Subgroups — #10695327

Order 768.

430 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2859	C₂	1	1	yes	maximal, characteristic
2	#69392	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#67634	C₂	1	2	yes	maximal
5	#67634	C₂	1	2	yes	maximal
6	#9901	—	3	1	no	maximal, Sylow
7	C₂×C₃×#125	C₄	1	2	yes	—
8	C₃×#389	—	2	1	no	—
9	C₃×#389	—	2	1	no	—
10	C₃×#389	—	2	1	no	—
11	C₃×#389	—	2	1	no	—
12	C₃×#384	C₂²	1	1	yes	characteristic
13	C₃×#384	C₂²	1	1	yes	characteristic
14	C₃×#382	C₄	1	2	yes	—
15	C₃×C₄³	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
17	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
18	C₂×C₄×C₃⋊C₈	—	2	1	no	—
19	#4524	—	2	1	no	—
20	#4524	—	2	1	no	—
21	#4524	—	2	1	no	—
22	#2859	D₃	1	1	yes	characteristic
23	#1831	—	3	1	no	—
24	C₄×C₄⋊C₈	—	3	1	no	—
25	#1296	—	3	2	no	—
26	#1296	—	3	2	no	—
27	C₃×#125	—	4	4	no	—
28	C₂×C₃×C₄⋊C₄	—	2	2	no	—
29	C₂×C₃×C₄⋊C₄	—	2	2	no	—
30	C₂×C₃×C₄⋊C₄	—	2	1	no	—
31	C₂×C₃×C₄⋊C₄	—	2	1	no	—
32	C₂×C₃×C₄⋊C₄	—	2	1	no	—
33	C₂×C₃×C₄⋊C₄	—	2	1	no	—
34	C₂×C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
35	C₂×C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
36	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
39	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
40	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
41	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
42	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
43	C₂²×C₃⋊C₈	—	2	2	no	—
44	C₂²×C₃⋊C₈	—	2	2	no	—
45	C₃×#95	—	2	2	no	—
46	C₃×#95	—	2	2	no	—
47	C₄⋊(C₃⋊C₈)	—	2	2	no	—
48	C₄⋊(C₃⋊C₈)	—	2	4	no	—
49	C₄⋊(C₃⋊C₈)	Q₈	1	4	yes	—
50	C₄×C₃⋊C₈	—	4	2	no	—
51	C₂×#125	Dic₃	1	2	yes	—
52	C₂×C₄⋊C₈	—	3	2	no	—
53	C₂×C₄⋊C₈	—	3	2	no	—
54	C₂×C₄×C₈	—	6	1	no	—
55	#389	—	2	1	no	—
56	#389	—	2	1	no	—
57	#389	—	2	1	no	—
58	#389	—	2	1	no	—
59	#384	C₂×D₃	1	1	yes	characteristic
60	#384	C₂×D₃	1	1	yes	characteristic
61	#382	Dic₃	1	2	yes	—
62	C₄³	C₂×D₃	1	1	yes	characteristic
63	#336	—	6	1	no	—
64	#336	—	6	1	no	—
65	#336	—	6	1	no	—
66	C₂²×C₃×C₄	—	2	2	no	—
67	C₂²×C₃×C₄	—	2	2	no	—
68	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
75	C₃×C₄⋊C₄	—	4	2	no	—
76	C₃×C₄⋊C₄	—	4	2	no	—
77	C₃×C₄⋊C₄	—	4	4	no	—
78	C₃×C₄⋊C₄	—	4	4	no	—
79	C₃×C₄⋊C₄	—	4	2	no	—
80	C₃×C₄⋊C₄	—	4	2	no	—
81	C₃×C₄⋊C₄	—	4	2	no	—
82	C₃×C₄⋊C₄	—	4	1	no	—
83	C₃×C₄⋊C₄	—	4	1	no	—
84	C₃×C₄²	—	2	2	no	—
85	C₃×C₄²	—	2	2	no	—
86	C₃×C₄²	—	2	2	no	—
87	C₃×C₄²	—	2	2	no	—
88	C₃×C₄²	—	2	2	no	—
89	C₃×C₄²	—	2	1	no	—
90	C₃×C₄²	—	2	1	no	—
91	C₃×C₄²	C₂×Q₈	1	2	yes	—
92	C₃×C₄²	#41	1	2	yes	—
93	C₂×C₃⋊C₈	—	2	4	no	—
