Subgroups — #10713282

Order 768.

342 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×C₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#4505	C₂	1	2	yes	maximal
3	#69765	C₂	1	1	yes	maximal, characteristic
4	C₄×#818	C₂	1	1	yes	maximal, characteristic
5	#67806	C₂	1	2	yes	maximal
6	#13177	—	3	1	no	maximal, Sylow
7	C₂²×C₃×C₁₆	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×#818	—	2	1	no	—
10	C₂×#818	C₂²	1	2	yes	—
11	C₂×#818	C₂²	1	1	yes	characteristic
12	C₂×#818	C₂²	1	1	yes	characteristic
13	#4935	—	2	1	no	—
14	C₄×C₄⋊Dic₃	—	2	1	no	—
15	C₃×C₄⋊C₁₆	C₄	1	4	yes	complemented
16	#4551	—	2	1	no	—
17	#4505	—	2	4	no	—
18	C₂×#367	—	3	2	no	—
19	C₂×C₄⋊C₁₆	D₃	1	1	yes	characteristic
20	#1845	—	3	1	no	—
21	C₄×#107	—	3	1	no	—
22	#1380	—	3	2	no	—
23	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
25	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
26	C₂×C₄⋊Dic₃	—	2	1	no	—
27	C₂×C₄⋊Dic₃	—	2	1	no	—
28	C₂×C₄⋊Dic₃	—	2	2	no	—
29	C₂×C₄⋊Dic₃	—	4	1	no	—
30	C₂×C₄⋊Dic₃	—	4	1	no	—
31	C₂×C₄×Dic₃	—	4	1	no	—
32	C₂×C₃×C₁₆	—	2	2	no	—
33	C₂×C₃×C₁₆	—	2	2	no	—
34	C₂×C₃×C₁₆	C₂×C₄	1	4	yes	—
35	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
36	C₃×C₄×C₈	Q₈	1	2	yes	—
37	C₃⋊#95	—	4	1	no	—
38	#818	—	2	1	no	—
39	#818	—	2	1	no	—
40	#818	—	2	2	no	—
41	#818	—	2	2	no	—
42	#818	—	4	2	no	—
43	#818	D₄	1	2	yes	—
44	#818	Q₈	1	2	yes	—
45	C₂²×C₁₆	C₂×D₃	1	2	yes	—
46	C₂×#107	—	3	1	no	—
47	C₂×#107	—	3	1	no	—
48	C₂×#107	—	3	2	no	—
49	C₂×#107	—	6	1	no	—
50	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
51	#384	—	6	1	no	—
52	C₄×C₄⋊C₄	—	6	1	no	—
53	#367	—	6	4	no	—
54	C₄⋊C₁₆	Dic₃	1	4	yes	complemented
55	#340	—	6	1	no	—
56	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
57	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
58	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₃	—	4	1	no	—
60	C₂²×Dic₃	—	4	1	no	—
61	C₂²×Dic₃	—	4	2	no	—
62	C₂×C₃×C₈	—	2	1	no	—
63	C₂×C₃×C₈	—	2	1	no	—
64	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
65	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₃×C₈	#41	1	1	yes	characteristic
70	C₂×C₃×C₈	#41	1	1	yes	characteristic
71	C₃×C₄²	C₄⋊C₄	1	2	yes	—
72	C₃×C₄²	Q₁₆	1	2	yes	—
73	C₄⋊Dic₃	—	2	2	no	—
74	C₄⋊Dic₃	—	2	2	no	—
75	C₄⋊Dic₃	—	4	1	no	—
76	C₄⋊Dic₃	—	4	1	no	—
77	C₄⋊Dic₃	—	4	2	no	—
78	C₄⋊Dic₃	—	4	2	no	—
79	C₄⋊Dic₃	—	8	2	no	—
80	C₄⋊Dic₃	—	8	2	no	—
81	C₄×Dic₃	—	8	2	no	—
82	C₃×C₁₆	—	2	4	no	—
83	C₂²×C₈	D₁₂	1	1	yes	characteristic
84	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
85	C₂×C₄⋊C₄	—	6	1	no	—
86	C₂×C₄⋊C₄	—	6	1	no	—
