Subgroups — #10694566

Order 768.

502 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3147	C₂	1	1	yes	maximal, characteristic
2	#71814	C₂	1	4	yes	maximal, complemented
3	#69766	C₂	1	1	yes	maximal, characteristic, complemented
4	#69724	C₂	1	1	yes	maximal, characteristic, complemented
5	#9756	—	3	1	no	maximal, Sylow
6	C₂×C₃×#135	—	2	1	no	—
7	C₂×C₃×#135	—	2	1	no	—
8	C₃×#532	C₂²	1	2	yes	—
9	C₃×#495	—	2	1	no	—
10	C₃×#495	—	2	1	no	—
11	C₃×#494	C₄	1	1	yes	characteristic, complemented
12	C₃×#493	C₄	1	1	yes	characteristic, complemented
13	C₃×#492	C₄	1	2	yes	complemented
14	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×#818	—	2	1	no	—
16	C₂×#817	—	2	1	no	—
17	#4936	C₂²	1	2	yes	—
18	#4925	C₂²	1	2	yes	—
19	#4537	—	2	8	no	—
20	#4455	—	2	1	no	—
21	C₈×C₃⋊C₈	—	2	1	no	—
22	#3147	D₃	1	1	yes	characteristic
23	#1846	—	3	1	no	—
24	#1517	—	3	4	no	—
25	#1450	—	3	1	no	—
26	C₂×C₃×#41	—	2	2	no	—
27	C₃×#135	—	4	2	no	—
28	C₃×#135	—	4	2	no	—
29	C₃×#135	—	4	2	no	—
30	C₃×#135	—	4	2	no	—
31	C₂×C₃×Q₁₆	—	2	1	no	—
32	C₂×C₃×Q₁₆	—	2	1	no	—
33	C₂×C₃×SD₁₆	—	2	2	no	—
34	C₂×C₃×SD₁₆	—	2	2	no	—
35	C₂×C₃×D₈	—	2	1	no	—
36	C₂×C₃×D₈	—	2	1	no	—
37	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
38	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
39	C₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
40	C₃×#127	C₂×C₄	1	2	yes	—
41	C₃×#124	—	2	2	no	—
42	C₃×#121	—	2	4	no	—
43	C₃×C₄×D₄	—	2	4	no	—
44	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
45	C₂×C₄⋊Dic₃	—	2	2	no	—
46	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
47	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₃×C₄×C₈	D₄	1	1	yes	characteristic
49	C₃×C₄×C₈	D₄	1	1	yes	characteristic
50	C₃⋊#95	—	2	4	no	—
51	#830	D₄	1	4	yes	—
52	#818	—	4	2	no	—
53	#817	—	4	2	no	—
54	C₄⋊(C₃⋊C₈)	—	2	2	no	—
55	C₄×C₃⋊C₈	—	2	2	no	—
56	C₂×#135	—	2	1	no	—
57	C₂×#135	—	2	1	no	—
58	#532	C₂×D₃	1	2	yes	—
59	#495	—	2	1	no	—
60	#495	—	2	1	no	—
61	#494	Dic₃	1	1	yes	characteristic, complemented
62	#493	Dic₃	1	1	yes	characteristic, complemented
63	#492	Dic₃	1	2	yes	complemented
64	#431	—	3	2	no	—
65	C₂×#107	—	6	1	no	—
66	C₂×#106	—	6	1	no	—
67	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
68	#382	—	3	2	no	—
69	#327	—	6	8	no	—
70	#322	—	6	1	no	—
71	C₈²	—	6	1	no	—
72	C₃×#41	—	4	2	no	—
73	C₃×#41	—	4	2	no	—
74	C₃×#41	—	4	2	no	—
75	C₃×#41	—	4	2	no	—
76	C₂×C₃×Q₈	—	2	2	no	—
77	C₂×C₃×D₄	—	2	2	no	—
78	C₂×C₃×D₄	—	2	4	no	—
79	C₂×C₃×D₄	—	4	2	no	—
80	C₂²×C₃×C₄	—	4	2	no	—
81	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
82	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂²×Dic₃	—	4	2	no	—
84	C₃×Q₁₆	—	4	2	no	—
85	C₃×Q₁₆	—	4	2	no	—
86	C₃×SD₁₆	—	4	4	no	—
87	C₃×SD₁₆	—	4	4	no	—
