Subgroups — #10688293

Order 768.

418 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#4679	C₂	1	1	yes	maximal, characteristic
2	C₃×#1834	C₂	1	1	yes	maximal, characteristic
3	C₂×#4458	C₂	1	1	yes	maximal, characteristic
4	#69588	C₂	1	2	yes	maximal
5	#67634	C₂	1	2	yes	maximal
6	#8655	—	3	1	no	maximal, Sylow
7	C₂×C₄×Dic₆	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₃×#384	C₂²	1	2	yes	—
10	C₂×#815	—	2	1	no	—
11	C₂×Dic₃⋊C₈	—	2	1	no	—
12	C₂×#810	—	2	2	no	—
13	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
15	#4679	—	2	2	no	—
16	#4679	C₄	1	4	yes	complemented
17	C₃×#340	—	2	1	no	—
18	C₃×#340	—	2	1	no	—
19	#4529	—	2	2	no	—
20	#4458	—	2	2	no	—
21	C₂×#446	—	3	1	no	—
22	#1834	D₃	1	1	yes	characteristic
23	#1803	—	3	2	no	—
24	C₂×#335	—	3	1	no	—
25	#1296	—	3	2	no	—
26	C₂²×Dic₆	D₄	1	1	yes	characteristic
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×C₄⋊C₄	—	2	2	no	—
31	C₂×C₃×C₄⋊C₄	—	2	2	no	—
32	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊Dic₃	D₄	1	1	yes	characteristic
34	C₂×Dic₃⋊C₄	—	2	1	no	—
35	C₂×C₄×Dic₃	—	2	1	no	—
36	C₂²×C₃⋊C₈	—	2	1	no	—
37	C₂²×C₃⋊C₈	—	2	2	no	—
38	C₄×Dic₆	—	2	1	no	—
39	C₄×Dic₆	—	2	1	no	—
40	C₄×Dic₆	C₂×C₄	1	2	yes	—
41	C₄×Dic₆	Q₈	1	2	yes	—
42	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
43	C₃×C₄×C₈	D₄	1	2	yes	—
44	#815	—	2	4	no	—
45	Dic₃⋊C₈	—	2	4	no	—
46	#810	—	4	4	no	—
47	C₄⋊(C₃⋊C₈)	—	2	1	no	—
48	C₄⋊(C₃⋊C₈)	—	2	2	no	—
49	C₄⋊(C₃⋊C₈)	—	2	2	no	—
50	C₄⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
51	C₂×C₄×Q₈	—	3	1	no	—
52	#446	—	3	4	no	—
53	#446	—	6	2	no	—
54	C₂×C₄⋊C₈	—	3	1	no	—
55	C₂×C₄⋊C₈	—	3	2	no	—
56	C₂×C₄⋊C₈	—	6	1	no	—
57	C₂×#103	—	6	2	no	—
58	C₂×#97	—	6	1	no	—
59	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
60	#384	C₂×D₃	1	2	yes	—
61	#340	—	2	1	no	—
62	#340	—	2	1	no	—
63	#340	—	6	2	no	—
64	#335	—	6	2	no	—
65	C₂²×C₃×C₄	—	4	2	no	—
66	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₃×C₄	#41	1	1	yes	characteristic
69	C₂²×Dic₃	—	2	1	no	—
70	C₂²×Dic₃	—	2	1	no	—
71	C₂×Dic₆	—	2	2	no	—
72	C₂×Dic₆	—	2	2	no	—
73	C₂×Dic₆	—	2	1	no	—
74	C₂×Dic₆	C₄⋊C₄	1	2	yes	—
75	C₂×C₃×C₈	—	2	2	no	—
76	C₂×C₃×C₈	—	2	2	no	—
77	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
78	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
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	2	no	—
83	C₃×C₄⋊C₄	—	8	2	no	—
84	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
85	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
86	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
