Subgroups — #10694564

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	#78467	C₂	1	1	yes	maximal, characteristic
2	C₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
3	#69724	C₂	1	1	yes	maximal, characteristic, complemented
4	#69545	C₂	1	4	yes	maximal, complemented
5	#9756	—	3	1	no	maximal, Sylow
6	C₂×#904	—	2	1	no	—
7	C₂×#904	—	2	1	no	—
8	#5477	C₂²	1	2	yes	—
9	C₂×C₃×#107	—	2	1	no	—
10	C₂×C₃×#106	—	2	1	no	—
11	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₃×#382	C₂²	1	2	yes	—
13	#4925	C₂²	1	2	yes	—
14	#4699	C₄	1	1	yes	characteristic, complemented
15	#4696	—	2	1	no	—
16	#4696	—	2	1	no	—
17	#4695	C₄	1	1	yes	characteristic, complemented
18	#4693	C₄	1	2	yes	complemented
19	#4455	—	2	1	no	—
20	C₈×C₃⋊C₈	—	2	1	no	—
21	#4451	—	2	8	no	—
22	#3147	—	3	1	no	—
23	#1846	D₃	1	1	yes	characteristic, complemented
24	#1517	—	3	4	no	—
25	#1450	—	3	1	no	—
26	C₂×#235	—	2	2	no	—
27	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄⋊C₄	—	2	2	no	—
30	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
31	#930	—	2	4	no	—
32	C₂×Dic₁₂	—	2	1	no	—
33	C₂×Dic₁₂	—	2	1	no	—
34	#904	—	4	2	no	—
35	#904	—	4	2	no	—
36	#904	—	4	2	no	—
37	#904	—	4	2	no	—
38	C₂×D₂₄	—	2	1	no	—
39	C₂×D₂₄	—	2	1	no	—
40	C₂×#204	—	2	2	no	—
41	C₂×#204	—	2	2	no	—
42	#875	—	2	2	no	—
43	#874	C₂×C₄	1	2	yes	—
44	C₄×D₁₂	—	2	4	no	—
45	#869	C₂×C₄	1	2	yes	—
46	C₃×#107	—	4	2	no	—
47	C₃×#106	—	4	2	no	—
48	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
49	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
50	C₃×C₄×C₈	D₄	1	1	yes	characteristic
51	C₃×C₄×C₈	D₄	1	1	yes	characteristic
52	C₃×#95	—	2	4	no	—
53	#830	D₄	1	4	yes	—
54	C₄⋊(C₃⋊C₈)	—	2	2	no	—
55	C₄×C₃⋊C₈	—	2	2	no	—
56	C₂×#135	—	6	1	no	—
57	C₂×#135	—	6	1	no	—
58	#532	—	3	2	no	—
59	#495	—	6	1	no	—
60	#495	—	6	1	no	—
61	#494	—	3	1	no	—
62	#493	—	3	1	no	—
63	#492	—	3	2	no	—
64	#431	—	3	2	no	—
65	C₂×#107	—	2	1	no	—
66	C₂×#106	—	2	1	no	—
67	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
68	#382	C₂×D₃	1	2	yes	—
69	#327	—	6	8	no	—
70	#322	—	6	1	no	—
71	C₈²	—	6	1	no	—
72	C₂²×C₃×C₄	—	4	2	no	—
73	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
74	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
75	C₂×#67	—	4	2	no	—
76	#235	—	4	2	no	—
77	#235	—	4	2	no	—
78	#235	—	4	2	no	—
79	#235	—	4	2	no	—
80	C₂×D₁₂	—	2	2	no	—
81	C₂×D₁₂	—	2	4	no	—
82	C₂×C₄×D₃	—	4	2	no	—
83	C₂×Dic₆	—	2	2	no	—
84	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₃×C₈	D₈	1	2	yes	—
89	C₂×C₃×C₈	D₈	1	2	yes	—
90	C₂×C₃×C₈	SD₁₆	1	2	yes	—
91	C₂×C₃×C₈	SD₁₆	1	2	yes	—
92	C₃×C₄⋊C₄	—	4	2	no	—
93	C₃×C₄⋊C₄	—	4	2	no	—
94	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
95	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
96	C₃×C₄²	#41	1	2	yes	—
97	(C₂×D₃)⋊C₄	—	4	4	no	—
98	C₄⋊Dic₃	—	2	4	no	—
99	C₂×C₃⋊C₈	—	2	4	no	—
100	Dic₁₂	—	4	2	no	—
101	Dic₁₂	—	4	2	no	—
102	D₂₄	—	4	2	no	—
103	D₂₄	—	4	2	no	—
104	#204	—	4	4	no	—
105	#204	—	4	4	no	—
106	C₂×#41	—	6	2	no	—
107	#135	—	12	2	no	—
108	#135	—	12	2	no	—
109	#135	—	12	2	no	—
110	#135	—	12	2	no	—
111	C₂×Q₁₆	—	6	1	no	—
112	C₂×Q₁₆	—	6	1	no	—
113	C₂×SD₁₆	—	6	2	no	—
114	C₂×SD₁₆	—	6	2	no	—
115	C₂×D₈	—	6	1	no	—
116	C₂×D₈	—	6	1	no	—
117	C₂²×C₈	#67	1	1	yes	characteristic
118	C₂²×C₈	#67	1	1	yes	characteristic
119	C₄⋊Q₈	—	3	2	no	—
120	#127	—	3	2	no	—
121	#124	—	6	2	no	—
122	#121	—	6	4	no	—
123	C₄×D₄	—	6	4	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
126	#107	—	4	2	no	—
127	#106	—	4	2	no	—
128	C₄⋊C₈	—	6	2	no	—
129	C₂²⋊C₈	—	3	4	no	—
130	C₄×C₈	—	6	2	no	—
131	C₄×C₈	C₄×D₃	1	1	yes	characteristic
132	C₄×C₈	C₄×D₃	1	1	yes	characteristic
133	C₄×C₈	D₁₂	1	1	yes	characteristic
134	C₄×C₈	D₁₂	1	1	yes	characteristic
135	#95	—	2	4	no	—
136	C₂³×C₃	#127	1	1	yes	characteristic
137	C₂²×D₃	—	4	2	no	—
138	C₂×C₃×C₄	—	4	2	no	—
