Subgroups — #10710549

Order 768.

334 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×#364	C₂	1	1	yes	maximal, characteristic
2	#71419	C₂	1	2	yes	maximal
3	#69779	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	#67815	C₂	1	2	yes	maximal
6	#12664	—	3	1	no	maximal, Sylow
7	C₂×C₃×M₃₂	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×#819	C₄	1	1	yes	characteristic
10	#5106	C₄	1	1	yes	characteristic
11	#5101	C₂²	1	2	yes	—
12	C₂²×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₃×#364	C₄	1	4	yes	complemented
15	#4523	C₄	1	2	yes	—
16	#4502	—	2	2	no	—
17	Dic₃×C₁₆	—	2	2	no	—
18	C₄×C₃⋊C₁₆	C₄	1	4	yes	complemented
19	C₂×#364	D₃	1	1	yes	characteristic
20	#2108	—	3	2	no	—
21	C₂×C₄×C₁₆	—	3	1	no	—
22	#1772	—	3	1	no	—
23	#1379	—	3	2	no	—
24	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
26	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
27	#926	C₂×C₄	1	2	yes	—
28	C₂×#208	C₂×C₄	1	2	yes	—
29	C₃×M₃₂	—	2	2	no	—
30	C₃×M₃₂	C₂×C₄	1	4	yes	—
31	C₂×C₃×C₁₆	—	2	2	no	—
32	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
33	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
34	#819	C₈	1	4	yes	—
35	#818	C₈	1	2	yes	—
36	#817	C₈	1	2	yes	—
37	#815	—	2	2	no	—
38	C₈×Dic₃	—	2	2	no	—
39	C₂×C₃⋊C₁₆	—	2	1	no	—
40	C₂×C₃⋊C₁₆	—	2	1	no	—
41	C₂×C₃⋊C₁₆	C₂×C₄	1	2	yes	—
42	C₂×C₃⋊C₁₆	C₂×C₄	1	2	yes	—
43	C₂×C₃⋊C₁₆	D₄	1	1	yes	characteristic
44	C₂×C₃⋊C₁₆	D₄	1	1	yes	characteristic
45	C₂×C₃⋊C₁₆	Q₈	1	2	yes	—
46	C₂×M₃₂	C₂×D₃	1	2	yes	—
47	C₂²×C₁₆	—	3	1	no	—
48	C₂²×C₁₆	—	3	1	no	—
49	C₂×#108	—	3	1	no	—
50	#427	—	3	1	no	—
51	#405	—	3	2	no	—
52	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
53	#364	Dic₃	1	4	yes	complemented
54	#346	—	6	2	no	—
55	C₄×C₁₆	—	3	4	no	—
56	C₄×C₁₆	—	6	2	no	—
57	#339	—	3	2	no	—
58	C₂²×C₃×C₄	—	2	1	no	—
59	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃×C₈	C₄²	1	2	yes	—
61	C₂×C₃×C₈	C₂×C₈	1	1	yes	characteristic
62	C₂×C₃×C₈	C₂×C₈	1	1	yes	characteristic
63	C₂×C₃×C₈	C₂×C₈	1	2	yes	—
64	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₃×C₈	#41	1	1	yes	characteristic
69	C₂×C₃×C₈	#41	1	1	yes	characteristic
70	C₃×C₄²	—	2	1	no	—
71	C₃×C₄²	C₄²	1	2	yes	—
72	#217	—	2	2	no	—
73	C₄⋊Dic₃	C₂×C₈	1	4	yes	—
74	C₄×Dic₃	—	2	2	no	—
75	#208	—	2	2	no	—
76	#208	C₂×C₈	1	4	yes	—
77	C₂×C₃⋊C₈	—	2	2	no	—
78	C₃×C₁₆	—	2	4	no	—
79	C₃⋊C₁₆	—	2	2	no	—
80	C₃⋊C₁₆	C₄⋊C₄	1	2	yes	—
81	C₃⋊C₁₆	C₄⋊C₄	1	2	yes	—
82	C₂×M₁₆	—	3	2	no	—
83	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
84	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
85	#117	—	3	2	no	—
86	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
87	M₃₂	—	2	2	no	—
88	M₃₂	C₂×Dic₃	1	4	yes	—
89	C₂×C₁₆	—	2	2	no	—
90	C₂×C₁₆	—	3	2	no	—
91	C₂×C₁₆	—	3	1	no	—
92	C₂×C₁₆	—	3	1	no	—
93	C₂×C₁₆	—	3	2	no	—
94	C₂×C₁₆	—	3	2	no	—
95	C₂×C₁₆	—	6	1	no	—
96	C₂×C₁₆	—	6	1	no	—
97	#108	—	3	4	no	—
98	#107	—	3	2	no	—
99	#106	—	3	2	no	—
100	#97	—	6	2	no	—
101	C₄×C₈	—	6	2	no	—
102	C₄×C₈	C₄×D₃	1	2	yes	—
103	C₄×C₈	C₂×Dic₃	1	2	yes	—
104	C₂³×C₃	#131	1	1	yes	characteristic
105	C₂×C₃×C₄	—	2	4	no	—
106	C₂×C₃×C₄	—	2	2	no	—
107	C₂×C₃×C₄	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
109	C₂×C₃×C₄	C₄×Q₈	1	1	yes	characteristic
