Subgroups — #10693929

Order 768.

310 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#1564	C₂	1	1	yes	maximal, characteristic
2	#70749	C₂	1	2	yes	maximal
3	#69792	C₂	1	1	yes	maximal, characteristic
4	#69757	C₂	1	1	yes	maximal, characteristic
5	#67746	C₂	1	2	yes	maximal
6	#9631	—	3	1	no	maximal, Sylow
7	C₃×#431	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×#819	—	2	1	no	—
10	C₂×#819	—	2	1	no	—
11	C₂×#815	—	2	1	no	—
12	C₄×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
13	#4924	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
15	C₃×#335	C₄	1	2	yes	complemented
16	C₃×#334	C₄	1	2	yes	complemented
17	#4550	—	2	1	no	—
18	#4457	—	2	4	no	—
19	#1848	—	3	1	no	—
20	#1770	—	3	1	no	—
21	#1570	—	3	2	no	—
22	#1564	D₃	1	1	yes	characteristic
23	#1303	—	3	2	no	—
24	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
25	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
26	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊Dic₃	—	2	1	no	—
28	C₂×C₄×Dic₃	—	2	1	no	—
29	C₂×#208	—	2	2	no	—
30	C₂²×C₃⋊C₈	—	2	1	no	—
31	C₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
32	C₃×C₂²⋊C₈	—	2	2	no	—
33	C₃×C₄×C₈	—	2	2	no	—
34	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄×C₈	Q₈	1	1	yes	characteristic
37	C₃×C₄×C₈	Q₈	1	1	yes	characteristic
38	C₃⋊#95	Q₈	1	2	yes	—
39	#819	—	4	2	no	—
40	#819	—	4	2	no	—
41	#815	—	4	2	no	—
42	C₄⋊(C₃⋊C₈)	—	2	1	no	—
43	C₄⋊(C₃⋊C₈)	—	2	2	no	—
44	C₄⋊(C₃⋊C₈)	—	2	2	no	—
45	C₄⋊(C₃⋊C₈)	D₄	1	2	yes	—
46	#431	C₂×D₃	1	2	yes	—
47	C₂×#108	—	6	1	no	—
48	C₂×#108	—	6	1	no	—
49	#423	—	3	2	no	—
50	C₂×C₄⋊C₈	—	3	1	no	—
51	C₂×#97	—	6	1	no	—
52	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
53	C₄×C₄⋊C₄	—	3	1	no	—
54	#336	—	6	1	no	—
55	#335	Dic₃	1	2	yes	complemented
56	#334	—	6	4	no	—
57	#334	Dic₃	1	2	yes	complemented
58	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₃×C₄	#41	1	1	yes	characteristic
61	C₂²×Dic₃	—	2	2	no	—
62	C₂×C₃×C₈	—	2	1	no	—
63	C₂×C₃×C₈	—	2	1	no	—
64	C₂×C₃×C₈	—	2	4	no	—
65	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
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	SD₁₆	1	2	yes	—
70	C₂×C₃×C₈	SD₁₆	1	2	yes	—
71	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
72	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
73	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
74	C₃×C₄²	#41	1	1	yes	characteristic
75	C₄⋊Dic₃	—	4	2	no	—
76	C₄×Dic₃	—	4	1	no	—
77	C₄×Dic₃	—	4	1	no	—
78	#208	—	4	2	no	—
79	#208	—	4	2	no	—
80	C₂×C₃⋊C₈	—	2	2	no	—
81	C₂×C₃⋊C₈	—	4	1	no	—
82	C₂×C₃⋊C₈	—	4	1	no	—
83	C₂×C₃⋊C₈	—	4	2	no	—
84	C₂×M₁₆	—	6	2	no	—
85	C₂²×C₈	—	6	1	no	—
86	C₂²×C₈	D₁₂	1	1	yes	characteristic
87	C₂²×C₈	D₁₂	1	1	yes	characteristic
88	C₂×C₄⋊C₄	—	6	1	no	—
89	C₂×C₄²	—	6	1	no	—
90	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
91	#108	—	12	2	no	—
92	#108	—	12	2	no	—
93	C₄⋊C₈	—	3	2	no	—
94	C₄⋊C₈	—	6	2	no	—
95	C₄⋊C₈	—	6	1	no	—
96	C₄⋊C₈	—	6	2	no	—
97	C₄⋊C₈	C₂×Dic₃	1	4	yes	—
98	C₂²⋊C₈	—	2	2	no	—
99	#97	—	12	2	no	—
100	C₄×C₈	—	2	2	no	—
101	C₄×C₈	Dic₆	1	1	yes	characteristic
102	C₄×C₈	Dic₆	1	1	yes	characteristic
103	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
104	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
105	#95	—	3	2	no	—
106	C₂³×C₃	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₃×C₄	—	2	2	no	—
108	C₂×C₃×C₄	—	4	1	no	—
109	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
111	C₂×C₃×C₄	C₄×Q₈	1	1	yes	characteristic
112	C₂×C₃×C₄	#121	1	1	yes	characteristic
113	C₂×C₃×C₄	#125	1	1	yes	characteristic
114	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
115	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
