Subgroups — #10704236

Order 768.

248 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
2	#69745	C₂	1	1	yes	maximal, characteristic
3	#69725	C₂	1	1	yes	maximal, characteristic
4	#67512	C₂	1	2	yes	maximal, complemented
5	#67511	C₂	1	2	yes	maximal, complemented
6	#11414	—	3	1	no	maximal, Sylow
7	C₂×C₃×M₃₂	C₄	1	2	yes	—
8	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	#4925	C₄	1	1	yes	characteristic
10	C₄×#208	C₄	1	1	yes	characteristic
11	C₃×#346	C₂²	1	1	yes	characteristic
12	C₃×#346	C₂²	1	1	yes	characteristic
13	C₃×C₄×C₁₆	C₄	1	2	yes	—
14	#4549	C₄	1	2	yes	—
15	C₄⋊(C₃⋊C₁₆)	C₄	1	2	yes	—
16	#4461	C₂²	1	2	yes	—
17	#4455	C₂²	1	2	yes	—
18	#4454	C₄	1	2	yes	—
19	#2163	—	3	1	no	—
20	C₄×M₃₂	D₃	1	1	yes	characteristic
21	#1451	—	3	1	no	—
22	#1313	—	3	2	no	—
23	#1313	—	3	2	no	—
24	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
27	C₂×#208	—	2	1	no	—
28	C₃×M₃₂	—	2	4	no	—
29	C₂×C₃×C₁₆	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₁₆	C₂×C₄	1	2	yes	—
31	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
33	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
34	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
35	#830	—	2	1	no	—
36	C₂×C₃⋊C₁₆	C₂×C₄	1	4	yes	—
37	C₄⋊(C₃⋊C₈)	—	2	1	no	—
38	#803	—	2	1	no	—
39	C₄×C₃⋊C₈	C₂×C₄	1	2	yes	—
40	C₄×C₃⋊C₈	C₂×C₄	1	2	yes	—
41	C₂×M₃₂	Dic₃	1	2	yes	—
42	#431	—	3	1	no	—
43	C₄×M₁₆	—	3	1	no	—
44	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
45	C₄⋊C₁₆	—	3	2	no	—
46	C₂²⋊C₁₆	—	3	2	no	—
47	#346	—	3	2	no	—
48	#346	C₂×D₃	1	1	yes	characteristic
49	#346	C₂×D₃	1	1	yes	characteristic
50	C₄×C₁₆	Dic₃	1	2	yes	—
51	#322	—	3	2	no	—
52	#322	—	3	2	no	—
53	C₂²×C₃×C₄	C₄²	1	1	yes	characteristic
54	C₂²×C₃×C₄	M₁₆	1	1	yes	characteristic
55	C₂²×C₃×C₄	M₁₆	1	1	yes	characteristic
56	C₂×C₃×C₈	—	2	2	no	—
57	C₂×C₃×C₈	—	2	2	no	—
58	C₂×C₃×C₈	C₄²	1	1	yes	characteristic
59	C₂×C₃×C₈	C₄²	1	1	yes	characteristic
60	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₃×C₄²	C₄²	1	1	yes	characteristic
63	C₃×C₄²	M₁₆	1	1	yes	characteristic
64	C₃×C₄²	M₁₆	1	1	yes	characteristic
65	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
66	#208	—	4	2	no	—
67	C₂×C₃⋊C₈	—	2	2	no	—
68	C₂×C₃⋊C₈	—	2	2	no	—
69	C₃×C₁₆	—	2	2	no	—
70	C₃×C₁₆	M₁₆	1	4	yes	complemented
71	C₃⋊C₁₆	—	2	4	no	—
72	C₂×M₁₆	—	6	1	no	—
73	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
74	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
75	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
76	M₃₂	—	2	4	no	—
77	C₂×C₁₆	—	3	4	no	—
78	C₂×C₁₆	C₂×Dic₃	1	2	yes	—
79	C₂×C₁₆	C₂×Dic₃	1	2	yes	—
80	C₄⋊C₈	—	6	1	no	—
81	C₂²⋊C₈	—	6	1	no	—
82	#97	—	6	1	no	—
83	C₄×C₈	—	3	2	no	—
84	C₄×C₈	—	3	2	no	—
85	C₄×C₈	C₄×D₃	1	1	yes	characteristic
86	C₄×C₈	C₄×D₃	1	1	yes	characteristic
87	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
88	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
89	C₂³×C₃	#97	1	1	yes	characteristic
90	C₂×C₃×C₄	—	2	2	no	—
91	C₂×C₃×C₄	—	2	1	no	—
92	C₂×C₃×C₄	—	2	1	no	—
