Subgroups — #10695016

Order 768.

526 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#4699	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#322	C₂	1	1	yes	maximal, characteristic
3	#69777	C₂	1	2	yes	maximal
4	#69777	C₂	1	2	yes	maximal
5	#69762	C₂	1	1	yes	maximal, characteristic
6	#9835	—	3	1	no	maximal, Sylow
7	C₂²×Dic₁₂	—	2	2	no	—
8	C₂×#869	C₂²	1	2	yes	—
9	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	#4935	C₂²	1	2	yes	—
13	#4699	—	2	2	no	—
14	#4699	C₄	1	4	yes	complemented
15	C₃×#322	—	2	2	no	—
16	#4551	—	2	2	no	—
17	#4551	—	2	4	no	—
18	#4551	—	2	4	no	—
19	C₂×#494	—	3	1	no	—
20	#1990	—	3	2	no	—
21	#1990	—	3	2	no	—
22	#1834	—	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	C₂×Dic₁₂	—	2	8	no	—
35	C₂×Dic₁₂	—	4	8	no	—
36	#869	—	2	2	no	—
37	#869	—	2	2	no	—
38	#869	—	2	2	no	—
39	#869	C₂×C₄	1	4	yes	—
40	C₃×C₄×C₈	—	2	1	no	—
41	C₃×C₄×C₈	—	2	1	no	—
42	C₃×C₄×C₈	—	2	1	no	—
43	C₃×C₄×C₈	—	2	1	no	—
44	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
45	C₃×C₄×C₈	D₄	1	2	yes	—
46	C₂²×Q₁₆	—	6	2	no	—
47	C₂×C₄⋊Q₈	—	3	2	no	—
48	#494	—	3	4	no	—
49	#494	—	6	2	no	—
50	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
51	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
52	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
53	#384	—	3	2	no	—
54	#340	—	6	2	no	—
55	#340	—	6	4	no	—
56	#340	—	6	4	no	—
57	#322	—	2	2	no	—
58	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
61	C₂²×Dic₃	—	4	2	no	—
62	C₂²×Dic₃	—	4	2	no	—
63	C₂×Dic₆	—	2	4	no	—
64	C₂×Dic₆	—	4	8	no	—
65	C₂×Dic₆	—	4	2	no	—
66	C₂×C₃×C₈	—	2	2	no	—
67	C₂×C₃×C₈	—	2	2	no	—
68	C₂×C₃×C₈	—	2	4	no	—
69	C₂×C₃×C₈	—	2	2	no	—
70	C₂×C₃×C₈	—	2	4	no	—
71	C₂×C₃×C₈	C₂²⋊C₄	1	4	yes	—
72	C₂×C₃×C₈	Q₁₆	1	4	yes	complemented
73	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
74	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
75	C₃×C₄²	#41	1	1	yes	characteristic
76	C₃×C₄²	#41	1	1	yes	characteristic
77	C₄⋊Dic₃	—	2	4	no	—
78	C₄⋊Dic₃	—	2	4	no	—
79	C₄⋊Dic₃	—	4	2	no	—
80	C₄⋊Dic₃	—	4	2	no	—
81	C₄⋊Dic₃	—	4	4	no	—
82	C₄⋊Dic₃	—	4	4	no	—
83	C₄⋊Dic₃	—	4	4	no	—
84	Dic₁₂	—	4	16	no	—
85	C₂²×Q₈	—	6	2	no	—
86	C₂×Q₁₆	—	6	8	no	—
87	C₂×Q₁₆	—	12	8	no	—
88	C₂²×C₈	D₁₂	1	2	yes	—
89	C₂²×C₈	D₁₂	1	2	yes	—
90	C₂²×C₈	D₁₂	1	2	yes	—
91	C₄⋊Q₈	—	3	4	no	—
92	C₄⋊Q₈	—	6	2	no	—
93	C₄⋊Q₈	—	6	2	no	—
94	C₄⋊Q₈	—	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×C₄⋊C₄	—	6	2	no	—
100	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
101	C₄×C₈	—	2	1	no	—
102	C₄×C₈	—	2	1	no	—
103	C₄×C₈	—	2	1	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	C₄×D₃	1	2	yes	—
106	C₄×C₈	#67	1	2	yes	—
107	C₂³×C₃	#127	1	1	yes	characteristic
108	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
110	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
111	C₂×C₃×C₄	#121	1	1	yes	characteristic
112	C₂×C₃×C₄	#121	1	1	yes	characteristic
113	C₂×C₃×C₄	#124	1	1	yes	characteristic
114	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
115	C₂×C₃×C₄	#135	1	2	yes	—
116	C₂×C₃×C₄	AGL(1,ℤ/8)	1	4	yes	—
117	C₂×C₃×C₄	#137	1	4	yes	—
