Subgroups — #10713232

Order 768.

318 conjugacy classes of subgroups.

The subgroup lattice has 192 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×#4505	C₂	1	1	yes	maximal, characteristic
3	C₂×#4504	C₂	1	1	yes	maximal, characteristic
4	#69770	C₂	1	2	yes	maximal
5	#67806	C₂	1	2	yes	maximal
6	#13168	—	3	1	no	maximal, Sylow
7	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×#818	C₂²	1	2	yes	—
11	C₂×#818	C₂²	1	2	yes	—
12	#4928	—	2	2	no	—
13	C₃×#346	—	2	2	no	—
14	#4551	—	2	2	no	—
15	#4505	—	2	2	no	—
16	#4504	C₄	1	4	yes	complemented
17	C₂×#367	—	3	1	no	—
18	C₂×#366	—	3	1	no	—
19	C₂×#346	D₃	1	1	yes	characteristic
20	#1942	—	3	2	no	—
21	#1380	—	3	2	no	—
22	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
24	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
25	C₂×C₄⋊Dic₃	—	2	2	no	—
26	C₂×C₄⋊Dic₃	—	2	2	no	—
27	C₂×Dic₃⋊C₄	—	4	2	no	—
28	C₂×C₃×C₁₆	—	2	1	no	—
29	C₂×C₃×C₁₆	—	2	1	no	—
30	C₂×C₃×C₁₆	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₁₆	D₄	1	2	yes	—
32	C₂×C₃×C₁₆	Q₈	1	2	yes	—
33	C₂×C₃×C₁₆	Q₈	1	2	yes	—
34	C₃×C₄×C₈	—	2	1	no	—
35	C₃×C₄×C₈	—	2	1	no	—
36	C₃⋊#95	—	4	2	no	—
37	#818	—	2	2	no	—
38	#818	—	2	2	no	—
39	#818	—	2	2	no	—
40	#818	C₂×C₄	1	4	yes	—
41	C₂²×C₁₆	C₂×D₃	1	1	yes	characteristic
42	C₂²×C₁₆	C₂×D₃	1	1	yes	characteristic
43	C₂×#107	—	3	2	no	—
44	C₂×#107	—	3	2	no	—
45	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
46	#389	—	6	2	no	—
47	#367	—	6	2	no	—
48	#366	—	3	4	no	—
49	#346	—	2	2	no	—
50	#340	—	6	2	no	—
51	C₂²×C₃×C₄	D₈	1	2	yes	—
52	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₃	—	4	2	no	—
54	C₂²×Dic₃	—	4	2	no	—
55	C₂×C₃×C₈	—	2	1	no	—
56	C₂×C₃×C₈	—	2	1	no	—
57	C₂×C₃×C₈	Q₁₆	1	2	yes	—
58	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₃×C₈	#41	1	1	yes	characteristic
63	C₂×C₃×C₈	#41	1	1	yes	characteristic
64	C₃×C₄²	—	2	1	no	—
65	C₃×C₄²	—	2	1	no	—
66	C₄⋊Dic₃	—	2	4	no	—
67	C₄⋊Dic₃	—	4	2	no	—
68	C₄⋊Dic₃	—	4	2	no	—
69	C₄⋊Dic₃	—	4	2	no	—
70	Dic₃⋊C₄	—	8	4	no	—
71	C₃×C₁₆	—	2	2	no	—
72	C₃×C₁₆	C₄⋊C₄	1	4	yes	complemented
73	C₂²×C₈	D₁₂	1	1	yes	characteristic
74	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	6	2	no	—
76	C₂×C₄⋊C₄	—	6	2	no	—
77	C₂×C₄⋊C₄	—	12	2	no	—
78	C₂×C₄²	D₁₂	1	1	yes	characteristic
79	C₂×C₁₆	—	2	1	no	—
80	C₂×C₁₆	—	2	1	no	—
81	C₂×C₁₆	Dic₆	1	2	yes	—
82	C₂×C₁₆	Dic₆	1	2	yes	—
83	C₂×C₁₆	C₄×D₃	1	2	yes	—
84	C₂×C₁₆	#67	1	2	yes	—
85	#107	—	3	4	no	—
86	#107	—	6	2	no	—
87	#107	—	6	2	no	—
88	#107	—	6	2	no	—
89	C₄×C₈	—	2	1	no	—
90	C₄×C₈	—	2	1	no	—
91	#95	—	12	2	no	—
92	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
93	C₂×C₃×C₄	—	2	2	no	—
94	C₂×C₃×C₄	—	2	2	no	—
95	C₂×C₃×C₄	—	2	2	no	—
96	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
97	C₂×C₃×C₄	#121	1	1	yes	characteristic
98	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
99	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
100	C₂×C₃×C₄	#135	1	1	yes	characteristic
101	C₂×C₃×C₄	#135	1	1	yes	characteristic
102	C₂×Dic₃	—	4	4	no	—
103	C₂×Dic₃	—	8	2	no	—
104	C₂×Dic₃	—	8	2	no	—
105	C₂×Dic₃	—	8	2	no	—
106	C₂×Dic₃	—	8	2	no	—
107	C₂×Dic₃	—	8	2	no	—
108	C₃×C₈	—	2	2	no	—
