Subgroups — #10693870

Order 768.

370 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#334	C₂	1	1	yes	maximal, characteristic
2	#70453	C₂	1	2	yes	maximal
3	#69765	C₂	1	1	yes	maximal, characteristic
4	#69757	C₂	1	1	yes	maximal, characteristic
5	#67702	C₂	1	2	yes	maximal
6	#9620	—	3	1	no	maximal, Sylow
7	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×#818	—	2	1	no	—
10	C₂×#818	—	2	1	no	—
11	C₂×#818	—	2	2	no	—
12	C₂×#815	—	2	1	no	—
13	#4935	C₂²	1	2	yes	—
14	C₄×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
16	C₃×#334	C₄	1	4	yes	complemented
17	#4551	—	2	2	no	—
18	#4550	—	2	1	no	—
19	#1947	—	3	2	no	—
20	#1845	—	3	1	no	—
21	#1770	—	3	1	no	—
22	C₂×#334	D₃	1	1	yes	characteristic
23	#1296	—	3	2	no	—
24	C₂²×C₃×C₈	—	2	2	no	—
25	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
27	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊Dic₃	—	2	2	no	—
29	C₂×C₄⋊Dic₃	—	2	2	no	—
30	C₂×C₄⋊Dic₃	—	2	2	no	—
31	C₂×C₄⋊Dic₃	D₄	1	1	yes	characteristic
32	C₂×C₄⋊Dic₃	D₄	1	1	yes	characteristic
33	C₂×C₄×Dic₃	—	2	1	no	—
34	C₂²×C₃⋊C₈	—	2	1	no	—
35	C₃×C₄⋊C₈	—	2	2	no	—
36	C₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
37	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
38	C₃×C₄×C₈	Q₈	1	2	yes	—
39	C₃⋊#95	—	2	1	no	—
40	#818	—	4	4	no	—
41	#818	—	4	2	no	—
42	#818	—	4	2	no	—
43	#815	—	4	2	no	—
44	C₄⋊(C₃⋊C₈)	—	2	1	no	—
45	C₄⋊(C₃⋊C₈)	Q₈	1	2	yes	—
46	C₂×#107	—	6	2	no	—
47	C₂×#107	—	6	1	no	—
48	C₂×#107	—	6	1	no	—
49	C₂×C₄⋊C₈	—	3	1	no	—
50	C₂×C₄⋊C₈	C₂×D₃	1	2	yes	—
51	C₂×#97	—	6	1	no	—
52	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
53	#384	—	3	2	no	—
54	C₄×C₄⋊C₄	—	3	1	no	—
55	#340	—	6	2	no	—
56	#336	—	6	1	no	—
57	#334	Dic₃	1	4	yes	complemented
58	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₃×C₄	#41	1	1	yes	characteristic
61	C₂²×Dic₃	—	2	1	no	—
62	C₂²×Dic₃	—	2	1	no	—
63	C₂²×Dic₃	—	4	2	no	—
64	C₂×C₃×C₈	—	2	1	no	—
65	C₂×C₃×C₈	—	2	1	no	—
66	C₂×C₃×C₈	—	2	4	no	—
67	C₂×C₃×C₈	—	4	2	no	—
68	C₂×C₃×C₈	—	4	2	no	—
69	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
70	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
71	C₂×C₃×C₈	D₈	1	2	yes	—
72	C₂×C₃×C₈	Q₁₆	1	2	yes	—
73	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
74	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
75	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
76	C₃×C₄²	#41	1	1	yes	characteristic
77	C₄⋊Dic₃	—	2	2	no	—
78	C₄⋊Dic₃	—	2	2	no	—
79	C₄⋊Dic₃	—	4	2	no	—
80	C₄⋊Dic₃	—	4	2	no	—
81	C₄⋊Dic₃	—	4	2	no	—
82	C₄⋊Dic₃	—	4	2	no	—
83	C₄⋊Dic₃	—	8	2	no	—
84	C₄×Dic₃	—	4	1	no	—
85	C₄×Dic₃	—	4	1	no	—
86	C₂×C₃⋊C₈	—	2	2	no	—
87	C₂×C₃⋊C₈	—	4	1	no	—
88	C₂×C₃⋊C₈	—	4	1	no	—
89	C₂²×C₈	—	2	2	no	—
