Subgroups — #10713602

Order 768.

410 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	C₂×C₃×#508	C₂	1	1	yes	maximal, characteristic
2	#72086	C₂	1	2	yes	maximal, complemented
3	#72076	C₂	1	2	yes	maximal, complemented
4	#71404	C₂	1	1	yes	maximal, characteristic, complemented
5	#71399	C₂	1	1	yes	maximal, characteristic, complemented
6	#13243	—	3	1	no	maximal, Sylow
7	C₂×C₃×#135	C₂²	1	2	yes	—
8	C₃×#508	—	2	1	no	—
9	C₃×#508	—	2	1	no	—
10	C₃×#508	—	2	1	no	—
11	C₃×#508	—	2	1	no	—
12	C₂×C₃×Q₃₂	C₄	1	1	yes	characteristic, complemented
13	C₂×C₃×SD₃₂	C₄	1	2	yes	complemented
14	C₂×C₃×D₁₆	C₄	1	1	yes	characteristic, complemented
15	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
16	#5240	—	2	2	no	—
17	#5236	—	2	2	no	—
18	#5106	C₂²	1	2	yes	—
19	#5101	C₂²	1	1	yes	characteristic
20	C₂×#812	C₂²	1	1	yes	characteristic
21	#4566	—	2	4	no	—
22	#4505	—	2	1	no	—
23	#4504	—	2	1	no	—
24	#4502	—	2	1	no	—
25	Dic₃×C₁₆	—	2	1	no	—
26	C₂×#508	D₃	1	1	yes	characteristic
27	#2158	—	3	1	no	—
28	#2145	—	3	2	no	—
29	#2108	—	3	1	no	—
30	#1976	—	3	2	no	—
31	C₂×C₃×#41	—	2	2	no	—
32	C₃×#135	—	2	2	no	—
33	C₃×#135	—	2	2	no	—
34	C₃×#135	—	2	2	no	—
35	C₃×#135	—	2	2	no	—
36	C₂×C₃×Q₁₆	C₂×C₄	1	2	yes	—
37	C₂×C₃×SD₁₆	—	2	2	no	—
38	C₂×C₃×D₈	C₂×C₄	1	2	yes	—
39	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
40	#926	—	2	2	no	—
41	#926	D₄	1	1	yes	characteristic
42	C₂×#208	D₄	1	1	yes	characteristic
43	C₃×Q₃₂	—	2	1	no	—
44	C₃×Q₃₂	—	2	1	no	—
45	C₃×SD₃₂	—	2	2	no	—
46	C₃×SD₃₂	—	2	2	no	—
47	C₃×D₁₆	—	2	1	no	—
48	C₃×D₁₆	—	2	1	no	—
49	C₂×C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
50	C₂×C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
51	C₂×C₃×C₁₆	D₄	1	1	yes	characteristic
52	C₂×C₃×C₁₆	D₄	1	1	yes	characteristic
53	C₂×C₃×C₁₆	D₄	1	1	yes	characteristic
54	C₂×C₃×C₁₆	D₄	1	1	yes	characteristic
55	#837	—	4	4	no	—
56	#834	—	4	4	no	—
57	#818	—	2	2	no	—
58	#817	—	2	2	no	—
59	#815	—	2	1	no	—
60	C₈×Dic₃	—	2	1	no	—
61	#812	—	2	1	no	—
62	#812	D₄	1	2	yes	—
63	C₂×C₃⋊C₁₆	—	2	1	no	—
64	C₂×#135	C₂×D₃	1	2	yes	—
65	#508	—	2	1	no	—
66	#508	—	2	1	no	—
67	#508	—	2	1	no	—
68	#508	—	2	1	no	—
69	C₂×Q₃₂	Dic₃	1	1	yes	characteristic, complemented
70	C₂×SD₃₂	Dic₃	1	2	yes	complemented
71	C₂×D₁₆	Dic₃	1	1	yes	characteristic, complemented
72	C₂×M₃₂	—	3	1	no	—
73	C₂²×C₁₆	C₂×D₃	1	1	yes	characteristic
74	#427	—	3	2	no	—
75	#421	—	6	2	no	—
76	#410	—	6	2	no	—
77	#405	—	3	1	no	—
78	#367	—	6	1	no	—
79	#366	—	6	1	no	—
80	#360	—	6	4	no	—
81	#346	—	6	1	no	—
82	C₄×C₁₆	—	6	1	no	—
83	C₃×#41	—	4	2	no	—
84	C₃×#41	—	4	2	no	—
85	C₃×#41	—	4	2	no	—
86	C₃×#41	—	4	2	no	—
87	C₂×C₃×Q₈	—	2	2	no	—
88	C₂×C₃×D₄	—	2	2	no	—
89	C₂×C₃×D₄	—	4	2	no	—
90	C₂²×C₃×C₄	—	4	2	no	—
91	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
92	C₃×Q₁₆	—	2	2	no	—
93	C₃×Q₁₆	—	2	2	no	—
94	C₃×SD₁₆	—	4	2	no	—
95	C₃×SD₁₆	—	4	2	no	—
96	C₃×D₈	—	2	2	no	—
