Subgroups — #10713321

Order 768.

306 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#1848	C₂	1	1	yes	maximal, characteristic
2	C₂×#4494	C₂	1	2	yes	maximal
3	C₄×#817	C₂	1	1	yes	maximal, characteristic
4	#69743	C₂	1	1	yes	maximal, characteristic
5	#67730	C₂	1	2	yes	maximal
6	#13185	—	3	1	no	maximal, Sylow
7	C₂×C₃×#108	—	2	1	no	—
8	C₂×C₃×#108	C₂²	1	2	yes	—
9	C₃×#423	—	2	1	no	—
10	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×#817	C₂²	1	1	yes	characteristic
12	C₂×#817	C₂²	1	1	yes	characteristic
13	C₂×#812	C₂²	1	2	yes	—
14	C₄×C₄⋊Dic₃	—	2	1	no	—
15	#4551	—	2	1	no	—
16	#4494	—	2	4	no	—
17	C₄⋊(C₃⋊C₁₆)	C₄	1	2	yes	complemented
18	C₄⋊(C₃⋊C₁₆)	C₄	1	2	yes	complemented
19	C₂×#365	—	3	2	no	—
20	#2150	—	3	1	no	—
21	#1848	D₃	1	1	yes	characteristic
22	C₄×#106	—	3	1	no	—
23	#1383	—	3	2	no	—
24	C₂×C₃×M₁₆	—	2	2	no	—
25	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
27	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
28	C₂×C₄⋊Dic₃	—	2	1	no	—
29	C₂×C₄⋊Dic₃	—	2	1	no	—
30	C₂×C₄×Dic₃	—	4	1	no	—
31	C₃×#108	—	2	2	no	—
32	C₃×#108	—	2	2	no	—
33	C₃×#108	—	4	2	no	—
34	C₃×C₄⋊C₈	—	4	1	no	—
35	C₃×C₄⋊C₈	—	4	1	no	—
36	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄×C₈	Q₈	1	1	yes	characteristic
39	C₃×C₄×C₈	Q₈	1	1	yes	characteristic
40	C₃⋊#95	—	4	1	no	—
41	#817	—	2	1	no	—
42	#817	—	2	1	no	—
43	#817	D₄	1	2	yes	—
44	#817	Q₈	1	2	yes	—
45	#812	—	2	2	no	—
46	#812	—	2	2	no	—
47	C₂×C₃⋊C₁₆	C₂×C₄	1	4	yes	—
48	C₂×M₃₂	—	3	2	no	—
49	C₂×#108	—	2	1	no	—
50	C₂×#108	C₂×D₃	1	2	yes	—
51	C₂×#106	—	3	1	no	—
52	C₂×#106	—	3	1	no	—
53	#423	—	2	1	no	—
54	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
55	C₄×C₄⋊C₄	—	6	1	no	—
56	#365	—	6	4	no	—
57	C₄⋊C₁₆	—	3	2	no	—
58	C₄⋊C₁₆	—	3	2	no	—
59	#340	—	6	1	no	—
60	C₂²×C₃×C₄	SD₁₆	1	1	yes	characteristic
61	C₂²×C₃×C₄	SD₁₆	1	1	yes	characteristic
62	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
63	C₂²×Dic₃	—	4	1	no	—
64	C₂²×Dic₃	—	4	1	no	—
65	C₃×M₁₆	—	4	2	no	—
66	C₃×M₁₆	—	4	2	no	—
67	C₂×C₃×C₈	—	2	1	no	—
68	C₂×C₃×C₈	—	2	1	no	—
69	C₂×C₃×C₈	—	4	2	no	—
70	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₃×C₈	#41	1	1	yes	characteristic
77	C₂×C₃×C₈	#41	1	1	yes	characteristic
78	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
79	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
80	C₃×C₄²	SD₁₆	1	1	yes	characteristic
81	C₃×C₄²	SD₁₆	1	1	yes	characteristic
82	C₄⋊Dic₃	—	2	2	no	—
83	C₄⋊Dic₃	—	2	2	no	—
84	C₄⋊Dic₃	—	4	1	no	—
85	C₄⋊Dic₃	—	4	1	no	—
86	C₄×Dic₃	—	8	2	no	—
87	C₃⋊C₁₆	—	2	4	no	—
88	C₂×M₁₆	—	2	2	no	—
