Subgroups — #10695021

Order 768.

510 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#4682	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#322	C₂	1	1	yes	maximal, characteristic
3	#69777	C₂	1	2	yes	maximal
4	#69771	C₂	1	2	yes	maximal
5	#69764	C₂	1	1	yes	maximal, characteristic
6	#9836	—	3	1	no	maximal, Sylow
7	C₂×#869	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×#818	—	2	2	no	—
12	C₂×#817	—	2	2	no	—
13	C₂×#816	—	2	4	no	—
14	#4935	C₂²	1	2	yes	—
15	#4682	—	2	2	no	—
16	#4682	C₄	1	4	yes	complemented
17	C₃×#322	—	2	2	no	—
18	#4551	—	2	4	no	—
19	C₂×#500	—	3	1	no	—
20	#1990	—	3	2	no	—
21	#1956	—	3	2	no	—
22	#1844	—	3	1	no	—
23	C₂×#322	D₃	1	1	yes	characteristic
24	C₂²×Dic₆	—	2	2	no	—
25	C₂²×C₃×C₈	D₄	1	2	yes	—
26	C₂²×C₃×C₈	D₄	1	2	yes	—
27	C₂²×C₃×C₈	D₄	1	2	yes	—
28	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₃	—	2	2	no	—
30	C₂×C₄⋊Dic₃	—	2	2	no	—
31	C₂×C₄⋊Dic₃	—	2	2	no	—
32	C₂×C₄⋊Dic₃	—	2	2	no	—
33	C₂×C₄⋊Dic₃	—	2	2	no	—
34	#869	—	2	2	no	—
35	#869	—	2	2	no	—
36	#869	—	2	2	no	—
37	#869	C₂×C₄	1	4	yes	—
38	C₃×C₄×C₈	—	2	1	no	—
39	C₃×C₄×C₈	—	2	1	no	—
40	C₃×C₄×C₈	—	2	1	no	—
41	C₃×C₄×C₈	—	2	1	no	—
42	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
43	C₃×C₄×C₈	D₄	1	2	yes	—
44	#818	—	2	8	no	—
45	#817	—	4	4	no	—
46	#816	—	4	8	no	—
47	C₂×C₄⋊Q₈	—	3	2	no	—
48	#500	—	3	4	no	—
49	#500	—	6	2	no	—
50	C₂×#107	—	6	2	no	—
51	C₂×#106	—	6	2	no	—
52	C₂×#103	—	6	4	no	—
53	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
54	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
56	#384	—	3	2	no	—
57	#340	—	6	4	no	—
58	#322	—	2	2	no	—
59	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₃	—	4	2	no	—
63	C₂²×Dic₃	—	4	2	no	—
64	C₂×Dic₆	—	2	4	no	—
65	C₂×Dic₆	—	4	4	no	—
66	C₂×Dic₆	—	4	4	no	—
67	C₂×Dic₆	—	4	2	no	—
68	C₂×C₃×C₈	—	2	2	no	—
69	C₂×C₃×C₈	—	2	2	no	—
70	C₂×C₃×C₈	—	2	4	no	—
71	C₂×C₃×C₈	—	2	4	no	—
72	C₂×C₃×C₈	C₂²⋊C₄	1	4	yes	—
73	C₂×C₃×C₈	SD₁₆	1	4	yes	—
74	C₂×C₃×C₈	Q₁₆	1	4	yes	—
75	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
76	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
77	C₃×C₄²	#41	1	1	yes	characteristic
78	C₃×C₄²	#41	1	1	yes	characteristic
79	C₄⋊Dic₃	—	2	4	no	—
80	C₄⋊Dic₃	—	2	4	no	—
81	C₄⋊Dic₃	—	4	2	no	—
82	C₄⋊Dic₃	—	4	2	no	—
83	C₄⋊Dic₃	—	4	4	no	—
84	C₄⋊Dic₃	—	4	4	no	—
85	C₄⋊Dic₃	—	4	4	no	—
86	C₂²×Q₈	—	6	2	no	—
87	C₂²×C₈	D₁₂	1	2	yes	—
88	C₂²×C₈	D₁₂	1	2	yes	—
89	C₂²×C₈	D₁₂	1	2	yes	—
90	C₄⋊Q₈	—	3	4	no	—
91	C₄⋊Q₈	—	6	2	no	—
92	C₄⋊Q₈	—	6	2	no	—
93	C₄⋊Q₈	—	6	2	no	—
94	C₂×C₄⋊C₄	—	6	2	no	—
95	C₂×C₄⋊C₄	—	6	2	no	—
96	C₂×C₄⋊C₄	—	6	2	no	—
97	C₂×C₄⋊C₄	—	6	2	no	—
98	C₂×C₄⋊C₄	—	6	2	no	—
99	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
100	#107	—	6	8	no	—
101	#106	—	12	4	no	—
102	#103	—	12	8	no	—
103	C₄×C₈	—	2	1	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	—	2	1	no	—
106	C₄×C₈	—	2	1	no	—
107	C₄×C₈	C₄×D₃	1	2	yes	—
108	C₄×C₈	#67	1	2	yes	—
109	C₂³×C₃	#127	1	1	yes	characteristic
110	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
112	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
113	C₂×C₃×C₄	#121	1	1	yes	characteristic
114	C₂×C₃×C₄	#121	1	1	yes	characteristic
115	C₂×C₃×C₄	#124	1	1	yes	characteristic
116	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
117	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
