Subgroups — #10694946

Order 768.

526 conjugacy classes of subgroups.

The subgroup lattice has 234 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×C₈²	C₂	1	1	yes	maximal, characteristic
3	#69777	C₂	1	2	yes	maximal
4	#69771	C₂	1	2	yes	maximal
5	#69765	C₂	1	1	yes	maximal, characteristic
6	#9823	—	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×#818	—	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×C₈²	—	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	#1845	—	3	1	no	—
23	C₂×C₈²	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	#818	—	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×#107	—	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	C₈²	—	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	2	no	—
71	C₂×C₃×C₈	—	2	2	no	—
72	C₂×C₃×C₈	C₂²⋊C₄	1	4	yes	—
73	C₂×C₃×C₈	D₈	1	4	yes	—
74	C₂×C₃×C₈	SD₁₆	1	4	yes	—
75	C₂×C₃×C₈	Q₁₆	1	4	yes	—
76	C₂×C₃×C₈	Q₁₆	1	4	yes	—
77	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
78	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
79	C₃×C₄²	#41	1	1	yes	characteristic
80	C₃×C₄²	#41	1	1	yes	characteristic
81	C₄⋊Dic₃	—	2	4	no	—
82	C₄⋊Dic₃	—	2	4	no	—
83	C₄⋊Dic₃	—	4	2	no	—
84	C₄⋊Dic₃	—	4	2	no	—
85	C₄⋊Dic₃	—	4	4	no	—
86	C₄⋊Dic₃	—	4	4	no	—
87	C₄⋊Dic₃	—	4	4	no	—
88	C₂²×Q₈	—	6	2	no	—
89	C₂²×C₈	D₁₂	1	2	yes	—
90	C₂²×C₈	D₁₂	1	2	yes	—
91	C₂²×C₈	D₁₂	1	2	yes	—
92	C₄⋊Q₈	—	3	4	no	—
93	C₄⋊Q₈	—	6	2	no	—
94	C₄⋊Q₈	—	6	2	no	—
95	C₄⋊Q₈	—	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×C₄⋊C₄	—	6	2	no	—
101	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
102	#107	—	6	8	no	—
103	#107	—	12	4	no	—
104	#103	—	12	8	no	—
105	C₄×C₈	—	2	1	no	—
106	C₄×C₈	—	2	1	no	—
107	C₄×C₈	—	2	1	no	—
108	C₄×C₈	—	2	1	no	—
109	C₄×C₈	C₄×D₃	1	2	yes	—
110	C₄×C₈	#67	1	2	yes	—
111	C₂³×C₃	#127	1	1	yes	characteristic
112	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
114	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
115	C₂×C₃×C₄	#121	1	1	yes	characteristic
116	C₂×C₃×C₄	#121	1	1	yes	characteristic
117	C₂×C₃×C₄	#124	1	1	yes	characteristic
118	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
119	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
120	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
121	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
122	C₂×C₃×C₄	#135	1	2	yes	—
123	C₂×C₃×C₄	#135	1	2	yes	—
124	C₂×Dic₃	—	4	4	no	—
125	C₂×Dic₃	—	8	2	no	—
126	C₂×Dic₃	—	8	2	no	—
127	C₂×Dic₃	—	8	2	no	—
128	C₂×Dic₃	—	8	2	no	—
129	C₂×Dic₃	—	8	2	no	—
130	Dic₆	—	4	8	no	—
131	Dic₆	—	8	4	no	—
132	C₃×C₈	—	2	4	no	—
133	C₃×C₈	—	2	4	no	—
134	C₃×C₈	#103	1	8	yes	complemented
135	C₂×Q₈	—	6	4	no	—
136	C₂×Q₈	—	12	2	no	—
137	C₂×Q₈	—	12	4	no	—
138	C₂×Q₈	—	12	4	no	—
139	C₂²×C₄	—	12	2	no	—
140	C₂²×C₄	—	12	2	no	—
141	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
142	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
143	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	#204	1	4	yes	—
149	C₂×C₈	D₂₄	1	4	yes	—
150	C₂×C₈	Dic₁₂	1	4	yes	—
151	C₂×C₈	Dic₁₂	1	4	yes	—
152	C₂×C₈	(C₂×D₃)⋊C₄	1	4	yes	—
