Subgroups — #10693902

Order 768.

290 conjugacy classes of subgroups.

The subgroup lattice has 204 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	#69731	C₂	1	1	yes	maximal, characteristic
4	#69725	C₂	1	1	yes	maximal, characteristic
5	#67707	C₂	1	2	yes	maximal
6	#9626	—	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	#4458	—	2	1	no	—
17	#4458	—	2	1	no	—
18	#4455	—	2	1	no	—
19	#4454	—	2	1	no	—
20	#1950	—	3	2	no	—
21	#1567	—	3	1	no	—
22	C₂×#334	D₃	1	1	yes	characteristic
23	#1451	—	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	1	yes	characteristic
31	C₂×#208	D₄	1	1	yes	characteristic
32	C₃×C₄⋊C₈	—	2	2	no	—
33	C₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
34	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄×C₈	D₄	1	1	yes	characteristic
37	C₃×C₄×C₈	D₄	1	1	yes	characteristic
38	#830	Q₈	1	2	yes	—
39	#819	—	4	4	no	—
40	C₄⋊(C₃⋊C₈)	—	2	1	no	—
41	C₄⋊(C₃⋊C₈)	—	2	2	no	—
42	C₄⋊(C₃⋊C₈)	—	2	2	no	—
43	#803	—	2	1	no	—
44	C₄×C₃⋊C₈	—	2	1	no	—
45	C₄×C₃⋊C₈	—	2	1	no	—
46	#431	—	3	1	no	—
47	C₂×#108	—	6	2	no	—
48	#423	—	3	2	no	—
49	C₂×C₄⋊C₈	C₂×D₃	1	2	yes	—
50	C₄×M₁₆	—	3	1	no	—
51	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
52	#341	—	6	2	no	—
53	#335	—	6	1	no	—
54	#335	—	6	1	no	—
55	#334	Dic₃	1	2	yes	—
56	#334	Dic₃	1	2	yes	—
57	#322	—	6	1	no	—
58	#322	—	6	1	no	—
59	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₃×C₈	—	2	2	no	—
63	C₂×C₃×C₈	—	2	4	no	—
64	C₂×C₃×C₈	—	4	1	no	—
65	C₂×C₃×C₈	—	4	2	no	—
66	C₂×C₃×C₈	—	4	2	no	—
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	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
71	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
72	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
73	C₃×C₄²	#41	1	1	yes	characteristic
74	C₃×C₄²	#41	1	1	yes	characteristic
75	#208	—	2	2	no	—
76	#208	—	2	2	no	—
77	#208	—	4	2	no	—
78	#208	—	4	2	no	—
79	C₂×C₃⋊C₈	—	2	1	no	—
80	C₂×C₃⋊C₈	—	2	1	no	—
81	C₂×C₃⋊C₈	—	2	2	no	—
82	C₂×C₃⋊C₈	—	4	2	no	—
83	C₂×M₁₆	—	3	1	no	—
84	C₂×M₁₆	—	3	1	no	—
85	C₂×M₁₆	—	6	2	no	—
86	C₂²×C₈	—	2	2	no	—
87	C₂²×C₈	Dic₆	1	1	yes	characteristic
88	C₂²×C₈	Dic₆	1	1	yes	characteristic
89	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
90	#108	—	12	4	no	—
91	C₄⋊C₈	—	2	2	no	—
92	C₄⋊C₈	—	6	2	no	—
93	C₄⋊C₈	—	6	1	no	—
94	C₄⋊C₈	—	6	2	no	—
95	C₄⋊C₈	C₂×Dic₃	1	4	yes	—
96	C₂²⋊C₈	—	3	2	no	—
97	#97	—	6	1	no	—
98	C₄×C₈	—	6	1	no	—
99	C₄×C₈	—	6	1	no	—
100	C₄×C₈	D₁₂	1	1	yes	characteristic
101	C₄×C₈	D₁₂	1	1	yes	characteristic
102	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
103	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
104	C₂³×C₃	C₄⋊Q₈	1	1	yes	characteristic
105	C₂×C₃×C₄	—	2	1	no	—
106	C₂×C₃×C₄	—	2	1	no	—
107	C₂×C₃×C₄	—	4	1	no	—
108	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
110	C₂×C₃×C₄	C₄×Q₈	1	1	yes	characteristic
111	C₂×C₃×C₄	#121	1	1	yes	characteristic
112	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₃×C₄	#125	1	1	yes	characteristic
