Subgroups — #10713339

Order 768.

280 conjugacy classes of subgroups.

The subgroup lattice has 212 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	#71427	C₂	1	2	yes	maximal
3	#69792	C₂	1	1	yes	maximal, characteristic
4	#69779	C₂	1	1	yes	maximal, characteristic
5	#67821	C₂	1	2	yes	maximal
6	#13188	—	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	—	2	1	no	—
10	C₂×#819	C₂²	1	2	yes	—
11	C₂×#819	C₂²	1	1	yes	characteristic
12	#5106	C₂²	1	1	yes	characteristic
13	#5101	—	2	1	no	—
14	#4924	—	2	1	no	—
15	C₃×#364	C₄	1	2	yes	complemented
16	C₃×#364	C₄	1	2	yes	complemented
17	#4523	—	2	1	no	—
18	#4513	—	2	4	no	—
19	#2155	—	3	2	no	—
20	C₂×#364	D₃	1	1	yes	characteristic
21	#1848	—	3	1	no	—
22	#1772	—	3	1	no	—
23	#1388	—	3	2	no	—
24	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
25	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
26	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
27	#926	—	2	1	no	—
28	C₂×#208	—	2	1	no	—
29	C₂×#208	—	2	2	no	—
30	C₃×M₃₂	—	2	2	no	—
31	C₃×M₃₂	C₂×C₄	1	4	yes	—
32	C₂×C₃×C₁₆	—	2	2	no	—
33	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
34	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄×C₈	Q₈	1	1	yes	characteristic
36	C₃×C₄×C₈	Q₈	1	1	yes	characteristic
37	#819	—	2	1	no	—
38	#819	—	2	2	no	—
39	#819	—	2	2	no	—
40	#819	—	4	1	no	—
41	#819	—	4	1	no	—
42	#819	Q₈	1	2	yes	—
43	#818	—	2	1	no	—
44	#817	D₄	1	2	yes	—
45	#815	—	4	1	no	—
46	C₈×Dic₃	—	4	1	no	—
47	C₄⋊(C₃⋊C₈)	—	4	2	no	—
48	C₂×M₃₂	C₂×D₃	1	2	yes	—
49	C₂×#108	—	3	1	no	—
50	C₂×#108	—	3	2	no	—
51	C₂×#108	—	6	1	no	—
52	#427	—	3	1	no	—
53	#423	—	6	1	no	—
54	#405	—	6	1	no	—
55	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
56	#365	—	6	4	no	—
57	#364	Dic₃	1	2	yes	complemented
58	#364	Dic₃	1	2	yes	complemented
59	#339	—	6	1	no	—
60	C₂²×C₃×C₄	—	2	1	no	—
61	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
65	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
66	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
67	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
68	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₃×C₈	#41	1	1	yes	characteristic
73	C₂×C₃×C₈	#41	1	1	yes	characteristic
74	C₃×C₄²	—	2	1	no	—
75	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
76	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
77	#217	—	4	1	no	—
78	C₄⋊Dic₃	—	2	2	no	—
79	C₄×Dic₃	—	4	1	no	—
80	#208	—	2	2	no	—
81	#208	—	4	1	no	—
82	#208	—	4	2	no	—
83	#208	—	4	2	no	—
84	C₂×C₃⋊C₈	—	4	1	no	—
85	C₂×C₃⋊C₈	—	4	2	no	—
86	C₃×C₁₆	—	2	4	no	—
87	C₂×M₁₆	—	6	1	no	—
88	C₂×M₁₆	—	6	2	no	—
89	C₂²×C₈	D₁₂	1	1	yes	characteristic
90	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
91	#117	—	6	1	no	—
92	C₂×C₄²	D₁₂	1	1	yes	characteristic
93	M₃₂	—	2	2	no	—
94	M₃₂	C₂×Dic₃	1	4	yes	—
95	C₂×C₁₆	—	2	2	no	—
96	#108	—	3	2	no	—
97	#108	—	6	2	no	—
98	#108	—	6	1	no	—
99	#108	—	6	2	no	—
100	#108	—	12	1	no	—
101	#108	—	12	1	no	—
102	#107	—	6	1	no	—
103	#106	—	3	2	no	—
104	C₄⋊C₈	—	12	2	no	—
105	#97	—	12	1	no	—
106	C₄×C₈	—	12	1	no	—
107	C₄×C₈	Dic₆	1	1	yes	characteristic
108	C₄×C₈	Dic₆	1	1	yes	characteristic
109	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
110	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
111	C₂³×C₃	#137	1	1	yes	characteristic
112	C₂×C₃×C₄	—	2	1	no	—
113	C₂×C₃×C₄	—	2	1	no	—
114	C₂×C₃×C₄	—	4	2	no	—
115	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₃×C₄	#121	1	1	yes	characteristic
117	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×Dic₃	—	4	2	no	—
