Subgroups — #10693887

Order 768.

298 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#334	C₂	1	1	yes	maximal, characteristic
2	#70474	C₂	1	2	yes	maximal
3	#69730	C₂	1	1	yes	maximal, characteristic
4	C₈×#208	C₂	1	1	yes	maximal, characteristic
5	#67707	C₂	1	2	yes	maximal
6	#9623	—	3	1	no	maximal, Sylow
7	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×#819	—	2	2	no	—
10	#4925	C₂²	1	1	yes	characteristic
11	#4924	C₂²	1	2	yes	—
12	C₄×#208	C₂²	1	1	yes	characteristic
13	C₃×#334	C₄	1	2	yes	—
14	C₃×#334	C₄	1	2	yes	—
15	#4552	—	2	2	no	—
16	#4457	—	2	1	no	—
17	#4457	—	2	1	no	—
18	#4454	—	2	1	no	—
19	C₈×C₃⋊C₈	—	2	1	no	—
20	#1950	—	3	2	no	—
21	#1566	—	3	1	no	—
22	C₂×#334	D₃	1	1	yes	characteristic
23	C₈×M₁₆	—	3	1	no	—
24	#1301	—	3	2	no	—
25	C₂²×C₃×C₈	—	2	2	no	—
26	C₂²×C₃×C₈	Q₈	1	1	yes	characteristic
27	C₂²×C₃×C₈	Q₈	1	1	yes	characteristic
28	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
29	C₂×#208	—	2	2	no	—
30	C₂×#208	D₄	1	2	yes	—
31	C₃×C₄⋊C₈	—	2	2	no	—
32	C₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
33	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
34	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄×C₈	D₄	1	1	yes	characteristic
36	C₃×C₄×C₈	D₄	1	1	yes	characteristic
37	#830	Q₈	1	2	yes	—
38	#819	—	4	4	no	—
39	C₄⋊(C₃⋊C₈)	—	2	1	no	—
40	C₄⋊(C₃⋊C₈)	—	2	2	no	—
41	C₄⋊(C₃⋊C₈)	—	2	2	no	—
42	#803	—	2	1	no	—
43	C₄×C₃⋊C₈	—	2	1	no	—
44	C₄×C₃⋊C₈	—	2	1	no	—
45	#431	—	3	1	no	—
46	C₂×#108	—	6	2	no	—
47	#423	—	3	2	no	—
48	C₂×C₄⋊C₈	C₂×D₃	1	2	yes	—
49	C₄×M₁₆	—	3	1	no	—
50	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
51	#341	—	6	2	no	—
52	#334	—	6	1	no	—
53	#334	—	6	1	no	—
54	#334	Dic₃	1	2	yes	—
55	#334	Dic₃	1	2	yes	—
56	#322	—	6	1	no	—
57	C₈²	—	6	1	no	—
58	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₃×C₈	—	2	2	no	—
62	C₂×C₃×C₈	—	2	4	no	—
63	C₂×C₃×C₈	—	4	1	no	—
64	C₂×C₃×C₈	—	4	2	no	—
65	C₂×C₃×C₈	—	4	2	no	—
66	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
70	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
71	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
72	C₃×C₄²	#41	1	1	yes	characteristic
73	C₃×C₄²	#41	1	1	yes	characteristic
74	#208	—	2	2	no	—
75	#208	—	4	2	no	—
76	#208	—	4	2	no	—
77	#208	SD₁₆	1	4	yes	—
78	C₂×C₃⋊C₈	—	2	2	no	—
79	C₂×C₃⋊C₈	—	2	2	no	—
80	C₂×C₃⋊C₈	—	4	2	no	—
81	C₂×M₁₆	—	3	2	no	—
82	C₂×M₁₆	—	6	2	no	—
83	C₂²×C₈	—	2	2	no	—
84	C₂²×C₈	Dic₆	1	1	yes	characteristic
85	C₂²×C₈	Dic₆	1	1	yes	characteristic
86	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
87	#108	—	12	4	no	—
88	C₄⋊C₈	—	2	2	no	—
89	C₄⋊C₈	—	6	2	no	—
90	C₄⋊C₈	—	6	1	no	—
91	C₄⋊C₈	—	6	2	no	—
92	C₄⋊C₈	C₂×Dic₃	1	4	yes	—
93	C₂²⋊C₈	—	3	2	no	—
94	#97	—	6	1	no	—
95	C₄×C₈	—	6	1	no	—
96	C₄×C₈	—	6	1	no	—
97	C₄×C₈	D₁₂	1	1	yes	characteristic
98	C₄×C₈	D₁₂	1	1	yes	characteristic
99	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
100	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
101	C₂³×C₃	C₄⋊Q₈	1	1	yes	characteristic
102	C₂×C₃×C₄	—	2	1	no	—
103	C₂×C₃×C₄	—	2	1	no	—
104	C₂×C₃×C₄	—	4	1	no	—
105	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
107	C₂×C₃×C₄	C₄×Q₈	1	1	yes	characteristic
108	C₂×C₃×C₄	#121	1	1	yes	characteristic