139	C₂×C₃×C₄	—	8	4	no	—
140	C₂×C₃×C₄	—	8	2	no	—
141	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
143	C₂×C₃×C₄	#121	1	2	yes	—
144	C₂×C₃×C₄	#124	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
146	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
149	#67	—	8	2	no	—
150	#67	—	8	2	no	—
151	C₂×Dic₃	—	4	2	no	—
152	D₁₂	—	4	2	no	—
153	D₁₂	—	4	2	no	—
154	D₁₂	—	8	4	no	—
155	C₄×D₃	—	8	2	no	—
156	C₄×D₃	—	8	2	no	—
157	Dic₆	—	4	2	no	—
158	Dic₆	—	4	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	—	12	2	no	—
165	#41	—	12	2	no	—
166	#41	—	12	2	no	—
167	#41	—	12	2	no	—
168	C₂×Q₈	—	6	2	no	—
169	C₂×D₄	—	6	4	no	—
170	C₂×D₄	—	6	2	no	—
171	C₂×D₄	—	12	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	—	12	2	no	—
177	Q₁₆	—	12	2	no	—
178	SD₁₆	—	12	4	no	—
179	SD₁₆	—	12	4	no	—
180	D₈	—	12	2	no	—
181	D₈	—	12	2	no	—
182	C₂×C₈	—	6	4	no	—
183	C₂×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
184	C₂×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
185	C₂×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
186	C₂×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
187	C₂×C₈	#213	1	2	yes	—
188	C₂×C₈	#213	1	2	yes	—
189	C₂×C₈	#215	1	2	yes	—
190	C₂×C₈	#215	1	2	yes	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	6	4	no	—
194	C₂²⋊C₄	—	12	4	no	—
195	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
196	C₄²	C₂×D₁₂	1	1	yes	characteristic
197	C₄²	#239	1	2	yes	—
198	C₂²×C₃	—	2	2	no	—
199	C₂²×C₃	#390	1	1	yes	characteristic
200	C₂²×C₃	#492	1	1	yes	characteristic
201	C₂²×C₃	#493	1	1	yes	characteristic
202	C₂×D₃	—	8	2	no	—
203	C₂×D₃	—	8	4	no	—
204	C₃×C₄	—	2	2	no	—
205	C₃×C₄	—	2	2	no	—
206	C₃×C₄	—	16	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	—	8	2	no	—
212	C₂³	—	12	2	no	—
213	C₂³	#940	1	1	yes	characteristic
214	Q₈	—	12	2	no	—
215	Q₈	—	12	2	no	—
216	D₄	—	12	2	no	—
217	D₄	—	12	2	no	—
218	D₄	—	24	4	no	—
219	D₄	—	24	2	no	—
220	D₄	—	24	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	8	4	no	—
223	C₂×C₄	—	8	2	no	—
224	C₂×C₄	—	12	2	no	—
225	C₂×C₄	—	24	2	no	—
226	C₂×C₄	—	24	2	no	—
227	C₂×C₄	#893	1	2	yes	—
228	C₂×C₄	#895	1	2	yes	—
229	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
230	C₂×C₄	C₂×#213	1	1	yes	characteristic
231	C₂×C₄	C₂×#213	1	1	yes	characteristic
232	C₂×C₄	C₂×#215	1	1	yes	characteristic
233	C₂×C₄	C₂×#215	1	1	yes	characteristic
234	C₂×C₄	#947	1	1	yes	characteristic
235	C₈	—	12	4	no	—
236	C₈	#809	1	2	yes	—
237	C₈	#809	1	2	yes	—
238	C₈	#809	1	2	yes	—
239	C₈	#809	1	2	yes	—
240	C₂×C₃	—	2	1	no	—
241	C₂×C₃	—	2	1	no	—
242	C₂×C₃	#1957	1	1	yes	characteristic
243	C₂×C₃	#1976	1	2	yes	—
244	D₃	—	16	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	24	4	no	—
247	C₂²	—	24	2	no	—
248	C₂²	#4988	1	1	yes	characteristic
249	C₂²	#5073	1	1	yes	characteristic
250	C₂²	#5085	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	2	2	no	—
253	C₄	—	16	2	no	—
254	C₄	—	24	2	no	—
255	C₄	C₂×#809	1	1	yes	characteristic
256	C₄	C₂×#809	1	1	yes	characteristic
257	C₄	#5088	1	1	yes	characteristic
258	C₄	#5088	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	—	48	2	no	—
263	C₂	#69580	1	1	yes	characteristic
264	C₂	#70658	1	2	yes	—

	subgroup	G/H
center	C₂²	#4988
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1846	C₂
socle	C₂²×C₃	#390
prime radical	#78467	C₂
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#1846	D₃
hypocenter	C₃	#9756
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1846

Complements

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

N	H ≅ G/N	type
2. C₃×#1846	C₂	semidirect
3. #69724	C₂	semidirect
4. #69545	C₂	semidirect
14. #4699	C₄	semidirect
17. #4695	C₄	semidirect
18. #4693	C₄	semidirect
23. #1846	D₃	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
#78467	2
C₃×#1846	2
#69724	2
#69545	2
#9756	3