Subgroups — #10706751

Order 768.

220 conjugacy classes of subgroups.

The subgroup lattice has 178 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×#4463	C₂	1	1	yes	maximal, characteristic
3	C₄×#812	C₂	1	1	yes	maximal, characteristic
4	#67567	C₂	1	4	yes	maximal, complemented
5	#12002	—	3	1	no	maximal, Sylow
6	C₂×C₃×M₃₂	—	2	1	no	—
7	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₂×#812	—	2	1	no	—
9	C₂×#812	C₄	1	1	yes	characteristic
10	C₂×#812	C₄	1	1	yes	characteristic
11	C₃×#364	—	2	1	no	—
12	C₃×#364	C₂²	1	2	yes	—
13	#4463	—	2	1	no	—
14	#4463	C₂²	1	2	yes	—
15	#4461	C₂²	1	2	yes	—
16	C₄×C₃⋊C₁₆	C₄	1	2	yes	—
17	C₂×#364	—	3	1	no	—
18	C₂×#364	D₃	1	1	yes	characteristic
19	C₄×M₃₂	—	3	1	no	—
20	#1369	—	3	4	no	—
21	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
23	C₂×C₃×C₄²	Q₈	1	1	yes	characteristic
24	C₃×M₃₂	—	2	2	no	—
25	C₃×M₃₂	—	4	1	no	—
26	C₂×C₃×C₁₆	—	4	1	no	—
27	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₃×C₄×C₈	D₄	1	1	yes	characteristic
29	C₃×C₄×C₈	Q₈	1	1	yes	characteristic
30	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
31	#812	—	2	1	no	—
32	#812	—	2	1	no	—
33	#812	—	2	2	no	—
34	#812	—	4	1	no	—
35	#812	C₈	1	2	yes	—
36	#812	C₈	1	2	yes	—
37	C₂×C₃⋊C₁₆	—	4	1	no	—
38	C₂×C₃⋊C₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₃⋊C₁₆	C₂×C₄	1	2	yes	—
40	C₂×M₃₂	—	2	1	no	—
41	C₂×M₃₂	—	3	1	no	—
42	C₂×M₃₂	—	3	1	no	—
43	C₂×M₃₂	—	6	1	no	—
44	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
45	#364	—	2	1	no	—
46	#364	—	3	2	no	—
47	#364	—	6	1	no	—
48	#364	C₂×D₃	1	2	yes	—
49	#346	—	3	2	no	—
50	C₄×C₁₆	—	3	2	no	—
51	C₂²×C₃×C₄	—	2	1	no	—
52	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₃×C₈	C₂×C₈	1	1	yes	characteristic
56	C₂×C₃×C₈	C₂×C₈	1	1	yes	characteristic
57	C₂×C₃×C₈	M₁₆	1	1	yes	characteristic
58	C₂×C₃×C₈	M₁₆	1	1	yes	characteristic
59	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
60	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
61	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
62	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
63	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
65	C₃×C₄²	—	2	1	no	—
66	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
67	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
68	C₃×C₁₆	—	4	2	no	—
69	C₃⋊C₁₆	—	2	2	no	—
70	C₃⋊C₁₆	—	4	2	no	—
71	C₃⋊C₁₆	C₂×C₈	1	4	yes	—
72	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
73	C₂²×C₈	#67	1	1	yes	characteristic
74	C₂×C₄²	Dic₆	1	1	yes	characteristic
75	M₃₂	—	2	2	no	—
76	M₃₂	—	3	2	no	—
77	M₃₂	—	3	2	no	—
78	M₃₂	—	4	1	no	—
79	M₃₂	—	6	1	no	—
80	M₃₂	—	6	1	no	—
81	M₃₂	—	6	2	no	—
82	M₃₂	—	12	1	no	—
83	C₂×C₁₆	—	3	2	no	—
84	C₂×C₁₆	—	3	2	no	—
85	C₂×C₁₆	—	4	1	no	—
86	C₂×C₁₆	—	12	1	no	—
87	C₄×C₈	Dic₆	1	1	yes	characteristic
88	C₄×C₈	C₄×D₃	1	1	yes	characteristic
89	C₄×C₈	#67	1	1	yes	characteristic
90	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
91	C₂³×C₃	#108	1	1	yes	characteristic
92	C₂×C₃×C₄	—	2	1	no	—
93	C₂×C₃×C₄	—	2	1	no	—
94	C₂×C₃×C₄	—	4	2	no	—
95	C₂×C₃×C₄	C₄⋊C₈	1	1	yes	characteristic
96	C₂×C₃×C₄	C₄⋊C₈	1	1	yes	characteristic
97	C₂×C₃×C₄	#106	1	1	yes	characteristic
98	C₂×C₃×C₄	#106	1	1	yes	characteristic
99	C₂×C₃×C₄	#108	1	1	yes	characteristic
