₃×Q₈	1	1	yes	characteristic
133	C₂×C₈	#239	1	1	yes	characteristic
134	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
135	C₄⋊C₄	—	6	2	no	—
136	C₄⋊C₄	—	12	2	no	—
137	C₄⋊C₄	—	12	2	no	—
138	C₂²⋊C₄	—	12	1	no	—
139	C₄²	—	2	1	no	—
140	C₄²	—	12	1	no	—
141	C₄²	C₄⋊Dic₃	1	2	yes	—
142	C₁₆	—	2	4	no	—
143	C₂²×C₃	—	2	2	no	—
144	C₂²×C₃	#428	1	1	yes	characteristic
145	C₂²×C₃	#460	1	1	yes	characteristic
146	C₂²×C₃	#476	1	1	yes	characteristic
147	C₃×C₄	—	4	2	no	—
148	C₃×C₄	#389	1	1	yes	characteristic
149	C₃×C₄	C₂×#106	1	1	yes	characteristic
150	C₃×C₄	#461	1	1	yes	characteristic
151	C₃×C₄	#475	1	1	yes	characteristic
152	Dic₃	—	8	2	no	—
153	Dic₃	—	16	2	no	—
154	C₂³	#908	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	12	2	no	—
158	C₂×C₄	—	24	4	no	—
159	C₂×C₄	—	24	2	no	—
160	C₂×C₄	#895	1	1	yes	characteristic
161	C₂×C₄	#897	1	1	yes	characteristic
162	C₂×C₄	C₂×#204	1	1	yes	characteristic
163	C₂×C₄	C₂×#204	1	1	yes	characteristic
164	C₂×C₄	#909	1	1	yes	characteristic
165	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
166	C₈	—	12	2	no	—
167	C₈	#817	1	2	yes	—
168	C₈	#817	1	2	yes	—
169	C₈	#888	1	1	yes	characteristic
170	C₈	#890	1	1	yes	characteristic
171	C₈	D₄×Dic₃	1	1	yes	characteristic
172	C₈	Q₈×Dic₃	1	1	yes	characteristic
173	C₂×C₃	—	2	1	no	—
174	C₂×C₃	—	2	1	no	—
175	C₂×C₃	#1948	1	1	yes	characteristic
176	C₂×C₃	#2231	1	2	yes	—
177	C₂²	—	2	2	no	—
178	C₂²	#4842	1	1	yes	characteristic
179	C₂²	#4848	1	1	yes	characteristic
180	C₂²	#5121	1	1	yes	characteristic
181	C₄	—	4	2	no	—
182	C₄	—	24	2	no	—
183	C₄	—	48	2	no	—
184	C₄	#4841	1	1	yes	characteristic
185	C₄	#4851	1	1	yes	characteristic
186	C₄	#4984	1	1	yes	characteristic
187	C₄	C₂×#817	1	1	yes	characteristic
188	C₃	#13187	1	1	yes	characteristic, complemented, Sylow
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	#69308	1	2	yes	—
192	C₂	#70454	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#4842
commutator	C₂×C₃×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₂×C₃×#364	C₂
socle	C₂²×C₃	#476
prime radical	C₂³×C₃	AGL(1,ℤ/8)
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₂×#364	D₃
hypocenter	C₃	#13187
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
15. C₃×#364	C₄	semidirect
53. #364	Dic₃	semidirect
188. C₃	#13187	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#364	2
C₂×#4513	2
#69779	2
#69766	2
#67819	2
#13187	3