Subgroups — #10713711

Order 768.

386 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×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₂×#4461	C₂	1	1	yes	maximal, characteristic
4	#67657	C₂	1	4	yes	maximal
5	#13264	—	3	1	no	maximal, Sylow
6	C₂×C₃×C₄×Q₈	C₄	1	1	yes	characteristic
7	C₃×C₈×Q₈	—	2	2	no	—
8	C₃×C₈×Q₈	—	2	2	no	—
9	C₂×C₃×C₄⋊C₈	C₄	1	2	yes	—
10	C₂×C₃×C₄⋊C₈	C₄	1	1	yes	characteristic
11	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
12	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃⋊C₁₆	C₂²	1	2	yes	—
14	C₂²×C₃⋊C₁₆	C₂²	1	2	yes	—
15	C₄⋊(C₃⋊C₁₆)	—	2	2	no	—
16	#4461	—	2	2	no	—
17	C₂×C₈×Q₈	D₃	1	1	yes	characteristic
18	C₂×C₄⋊C₁₆	—	3	1	no	—
19	C₂×#346	—	3	1	no	—
20	#1374	—	3	4	no	—
21	C₂²×C₃×Q₈	C₈	1	1	yes	characteristic
22	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
25	C₃×C₄×Q₈	—	2	2	no	—
26	C₃×C₄×Q₈	—	2	2	no	—
27	C₂×C₃×C₄⋊C₄	C₈	1	1	yes	characteristic
28	C₂×C₃×C₄⋊C₄	C₈	1	2	yes	—
29	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
31	C₃×C₄⋊C₈	—	2	4	no	—
32	C₃×C₄⋊C₈	—	2	2	no	—
33	C₃×C₄×C₈	—	2	2	no	—
34	C₃×C₄×C₈	—	2	2	no	—
35	C₃×C₄×C₈	D₄	1	2	yes	—
36	C₃×C₄×C₈	D₄	1	2	yes	—
37	C₂×C₃⋊C₁₆	—	2	2	no	—
38	C₂×C₃⋊C₁₆	—	2	2	no	—
39	C₂×C₃⋊C₁₆	—	2	2	no	—
40	C₂×C₃⋊C₁₆	—	2	4	no	—
41	C₂×C₃⋊C₁₆	Q₈	1	4	yes	complemented
42	C₂×C₄×Q₈	Dic₃	1	1	yes	characteristic
43	C₂²×C₁₆	—	3	2	no	—
44	C₂²×C₁₆	—	3	2	no	—
45	C₈×Q₈	—	2	2	no	—
46	C₈×Q₈	—	2	2	no	—
47	C₂×C₄⋊C₈	Dic₃	1	2	yes	—
48	C₂×C₄⋊C₈	Dic₃	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂×D₃	1	2	yes	—
50	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
51	C₄⋊C₁₆	—	6	2	no	—
52	#346	—	6	2	no	—
53	C₂×C₃×Q₈	—	2	2	no	—
54	C₂×C₃×Q₈	—	2	4	no	—
55	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
56	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
59	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃×C₈	—	2	2	no	—
61	C₂×C₃×C₈	—	2	2	no	—
62	C₂×C₃×C₈	—	2	4	no	—
63	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
64	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
65	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
67	C₂×C₃×C₈	#41	1	2	yes	—
68	C₃×C₄⋊C₄	—	2	4	no	—
69	C₃×C₄⋊C₄	—	2	2	no	—
70	C₃×C₄²	—	2	2	no	—
71	C₃×C₄²	—	2	2	no	—
72	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
73	C₃×C₄²	M₁₆	1	2	yes	—
74	C₃⋊C₁₆	—	2	4	no	—
75	C₃⋊C₁₆	—	4	2	no	—
76	C₂²×Q₈	C₃⋊C₈	1	1	yes	characteristic
77	C₂²×C₈	C₂×Dic₃	1	2	yes	—
78	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
79	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
80	C₄×Q₈	—	2	2	no	—
81	C₄×Q₈	—	2	2	no	—
82	C₂×C₄⋊C₄	C₃⋊C₈	1	1	yes	characteristic
83	C₂×C₄⋊C₄	C₃⋊C₈	1	2	yes	—
84	C₂×C₄²	C₂×Dic₃	1	2	yes	—
85	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
86	C₂×C₁₆	—	3	4	no	—
87	C₂×C₁₆	—	6	2	no	—
88	C₂×C₁₆	—	6	2	no	—
89	C₂×C₁₆	—	6	4	no	—
90	C₂×C₁₆	—	6	2	no	—
91	C₄⋊C₈	—	2	4	no	—
92	C₄⋊C₈	—	2	2	no	—
93	C₄×C₈	—	2	2	no	—
94	C₄×C₈	—	2	2	no	—
95	C₄×C₈	#67	1	2	yes	—
96	C₄×C₈	#67	1	2	yes	—
97	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
98	C₃×Q₈	—	2	8	no	—