87	C₃×C₄²	#41	1	1	yes	characteristic
88	C₄⋊Dic₃	—	2	1	no	—
89	C₄⋊Dic₃	C₄⋊C₄	1	2	yes	—
90	Dic₃⋊C₄	—	2	2	no	—
91	Dic₃⋊C₄	—	2	2	no	—
92	C₄×Dic₃	—	2	2	no	—
93	C₄×Dic₃	—	2	2	no	—
94	C₂×C₃⋊C₈	—	2	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×Q₈	—	3	1	no	—
100	C₂²×C₈	—	6	1	no	—
101	C₂²×C₈	—	6	2	no	—
102	C₂²×C₈	#67	1	1	yes	characteristic
103	C₂²×C₈	#67	1	1	yes	characteristic
104	C₄×Q₈	—	3	2	no	—
105	C₄×Q₈	—	3	2	no	—
106	C₄×Q₈	—	6	1	no	—
107	C₄×Q₈	—	6	1	no	—
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄⋊C₄	—	2	2	no	—
111	C₂×C₄⋊C₄	—	3	1	no	—
112	C₂×C₄⋊C₄	—	6	1	no	—
113	C₂×C₄²	—	6	1	no	—
114	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
115	C₄⋊C₈	—	3	2	no	—
116	C₄⋊C₈	—	6	4	no	—
117	C₄⋊C₈	—	6	1	no	—
118	C₄⋊C₈	—	6	2	no	—
119	C₄⋊C₈	—	6	2	no	—
120	#103	—	12	4	no	—
121	#97	—	6	4	no	—
122	C₄×C₈	C₄×D₃	1	2	yes	—
123	C₄×C₈	D₁₂	1	2	yes	—
124	C₂³×C₃	#121	1	1	yes	characteristic
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	—	8	2	no	—
128	C₂×C₃×C₄	—	8	2	no	—
129	C₂×C₃×C₄	—	8	2	no	—
130	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₃×C₄	#117	1	1	yes	characteristic
133	C₂×C₃×C₄	#121	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
136	C₂×C₃×C₄	#135	1	2	yes	—
137	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
138	C₂×C₃×C₄	#137	1	2	yes	—
139	C₂×C₃×C₄	#137	1	2	yes	—
140	C₂×Dic₃	—	2	2	no	—
141	C₂×Dic₃	—	2	2	no	—
142	C₂×Dic₃	—	4	2	no	—
143	C₂×Dic₃	—	4	1	no	—
144	C₂×Dic₃	—	4	1	no	—
145	Dic₆	—	2	4	no	—
146	Dic₆	—	4	1	no	—
147	Dic₆	—	4	1	no	—
148	C₃×C₈	—	2	2	no	—
149	C₃×C₈	—	2	2	no	—
150	C₃⋊C₈	—	4	2	no	—
151	C₃⋊C₈	—	8	2	no	—
152	C₂×Q₈	—	3	2	no	—
153	C₂×Q₈	—	6	1	no	—
154	C₂×Q₈	—	6	2	no	—
155	C₂×Q₈	—	6	2	no	—
156	C₂²×C₄	—	4	2	no	—
157	C₂²×C₄	—	6	1	no	—
158	C₂²×C₄	—	6	1	no	—
159	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
160	C₂²×C₄	#237	1	1	yes	characteristic
161	C₂²×C₄	C₂×#67	1	1	yes	characteristic
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	6	2	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	(C₂×D₃)⋊C₄	1	2	yes	—
170	C₂×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
171	C₄⋊C₄	—	3	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	6	2	no	—
177	C₄⋊C₄	—	6	1	no	—
178	C₄⋊C₄	—	6	2	no	—
179	C₄⋊C₄	—	8	2	no	—
180	C₄²	—	6	2	no	—
181	C₄²	—	6	2	no	—
182	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
183	C₄²	C₂×D₁₂	1	1	yes	characteristic
184	C₄²	D₃×Q₈	1	1	yes	characteristic
185	C₄²	#239	1	1	yes	characteristic
186	C₂²×C₃	#380	1	1	yes	characteristic