118	C₂×Dic₃	—	4	4	no	—
119	C₂×Dic₃	—	8	4	no	—
120	C₂×Dic₃	—	8	2	no	—
121	C₂×Dic₃	—	8	2	no	—
122	C₂×Dic₃	—	8	2	no	—
123	Dic₆	—	4	8	no	—
124	Dic₆	—	8	4	no	—
125	C₃×C₈	—	2	4	no	—
126	C₃×C₈	—	2	4	no	—
127	C₃×C₈	—	4	2	no	—
128	C₂×Q₈	—	6	4	no	—
129	C₂×Q₈	—	12	2	no	—
130	C₂×Q₈	—	12	8	no	—
131	C₂²×C₄	—	12	2	no	—
132	C₂²×C₄	—	12	2	no	—
133	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
134	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
135	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
136	Q₁₆	—	12	16	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	Dic₁₂	1	4	yes	complemented
143	C₂×C₈	(C₂×D₃)⋊C₄	1	4	yes	—
144	C₄⋊C₄	—	6	4	no	—
145	C₄⋊C₄	—	6	4	no	—
146	C₄⋊C₄	—	12	2	no	—
147	C₄⋊C₄	—	12	2	no	—
148	C₄⋊C₄	—	12	4	no	—
149	C₄⋊C₄	—	12	4	no	—
150	C₄⋊C₄	—	12	4	no	—
151	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
152	C₄²	#235	1	1	yes	characteristic
153	C₄²	#235	1	1	yes	characteristic
154	C₄²	C₂×#67	1	1	yes	characteristic
155	C₂²×C₃	#390	1	1	yes	characteristic
156	C₂²×C₃	#494	1	1	yes	characteristic
157	C₂²×C₃	#495	1	1	yes	characteristic
158	C₂²×C₃	#496	1	2	yes	—
159	C₂²×C₃	#497	1	2	yes	—
160	C₃×C₄	#418	1	2	yes	—
161	C₃×C₄	C₄×Q₁₆	1	2	yes	—
162	C₃×C₄	#441	1	2	yes	—
163	C₃×C₄	#467	1	2	yes	—
164	C₃×C₄	#470	1	2	yes	—
165	C₃×C₄	#490	1	2	yes	—
166	Dic₃	—	8	4	no	—
167	Dic₃	—	16	2	no	—
168	C₂³	#874	1	1	yes	characteristic
169	Q₈	—	12	8	no	—
170	Q₈	—	24	4	no	—
171	C₂×C₄	—	12	4	no	—
172	C₂×C₄	—	24	4	no	—
173	C₂×C₄	—	24	2	no	—
174	C₂×C₄	—	24	2	no	—
175	C₂×C₄	—	24	2	no	—
176	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
177	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
178	C₂×C₄	#875	1	1	yes	characteristic
179	C₂×C₄	#904	1	2	yes	—
180	C₂×C₄	C₂×Dic₁₂	1	2	yes	—
181	C₂×C₄	#908	1	4	yes	—
182	C₂×C₄	#909	1	4	yes	—
183	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
184	C₂×C₄	#930	1	1	yes	characteristic
185	C₂×C₄	#930	1	1	yes	characteristic
186	C₈	—	2	4	no	—
187	C₈	—	2	4	no	—
188	C₈	—	4	2	no	—
189	C₂×C₃	#1720	1	2	yes	—
190	C₂×C₃	#1723	1	2	yes	—
191	C₂×C₃	#1961	1	1	yes	characteristic
192	C₂×C₃	#1969	1	1	yes	characteristic
193	C₂×C₃	#1971	1	1	yes	characteristic
194	C₂²	#4696	1	1	yes	characteristic
195	C₂²	#4699	1	1	yes	characteristic
196	C₂²	#4712	1	2	yes	—
197	C₂²	#4715	1	2	yes	—
198	C₂²	#4939	1	1	yes	characteristic
199	C₄	—	24	4	no	—
200	C₄	—	48	2	no	—
201	C₄	C₄×Dic₁₂	1	2	yes	—
202	C₄	#4713	1	2	yes	—
203	C₄	#4716	1	2	yes	—
204	C₄	#5116	1	2	yes	—
205	C₄	#5131	1	2	yes	—
206	C₄	#5137	1	2	yes	—
207	C₃	#9835	1	1	yes	characteristic, complemented, Sylow
208	C₂	#68097	1	2	yes	—
209	C₂	#68099	1	2	yes	—
210	C₂	#69775	1	1	yes	characteristic
211	C₂	#69823	1	1	yes	characteristic
212	C₂	#69824	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	#9835
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
14. #4699	C₄	semidirect
72. C₂×C₃×C₈	Q₁₆	semidirect
142. C₂×C₈	Dic₁₂	semidirect
207. C₃	#9835	semidirect

Maximal subgroups

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

subgroup	index
C₂×#4699	2
C₂×C₃×#322	2
#69777	2
#69762	2
#9835	3