93	C₂×C₃×C₄	—	4	1	no	—
94	C₂×C₃×C₄	#97	1	1	yes	characteristic
95	C₂×C₃×C₄	#97	1	1	yes	characteristic
96	C₂×C₃×C₄	#97	1	1	yes	characteristic
97	C₂×C₃×C₄	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
100	C₃×C₈	—	2	1	no	—
101	C₃×C₈	—	2	1	no	—
102	C₃×C₈	C₂×M₁₆	1	2	yes	—
103	C₃×C₈	#131	1	2	yes	—
104	C₃⋊C₈	—	4	2	no	—
105	C₃⋊C₈	—	4	2	no	—
106	C₂²×C₄	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
107	C₂²×C₄	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
108	C₂²×C₄	C₄×Dic₃	1	1	yes	characteristic
109	M₁₆	—	12	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	6	2	no	—
113	C₂×C₈	—	6	2	no	—
114	C₂×C₈	C₄×Dic₃	1	1	yes	characteristic
115	C₂×C₈	C₄×Dic₃	1	1	yes	characteristic
116	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
117	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
118	C₄²	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
119	C₄²	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
120	C₄²	C₄×Dic₃	1	1	yes	characteristic
121	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
122	C₁₆	—	2	2	no	—
123	C₁₆	—	6	4	no	—
124	C₁₆	#208	1	4	yes	complemented
125	C₂²×C₃	—	2	1	no	—
126	C₂²×C₃	—	2	1	no	—
127	C₂²×C₃	—	2	1	no	—
128	C₂²×C₃	C₂×#97	1	1	yes	characteristic
129	C₃×C₄	—	2	1	no	—
130	C₃×C₄	—	2	1	no	—
131	C₃×C₄	—	2	1	no	—
132	C₃×C₄	—	2	1	no	—
133	C₃×C₄	#347	1	2	yes	—
134	C₃×C₄	C₄×M₁₆	1	1	yes	characteristic
135	C₃×C₄	#405	1	1	yes	characteristic
136	C₂³	#815	1	1	yes	characteristic
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	#815	1	1	yes	characteristic
142	C₂×C₄	#815	1	1	yes	characteristic
143	C₂×C₄	#815	1	1	yes	characteristic
144	C₂×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
145	C₂×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
146	C₂×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	12	2	no	—
150	C₈	—	12	2	no	—
151	C₈	#901	1	2	yes	—
152	C₈	C₂×#208	1	2	yes	—
153	C₂×C₃	—	4	1	no	—
154	C₂×C₃	#1451	1	1	yes	characteristic
155	C₂×C₃	C₂×#347	1	1	yes	characteristic
156	C₂×C₃	#2110	1	1	yes	characteristic
157	C₂²	—	2	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	C₂×#815	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	#4512	1	2	yes	—
166	C₄	C₄×#208	1	1	yes	characteristic
167	C₄	#5101	1	1	yes	characteristic
168	C₃	#11414	1	1	yes	characteristic, complemented, Sylow
169	C₂	—	4	1	no	—
170	C₂	#69725	1	1	yes	characteristic
171	C₂	C₂×#4512	1	1	yes	characteristic
172	C₂	#71425	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₃
commutator	C₂×C₃×C₄	C₂×C₄²
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₃×C₄×M₃₂	C₂
socle	C₂²×C₃	C₂×#97
prime radical	C₂³×C₃	#97
cosocle	C₃×C₄×C₈	C₂³
hypercenter	C₄×M₃₂	D₃
hypocenter	C₃	#11414
LCS	G ▷ C₂×C₃×C₄ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
4. #67512	C₂	semidirect
5. #67511	C₂	semidirect
70. C₃×C₁₆	M₁₆	semidirect
124. C₁₆	#208	semidirect
168. C₃	#11414	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×M₃₂	2
#69745	2
#69725	2
#67512	2
#67511	2
#11414	3