114	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
115	C₂×C₃×C₄	#137	1	2	yes	—
116	C₃×C₈	—	2	1	no	—
117	C₃×C₈	—	2	1	no	—
118	C₃×C₈	—	2	1	no	—
119	C₃×C₈	—	4	4	no	—
120	C₃×C₈	#108	1	2	yes	complemented
121	C₃⋊C₈	—	4	2	no	—
122	C₃⋊C₈	—	4	1	no	—
123	C₃⋊C₈	—	4	1	no	—
124	C₃⋊C₈	—	8	2	no	—
125	C₂²×C₄	C₂×Dic₆	1	1	yes	characteristic
126	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
127	C₂²×C₄	D₃×Q₈	1	1	yes	characteristic
128	M₁₆	—	6	2	no	—
129	M₁₆	—	6	2	no	—
130	M₁₆	—	12	2	no	—
131	M₁₆	—	12	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	4	1	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	6	1	no	—
138	C₂×C₈	—	6	1	no	—
139	C₂×C₈	—	6	2	no	—
140	C₂×C₈	—	12	2	no	—
141	C₂×C₈	C₄⋊Dic₃	1	1	yes	characteristic
142	C₂×C₈	C₄⋊Dic₃	1	1	yes	characteristic
143	C₂×C₈	C₄⋊Dic₃	1	1	yes	characteristic
144	C₂×C₈	C₄⋊Dic₃	1	1	yes	characteristic
145	C₄²	C₂×D₁₂	1	1	yes	characteristic
146	C₄²	#237	1	1	yes	characteristic
147	C₄²	#239	1	1	yes	characteristic
148	C₄²	C₂²×Dic₃	1	1	yes	characteristic
149	C₂²×C₃	—	2	1	no	—
150	C₂²×C₃	—	2	1	no	—
151	C₂²×C₃	#389	1	1	yes	characteristic
152	C₂²×C₃	#501	1	2	yes	—
153	C₃×C₄	—	2	1	no	—
154	C₃×C₄	—	2	1	no	—
155	C₃×C₄	—	2	1	no	—
156	C₃×C₄	C₂×#108	1	1	yes	characteristic
157	C₃×C₄	#430	1	1	yes	characteristic
158	C₃×C₄	#440	1	2	yes	—
159	C₃×C₄	#469	1	1	yes	characteristic
160	C₃×C₄	#470	1	1	yes	characteristic
161	C₂³	#888	1	1	yes	characteristic
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	#890	1	1	yes	characteristic
166	C₂×C₄	#895	1	1	yes	characteristic
167	C₂×C₄	#897	1	1	yes	characteristic
168	C₂×C₄	#914	1	2	yes	—
169	C₂×C₄	#915	1	2	yes	—
170	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
171	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
172	C₂×C₄	Q₈×Dic₃	1	1	yes	characteristic
173	C₈	—	2	1	no	—
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	4	4	no	—
177	C₈	—	12	2	no	—
178	C₈	—	12	1	no	—
179	C₈	—	12	1	no	—
180	C₈	—	24	2	no	—
181	C₈	#819	1	2	yes	complemented
182	C₂×C₃	—	2	2	no	—
183	C₂×C₃	#1944	1	1	yes	characteristic
184	C₂×C₃	#1950	1	1	yes	characteristic
185	C₂×C₃	#1953	1	1	yes	characteristic
186	C₂²	—	2	1	no	—
187	C₂²	—	2	1	no	—
188	C₂²	#4856	1	2	yes	—
189	C₂²	#4984	1	1	yes	characteristic
190	C₄	—	2	1	no	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	#4865	1	1	yes	characteristic
194	C₄	#4867	1	1	yes	characteristic
195	C₄	C₂×#819	1	1	yes	characteristic
196	C₄	#5125	1	1	yes	characteristic
197	C₄	#5164	1	2	yes	—
198	C₃	#9626	1	1	yes	characteristic, complemented, Sylow
199	C₂	—	2	2	no	—
200	C₂	#70474	1	1	yes	characteristic
201	C₂	#70562	1	1	yes	characteristic
202	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	#9626
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
120. C₃×C₈	#108	semidirect
181. C₈	#819	semidirect
198. C₃	#9626	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
#69731	2
#69725	2
#67707	2
#9626	3