87	C₂×C₄⋊C₄	—	6	2	no	—
88	C₂×C₄⋊C₄	—	12	1	no	—
89	C₂×C₄⋊C₄	—	12	1	no	—
90	C₂×C₄²	—	12	1	no	—
91	C₂×C₄²	D₁₂	1	1	yes	characteristic
92	C₂×C₁₆	—	2	2	no	—
93	C₂×C₁₆	—	2	2	no	—
94	C₂×C₁₆	C₂×Dic₃	1	4	yes	—
95	#107	—	3	2	no	—
96	#107	—	3	2	no	—
97	#107	—	6	1	no	—
98	#107	—	6	1	no	—
99	#107	—	6	2	no	—
100	#107	—	6	2	no	—
101	#107	—	12	2	no	—
102	C₄×C₈	Dic₆	1	2	yes	—
103	C₄×C₈	C₂×Dic₃	1	2	yes	—
104	#95	—	12	1	no	—
105	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
106	C₂×C₃×C₄	—	2	1	no	—
107	C₂×C₃×C₄	—	2	1	no	—
108	C₂×C₃×C₄	#107	1	2	yes	—
109	C₂×C₃×C₄	#107	1	2	yes	—
110	C₂×C₃×C₄	SD₃₂	1	2	yes	—
111	C₂×C₃×C₄	SD₃₂	1	2	yes	—
112	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₃×C₄	#121	1	1	yes	characteristic
114	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
116	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₃×C₄	#137	1	1	yes	characteristic
118	C₂×Dic₃	—	4	2	no	—
119	C₂×Dic₃	—	4	2	no	—
120	C₂×Dic₃	—	8	1	no	—
121	C₂×Dic₃	—	8	1	no	—
122	C₂×Dic₃	—	8	1	no	—
123	C₂×Dic₃	—	8	1	no	—
124	C₂×Dic₃	—	8	2	no	—
125	C₂×Dic₃	—	8	2	no	—
126	C₂×Dic₃	—	8	2	no	—
127	C₃×C₈	—	2	2	no	—
128	C₃×C₈	C₄×D₄	1	1	yes	characteristic
129	C₃×C₈	C₄×Q₈	1	1	yes	characteristic
130	C₃×C₈	#125	1	1	yes	characteristic
131	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
132	C₂²×C₄	—	12	1	no	—
133	C₂²×C₄	—	12	1	no	—
134	C₂²×C₄	—	12	2	no	—
135	C₂²×C₄	D₂₄	1	1	yes	characteristic
136	C₂²×C₄	D₂₄	1	1	yes	characteristic
137	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	C₄⋊Dic₃	1	2	yes	—
141	C₂×C₈	C₂×Dic₆	1	1	yes	characteristic
142	C₂×C₈	D₃×D₄	1	1	yes	characteristic
143	C₂×C₈	#237	1	1	yes	characteristic
144	C₂×C₈	D₃×Q₈	1	1	yes	characteristic
145	C₂×C₈	#239	1	1	yes	characteristic
146	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
147	C₄⋊C₄	—	6	2	no	—
148	C₄⋊C₄	—	6	2	no	—
149	C₄⋊C₄	—	12	1	no	—
150	C₄⋊C₄	—	12	1	no	—
151	C₄⋊C₄	—	12	2	no	—
152	C₄⋊C₄	—	12	2	no	—
153	C₄⋊C₄	—	24	2	no	—
154	C₄⋊C₄	—	24	2	no	—
155	C₄²	—	24	2	no	—
156	C₄²	Dic₁₂	1	2	yes	—
157	C₄²	C₄⋊Dic₃	1	2	yes	—
158	C₁₆	—	2	4	no	—
159	C₂²×C₃	C₂×#107	1	1	yes	characteristic
160	C₂²×C₃	#459	1	1	yes	characteristic
161	C₂²×C₃	#474	1	1	yes	characteristic
162	C₂²×C₃	C₂×SD₃₂	1	1	yes	characteristic
163	C₂²×C₃	C₂×SD₃₂	1	1	yes	characteristic
164	C₂²×C₃	#509	1	1	yes	characteristic
165	C₂²×C₃	#510	1	1	yes	characteristic
166	C₃×C₄	—	2	2	no	—
167	C₃×C₄	#367	1	4	yes	complemented
168	C₃×C₄	#389	1	1	yes	characteristic
169	C₃×C₄	#428	1	1	yes	characteristic
170	C₃×C₄	#462	1	1	yes	characteristic
171	C₃×C₄	#477	1	1	yes	characteristic
