Subgroups — #10694994

Order 768.

494 conjugacy classes of subgroups.

The subgroup lattice has 216 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	#69762	C₂	1	1	yes	maximal, characteristic
6	#9831	—	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×#816	—	2	4	no	—
13	#4935	C₂²	1	2	yes	—
14	#4682	—	2	2	no	—
15	#4682	C₄	1	4	yes	complemented
16	C₃×#322	—	2	2	no	—
17	#4551	—	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	#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	#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	#816	—	4	8	no	—
46	C₂×C₄⋊Q₈	—	3	2	no	—
47	#500	—	3	4	no	—
48	#500	—	6	2	no	—
49	C₂×#107	—	6	2	no	—
50	C₂×#103	—	6	4	no	—
51	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
52	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
54	#384	—	3	2	no	—
55	#340	—	6	2	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	4	no	—
65	C₂×Dic₆	—	4	4	no	—
66	C₂×Dic₆	—	4	2	no	—
67	C₂×C₃×C₈	—	2	2	no	—
68	C₂×C₃×C₈	—	2	2	no	—
69	C₂×C₃×C₈	—	2	4	no	—
70	C₂×C₃×C₈	—	2	2	no	—
71	C₂×C₃×C₈	—	2	4	no	—
72	C₂×C₃×C₈	C₂²⋊C₄	1	4	yes	—
73	C₂×C₃×C₈	D₈	1	4	yes	—
74	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
75	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
76	C₃×C₄²	#41	1	1	yes	characteristic
77	C₃×C₄²	#41	1	1	yes	characteristic
78	C₄⋊Dic₃	—	2	4	no	—
79	C₄⋊Dic₃	—	2	4	no	—
80	C₄⋊Dic₃	—	4	2	no	—
81	C₄⋊Dic₃	—	4	2	no	—
82	C₄⋊Dic₃	—	4	4	no	—
83	C₄⋊Dic₃	—	4	4	no	—
84	C₄⋊Dic₃	—	4	4	no	—
85	C₂²×Q₈	—	6	2	no	—
86	C₂²×C₈	D₁₂	1	2	yes	—
87	C₂²×C₈	D₁₂	1	2	yes	—
88	C₂²×C₈	D₁₂	1	2	yes	—
89	C₄⋊Q₈	—	3	4	no	—
90	C₄⋊Q₈	—	6	2	no	—
91	C₄⋊Q₈	—	6	2	no	—
92	C₄⋊Q₈	—	6	2	no	—
93	C₂×C₄⋊C₄	—	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×C₄²	C₂²×D₃	1	1	yes	characteristic
99	#107	—	6	8	no	—
100	#103	—	12	8	no	—
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×D₈	1	2	yes	—
115	C₂×C₃×C₄	#135	1	2	yes	—
116	C₂×C₃×C₄	#137	1	4	yes	—
117	C₂×C₃×C₄	#137	1	4	yes	—
118	C₂×Dic₃	—	4	4	no	—
119	C₂×Dic₃	—	8	2	no	—
120	C₂×Dic₃	—	8	2	no	—
121	C₂×Dic₃	—	8	2	no	—
122	C₂×Dic₃	—	8	2	no	—
123	C₂×Dic₃	—	8	2	no	—
124	Dic₆	—	4	8	no	—
125	Dic₆	—	8	4	no	—
126	C₃×C₈	—	2	4	no	—
127	C₃×C₈	—	2	4	no	—
128	C₃×C₈	—	4	2	no	—
129	C₂×Q₈	—	6	4	no	—
130	C₂×Q₈	—	12	2	no	—
131	C₂×Q₈	—	12	4	no	—
132	C₂×Q₈	—	12	4	no	—
133	C₂²×C₄	—	12	2	no	—
134	C₂²×C₄	—	12	2	no	—
135	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
136	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
137	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	D₂₄	1	4	yes	—
144	C₂×C₈	(C₂×D₃)⋊C₄	1	4	yes	—
145	C₄⋊C₄	—	6	4	no	—
146	C₄⋊C₄	—	6	4	no	—
147	C₄⋊C₄	—	12	2	no	—
148	C₄⋊C₄	—	12	2	no	—
149	C₄⋊C₄	—	12	4	no	—
150	C₄⋊C₄	—	12	4	no	—
151	C₄⋊C₄	—	12	4	no	—
152	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
153	C₄²	#235	1	1	yes	characteristic
154	C₄²	#235	1	1	yes	characteristic
155	C₄²	C₂×#67	1	1	yes	characteristic
156	C₂²×C₃	#390	1	1	yes	characteristic
157	C₂²×C₃	#493	1	1	yes	characteristic
158	C₂²×C₃	#495	1	1	yes	characteristic
159	C₂²×C₃	#497	1	2	yes	—
160	C₂²×C₃	#497	1	2	yes	—
161	C₃×C₄	#419	1	2	yes	—
162	C₃×C₄	C₄×D₈	1	2	yes	—
163	C₃×C₄	#441	1	2	yes	—
164	C₃×C₄	#466	1	2	yes	—
165	C₃×C₄	#470	1	2	yes	—
166	C₃×C₄	#491	1	2	yes	—
167	Dic₃	—	8	4	no	—
168	Dic₃	—	16	2	no	—
169	C₂³	#874	1	1	yes	characteristic
170	Q₈	—	12	8	no	—
171	Q₈	—	24	4	no	—
172	C₂×C₄	—	12	4	no	—
173	C₂×C₄	—	24	2	no	—
174	C₂×C₄	—	24	2	no	—
175	C₂×C₄	—	24	2	no	—
176	C₂×C₄	—	24	2	no	—
177	C₂×C₄	—	24	2	no	—
178	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
179	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
180	C₂×C₄	#875	1	1	yes	characteristic
181	C₂×C₄	C₂×D₂₄	1	2	yes	—
182	C₂×C₄	#904	1	2	yes	—
183	C₂×C₄	#909	1	4	yes	—
184	C₂×C₄	#909	1	4	yes	—
185	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
186	C₂×C₄	#930	1	1	yes	characteristic
187	C₂×C₄	#930	1	1	yes	characteristic
188	C₈	—	2	4	no	—
189	C₈	—	2	4	no	—
190	C₈	—	4	2	no	—
191	C₂×C₃	#1718	1	2	yes	—
192	C₂×C₃	#1722	1	2	yes	—
193	C₂×C₃	#1960	1	1	yes	characteristic
194	C₂×C₃	#1970	1	1	yes	characteristic
195	C₂×C₃	#1971	1	1	yes	characteristic
196	C₂²	#4695	1	1	yes	characteristic
197	C₂²	#4696	1	1	yes	characteristic
198	C₂²	#4715	1	2	yes	—
199	C₂²	#4715	1	2	yes	—
200	C₂²	#4939	1	1	yes	characteristic
201	C₄	—	24	4	no	—
202	C₄	—	48	2	no	—
203	C₄	C₄×D₂₄	1	2	yes	—
204	C₄	#4703	1	2	yes	—
205	C₄	#4716	1	2	yes	—
206	C₄	#5115	1	2	yes	—
207	C₄	#5120	1	2	yes	—
208	C₄	#5137	1	2	yes	—
209	C₃	#9831	1	1	yes	characteristic, complemented, Sylow
210	C₂	#68096	1	2	yes	—
211	C₂	#68098	1	2	yes	—
212	C₂	#69776	1	1	yes	characteristic
213	C₂	#69822	1	1	yes	characteristic
214	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	#9831
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
15. #4682	C₄	semidirect
209. C₃	#9831	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
#69762	2
#9831	3