110	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₃×C₄	#131	1	1	yes	characteristic
113	C₂×Dic₃	—	2	4	no	—
114	C₃×C₈	C₄×C₈	1	2	yes	—
115	C₃×C₈	C₄×C₈	1	2	yes	—
116	C₃×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₃×C₈	#117	1	1	yes	characteristic
118	C₃×C₈	C₄×D₄	1	1	yes	characteristic
119	C₃×C₈	C₄×Q₈	1	1	yes	characteristic
120	C₃⋊C₈	—	2	4	no	—
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
123	M₁₆	—	3	4	no	—
124	M₁₆	—	6	2	no	—
125	C₂×C₈	—	6	2	no	—
126	C₂×C₈	D₃×C₈	1	1	yes	characteristic
127	C₂×C₈	D₃×C₈	1	1	yes	characteristic
128	C₂×C₈	D₃×C₈	1	2	yes	—
129	C₂×C₈	C₄×Dic₃	1	2	yes	—
130	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
131	C₂×C₈	D₃×D₄	1	1	yes	characteristic
132	C₂×C₈	#237	1	1	yes	characteristic
133	C₂×C₈	D₃×Q₈	1	1	yes	characteristic
134	C₂×C₈	#239	1	1	yes	characteristic
135	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
136	C₄⋊C₄	—	3	4	no	—
137	C₂²⋊C₄	—	6	2	no	—
138	C₄²	—	2	1	no	—
139	C₄²	—	6	2	no	—
140	C₄²	C₄×Dic₃	1	2	yes	—
141	C₁₆	—	2	4	no	—
142	C₁₆	—	3	2	no	—
143	C₁₆	—	3	2	no	—
144	C₁₆	—	6	2	no	—
145	C₂²×C₃	—	2	2	no	—
146	C₂²×C₃	#405	1	1	yes	characteristic
147	C₂²×C₃	C₈×D₄	1	1	yes	characteristic
148	C₂²×C₃	C₈×Q₈	1	1	yes	characteristic
149	C₃×C₄	—	2	4	no	—
150	C₃×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
151	C₃×C₄	C₂×C₄×C₈	1	1	yes	characteristic
152	C₃×C₄	C₈×D₄	1	1	yes	characteristic
153	C₃×C₄	C₈×Q₈	1	1	yes	characteristic
154	Dic₃	—	4	4	no	—
155	C₂³	#907	1	1	yes	characteristic
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	6	4	no	—
159	C₂×C₄	#887	1	1	yes	characteristic
160	C₂×C₄	#893	1	1	yes	characteristic
161	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
162	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
163	C₂×C₄	#907	1	1	yes	characteristic
164	C₂×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
165	C₈	—	6	4	no	—
166	C₈	C₈×Dic₃	1	2	yes	—
167	C₈	C₈×Dic₃	1	2	yes	—
168	C₈	D₃×C₄⋊C₄	1	1	yes	characteristic
169	C₈	#892	1	1	yes	characteristic
170	C₈	D₄×Dic₃	1	1	yes	characteristic
171	C₈	Q₈×Dic₃	1	1	yes	characteristic
172	C₂×C₃	—	2	1	no	—
173	C₂×C₃	—	2	1	no	—
174	C₂×C₃	C₈×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₃	#2170	1	2	yes	—
176	C₂²	—	2	2	no	—
177	C₂²	#4830	1	1	yes	characteristic
178	C₂²	#4834	1	1	yes	characteristic
179	C₂²	#5122	1	1	yes	characteristic
180	C₄	—	2	4	no	—
181	C₄	—	12	4	no	—
182	C₄	#4830	1	1	yes	characteristic
183	C₄	#4834	1	1	yes	characteristic
184	C₄	Dic₃×C₄⋊C₄	1	1	yes	characteristic
185	C₄	C₂×C₈×Dic₃	1	1	yes	characteristic
186	C₃	#12664	1	1	yes	characteristic, complemented, Sylow
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	#69300	1	2	yes	—
190	C₂	#70465	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₃×D₄
commutator	C₃×C₄	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₂×C₃×#364	C₂
socle	C₂²×C₃	C₈×D₄
prime radical	C₂³×C₃	#131
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₂×#364	D₃
hypocenter	C₃	#12664
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
14. C₃×#364	C₄	semidirect
18. C₄×C₃⋊C₁₆	C₄	semidirect
53. #364	Dic₃	semidirect
186. C₃	#12664	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
#71419	2
#69779	2
C₂×C₄×C₃⋊C₁₆	2
#67815	2
#12664	3