118	C₂×C₃×C₄	AGL(1,ℤ/8)	1	4	yes	—
119	C₂×C₃×C₄	#137	1	4	yes	—
120	C₂×Dic₃	—	4	4	no	—
121	C₂×Dic₃	—	8	2	no	—
122	C₂×Dic₃	—	8	2	no	—
123	C₂×Dic₃	—	8	2	no	—
124	C₂×Dic₃	—	8	2	no	—
125	C₂×Dic₃	—	8	2	no	—
126	Dic₆	—	4	8	no	—
127	Dic₆	—	8	4	no	—
128	C₃×C₈	—	4	2	no	—
129	C₃×C₈	—	4	2	no	—
130	C₃×C₈	#103	1	8	yes	complemented
131	C₂×Q₈	—	6	4	no	—
132	C₂×Q₈	—	12	2	no	—
133	C₂×Q₈	—	12	4	no	—
134	C₂×Q₈	—	12	4	no	—
135	C₂²×C₄	—	12	2	no	—
136	C₂²×C₄	—	12	2	no	—
137	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
139	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	#204	1	4	yes	—
145	C₂×C₈	Dic₁₂	1	4	yes	—
146	C₂×C₈	(C₂×D₃)⋊C₄	1	4	yes	—
147	C₄⋊C₄	—	6	4	no	—
148	C₄⋊C₄	—	6	4	no	—
149	C₄⋊C₄	—	12	2	no	—
150	C₄⋊C₄	—	12	2	no	—
151	C₄⋊C₄	—	12	4	no	—
152	C₄⋊C₄	—	12	4	no	—
153	C₄⋊C₄	—	12	4	no	—
154	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
155	C₄²	#235	1	1	yes	characteristic
156	C₄²	#235	1	1	yes	characteristic
157	C₄²	C₂×#67	1	1	yes	characteristic
158	C₂²×C₃	#390	1	1	yes	characteristic
159	C₂²×C₃	#492	1	1	yes	characteristic
160	C₂²×C₃	#494	1	1	yes	characteristic
161	C₂²×C₃	#496	1	2	yes	—
162	C₂²×C₃	#497	1	2	yes	—
163	C₃×C₄	C₂×#103	1	2	yes	—
164	C₃×C₄	#441	1	2	yes	—
165	C₃×C₄	#442	1	2	yes	—
166	C₃×C₄	#469	1	2	yes	—
167	C₃×C₄	#470	1	2	yes	—
168	C₃×C₄	#487	1	2	yes	—
169	Dic₃	—	8	4	no	—
170	Dic₃	—	16	2	no	—
171	C₂³	#874	1	1	yes	characteristic
172	Q₈	—	12	8	no	—
173	Q₈	—	24	4	no	—
174	C₂×C₄	—	12	4	no	—
175	C₂×C₄	—	24	2	no	—
176	C₂×C₄	—	24	2	no	—
177	C₂×C₄	—	24	2	no	—
178	C₂×C₄	—	24	2	no	—
179	C₂×C₄	—	24	2	no	—
180	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
181	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
182	C₂×C₄	#875	1	1	yes	characteristic
183	C₂×C₄	C₂×#204	1	2	yes	—
184	C₂×C₄	C₂×Dic₁₂	1	2	yes	—
185	C₂×C₄	#908	1	4	yes	—
186	C₂×C₄	#909	1	4	yes	—
187	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
188	C₂×C₄	#930	1	1	yes	characteristic
189	C₂×C₄	#930	1	1	yes	characteristic
190	C₈	—	4	2	no	—
191	C₈	—	4	2	no	—
192	C₈	#816	1	8	yes	complemented
193	C₂×C₃	#1714	1	2	yes	—
194	C₂×C₃	#1723	1	2	yes	—
195	C₂×C₃	#1956	1	1	yes	characteristic
196	C₂×C₃	#1971	1	1	yes	characteristic
197	C₂×C₃	#1972	1	1	yes	characteristic
198	C₂²	#4693	1	1	yes	characteristic
199	C₂²	#4699	1	1	yes	characteristic
200	C₂²	#4712	1	2	yes	—
201	C₂²	#4715	1	2	yes	—
202	C₂²	#4939	1	1	yes	characteristic
203	C₄	—	24	4	no	—
204	C₄	—	48	2	no	—
205	C₄	#4681	1	2	yes	—
206	C₄	#4711	1	2	yes	—
207	C₄	#4716	1	2	yes	—
208	C₄	C₂×#816	1	2	yes	—
209	C₄	#5135	1	2	yes	—
210	C₄	#5137	1	2	yes	—
211	C₃	#9836	1	1	yes	characteristic, complemented, Sylow
212	C₂	#68090	1	2	yes	—
213	C₂	#68099	1	2	yes	—
214	C₂	#69771	1	1	yes	characteristic
215	C₂	#69824	1	1	yes	characteristic
216	C₂	#69825	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#874
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂×C₃×#322	C₂
socle, prime radical	C₂³×C₃	#127
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×#322	D₃
hypocenter	C₃	#9836
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
16. #4682	C₄	semidirect
130. C₃×C₈	#103	semidirect
192. C₈	#816	semidirect
211. C₃	#9836	semidirect

Maximal subgroups

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

subgroup	index
C₂×#4682	2
C₂×C₃×#322	2
#69777	2
#69771	2
#69764	2
#9836	3