94	C₂×C₃⋊C₈	—	4	2	no	—
95	C₂×C₃⋊C₈	—	4	2	no	—
96	C₂×C₃⋊C₈	—	4	2	no	—
97	C₂×C₃⋊C₈	—	4	2	no	—
98	C₂×C₃⋊C₈	—	4	2	no	—
99	C₂²×C₈	—	6	2	no	—
100	C₂²×C₈	—	6	2	no	—
101	#125	—	4	4	no	—
102	C₂×C₄⋊C₄	—	2	2	no	—
103	C₂×C₄⋊C₄	—	2	2	no	—
104	C₂×C₄⋊C₄	—	2	1	no	—
105	C₂×C₄⋊C₄	—	2	1	no	—
106	C₂×C₄⋊C₄	—	2	1	no	—
107	C₂×C₄⋊C₄	—	2	1	no	—
108	C₂×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
109	C₂×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
110	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
111	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
112	C₂×C₄²	#67	1	1	yes	characteristic
113	C₂×C₄²	#67	1	1	yes	characteristic
114	C₂×C₄²	#67	1	1	yes	characteristic
115	C₂×C₄²	#67	1	1	yes	characteristic
116	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
117	C₄⋊C₈	—	3	4	no	—
118	C₄⋊C₈	—	6	2	no	—
119	C₄⋊C₈	—	6	4	no	—
120	C₄×C₈	—	12	2	no	—
121	#95	—	2	2	no	—
122	#95	—	2	2	no	—
123	C₂³×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	4	2	no	—
137	C₂×C₃×C₄	—	4	2	no	—
138	C₂×C₃×C₄	—	4	2	no	—
139	C₂×C₃×C₄	—	4	2	no	—
140	C₂×C₃×C₄	—	4	2	no	—
141	C₂×C₃×C₄	—	4	2	no	—
142	C₂×C₃×C₄	#100	1	2	yes	—
143	C₂×C₃×C₄	#101	1	2	yes	—
144	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
145	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
146	C₂×C₃×C₄	#124	1	1	yes	characteristic
147	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
148	C₂×C₃×C₄	#135	1	2	yes	—
149	C₂×C₃×C₄	#135	1	2	yes	—
150	C₂×C₃×C₄	#135	1	2	yes	—
151	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×C₃×C₄	#137	1	1	yes	characteristic
153	C₃⋊C₈	—	4	4	no	—
154	C₃⋊C₈	—	8	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	#217	1	1	yes	characteristic
158	C₂²×C₄	#217	1	1	yes	characteristic
159	C₂²×C₄	#217	1	1	yes	characteristic
160	C₂²×C₄	#217	1	1	yes	characteristic
161	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
162	C₂²×C₄	C₂×#67	1	1	yes	characteristic
163	C₂²×C₄	C₂×#67	1	1	yes	characteristic
164	C₂×C₈	—	6	4	no	—
165	C₂×C₈	—	12	2	no	—
166	C₂×C₈	—	12	2	no	—
167	C₂×C₈	—	12	2	no	—
168	C₂×C₈	—	12	2	no	—
169	C₂×C₈	—	12	2	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	4	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄⋊C₄	—	4	1	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	1	no	—
179	C₄²	—	2	2	no	—
180	C₄²	—	2	2	no	—
181	C₄²	—	2	1	no	—
182	C₄²	—	2	2	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	1	no	—
186	C₄²	D₃×Q₈	1	2	yes	—
187	C₄²	#239	1	2	yes	—
188	C₂²×C₃	#391	1	1	yes	characteristic
189	C₂²×C₃	C₂×#100	1	1	yes	characteristic