97	C₃×D₈	—	2	2	no	—
98	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₃×C₈	#41	1	1	yes	characteristic
103	C₂×C₃×C₈	#41	1	1	yes	characteristic
104	#217	—	4	2	no	—
105	#217	D₈	1	2	yes	—
106	C₄⋊Dic₃	—	2	1	no	—
107	C₄⋊Dic₃	—	4	2	no	—
108	C₄×Dic₃	—	2	1	no	—
109	C₄×Dic₃	—	4	2	no	—
110	#208	—	2	1	no	—
111	#208	D₈	1	2	yes	—
112	C₂×C₃⋊C₈	—	2	1	no	—
113	C₃×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
114	C₃×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
115	C₃×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
116	C₃×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
117	C₃⋊C₁₆	—	2	2	no	—
118	C₂×#41	—	2	2	no	—
119	#135	—	2	2	no	—
120	#135	—	2	2	no	—
121	#135	—	2	2	no	—
122	#135	—	2	2	no	—
123	C₂×Q₁₆	C₂×Dic₃	1	2	yes	—
124	C₂×SD₁₆	—	2	2	no	—
125	C₂×D₈	C₂×Dic₃	1	2	yes	—
126	C₂×M₁₆	—	3	1	no	—
127	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
128	#117	—	3	1	no	—
129	#117	—	6	2	no	—
130	Q₃₂	—	2	1	no	—
131	Q₃₂	—	2	1	no	—
132	SD₃₂	—	2	2	no	—
133	SD₃₂	—	2	2	no	—
134	D₁₆	—	2	1	no	—
135	D₁₆	—	2	1	no	—
136	M₃₂	—	3	2	no	—
137	M₃₂	—	6	1	no	—
138	C₂×C₁₆	—	6	1	no	—
139	C₂×C₁₆	C₂×Dic₃	1	1	yes	characteristic
140	C₂×C₁₆	C₂×Dic₃	1	1	yes	characteristic
141	C₂×C₁₆	#67	1	1	yes	characteristic
142	C₂×C₁₆	#67	1	1	yes	characteristic
143	C₂×C₁₆	#67	1	1	yes	characteristic
144	C₂×C₁₆	#67	1	1	yes	characteristic
145	#107	—	6	2	no	—
146	#106	—	6	2	no	—
147	#104	—	12	4	no	—
148	#99	—	12	4	no	—
149	#97	—	6	1	no	—
150	C₄×C₈	—	6	1	no	—
151	C₂³×C₃	—	4	2	no	—
152	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
153	C₃×Q₈	—	4	2	no	—
154	C₃×Q₈	—	4	2	no	—
155	C₃×D₄	—	4	2	no	—
156	C₃×D₄	—	4	2	no	—
157	C₃×D₄	—	8	2	no	—
158	C₃×D₄	—	8	2	no	—
159	C₂×C₃×C₄	—	4	2	no	—
160	C₂×C₃×C₄	—	8	2	no	—
161	C₂×C₃×C₄	—	8	2	no	—
162	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
163	C₂×C₃×C₄	#121	1	1	yes	characteristic
164	C₂×C₃×C₄	#127	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
166	C₂×C₃×C₄	#135	1	1	yes	characteristic
167	C₂×C₃×C₄	#135	1	1	yes	characteristic
168	C₂×Dic₃	—	2	2	no	—
169	C₂×Dic₃	—	8	2	no	—
170	C₃×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
171	C₃×C₈	C₄×D₄	1	1	yes	characteristic
172	C₃×C₈	#121	1	1	yes	characteristic
173	C₃×C₈	#124	1	1	yes	characteristic
174	C₃⋊C₈	—	2	2	no	—
175	#41	—	4	2	no	—
176	#41	—	4	2	no	—
177	#41	—	4	2	no	—
178	#41	—	4	2	no	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	—	4	2	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
184	Q₁₆	—	2	2	no	—
185	Q₁₆	—	2	2	no	—
186	SD₁₆	—	4	2	no	—
187	SD₁₆	—	4	2	no	—
188	D₈	—	2	2	no	—
189	D₈	—	2	2	no	—
190	M₁₆	—	3	2	no	—
191	M₁₆	—	6	1	no	—
192	C₂×C₈	—	6	1	no	—
193	C₂×C₈	D₃×D₄	1	1	yes	characteristic
194	C₂×C₈	#237	1	1	yes	characteristic
195	C₂×C₈	#237	1	1	yes	characteristic
196	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
197	C₂×C₈	C₂×#67	1	1	yes	characteristic