153	C₄⋊C₄	—	6	4	no	—
154	C₄⋊C₄	—	6	4	no	—
155	C₄⋊C₄	—	12	2	no	—
156	C₄⋊C₄	—	12	2	no	—
157	C₄⋊C₄	—	12	4	no	—
158	C₄⋊C₄	—	12	4	no	—
159	C₄⋊C₄	—	12	4	no	—
160	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
161	C₄²	#235	1	1	yes	characteristic
162	C₄²	#235	1	1	yes	characteristic
163	C₄²	C₂×#67	1	1	yes	characteristic
164	C₂²×C₃	#390	1	1	yes	characteristic
165	C₂²×C₃	#492	1	1	yes	characteristic
166	C₂²×C₃	#493	1	1	yes	characteristic
167	C₂²×C₃	#494	1	1	yes	characteristic
168	C₂²×C₃	#494	1	1	yes	characteristic
169	C₂²×C₃	#495	1	1	yes	characteristic
170	C₂²×C₃	#495	1	1	yes	characteristic
171	C₃×C₄	C₂×#103	1	2	yes	—
172	C₃×C₄	C₄×D₈	1	2	yes	—
173	C₃×C₄	C₄×Q₁₆	1	2	yes	—
174	C₃×C₄	#466	1	2	yes	—
175	C₃×C₄	#467	1	2	yes	—
176	C₃×C₄	#487	1	2	yes	—
177	Dic₃	—	8	4	no	—
178	Dic₃	—	16	2	no	—
179	C₂³	#874	1	1	yes	characteristic
180	Q₈	—	12	8	no	—
181	Q₈	—	24	4	no	—
182	C₂×C₄	—	12	4	no	—
183	C₂×C₄	—	24	2	no	—
184	C₂×C₄	—	24	2	no	—
185	C₂×C₄	—	24	2	no	—
186	C₂×C₄	—	24	2	no	—
187	C₂×C₄	—	24	2	no	—
188	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
189	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
190	C₂×C₄	#875	1	1	yes	characteristic
191	C₂×C₄	C₂×#204	1	2	yes	—
192	C₂×C₄	C₂×D₂₄	1	2	yes	—
193	C₂×C₄	#904	1	2	yes	—
194	C₂×C₄	#904	1	2	yes	—
195	C₂×C₄	C₂×Dic₁₂	1	2	yes	—
196	C₂×C₄	C₂×Dic₁₂	1	2	yes	—
197	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
198	C₂×C₄	#930	1	1	yes	characteristic
199	C₂×C₄	#930	1	1	yes	characteristic
200	C₈	—	2	4	no	—
201	C₈	—	2	4	no	—
202	C₈	#816	1	8	yes	complemented
203	C₂×C₃	#1706	1	1	yes	characteristic
204	C₂×C₃	#1707	1	1	yes	characteristic
205	C₂×C₃	#1710	1	1	yes	characteristic
206	C₂×C₃	#1713	1	1	yes	characteristic
207	C₂×C₃	#1956	1	1	yes	characteristic
208	C₂×C₃	#1969	1	1	yes	characteristic
209	C₂×C₃	#1970	1	1	yes	characteristic
210	C₂²	#4693	1	1	yes	characteristic
211	C₂²	#4695	1	1	yes	characteristic
212	C₂²	#4696	1	1	yes	characteristic
213	C₂²	#4696	1	1	yes	characteristic
214	C₂²	#4699	1	1	yes	characteristic
215	C₂²	#4699	1	1	yes	characteristic
216	C₂²	#4939	1	1	yes	characteristic
217	C₄	—	24	4	no	—
218	C₄	—	48	2	no	—
219	C₄	#4681	1	2	yes	—
220	C₄	C₄×D₂₄	1	2	yes	—
221	C₄	C₄×Dic₁₂	1	2	yes	—
222	C₄	C₂×#816	1	2	yes	—
223	C₄	#5115	1	2	yes	—
224	C₄	#5116	1	2	yes	—
225	C₃	#9823	1	1	yes	characteristic, complemented, Sylow
226	C₂	#68058	1	1	yes	characteristic
227	C₂	#68063	1	1	yes	characteristic
228	C₂	#68064	1	1	yes	characteristic
229	C₂	#68065	1	1	yes	characteristic
230	C₂	#69771	1	1	yes	characteristic
231	C₂	#69775	1	1	yes	characteristic
232	C₂	#69776	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#874
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂×C₃×C₈²	C₂
socle, prime radical	C₂³×C₃	#127
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×C₈²	D₃
hypocenter	C₃	#9823
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
16. #4682	C₄	semidirect
134. C₃×C₈	#103	semidirect
202. C₈	#816	semidirect
225. C₃	#9823	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×C₈²	2
#69777	2
#69771	2
#69765	2
#9823	3