172	Dic₃	—	8	2	no	—
173	Dic₃	—	8	2	no	—
174	Dic₃	—	16	2	no	—
175	C₂³	C₂×D₂₄	1	1	yes	characteristic
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	12	2	no	—
179	C₂×C₄	—	12	2	no	—
180	C₂×C₄	—	24	1	no	—
181	C₂×C₄	—	24	1	no	—
182	C₂×C₄	—	24	1	no	—
183	C₂×C₄	—	24	1	no	—
184	C₂×C₄	—	24	2	no	—
185	C₂×C₄	—	24	2	no	—
186	C₂×C₄	—	24	2	no	—
187	C₂×C₄	#800	1	2	yes	—
188	C₂×C₄	#800	1	2	yes	—
189	C₂×C₄	#818	1	2	yes	—
190	C₂×C₄	#818	1	2	yes	—
191	C₂×C₄	#895	1	1	yes	characteristic
192	C₂×C₄	#897	1	1	yes	characteristic
193	C₂×C₄	C₂×Dic₁₂	1	1	yes	characteristic
194	C₂×C₄	#908	1	1	yes	characteristic
195	C₂×C₄	#909	1	1	yes	characteristic
196	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
197	C₈	—	2	2	no	—
198	C₈	#888	1	1	yes	characteristic
199	C₈	#890	1	1	yes	characteristic
200	C₈	D₄×Dic₃	1	1	yes	characteristic
201	C₈	Q₈×Dic₃	1	1	yes	characteristic
202	C₂×C₃	#1947	1	1	yes	characteristic
203	C₂×C₃	C₂×#367	1	1	yes	characteristic
204	C₂×C₃	#2159	1	1	yes	characteristic
205	C₂×C₃	#2207	1	1	yes	characteristic
206	C₂×C₃	#2208	1	1	yes	characteristic
207	C₂×C₃	#2225	1	1	yes	characteristic
208	C₂×C₃	#2226	1	1	yes	characteristic
209	C₂²	#4843	1	1	yes	characteristic
210	C₂²	#4846	1	1	yes	characteristic
211	C₂²	C₂×#800	1	1	yes	characteristic
212	C₂²	C₂×#800	1	1	yes	characteristic
213	C₂²	#4908	1	1	yes	characteristic
214	C₂²	#4909	1	1	yes	characteristic
215	C₂²	C₂×#818	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	24	2	no	—
218	C₄	—	24	2	no	—
219	C₄	—	48	2	no	—
220	C₄	#4505	1	4	yes	complemented
221	C₄	#4849	1	1	yes	characteristic
222	C₄	#4850	1	1	yes	characteristic
223	C₄	#4984	1	1	yes	characteristic
224	C₄	#5121	1	1	yes	characteristic
225	C₃	#13177	1	1	yes	characteristic, complemented, Sylow
226	C₂	#69287	1	1	yes	characteristic
227	C₂	#69290	1	1	yes	characteristic
228	C₂	#69292	1	1	yes	characteristic
229	C₂	#69295	1	1	yes	characteristic
230	C₂	#70453	1	1	yes	characteristic
231	C₂	C₂×#4505	1	1	yes	characteristic
232	C₂	#71428	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₄
commutator	C₂×C₃×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₂×C₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₃	C₂×D₈
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₃
hypocenter	C₃	#13177
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
15. C₃×C₄⋊C₁₆	C₄	semidirect
54. C₄⋊C₁₆	Dic₃	semidirect
167. C₃×C₄	#367	semidirect
220. C₄	#4505	semidirect
225. C₃	#13177	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄⋊C₁₆	2
C₂×#4505	2
#69765	2
C₄×#818	2
#67806	2
#13177	3