90	C₂²×C₈	—	6	1	no	—
91	C₂²×C₈	D₁₂	1	1	yes	characteristic
92	C₂²×C₈	D₁₂	1	1	yes	characteristic
93	C₂×C₄⋊C₄	—	3	1	no	—
94	C₂×C₄⋊C₄	—	3	1	no	—
95	C₂×C₄⋊C₄	—	6	2	no	—
96	C₂×C₄⋊C₄	—	6	2	no	—
97	C₂×C₄⋊C₄	—	6	2	no	—
98	C₂×C₄²	—	6	1	no	—
99	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
100	#107	—	12	4	no	—
101	#107	—	12	2	no	—
102	#107	—	12	2	no	—
103	C₄⋊C₈	—	2	2	no	—
104	C₄⋊C₈	—	3	2	no	—
105	C₄⋊C₈	—	6	1	no	—
106	C₄⋊C₈	C₂×Dic₃	1	4	yes	—
107	#97	—	12	2	no	—
108	C₄×C₈	Dic₆	1	2	yes	—
109	C₄×C₈	C₂×Dic₃	1	2	yes	—
110	#95	—	6	1	no	—
111	C₂³×C₃	#121	1	1	yes	characteristic
112	C₂×C₃×C₄	—	2	1	no	—
113	C₂×C₃×C₄	—	2	1	no	—
114	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
116	C₂×C₃×C₄	C₄×Q₈	1	1	yes	characteristic
117	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₃×C₄	#125	1	1	yes	characteristic
119	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
120	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
122	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
124	C₂×C₃×C₄	#137	1	1	yes	characteristic
125	C₂×C₃×C₄	#137	1	2	yes	—
126	C₂×Dic₃	—	4	1	no	—
127	C₂×Dic₃	—	4	1	no	—
128	C₂×Dic₃	—	4	1	no	—
129	C₂×Dic₃	—	4	1	no	—
130	C₂×Dic₃	—	4	1	no	—
131	C₂×Dic₃	—	4	1	no	—
132	C₂×Dic₃	—	8	2	no	—
133	C₂×Dic₃	—	8	2	no	—
134	C₂×Dic₃	—	8	2	no	—
135	C₃×C₈	—	2	2	no	—
136	C₃×C₈	—	4	4	no	—
137	C₃×C₈	#107	1	4	yes	complemented
138	C₃⋊C₈	—	4	2	no	—
139	C₂²×C₄	—	6	1	no	—
140	C₂²×C₄	—	6	1	no	—
141	C₂²×C₄	—	12	2	no	—
142	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
143	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
144	C₂²×C₄	#239	1	1	yes	characteristic
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	6	2	no	—
151	C₂×C₈	—	12	1	no	—
152	C₂×C₈	—	12	1	no	—
153	C₂×C₈	D₂₄	1	2	yes	—
154	C₂×C₈	Dic₁₂	1	2	yes	—
155	C₂×C₈	C₄⋊Dic₃	1	2	yes	—
156	C₂×C₈	C₄⋊Dic₃	1	2	yes	—
157	C₄⋊C₄	—	6	2	no	—
158	C₄⋊C₄	—	6	2	no	—
159	C₄⋊C₄	—	12	2	no	—
160	C₄⋊C₄	—	12	2	no	—
161	C₄⋊C₄	—	12	2	no	—
162	C₄⋊C₄	—	12	2	no	—
163	C₄⋊C₄	—	24	2	no	—
164	C₄²	—	12	1	no	—
165	C₄²	—	12	1	no	—
166	C₄²	C₂×Dic₆	1	1	yes	characteristic
167	C₄²	#237	1	1	yes	characteristic
168	C₄²	D₃×Q₈	1	1	yes	characteristic
169	C₄²	C₂²×Dic₃	1	1	yes	characteristic
170	C₂²×C₃	#389	1	1	yes	characteristic
171	C₂²×C₃	#459	1	1	yes	characteristic
172	C₂²×C₃	#462	1	1	yes	characteristic
173	C₂²×C₃	#469	1	1	yes	characteristic
174	C₂²×C₃	#470	1	1	yes	characteristic
175	C₂²×C₃	#472	1	1	yes	characteristic
176	C₂²×C₃	#473	1	1	yes	characteristic
177	C₃×C₄	—	2	1	no	—
178	C₃×C₄	—	2	1	no	—
179	C₃×C₄	C₂×#107	1	1	yes	characteristic
180	C₃×C₄	#428	1	1	yes	characteristic
181	C₃×C₄	#440	1	2	yes	—
182	C₃×C₄	#474	1	1	yes	characteristic
