Subgroups — #10682016

Order 768.

266 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#346	C₂	1	1	yes	maximal, characteristic
2	C₂×#4461	C₂	1	1	yes	maximal, characteristic
3	C₂×#4455	C₂	1	1	yes	maximal, characteristic
4	#7137	—	3	1	no	maximal, Sylow
5	C₂²×C₃×C₁₆	C₄	1	2	yes	—
6	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₃⋊C₁₆	C₄	1	2	yes	—
8	C₂×C₄×C₃⋊C₈	C₄	1	2	yes	—
9	C₃×#346	—	2	1	no	—
10	C₃×#346	—	2	1	no	—
11	#4461	—	2	1	no	—
12	#4461	—	2	1	no	—
13	#4455	—	2	1	no	—
14	#4455	—	2	1	no	—
15	C₂×#346	—	3	1	no	—
16	C₂×#346	D₃	1	1	yes	characteristic
17	C₂×#322	—	3	1	no	—
18	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₃⋊C₈	—	2	2	no	—
22	C₂×C₃×C₁₆	—	2	2	no	—
23	C₂×C₃×C₁₆	—	2	2	no	—
24	C₂×C₃×C₁₆	C₈	1	4	yes	complemented
25	C₃×C₄×C₈	D₄	1	1	yes	characteristic
26	C₃×C₄×C₈	D₄	1	1	yes	characteristic
27	C₃×C₄×C₈	D₄	1	1	yes	characteristic
28	C₃×C₄×C₈	Q₈	1	1	yes	characteristic
29	C₂×C₃⋊C₁₆	—	2	2	no	—
30	C₂×C₃⋊C₁₆	—	2	2	no	—
31	C₂×C₃⋊C₁₆	C₈	1	4	yes	complemented
32	C₄×C₃⋊C₈	—	2	2	no	—
33	C₄×C₃⋊C₈	C₈	1	4	yes	—
34	C₂²×C₁₆	—	3	2	no	—
35	C₂²×C₁₆	Dic₃	1	2	yes	—
36	C₂×C₄×C₈	—	3	2	no	—
37	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
38	#346	—	2	1	no	—
39	#346	—	2	1	no	—
40	#346	—	6	1	no	—
41	#346	—	6	1	no	—
42	#322	—	6	1	no	—
43	#322	—	6	1	no	—
44	C₂²×C₃×C₄	C₄²	1	1	yes	characteristic
45	C₂²×C₃×C₄	M₁₆	1	2	yes	—
46	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₃×C₈	C₂×C₈	1	2	yes	—
51	C₂×C₃×C₈	C₂×C₈	1	2	yes	—
52	C₂×C₃×C₈	M₁₆	1	2	yes	—
53	C₂×C₃×C₈	M₁₆	1	2	yes	—
54	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
55	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
56	C₃×C₄²	C₂×C₈	1	2	yes	—
57	C₂×C₃⋊C₈	—	2	4	no	—
58	C₂×C₃⋊C₈	—	4	4	no	—
59	C₃×C₁₆	—	2	4	no	—
60	C₃⋊C₁₆	—	2	4	no	—
61	C₂²×C₈	—	6	2	no	—
62	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
63	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
64	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
65	C₂×C₁₆	—	2	2	no	—
66	C₂×C₁₆	—	2	2	no	—
67	C₂×C₁₆	—	3	4	no	—
68	C₂×C₁₆	—	6	2	no	—
69	C₂×C₁₆	—	6	2	no	—
70	C₂×C₁₆	C₃⋊C₈	1	4	yes	complemented
71	C₄×C₈	—	3	4	no	—
72	C₄×C₈	—	6	2	no	—
73	C₄×C₈	Dic₆	1	1	yes	characteristic
74	C₄×C₈	D₁₂	1	1	yes	characteristic
75	C₄×C₈	#67	1	1	yes	characteristic
76	C₄×C₈	#67	1	1	yes	characteristic
77	C₂³×C₃	#97	1	1	yes	characteristic
78	C₂×C₃×C₄	—	2	4	no	—
79	C₂×C₃×C₄	#95	1	1	yes	characteristic
80	C₂×C₃×C₄	C₄×C₈	1	1	yes	characteristic
81	C₂×C₃×C₄	C₄×C₈	1	1	yes	characteristic
82	C₂×C₃×C₄	C₄×C₈	1	1	yes	characteristic
83	C₂×C₃×C₄	#97	1	1	yes	characteristic
84	C₂×C₃×C₄	#97	1	1	yes	characteristic
85	C₂×C₃×C₄	C₂²⋊C₈	1	2	yes	—
86	C₂×C₃×C₄	C₄⋊C₈	1	2	yes	—
87	C₃×C₈	C₂²⋊C₈	1	2	yes	—