109	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₃×C₄	#125	1	1	yes	characteristic
111	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
112	C₂×C₃×C₄	#135	1	2	yes	—
113	C₃×C₈	—	2	1	no	—
114	C₃×C₈	—	2	1	no	—
115	C₃×C₈	—	2	1	no	—
116	C₃×C₈	—	4	4	no	—
117	C₃×C₈	#108	1	2	yes	complemented
118	C₃⋊C₈	—	2	4	no	—
119	C₃⋊C₈	—	4	2	no	—
120	C₃⋊C₈	—	8	2	no	—
121	C₂²×C₄	C₂×Dic₆	1	1	yes	characteristic
122	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
123	C₂²×C₄	D₃×Q₈	1	1	yes	characteristic
124	M₁₆	—	3	4	no	—
125	M₁₆	—	6	2	no	—
126	M₁₆	—	12	2	no	—
127	M₁₆	—	12	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	—	4	1	no	—
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	6	2	no	—
134	C₂×C₈	—	6	2	no	—
135	C₂×C₈	—	12	2	no	—
136	C₂×C₈	C₄⋊Dic₃	1	1	yes	characteristic
137	C₂×C₈	C₄⋊Dic₃	1	1	yes	characteristic
138	C₂×C₈	C₄⋊Dic₃	1	1	yes	characteristic
139	C₂×C₈	C₄⋊Dic₃	1	1	yes	characteristic
140	C₄²	C₂×D₁₂	1	1	yes	characteristic
141	C₄²	#237	1	1	yes	characteristic
142	C₄²	#239	1	1	yes	characteristic
143	C₄²	C₂²×Dic₃	1	1	yes	characteristic
144	C₂²×C₃	—	2	1	no	—
145	C₂²×C₃	—	2	1	no	—
146	C₂²×C₃	#389	1	1	yes	characteristic
147	C₂²×C₃	#498	1	1	yes	characteristic
148	C₂²×C₃	#499	1	1	yes	characteristic
149	C₃×C₄	—	2	1	no	—
150	C₃×C₄	—	2	1	no	—
151	C₃×C₄	—	2	1	no	—
152	C₃×C₄	C₂×#108	1	1	yes	characteristic
153	C₃×C₄	#430	1	1	yes	characteristic
154	C₃×C₄	C₄×SD₁₆	1	2	yes	—
155	C₃×C₄	#465	1	2	yes	—
156	C₂³	#888	1	1	yes	characteristic
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	#890	1	1	yes	characteristic
161	C₂×C₄	#895	1	1	yes	characteristic
162	C₂×C₄	#897	1	1	yes	characteristic
163	C₂×C₄	D₃×SD₁₆	1	2	yes	—
164	C₂×C₄	#916	1	2	yes	—
165	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
166	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
167	C₂×C₄	Q₈×Dic₃	1	1	yes	characteristic
168	C₈	—	2	1	no	—
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	4	4	no	—
172	C₈	—	6	4	no	—
173	C₈	—	12	2	no	—
174	C₈	—	24	2	no	—
175	C₈	#819	1	2	yes	complemented
176	C₂×C₃	—	2	1	no	—
177	C₂×C₃	—	2	1	no	—
178	C₂×C₃	#1941	1	1	yes	characteristic
179	C₂×C₃	#1950	1	1	yes	characteristic
180	C₂×C₃	#1953	1	1	yes	characteristic
181	C₂²	—	2	1	no	—
182	C₂²	—	2	1	no	—
183	C₂²	#4855	1	1	yes	characteristic
184	C₂²	#4858	1	1	yes	characteristic
185	C₂²	#4984	1	1	yes	characteristic
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	#4864	1	2	yes	—
190	C₄	C₂×#819	1	1	yes	characteristic
191	C₄	#5125	1	1	yes	characteristic
192	C₄	Dic₃×SD₁₆	1	2	yes	—
193	C₃	#9623	1	1	yes	characteristic, complemented, Sylow
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	#70474	1	1	yes	characteristic
197	C₂	#70560	1	1	yes	characteristic
198	C₂	#70707	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#895
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂×C₃×#334	C₂
socle	C₂²×C₃	#389
prime radical	C₂³×C₃	C₄⋊Q₈
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×#334	D₃
hypocenter	C₃	#9623
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
117. C₃×C₈	#108	semidirect
175. C₈	#819	semidirect
193. C₃	#9623	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#334	2
#70474	2
#69730	2
C₈×#208	2
#67707	2
#9623	3