88	C₃×D₈	—	4	2	no	—
89	C₃×D₈	—	4	2	no	—
90	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₃×C₈	D₈	1	2	yes	—
95	C₂×C₃×C₈	D₈	1	2	yes	—
96	C₂×C₃×C₈	SD₁₆	1	2	yes	—
97	C₂×C₃×C₈	SD₁₆	1	2	yes	—
98	C₃×C₄⋊C₄	—	2	4	no	—
99	C₃×C₂²⋊C₄	—	4	4	no	—
100	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
101	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
102	C₃×C₄²	#41	1	2	yes	—
103	C₄⋊Dic₃	—	4	2	no	—
104	C₄⋊Dic₃	—	4	2	no	—
105	C₂×C₃⋊C₈	—	2	4	no	—
106	C₂×#41	—	2	2	no	—
107	#135	—	4	2	no	—
108	#135	—	4	2	no	—
109	#135	—	4	2	no	—
110	#135	—	4	2	no	—
111	C₂×Q₁₆	—	2	1	no	—
112	C₂×Q₁₆	—	2	1	no	—
113	C₂×SD₁₆	—	2	2	no	—
114	C₂×SD₁₆	—	2	2	no	—
115	C₂×D₈	—	2	1	no	—
116	C₂×D₈	—	2	1	no	—
117	C₂²×C₈	#67	1	1	yes	characteristic
118	C₂²×C₈	#67	1	1	yes	characteristic
119	C₄⋊Q₈	C₂×Dic₃	1	2	yes	—
120	#127	C₂×Dic₃	1	2	yes	—
121	#124	—	2	2	no	—
122	#121	—	2	4	no	—
123	C₄×D₄	—	2	4	no	—
124	C₂×C₄⋊C₄	—	6	2	no	—
125	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
126	#107	—	12	2	no	—
127	#106	—	12	2	no	—
128	C₄⋊C₈	—	6	2	no	—
129	C₂²⋊C₈	—	3	4	no	—
130	C₄×C₈	—	6	2	no	—
131	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
132	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
133	C₄×C₈	#67	1	1	yes	characteristic
134	C₄×C₈	#67	1	1	yes	characteristic
135	#95	—	6	4	no	—
136	C₂³×C₃	—	4	2	no	—
137	C₂³×C₃	#127	1	1	yes	characteristic
138	C₃×Q₈	—	4	2	no	—
139	C₃×Q₈	—	4	2	no	—
140	C₃×D₄	—	4	2	no	—
141	C₃×D₄	—	4	2	no	—
142	C₃×D₄	—	8	4	no	—
143	C₃×D₄	—	8	2	no	—
144	C₃×D₄	—	8	2	no	—
145	C₂×C₃×C₄	—	4	2	no	—
146	C₂×C₃×C₄	—	4	2	no	—
147	C₂×C₃×C₄	—	8	2	no	—
148	C₂×C₃×C₄	—	8	2	no	—
149	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
151	C₂×C₃×C₄	#121	1	2	yes	—
152	C₂×C₃×C₄	#124	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
157	C₂×Dic₃	—	8	4	no	—
158	C₂×Dic₃	—	8	2	no	—
159	C₃×C₈	#102	1	2	yes	—
160	C₃×C₈	#102	1	2	yes	—
161	C₃×C₈	#102	1	2	yes	—
162	C₃×C₈	#102	1	2	yes	—
163	C₃⋊C₈	—	4	4	no	—
164	#41	—	4	2	no	—
165	#41	—	4	2	no	—
166	#41	—	4	2	no	—
167	#41	—	4	2	no	—
168	C₂×Q₈	—	2	2	no	—
169	C₂×D₄	—	2	4	no	—
170	C₂×D₄	—	2	2	no	—
171	C₂×D₄	—	4	2	no	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	—	12	2	no	—
174	C₂²×C₄	D₃×D₄	1	2	yes	—
175	C₂²×C₄	C₂×#67	1	1	yes	characteristic
176	Q₁₆	—	4	2	no	—
177	Q₁₆	—	4	2	no	—
178	SD₁₆	—	4	4	no	—
179	SD₁₆	—	4	4	no	—
180	D₈	—	4	2	no	—
181	D₈	—	4	2	no	—
182	C₂×C₈	—	6	4	no	—
183	C₂×C₈	#213	1	2	yes	—
184	C₂×C₈	#213	1	2	yes	—
185	C₂×C₈	#214	1	2	yes	—
186	C₂×C₈	#214	1	2	yes	—
187	C₂×C₈	#217	1	1	yes	characteristic
188	C₂×C₈	#217	1	1	yes	characteristic
189	C₂×C₈	#217	1	1	yes	characteristic
190	C₂×C₈	#217	1	1	yes	characteristic
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	12	2	no	—