88	C₃×C₈	C₂²⋊C₈	1	2	yes	—
89	C₃×C₈	C₄⋊C₈	1	2	yes	—
90	C₃×C₈	C₄⋊C₈	1	2	yes	—
91	C₃⋊C₈	—	4	4	no	—
92	C₂²×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
93	C₂²×C₄	C₄×Dic₃	1	1	yes	characteristic
94	C₂×C₈	—	6	4	no	—
95	C₂×C₈	—	12	4	no	—
96	C₂×C₈	D₃×C₈	1	2	yes	—
97	C₂×C₈	ΓL(1,𝔽₂₅)	1	2	yes	—
98	C₂×C₈	C₂×C₃⋊C₈	1	2	yes	—
99	C₂×C₈	#208	1	2	yes	—
100	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
101	C₂×C₈	C₄⋊Dic₃	1	1	yes	characteristic
102	C₂×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
103	C₂×C₈	#217	1	1	yes	characteristic
104	C₄²	D₃×C₈	1	2	yes	—
105	C₄²	Dic₃⋊C₄	1	1	yes	characteristic
106	C₄²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
107	C₁₆	—	2	4	no	—
108	C₁₆	—	6	4	no	—
109	C₂²×C₃	#322	1	1	yes	characteristic
110	C₂²×C₃	#322	1	1	yes	characteristic
111	C₂²×C₃	#336	1	1	yes	characteristic
112	C₂²×C₃	#347	1	4	yes	—
113	C₃×C₄	—	2	2	no	—
114	C₃×C₄	—	2	2	no	—
115	C₃×C₄	C₈²	1	1	yes	characteristic
116	C₃×C₄	#322	1	1	yes	characteristic
117	C₃×C₄	#336	1	1	yes	characteristic
118	C₃×C₄	#336	1	1	yes	characteristic
119	C₂³	#815	1	1	yes	characteristic
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	C₄×C₃⋊C₈	1	1	yes	characteristic
122	C₂×C₄	#803	1	1	yes	characteristic
123	C₂×C₄	C₈×Dic₃	1	1	yes	characteristic
124	C₂×C₄	C₈×Dic₃	1	1	yes	characteristic
125	C₂×C₄	Dic₃⋊C₈	1	2	yes	—
126	C₂×C₄	#815	1	1	yes	characteristic
127	C₂×C₄	(C₂×D₃)⋊C₈	1	2	yes	—
128	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
129	C₈	—	12	4	no	—
130	C₈	C₄⋊(C₃⋊C₈)	1	2	yes	—
131	C₈	Dic₃⋊C₈	1	2	yes	—
132	C₈	(C₂×D₃)⋊C₈	1	2	yes	—
133	C₈	#830	1	2	yes	—
134	C₂×C₃	#1273	1	1	yes	characteristic
135	C₂×C₃	#1313	1	2	yes	—
136	C₂×C₃	#1313	1	2	yes	—
137	C₂×C₃	#1375	1	2	yes	—
138	C₂²	#4454	1	1	yes	characteristic
139	C₂²	#4455	1	1	yes	characteristic
140	C₂²	#4512	1	4	yes	—
141	C₂²	#4550	1	1	yes	characteristic
142	C₄	—	2	2	no	—
143	C₄	—	2	2	no	—
144	C₄	C₈×C₃⋊C₈	1	1	yes	characteristic
145	C₄	#4454	1	1	yes	characteristic
146	C₄	#4524	1	1	yes	characteristic
147	C₄	#4550	1	1	yes	characteristic
148	C₃	#7137	1	1	yes	characteristic, complemented, Sylow
149	C₂	#67511	1	2	yes	—
150	C₂	#67512	1	2	yes	—
151	C₂	C₃⋊#1273	1	1	yes	characteristic
152	C₂	#67811	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₄×Dic₃
commutator	C₃×C₄	C₈²
Frattini	C₂×C₄×C₈	C₂×D₃
Fitting	C₂×C₃×#346	C₂
socle, prime radical	C₂³×C₃	#97
cosocle	C₂×C₃×C₄×C₈	C₂²
hypercenter	C₂×#346	D₃
hypocenter	C₃	#7137
LCS	G ▷ C₃×C₄ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
24. C₂×C₃×C₁₆	C₈	semidirect
31. C₂×C₃⋊C₁₆	C₈	semidirect
70. C₂×C₁₆	C₃⋊C₈	semidirect
148. C₃	#7137	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#346	2
C₂×#4461	2
C₂×#4455	2
#7137	3