198	C₂×C₈	C₂×#67	1	1	yes	characteristic
199	C₄⋊C₄	—	6	1	no	—
200	C₄⋊C₄	—	12	2	no	—
201	C₂²⋊C₄	—	3	2	no	—
202	C₂²⋊C₄	—	12	2	no	—
203	C₄²	—	6	1	no	—
204	C₄²	—	12	2	no	—
205	C₁₆	—	6	2	no	—
206	C₁₆	#217	1	1	yes	characteristic
207	C₁₆	#217	1	1	yes	characteristic
208	C₁₆	#217	1	1	yes	characteristic
209	C₁₆	#217	1	1	yes	characteristic
210	C₂²×C₃	—	4	1	no	—
211	C₂²×C₃	—	8	2	no	—
212	C₂²×C₃	—	16	2	no	—
213	C₂²×C₃	C₄×D₈	1	1	yes	characteristic
214	C₂²×C₃	#466	1	1	yes	characteristic
215	C₂²×C₃	#493	1	1	yes	characteristic
216	C₃×C₄	—	8	2	no	—
217	C₃×C₄	#390	1	1	yes	characteristic
218	C₃×C₄	C₄×D₈	1	1	yes	characteristic
219	C₃×C₄	#466	1	1	yes	characteristic
220	C₃×C₄	#495	1	1	yes	characteristic
221	Dic₃	—	4	2	no	—
222	Dic₃	—	16	2	no	—
223	C₂³	—	4	2	no	—
224	C₂³	D₃×D₈	1	1	yes	characteristic
225	Q₈	—	4	2	no	—
226	Q₈	—	4	2	no	—
227	D₄	—	4	2	no	—
228	D₄	—	4	2	no	—
229	D₄	—	8	2	no	—
230	D₄	—	8	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	6	2	no	—
233	C₂×C₄	—	8	2	no	—
234	C₂×C₄	—	8	2	no	—
235	C₂×C₄	—	24	2	no	—
236	C₂×C₄	D₃×D₈	1	1	yes	characteristic
237	C₂×C₄	#912	1	1	yes	characteristic
238	C₂×C₄	#912	1	1	yes	characteristic
239	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
240	C₂×C₄	#938	1	1	yes	characteristic
241	C₂×C₄	#940	1	1	yes	characteristic
242	C₈	—	6	2	no	—
243	C₈	D₄×Dic₃	1	1	yes	characteristic
244	C₈	#936	1	1	yes	characteristic
245	C₈	#938	1	1	yes	characteristic
246	C₈	C₂×#217	1	1	yes	characteristic
247	C₂×C₃	—	2	1	no	—
248	C₂×C₃	—	2	1	no	—
249	C₂×C₃	—	2	1	no	—
250	C₂×C₃	—	16	2	no	—
251	C₂×C₃	#1970	1	1	yes	characteristic
252	C₂²	—	4	1	no	—
253	C₂²	—	8	2	no	—
254	C₂²	—	16	2	no	—
255	C₂²	Dic₃×D₈	1	1	yes	characteristic
256	C₂²	#5151	1	1	yes	characteristic
257	C₂²	#5157	1	1	yes	characteristic
258	C₄	—	8	2	no	—
259	C₄	—	12	2	no	—
260	C₄	—	48	2	no	—
261	C₄	Dic₃×D₈	1	1	yes	characteristic
262	C₄	#5155	1	1	yes	characteristic
263	C₄	#5157	1	1	yes	characteristic
264	C₄	#5221	1	1	yes	characteristic
265	C₃	#13243	1	1	yes	characteristic, complemented, Sylow
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	2	1	no	—
269	C₂	—	16	2	no	—
270	C₂	#72038	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#5157
commutator	C₂×C₃×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₃
Fitting, prime radical	C₂×C₃×#508	C₂
socle	C₂×C₃	#1970
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₂×#508	D₃
hypocenter	C₃	#13243
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
2. #72086	C₂	semidirect
3. #72076	C₂	semidirect
4. #71404	C₂	semidirect
5. #71399	C₂	semidirect
12. C₂×C₃×Q₃₂	C₄	semidirect
13. C₂×C₃×SD₃₂	C₄	semidirect
14. C₂×C₃×D₁₆	C₄	semidirect
69. C₂×Q₃₂	Dic₃	semidirect
70. C₂×SD₃₂	Dic₃	semidirect
71. C₂×D₁₆	Dic₃	semidirect
265. C₃	#13243	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#508	2
#72086	2
#72076	2
#71404	2
#71399	2
#13243	3