99	C₂×C₃×C₄	—	2	2	no	—
100	C₂×C₃×C₄	—	2	2	no	—
101	C₂×C₃×C₄	—	2	2	no	—
102	C₂×C₃×C₄	—	2	2	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	—	2	4	no	—
105	C₂×C₃×C₄	C₂²⋊C₈	1	2	yes	—
106	C₂×C₃×C₄	C₂²⋊C₈	1	2	yes	—
107	C₂×C₃×C₄	M₃₂	1	4	yes	—
108	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
110	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₃×C₄	#131	1	2	yes	—
112	C₃×C₈	—	2	2	no	—
113	C₃×C₈	—	4	2	no	—
114	C₃×C₈	C₂²⋊Q₈	1	2	yes	—
115	C₃×C₈	#124	1	1	yes	characteristic
116	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
117	C₂×Q₈	—	2	4	no	—
118	C₂×Q₈	—	2	2	no	—
119	C₂²×C₄	C₂×C₃⋊C₈	1	2	yes	—
120	C₂²×C₄	C₂×C₃⋊C₈	1	2	yes	—
121	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
122	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
123	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	4	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	#217	1	2	yes	—
128	C₂×C₈	D₃×Q₈	1	2	yes	—
129	C₂×C₈	#239	1	2	yes	—
130	C₂×C₈	C₂×#67	1	1	yes	characteristic
131	C₂×C₈	C₂×#67	1	1	yes	characteristic
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄²	—	2	2	no	—
135	C₄²	—	2	2	no	—
136	C₄²	#208	1	2	yes	—
137	C₄²	#217	1	2	yes	—
138	C₁₆	—	6	4	no	—
139	C₁₆	—	12	2	no	—
140	C₂²×C₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
141	C₂²×C₃	C₈×Q₈	1	2	yes	—
142	C₂²×C₃	C₂×M₃₂	1	2	yes	—
143	C₂²×C₃	#504	1	2	yes	—
144	C₃×C₄	—	2	2	no	—
145	C₃×C₄	—	2	4	no	—
146	C₃×C₄	—	2	2	no	—
147	C₃×C₄	—	4	2	no	—
148	C₃×C₄	#391	1	1	yes	characteristic
149	C₃×C₄	#408	1	1	yes	characteristic
150	C₃×C₄	#446	1	2	yes	—
151	C₂³	C₂²×C₃⋊C₈	1	1	yes	characteristic
152	Q₈	—	2	8	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	#812	1	4	yes	—
160	C₂×C₄	#830	1	2	yes	—
161	C₂×C₄	#830	1	2	yes	—
162	C₂×C₄	C₂×#208	1	1	yes	characteristic
163	C₂×C₄	Q₈×Dic₃	1	2	yes	—
164	C₂×C₄	#948	1	2	yes	—
165	C₂×C₄	C₂×#217	1	1	yes	characteristic
166	C₈	—	2	2	no	—
167	C₈	—	4	2	no	—
168	C₈	#944	1	1	yes	characteristic
169	C₈	#946	1	2	yes	—
170	C₈	#947	1	1	yes	characteristic
171	C₂×C₃	#1984	1	1	yes	characteristic
172	C₂×C₃	#2112	1	1	yes	characteristic
173	C₂×C₃	#2113	1	1	yes	characteristic
174	C₂×C₃	#2183	1	4	yes	—
175	C₂²	Q₈×C₃⋊C₈	1	2	yes	—
176	C₂²	C₂×#812	1	2	yes	—
177	C₂²	#5140	1	2	yes	—
178	C₂²	C₂×#830	1	1	yes	characteristic
179	C₄	—	2	4	no	—
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	#5024	1	2	yes	—
184	C₄	#5229	1	1	yes	characteristic
185	C₄	#5234	1	1	yes	characteristic
186	C₃	#13264	1	1	yes	characteristic, complemented, Sylow
187	C₂	#70949	1	4	yes	—
188	C₂	#71634	1	1	yes	characteristic
189	C₂	#71963	1	1	yes	characteristic
190	C₂	#72004	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₃
commutator, socle, prime radical	C₂³×C₃	C₂²×C₈
Fitting	C₂×C₃×C₈×Q₈	C₂
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₂×C₈×Q₈	D₃
hypocenter	C₃	#13264
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
41. C₂×C₃⋊C₁₆	Q₈	semidirect
186. C₃	#13264	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₈×Q₈	2
C₂×C₄⋊(C₃⋊C₁₆)	2
C₂×#4461	2
#67657	2
#13264	3