89	C₂²×C₈	#67	1	1	yes	characteristic
90	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
91	C₂×C₄⋊C₄	—	6	1	no	—
92	C₂×C₄⋊C₄	—	6	1	no	—
93	C₂×C₄²	—	12	1	no	—
94	C₂×C₄²	#67	1	1	yes	characteristic
95	M₃₂	—	6	2	no	—
96	M₃₂	—	6	2	no	—
97	C₂×C₁₆	—	3	4	no	—
98	#108	—	2	2	no	—
99	#108	—	2	2	no	—
100	#108	—	4	2	no	—
101	#106	—	3	2	no	—
102	#106	—	3	2	no	—
103	#106	—	6	1	no	—
104	#106	—	6	1	no	—
105	C₄⋊C₈	—	4	1	no	—
106	C₄⋊C₈	—	4	1	no	—
107	C₄×C₈	Dic₆	1	1	yes	characteristic
108	C₄×C₈	Dic₆	1	1	yes	characteristic
109	C₄×C₈	C₄×D₃	1	1	yes	characteristic
110	C₄×C₈	C₄×D₃	1	1	yes	characteristic
111	#95	—	12	1	no	—
112	C₂³×C₃	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₃×C₄	—	2	2	no	—
114	C₂×C₃×C₄	—	4	1	no	—
115	C₂×C₃×C₄	#106	1	1	yes	characteristic
116	C₂×C₃×C₄	#106	1	1	yes	characteristic
117	C₂×C₃×C₄	#106	1	1	yes	characteristic
118	C₂×C₃×C₄	#106	1	1	yes	characteristic
119	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₃×C₄	#121	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₃×C₄	#137	1	1	yes	characteristic
125	C₂×Dic₃	—	4	2	no	—
126	C₂×Dic₃	—	4	2	no	—
127	C₂×Dic₃	—	8	1	no	—
128	C₂×Dic₃	—	8	1	no	—
129	C₂×Dic₃	—	8	1	no	—
130	C₂×Dic₃	—	8	1	no	—
131	C₃×C₈	—	2	1	no	—
132	C₃×C₈	—	2	1	no	—
133	C₃×C₈	—	8	2	no	—
134	C₃×C₈	C₄×D₄	1	1	yes	characteristic
135	C₃×C₈	C₄×Q₈	1	1	yes	characteristic
136	C₃×C₈	#125	1	1	yes	characteristic
137	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
138	C₂²×C₄	—	12	1	no	—
139	C₂²×C₄	—	12	1	no	—
140	C₂²×C₄	#215	1	1	yes	characteristic
141	C₂²×C₄	#215	1	1	yes	characteristic
142	C₂²×C₄	C₂×#67	1	1	yes	characteristic
143	M₁₆	—	4	2	no	—
144	M₁₆	—	4	2	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	4	2	no	—
148	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
149	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
150	C₂×C₈	C₂×Dic₆	1	1	yes	characteristic
151	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
152	C₂×C₈	D₃×D₄	1	1	yes	characteristic
153	C₂×C₈	#237	1	1	yes	characteristic
154	C₂×C₈	D₃×Q₈	1	1	yes	characteristic
155	C₂×C₈	#239	1	1	yes	characteristic
156	C₄⋊C₄	—	6	2	no	—
157	C₄⋊C₄	—	6	2	no	—
158	C₄⋊C₄	—	12	1	no	—
159	C₄⋊C₄	—	12	1	no	—
160	C₄²	—	24	2	no	—
161	C₄²	Dic₃⋊C₄	1	1	yes	characteristic
162	C₄²	Dic₃⋊C₄	1	1	yes	characteristic
163	C₄²	#214	1	1	yes	characteristic
164	C₄²	#214	1	1	yes	characteristic
165	C₁₆	—	6	4	no	—
166	C₂²×C₃	—	2	1	no	—
167	C₂²×C₃	—	2	1	no	—
168	C₂²×C₃	C₂×#106	1	1	yes	characteristic
169	C₂²×C₃	#461	1	1	yes	characteristic
170	C₂²×C₃	#476	1	1	yes	characteristic