109	C₃×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₃×C₈	C₄×Q₈	1	1	yes	characteristic
111	C₃×C₈	C₂²⋊Q₈	1	1	yes	characteristic
112	C₃×C₈	#125	1	1	yes	characteristic
113	C₂²×C₄	—	12	2	no	—
114	C₂²×C₄	—	12	2	no	—
115	C₂²×C₄	D₂₄	1	2	yes	—
116	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	Dic₁₂	1	2	yes	—
120	C₂×C₈	C₂×Dic₆	1	1	yes	characteristic
121	C₂×C₈	C₂×Dic₆	1	1	yes	characteristic
122	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
123	C₂×C₈	#235	1	1	yes	characteristic
124	C₂×C₈	#235	1	1	yes	characteristic
125	C₂×C₈	C₂×#67	1	1	yes	characteristic
126	C₄⋊C₄	—	6	4	no	—
127	C₄⋊C₄	—	12	2	no	—
128	C₄⋊C₄	—	12	2	no	—
129	C₄⋊C₄	—	12	2	no	—
130	C₄⋊C₄	—	24	4	no	—
131	C₄²	—	2	1	no	—
132	C₄²	—	2	1	no	—
133	C₁₆	—	2	2	no	—
134	C₁₆	Dic₃⋊C₄	1	4	yes	complemented
135	C₂²×C₃	C₄×D₈	1	1	yes	characteristic
136	C₂²×C₃	#466	1	1	yes	characteristic
137	C₂²×C₃	#500	1	1	yes	characteristic
138	C₂²×C₃	#509	1	2	yes	—
139	C₂²×C₃	#510	1	2	yes	—
140	C₃×C₄	—	4	2	no	—
141	C₃×C₄	#389	1	1	yes	characteristic
142	C₃×C₄	C₄×Q₁₆	1	1	yes	characteristic
143	C₃×C₄	#467	1	1	yes	characteristic
144	C₃×C₄	#499	1	1	yes	characteristic
145	Dic₃	—	8	4	no	—
146	Dic₃	—	16	2	no	—
147	C₂³	C₂×D₂₄	1	1	yes	characteristic
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	12	4	no	—
152	C₂×C₄	—	24	2	no	—
153	C₂×C₄	—	24	2	no	—
154	C₂×C₄	—	24	2	no	—
155	C₂×C₄	—	24	2	no	—
156	C₂×C₄	—	24	2	no	—
157	C₂×C₄	#869	1	1	yes	characteristic
158	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
159	C₂×C₄	#904	1	1	yes	characteristic
160	C₂×C₄	#904	1	1	yes	characteristic
161	C₂×C₄	C₂×Dic₁₂	1	1	yes	characteristic
162	C₂×C₄	#930	1	1	yes	characteristic
163	C₈	—	2	2	no	—
164	C₈	C₄×Dic₆	1	1	yes	characteristic
165	C₈	#870	1	1	yes	characteristic
166	C₈	C₂×Dic₃⋊C₄	1	1	yes	characteristic
167	C₈	#924	1	1	yes	characteristic
168	C₂×C₃	#1942	1	1	yes	characteristic
169	C₂×C₃	#2176	1	1	yes	characteristic
170	C₂×C₃	#2177	1	1	yes	characteristic
171	C₂×C₃	#2220	1	2	yes	—
172	C₂×C₃	#2255	1	2	yes	—
173	C₂²	#4682	1	1	yes	characteristic
174	C₂²	C₄×D₂₄	1	1	yes	characteristic
175	C₂²	#4908	1	2	yes	—
176	C₂²	#4909	1	2	yes	—
177	C₂²	#5115	1	1	yes	characteristic
178	C₄	—	4	2	no	—
179	C₄	—	24	4	no	—
180	C₄	—	48	2	no	—
181	C₄	#4683	1	1	yes	characteristic
182	C₄	C₄×Dic₁₂	1	1	yes	characteristic
183	C₄	#4928	1	1	yes	characteristic
184	C₄	#5116	1	1	yes	characteristic
185	C₃	#13168	1	1	yes	characteristic, complemented, Sylow
186	C₂	#69069	1	2	yes	—
187	C₂	#69078	1	1	yes	characteristic
188	C₂	#69082	1	1	yes	characteristic
189	C₂	#69770	1	1	yes	characteristic
190	C₂	#71444	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₂₄
commutator	C₂×C₃×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₂×C₃×#346	C₂
socle, prime radical	C₂³×C₃	C₂×D₈
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₂×#346	D₃
hypocenter	C₃	#13168
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
16. #4504	C₄	semidirect
72. C₃×C₁₆	C₄⋊C₄	semidirect
134. C₁₆	Dic₃⋊C₄	semidirect
185. C₃	#13168	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×#4505	2
C₂×#4504	2
#69770	2
#67806	2
#13168	3