116	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₃×C₄	#137	1	1	yes	characteristic
119	C₂×Dic₃	—	4	2	no	—
120	C₂×Dic₃	—	4	2	no	—
121	C₂×Dic₃	—	4	2	no	—
122	C₃×C₈	—	2	1	no	—
123	C₃×C₈	—	2	1	no	—
124	C₃×C₈	—	4	4	no	—
125	C₃×C₈	#106	1	2	yes	—
126	C₃×C₈	#106	1	2	yes	—
127	C₃⋊C₈	—	4	2	no	—
128	C₃⋊C₈	—	8	2	no	—
129	C₂²×C₄	—	6	2	no	—
130	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
131	C₂²×C₄	#237	1	1	yes	characteristic
132	C₂²×C₄	D₃×Q₈	1	1	yes	characteristic
133	M₁₆	—	12	2	no	—
134	M₁₆	—	12	2	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	6	2	no	—
139	C₂×C₈	—	12	1	no	—
140	C₂×C₈	—	12	1	no	—
141	C₂×C₈	—	12	2	no	—
142	C₂×C₈	#204	1	2	yes	—
143	C₂×C₈	#204	1	2	yes	—
144	C₂×C₈	C₄⋊Dic₃	1	1	yes	characteristic
145	C₂×C₈	C₄⋊Dic₃	1	1	yes	characteristic
146	C₂×C₈	C₄⋊Dic₃	1	1	yes	characteristic
147	C₂×C₈	C₄⋊Dic₃	1	1	yes	characteristic
148	C₄⋊C₄	—	12	2	no	—
149	C₄²	—	12	1	no	—
150	C₄²	—	12	1	no	—
151	C₄²	C₂×Dic₆	1	1	yes	characteristic
152	C₄²	D₃×D₄	1	1	yes	characteristic
153	C₄²	#239	1	1	yes	characteristic
154	C₄²	C₂²×Dic₃	1	1	yes	characteristic
155	C₂²×C₃	—	2	1	no	—
156	C₂²×C₃	—	2	1	no	—
157	C₂²×C₃	#389	1	1	yes	characteristic
158	C₂²×C₃	#475	1	1	yes	characteristic
159	C₂²×C₃	#476	1	1	yes	characteristic
160	C₃×C₄	—	2	1	no	—
161	C₃×C₄	—	2	1	no	—
162	C₃×C₄	—	2	1	no	—
163	C₃×C₄	C₂×#106	1	1	yes	characteristic
164	C₃×C₄	#428	1	1	yes	characteristic
165	C₃×C₄	#444	1	2	yes	—
166	C₃×C₄	#460	1	1	yes	characteristic
167	C₃×C₄	#461	1	1	yes	characteristic
168	Dic₃	—	8	2	no	—
169	C₂³	#897	1	1	yes	characteristic
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	12	2	no	—
173	C₂×C₄	—	12	2	no	—
174	C₂×C₄	—	12	2	no	—
175	C₂×C₄	#888	1	1	yes	characteristic
176	C₂×C₄	#890	1	1	yes	characteristic
177	C₂×C₄	#895	1	1	yes	characteristic
178	C₂×C₄	C₂×#204	1	1	yes	characteristic
179	C₂×C₄	C₂×#204	1	1	yes	characteristic
180	C₂×C₄	#908	1	1	yes	characteristic
181	C₂×C₄	#909	1	1	yes	characteristic
182	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
183	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
184	C₂×C₄	Q₈×Dic₃	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	2	1	no	—
187	C₈	—	4	4	no	—
188	C₈	—	12	2	no	—
189	C₈	—	24	2	no	—
190	C₈	#817	1	2	yes	—
191	C₈	#817	1	2	yes	—
192	C₂×C₃	—	2	1	no	—
193	C₂×C₃	—	2	1	no	—
194	C₂×C₃	#1946	1	1	yes	characteristic
195	C₂×C₃	#1948	1	1	yes	characteristic
196	C₂×C₃	#1953	1	1	yes	characteristic
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	#4842	1	1	yes	characteristic
200	C₂²	#4851	1	1	yes	characteristic
201	C₂²	#4984	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	24	2	no	—
206	C₄	#4841	1	1	yes	characteristic
207	C₄	#4848	1	1	yes	characteristic
208	C₄	C₂×#817	1	1	yes	characteristic
209	C₄	#5121	1	1	yes	characteristic
210	C₄	#5197	1	2	yes	—
211	C₃	#9631	1	1	yes	characteristic, complemented, Sylow
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	#70454	1	1	yes	characteristic
215	C₂	#70663	1	1	yes	characteristic
216	C₂	#70707	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#895
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1564	C₂
socle	C₂²×C₃	#389
prime radical	C₂³×C₃	C₂²⋊Q₈
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#1564	D₃
hypocenter	C₃	#9631
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
15. C₃×#335	C₄	semidirect
16. C₃×#334	C₄	semidirect
55. #335	Dic₃	semidirect
57. #334	Dic₃	semidirect
211. C₃	#9631	semidirect

Maximal subgroups

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

subgroup	index
C₃×#1564	2
#70749	2
#69792	2
#69757	2
#67746	2
#9631	3