122	C₃×C₈	#106	1	1	yes	characteristic
123	C₃×C₈	#106	1	1	yes	characteristic
124	C₃×C₈	#106	1	1	yes	characteristic
125	C₃×C₈	#106	1	1	yes	characteristic
126	C₃×C₈	C₄×D₄	1	1	yes	characteristic
127	C₃×C₈	C₄×Q₈	1	1	yes	characteristic
128	C₃×C₈	#125	1	1	yes	characteristic
129	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
130	C₃⋊C₈	—	4	2	no	—
131	C₃⋊C₈	—	8	2	no	—
132	C₂²×C₄	—	2	1	no	—
133	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
134	M₁₆	—	6	2	no	—
135	M₁₆	—	12	1	no	—
136	M₁₆	—	12	2	no	—
137	M₁₆	—	12	2	no	—
138	C₂×C₈	—	12	1	no	—
139	C₂×C₈	—	12	2	no	—
140	C₂×C₈	#204	1	1	yes	characteristic
141	C₂×C₈	#204	1	1	yes	characteristic
142	C₂×C₈	#204	1	1	yes	characteristic
143	C₂×C₈	#204	1	1	yes	characteristic
144	C₂×C₈	C₄⋊Dic₃	1	1	yes	characteristic
145	C₂×C₈	C₄⋊Dic₃	1	1	yes	characteristic
146	C₂×C₈	C₂×Dic₆	1	1	yes	characteristic
147	C₂×C₈	D₃×D₄	1	1	yes	characteristic
148	C₂×C₈	#237	1	1	yes	characteristic
149	C₂×C₈	D₃×Q₈	1	1	yes	characteristic
150	C₂×C₈	#239	1	1	yes	characteristic
151	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
152	C₄⋊C₄	—	6	2	no	—
153	C₂²⋊C₄	—	12	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	12	1	no	—
156	C₄²	C₄⋊Dic₃	1	1	yes	characteristic
157	C₄²	C₄⋊Dic₃	1	1	yes	characteristic
158	C₁₆	—	2	4	no	—
159	C₂²×C₃	—	4	1	no	—
160	C₂²×C₃	#428	1	1	yes	characteristic
161	C₂²×C₃	#461	1	1	yes	characteristic
162	C₂²×C₃	#475	1	1	yes	characteristic
163	C₃×C₄	—	2	2	no	—
164	C₃×C₄	—	4	1	no	—
165	C₃×C₄	#389	1	1	yes	characteristic
166	C₃×C₄	C₂×#106	1	1	yes	characteristic
167	C₃×C₄	#460	1	1	yes	characteristic
168	C₃×C₄	#476	1	1	yes	characteristic
169	Dic₃	—	8	2	no	—
170	C₂³	#909	1	1	yes	characteristic
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	12	2	no	—
175	C₂×C₄	#895	1	1	yes	characteristic
176	C₂×C₄	#897	1	1	yes	characteristic
177	C₂×C₄	C₂×#204	1	1	yes	characteristic
178	C₂×C₄	C₂×#204	1	1	yes	characteristic
179	C₂×C₄	#908	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
181	C₈	—	12	2	no	—
182	C₈	—	24	2	no	—
183	C₈	#817	1	1	yes	characteristic
184	C₈	#817	1	1	yes	characteristic
185	C₈	#817	1	1	yes	characteristic
186	C₈	#817	1	1	yes	characteristic
187	C₈	#888	1	1	yes	characteristic
188	C₈	#890	1	1	yes	characteristic
189	C₈	D₄×Dic₃	1	1	yes	characteristic
190	C₈	Q₈×Dic₃	1	1	yes	characteristic
191	C₂×C₃	—	2	1	no	—
192	C₂×C₃	—	2	1	no	—
193	C₂×C₃	—	2	1	no	—
194	C₂×C₃	#1948	1	1	yes	characteristic
195	C₂²	—	4	1	no	—
196	C₂²	#4841	1	1	yes	characteristic
197	C₂²	#4851	1	1	yes	characteristic
198	C₂²	#5121	1	1	yes	characteristic
199	C₄	—	2	2	no	—
200	C₄	—	4	1	no	—
201	C₄	—	24	2	no	—
202	C₄	#4842	1	1	yes	characteristic
203	C₄	#4848	1	1	yes	characteristic
204	C₄	#4984	1	1	yes	characteristic
205	C₄	C₂×#817	1	1	yes	characteristic
206	C₃	#13188	1	1	yes	characteristic, complemented, Sylow
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	#70454	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#4842
commutator	C₂×C₃×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₂×C₃×#364	C₂
socle	C₂×C₃	#1948
prime radical	C₂³×C₃	#137
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₂×#364	D₃
hypocenter	C₃	#13188
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
15. C₃×#364	C₄	semidirect
16. C₃×#364	C₄	semidirect
57. #364	Dic₃	semidirect
58. #364	Dic₃	semidirect
206. C₃	#13188	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
#71427	2
#69792	2
#69779	2
#67821	2
#13188	3