100	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₃×C₈	C₄⋊C₈	1	1	yes	characteristic
102	C₃×C₈	C₄⋊C₈	1	1	yes	characteristic
103	C₃×C₈	#106	1	1	yes	characteristic
104	C₃×C₈	#106	1	1	yes	characteristic
105	C₃×C₈	C₂²×C₈	1	1	yes	characteristic
106	C₃×C₈	C₂×M₁₆	1	1	yes	characteristic
107	C₃×C₈	C₂×SD₁₆	1	1	yes	characteristic
108	C₃×C₈	C₂×SD₁₆	1	1	yes	characteristic
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
111	C₂×C₈	D₃×C₈	1	1	yes	characteristic
112	C₂×C₈	D₃×C₈	1	1	yes	characteristic
113	C₂×C₈	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
114	C₂×C₈	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
115	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
116	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
117	C₂×C₈	#214	1	1	yes	characteristic
118	C₂×C₈	#214	1	1	yes	characteristic
119	C₂×C₈	#215	1	1	yes	characteristic
120	C₂×C₈	#215	1	1	yes	characteristic
121	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
122	C₂×C₈	C₂×#67	1	1	yes	characteristic
123	C₄²	—	2	1	no	—
124	C₄²	Dic₃⋊C₄	1	1	yes	characteristic
125	C₄²	C₂×Dic₆	1	1	yes	characteristic
126	C₁₆	—	3	4	no	—
127	C₁₆	—	4	2	no	—
128	C₁₆	—	6	2	no	—
129	C₁₆	—	12	2	no	—
130	C₂²×C₃	—	4	1	no	—
131	C₂²×C₃	#334	1	1	yes	characteristic
132	C₂²×C₃	#334	1	1	yes	characteristic
133	C₂²×C₃	C₂×#108	1	1	yes	characteristic
134	C₃×C₄	—	2	2	no	—
135	C₃×C₄	—	4	1	no	—
136	C₃×C₄	#334	1	1	yes	characteristic
137	C₃×C₄	#334	1	1	yes	characteristic
138	C₃×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
139	C₃×C₄	C₂×#106	1	1	yes	characteristic
140	C₂³	#822	1	1	yes	characteristic
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	#808	1	1	yes	characteristic
145	C₂×C₄	#808	1	1	yes	characteristic
146	C₂×C₄	Dic₃⋊C₈	1	1	yes	characteristic
147	C₂×C₄	Dic₃⋊C₈	1	1	yes	characteristic
148	C₂×C₄	#822	1	1	yes	characteristic
149	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
150	C₈	#808	1	1	yes	characteristic
151	C₈	#808	1	1	yes	characteristic
152	C₈	Dic₃⋊C₈	1	1	yes	characteristic
153	C₈	Dic₃⋊C₈	1	1	yes	characteristic
154	C₈	C₂×D₃×C₈	1	1	yes	characteristic
155	C₈	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
156	C₈	C₂×#214	1	1	yes	characteristic
157	C₈	C₂×#215	1	1	yes	characteristic
158	C₂×C₃	—	2	1	no	—
159	C₂×C₃	—	2	1	no	—
160	C₂×C₃	—	2	1	no	—
161	C₂×C₃	C₂×#334	1	1	yes	characteristic
162	C₂²	—	4	1	no	—
163	C₂²	#4480	1	1	yes	characteristic
164	C₂²	#4480	1	1	yes	characteristic
165	C₂²	C₂×#822	1	1	yes	characteristic
166	C₄	—	2	2	no	—
167	C₄	—	4	1	no	—
168	C₄	#4480	1	1	yes	characteristic
169	C₄	#4480	1	1	yes	characteristic
170	C₄	C₂×#808	1	1	yes	characteristic
171	C₄	C₂×Dic₃⋊C₈	1	1	yes	characteristic
172	C₃	#12002	1	1	yes	characteristic, complemented, Sylow
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	C₂×#4480	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#808
commutator	C₃×C₈	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₂×C₃×#364	C₂
socle	C₂×C₃	C₂×#334
prime radical	C₂³×C₃	#108
cosocle	C₃×C₄×C₈	C₂³
hypercenter	C₂×#364	D₃
hypocenter	C₃	#12002
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
4. #67567	C₂	semidirect
172. C₃	#12002	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×#4463	2
C₄×#812	2
#67567	2
#12002	3