190	C₂²×C₃	C₂×#101	1	1	yes	characteristic
191	C₂²×C₃	#416	1	1	yes	characteristic
192	C₂²×C₃	#416	1	1	yes	characteristic
193	C₂²×C₃	#416	1	1	yes	characteristic
194	C₂²×C₃	#417	1	1	yes	characteristic
195	C₃×C₄	—	2	2	no	—
196	C₃×C₄	—	2	2	no	—
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	—	4	1	no	—
199	C₃×C₄	—	4	1	no	—
200	C₃×C₄	—	8	2	no	—
201	C₃×C₄	—	8	2	no	—
202	C₃×C₄	#478	1	2	yes	—
203	C₃×C₄	#479	1	2	yes	—
204	C₃×C₄	#488	1	2	yes	—
205	C₃×C₄	#499	1	2	yes	—
206	C₂³	C₂×#217	1	1	yes	characteristic
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	#833	1	2	yes	—
226	C₂×C₄	#836	1	2	yes	—
227	C₂×C₄	#944	1	1	yes	characteristic
228	C₂×C₄	Q₈×Dic₃	1	2	yes	—
229	C₂×C₄	#946	1	2	yes	—
230	C₂×C₄	#947	1	1	yes	characteristic
231	C₂×C₄	#949	1	1	yes	characteristic
232	C₂×C₄	#950	1	2	yes	—
233	C₂×C₄	#950	1	2	yes	—
234	C₂×C₄	#950	1	2	yes	—
235	C₂×C₄	#951	1	1	yes	characteristic
236	C₈	—	12	4	no	—
237	C₈	—	24	2	no	—
238	C₂×C₃	#1543	1	1	yes	characteristic
239	C₂×C₃	#1544	1	1	yes	characteristic
240	C₂×C₃	#1551	1	1	yes	characteristic
241	C₂×C₃	#1552	1	1	yes	characteristic
242	C₂×C₃	#1989	1	1	yes	characteristic
243	C₂×C₃	#1991	1	1	yes	characteristic
244	C₂×C₃	#1997	1	1	yes	characteristic
245	C₂²	C₂×#833	1	1	yes	characteristic
246	C₂²	C₂×#836	1	1	yes	characteristic
247	C₂²	#5229	1	1	yes	characteristic
248	C₂²	#5232	1	1	yes	characteristic
249	C₂²	#5233	1	1	yes	characteristic
250	C₂²	#5233	1	1	yes	characteristic
251	C₂²	#5233	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	4	1	no	—
256	C₄	—	4	1	no	—
257	C₄	—	8	2	no	—
258	C₄	—	8	2	no	—
259	C₄	#5063	1	2	yes	—
260	C₄	#5065	1	2	yes	—
261	C₄	#5066	1	2	yes	—
262	C₄	#5068	1	2	yes	—
263	C₃	#9901	1	1	yes	characteristic, complemented, Sylow
264	C₂	#71738	1	1	yes	characteristic
265	C₂	#71740	1	1	yes	characteristic
266	C₂	#71790	1	1	yes	characteristic
267	C₂	#71805	1	1	yes	characteristic
268	C₂	#71807	1	1	yes	characteristic
269	C₂	#71829	1	1	yes	characteristic
270	C₂	#71831	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#217
commutator	C₂²×C₃×C₄	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#2859	C₂
socle, prime radical	C₂³×C₃	C₂×C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#2859	D₃
hypocenter	C₃	#9901
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

Complements

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

N	H ≅ G/N	type
263. C₃	#9901	semidirect

Maximal subgroups

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

subgroup	index
C₃×#2859	2
#69392	2
C₄×C₄⋊(C₃⋊C₈)	2
#67634	2
#9901	3