187	C₂²×C₃	#464	1	2	yes	—
188	C₂²×C₃	#468	1	2	yes	—
189	C₂²×C₃	#472	1	1	yes	characteristic
190	C₂²×C₃	#473	1	1	yes	characteristic
191	C₃×C₄	—	2	1	no	—
192	C₃×C₄	—	2	1	no	—
193	C₃×C₄	—	16	2	no	—
194	C₃×C₄	#416	1	1	yes	characteristic
195	C₃×C₄	#419	1	1	yes	characteristic
196	C₃×C₄	#428	1	2	yes	—
197	C₃×C₄	#469	1	1	yes	characteristic
198	C₃×C₄	#470	1	1	yes	characteristic
199	C₃×C₄	#479	1	2	yes	—
200	Dic₃	—	4	2	no	—
201	Dic₃	—	4	2	no	—
202	C₂³	#938	1	1	yes	characteristic
203	Q₈	—	6	4	no	—
204	Q₈	—	12	1	no	—
205	Q₈	—	12	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	6	2	no	—
209	C₂×C₄	—	6	2	no	—
210	C₂×C₄	—	8	2	no	—
211	C₂×C₄	—	8	2	no	—
212	C₂×C₄	—	8	2	no	—
213	C₂×C₄	—	12	1	no	—
214	C₂×C₄	—	12	2	no	—
215	C₂×C₄	—	12	1	no	—
216	C₂×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
217	C₂×C₄	#892	1	1	yes	characteristic
218	C₂×C₄	#895	1	1	yes	characteristic
219	C₂×C₄	#897	1	1	yes	characteristic
220	C₂×C₄	#914	1	2	yes	—
221	C₂×C₄	#915	1	2	yes	—
222	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
223	C₂×C₄	#946	1	1	yes	characteristic
224	C₂×C₄	#950	1	2	yes	—
225	C₂×C₄	#951	1	2	yes	—
226	C₈	—	2	2	no	—
227	C₈	—	2	2	no	—
228	C₈	—	12	2	no	—
229	C₈	—	24	2	no	—
230	C₂×C₃	#1698	1	2	yes	—
231	C₂×C₃	#1700	1	2	yes	—
232	C₂×C₃	#1805	1	1	yes	characteristic
233	C₂×C₃	#1809	1	1	yes	characteristic
234	C₂×C₃	#1816	1	1	yes	characteristic
235	C₂²	#4987	1	1	yes	characteristic
236	C₂²	#5054	1	2	yes	—
237	C₂²	#5160	1	1	yes	characteristic
238	C₂²	#5174	1	2	yes	—
239	C₂²	#5192	1	1	yes	characteristic
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	12	2	no	—
243	C₄	—	12	2	no	—
244	C₄	—	16	2	no	—
245	C₄	#4861	1	2	yes	—
246	C₄	#4865	1	1	yes	characteristic
247	C₄	#4867	1	1	yes	characteristic
248	C₄	#5002	1	1	yes	characteristic
249	C₄	#5004	1	1	yes	characteristic
250	C₄	#5063	1	2	yes	—
251	C₃	#8655	1	1	yes	characteristic, complemented, Sylow
252	C₂	#69559	1	1	yes	characteristic
253	C₂	#70578	1	1	yes	characteristic
254	C₂	#70646	1	1	yes	characteristic
255	C₂	#71210	1	2	yes	—
256	C₂	#71242	1	2	yes	—

	subgroup	G/H
center	C₂³	#938
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1834	C₂
socle, prime radical	C₂³×C₃	#121
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#1834	D₃
hypocenter	C₃	#8655
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
16. #4679	C₄	semidirect
251. C₃	#8655	semidirect

Maximal subgroups

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

subgroup	index
C₂×#4679	2
C₃×#1834	2
C₂×#4458	2
#69588	2
#67634	2
#8655	3