171	C₃×C₄	—	2	1	no	—
172	C₃×C₄	—	2	1	no	—
173	C₃×C₄	—	2	1	no	—
174	C₃×C₄	#365	1	2	yes	complemented
175	C₃×C₄	#389	1	1	yes	characteristic
176	C₃×C₄	#428	1	1	yes	characteristic
177	C₃×C₄	#460	1	1	yes	characteristic
178	C₃×C₄	#475	1	1	yes	characteristic
179	Dic₃	—	8	2	no	—
180	Dic₃	—	8	2	no	—
181	C₂³	C₂×#215	1	1	yes	characteristic
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	—	12	2	no	—
185	C₂×C₄	—	12	2	no	—
186	C₂×C₄	—	24	1	no	—
187	C₂×C₄	—	24	1	no	—
188	C₂×C₄	—	24	1	no	—
189	C₂×C₄	—	24	1	no	—
190	C₂×C₄	#808	1	1	yes	characteristic
191	C₂×C₄	#808	1	1	yes	characteristic
192	C₂×C₄	#808	1	1	yes	characteristic
193	C₂×C₄	#808	1	1	yes	characteristic
194	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
195	C₂×C₄	#932	1	1	yes	characteristic
196	C₂×C₄	C₂×#214	1	1	yes	characteristic
197	C₂×C₄	#938	1	1	yes	characteristic
198	C₂×C₄	#942	1	1	yes	characteristic
199	C₂×C₄	#946	1	1	yes	characteristic
200	C₈	—	2	1	no	—
201	C₈	—	2	1	no	—
202	C₈	—	8	2	no	—
203	C₈	#887	1	1	yes	characteristic
204	C₈	#888	1	1	yes	characteristic
205	C₈	#890	1	1	yes	characteristic
206	C₈	#893	1	1	yes	characteristic
207	C₂×C₃	—	2	1	no	—
208	C₂×C₃	—	2	1	no	—
209	C₂×C₃	#1948	1	1	yes	characteristic
210	C₂×C₃	C₂×#365	1	1	yes	characteristic
211	C₂×C₃	#2155	1	1	yes	characteristic
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	C₂×#808	1	1	yes	characteristic
215	C₂²	#5083	1	1	yes	characteristic
216	C₂²	#5084	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	24	2	no	—
221	C₄	—	24	2	no	—
222	C₄	#4494	1	2	yes	complemented
223	C₄	#4964	1	1	yes	characteristic
224	C₄	#4972	1	1	yes	characteristic
225	C₄	#5075	1	1	yes	characteristic
226	C₄	#5094	1	1	yes	characteristic
227	C₃	#13185	1	1	yes	characteristic, complemented, Sylow
228	C₂	—	2	1	no	—
229	C₂	—	2	1	no	—
230	C₂	#69568	1	1	yes	characteristic
231	C₂	C₂×#4494	1	1	yes	characteristic
232	C₂	#70685	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#808
commutator	C₂×C₃×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₃×#1848	C₂
socle	C₂²×C₃	C₂×#106
prime radical	C₂³×C₃	C₂×SD₁₆
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	#1848	D₃
hypocenter	C₃	#13185
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
17. C₄⋊(C₃⋊C₁₆)	C₄	semidirect
18. C₄⋊(C₃⋊C₁₆)	C₄	semidirect
174. C₃×C₄	#365	semidirect
222. C₄	#4494	semidirect
227. C₃	#13185	semidirect

Maximal subgroups

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

subgroup	index
C₃×#1848	2
C₂×#4494	2
C₄×#817	2
#69743	2
#67730	2
#13185	3