193	C₄⋊C₄	—	12	2	no	—
194	C₂²⋊C₄	—	4	4	no	—
195	C₄²	#237	1	2	yes	—
196	C₄²	C₂²×Dic₃	1	1	yes	characteristic
197	C₄²	C₂×#67	1	1	yes	characteristic
198	C₂²×C₃	—	2	2	no	—
199	C₂²×C₃	—	8	2	no	—
200	C₂²×C₃	—	8	4	no	—
201	C₂²×C₃	#390	1	1	yes	characteristic
202	C₂²×C₃	#492	1	1	yes	characteristic
203	C₂²×C₃	#493	1	1	yes	characteristic
204	C₃×C₄	—	2	2	no	—
205	C₃×C₄	—	2	2	no	—
206	C₃×C₄	—	8	2	no	—
207	C₃×C₄	C₂×#102	1	1	yes	characteristic
208	C₃×C₄	C₂×#102	1	1	yes	characteristic
209	C₃×C₄	#486	1	1	yes	characteristic
210	C₃×C₄	#486	1	1	yes	characteristic
211	Dic₃	—	16	2	no	—
212	C₂³	—	4	2	no	—
213	C₂³	#940	1	1	yes	characteristic
214	Q₈	—	4	2	no	—
215	Q₈	—	4	2	no	—
216	D₄	—	4	2	no	—
217	D₄	—	4	2	no	—
218	D₄	—	8	4	no	—
219	D₄	—	8	2	no	—
220	D₄	—	8	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	8	2	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	—	24	4	no	—
226	C₂×C₄	—	24	2	no	—
227	C₂×C₄	C₂×#213	1	1	yes	characteristic
228	C₂×C₄	C₂×#213	1	1	yes	characteristic
229	C₂×C₄	C₂×#214	1	1	yes	characteristic
230	C₂×C₄	C₂×#214	1	1	yes	characteristic
231	C₂×C₄	D₄×Dic₃	1	2	yes	—
232	C₂×C₄	#936	1	1	yes	characteristic
233	C₂×C₄	#938	1	2	yes	—
234	C₂×C₄	C₂×#217	1	1	yes	characteristic
235	C₈	—	12	4	no	—
236	C₈	#832	1	2	yes	—
237	C₈	#832	1	2	yes	—
238	C₈	#832	1	2	yes	—
239	C₈	#832	1	2	yes	—
240	C₂×C₃	—	2	1	no	—
241	C₂×C₃	—	2	1	no	—
242	C₂×C₃	—	16	2	no	—
243	C₂×C₃	#1957	1	1	yes	characteristic
244	C₂×C₃	#1976	1	2	yes	—
245	C₂²	—	2	2	no	—
246	C₂²	—	8	2	no	—
247	C₂²	—	8	4	no	—
248	C₂²	#5073	1	1	yes	characteristic
249	C₂²	#5076	1	1	yes	characteristic
250	C₂²	#5221	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	2	2	no	—
253	C₄	—	8	2	no	—
254	C₄	—	48	2	no	—
255	C₄	#5070	1	1	yes	characteristic
256	C₄	#5070	1	1	yes	characteristic
257	C₄	C₂×#832	1	1	yes	characteristic
258	C₄	C₂×#832	1	1	yes	characteristic
259	C₃	#9756	1	1	yes	characteristic, complemented, Sylow
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	16	2	no	—
263	C₂	#71764	1	1	yes	characteristic
264	C₂	#72086	1	2	yes	—

	subgroup	G/H
center	C₂²	#5221
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting, prime radical	C₃×#3147	C₂
socle	C₂²×C₃	#390
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#3147	D₃
hypocenter	C₃	#9756
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
2. #71814	C₂	semidirect
3. #69766	C₂	semidirect
4. #69724	C₂	semidirect
11. C₃×#494	C₄	semidirect
12. C₃×#493	C₄	semidirect
13. C₃×#492	C₄	semidirect
61. #494	Dic₃	semidirect
62. #493	Dic₃	semidirect
63. #492	Dic₃	semidirect
259. C₃	#9756	semidirect

Maximal subgroups

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

subgroup	index
C₃×#3147	2
#71814	2
#69766	2
#69724	2
#9756	3