183	C₃×C₄	#477	1	1	yes	characteristic
184	C₃×C₄	#501	1	2	yes	—
185	Dic₃	—	8	1	no	—
186	Dic₃	—	8	1	no	—
187	Dic₃	—	16	2	no	—
188	C₂³	#895	1	1	yes	characteristic
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	12	1	no	—
192	C₂×C₄	—	12	1	no	—
193	C₂×C₄	—	12	1	no	—
194	C₂×C₄	—	12	1	no	—
195	C₂×C₄	—	12	1	no	—
196	C₂×C₄	—	12	1	no	—
197	C₂×C₄	—	24	2	no	—
198	C₂×C₄	—	24	2	no	—
199	C₂×C₄	—	24	2	no	—
200	C₂×C₄	#888	1	1	yes	characteristic
201	C₂×C₄	#890	1	1	yes	characteristic
202	C₂×C₄	#897	1	1	yes	characteristic
203	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
204	C₂×C₄	C₂×Dic₁₂	1	1	yes	characteristic
205	C₂×C₄	#908	1	1	yes	characteristic
206	C₂×C₄	#909	1	1	yes	characteristic
207	C₂×C₄	#914	1	2	yes	—
208	C₂×C₄	#915	1	2	yes	—
209	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
210	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
211	C₂×C₄	Q₈×Dic₃	1	1	yes	characteristic
212	C₈	—	2	2	no	—
213	C₈	—	4	4	no	—
214	C₈	—	12	2	no	—
215	C₈	#818	1	4	yes	complemented
216	C₂×C₃	#1687	1	1	yes	characteristic
217	C₂×C₃	#1689	1	1	yes	characteristic
218	C₂×C₃	#1694	1	1	yes	characteristic
219	C₂×C₃	#1696	1	1	yes	characteristic
220	C₂×C₃	#1944	1	1	yes	characteristic
221	C₂×C₃	#1947	1	1	yes	characteristic
222	C₂×C₃	#1952	1	1	yes	characteristic
223	C₂²	#4843	1	1	yes	characteristic
224	C₂²	#4849	1	1	yes	characteristic
225	C₂²	#4865	1	1	yes	characteristic
226	C₂²	#4867	1	1	yes	characteristic
227	C₂²	#4896	1	1	yes	characteristic
228	C₂²	#4897	1	1	yes	characteristic
229	C₂²	#4984	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	24	1	no	—
233	C₄	—	24	1	no	—
234	C₄	—	48	2	no	—
235	C₄	#4846	1	1	yes	characteristic
236	C₄	#4850	1	1	yes	characteristic
237	C₄	#4856	1	2	yes	—
238	C₄	C₂×#818	1	1	yes	characteristic
239	C₄	#5121	1	1	yes	characteristic
240	C₄	#5164	1	2	yes	—
241	C₃	#9620	1	1	yes	characteristic, complemented, Sylow
242	C₂	#68540	1	1	yes	characteristic
243	C₂	#68541	1	1	yes	characteristic
244	C₂	#68552	1	1	yes	characteristic
245	C₂	#68553	1	1	yes	characteristic
246	C₂	#70453	1	1	yes	characteristic
247	C₂	#70562	1	1	yes	characteristic
248	C₂	#70701	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#895
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂×C₃×#334	C₂
socle, prime radical	C₂³×C₃	#121
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×#334	D₃
hypocenter	C₃	#9620
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
16. C₃×#334	C₄	semidirect
57. #334	Dic₃	semidirect
137. C₃×C₈	#107	semidirect
215. C₈	#818	semidirect
241. C₃	#9620	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#334	2
#70453	2
#69765	2
#69757	2
#67702	2
#9620	3