Subgroups — #10713294

Order 768.

286 conjugacy classes of subgroups.

The subgroup lattice has 220 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×#4506	C₂	1	2	yes	maximal
3	#69792	C₂	1	1	yes	maximal, characteristic
4	C₄×#819	C₂	1	1	yes	maximal, characteristic
5	#67808	C₂	1	2	yes	maximal
6	#13179	—	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×#819	—	2	1	no	—
10	C₂×#819	C₂²	1	2	yes	—
11	C₂×#819	C₂²	1	1	yes	characteristic
12	C₂×#819	C₂²	1	1	yes	characteristic
13	#4924	—	2	1	no	—
14	C₄×#208	—	2	1	no	—
15	C₃×C₄⋊C₁₆	C₄	1	2	yes	—
16	C₃×C₄⋊C₁₆	C₄	1	2	yes	—
17	#4552	—	2	1	no	—
18	#4506	—	2	4	no	—
19	C₂×#368	—	3	2	no	—
20	C₂×C₄⋊C₁₆	D₃	1	1	yes	characteristic
21	#1848	—	3	1	no	—
22	C₄×#108	—	3	1	no	—
23	#1381	—	3	2	no	—
24	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
25	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
26	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
27	C₂×#208	—	2	1	no	—
28	C₂×#208	—	2	1	no	—
29	C₂×#208	—	2	2	no	—
30	C₂×C₃×C₁₆	—	2	2	no	—
31	C₂×C₃×C₁₆	—	2	2	no	—
32	C₂×C₃×C₁₆	C₂×C₄	1	4	yes	—
33	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
34	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄×C₈	Q₈	1	1	yes	characteristic
36	C₃×C₄×C₈	Q₈	1	1	yes	characteristic
37	#819	—	2	1	no	—
38	#819	—	2	1	no	—
39	#819	—	2	2	no	—
40	#819	—	2	2	no	—
41	#819	—	4	2	no	—
42	#819	D₄	1	2	yes	—
43	#819	Q₈	1	2	yes	—
44	C₄⋊(C₃⋊C₈)	—	4	1	no	—
45	C₄⋊(C₃⋊C₈)	—	4	1	no	—
46	#803	—	4	1	no	—
47	C₄×C₃⋊C₈	—	4	1	no	—
48	C₂²×C₁₆	C₂×D₃	1	2	yes	—
49	C₂×#108	—	3	1	no	—
50	C₂×#108	—	3	1	no	—
51	C₂×#108	—	3	2	no	—
52	C₂×#108	—	6	1	no	—
53	#423	—	6	1	no	—
54	C₄×M₁₆	—	6	1	no	—
55	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
56	#368	—	6	4	no	—
57	C₄⋊C₁₆	Dic₃	1	2	yes	—
58	C₄⋊C₁₆	Dic₃	1	2	yes	—
59	#341	—	6	1	no	—
60	C₂²×C₃×C₄	Q₁₆	1	1	yes	characteristic
61	C₂²×C₃×C₄	Q₁₆	1	1	yes	characteristic
62	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×C₈	—	2	1	no	—
64	C₂×C₃×C₈	—	2	1	no	—
65	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₃×C₈	#41	1	1	yes	characteristic
72	C₂×C₃×C₈	#41	1	1	yes	characteristic
73	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
74	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
75	C₃×C₄²	D₈	1	1	yes	characteristic
76	C₃×C₄²	D₈	1	1	yes	characteristic
77	#208	—	2	2	no	—
78	#208	—	2	2	no	—
79	#208	—	4	1	no	—
80	#208	—	4	1	no	—
81	#208	—	4	2	no	—
82	#208	—	4	2	no	—
83	C₂×C₃⋊C₈	—	4	1	no	—
84	C₂×C₃⋊C₈	—	4	1	no	—
85	C₂×C₃⋊C₈	—	4	2	no	—
86	C₃×C₁₆	—	2	4	no	—
87	C₂×M₁₆	—	6	1	no	—
88	C₂×M₁₆	—	6	1	no	—
89	C₂×M₁₆	—	6	2	no	—
90	C₂²×C₈	D₁₂	1	1	yes	characteristic
91	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
92	C₂×C₄²	D₁₂	1	1	yes	characteristic
93	C₂×C₁₆	—	2	2	no	—
94	C₂×C₁₆	—	2	2	no	—
95	C₂×C₁₆	C₂×Dic₃	1	4	yes	—
96	#108	—	3	2	no	—
97	#108	—	3	2	no	—
98	#108	—	6	2	no	—
99	#108	—	6	1	no	—
100	#108	—	6	1	no	—
101	#108	—	6	2	no	—
102	#108	—	12	2	no	—
103	C₄⋊C₈	—	12	1	no	—
104	C₄⋊C₈	—	12	1	no	—
105	#97	—	12	1	no	—
106	C₄×C₈	—	12	1	no	—
107	C₄×C₈	Dic₆	1	1	yes	characteristic
108	C₄×C₈	Dic₆	1	1	yes	characteristic
109	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
110	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
111	C₂³×C₃	C₂×Q₁₆	1	1	yes	characteristic
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	—	4	1	no	—
114	C₂×C₃×C₄	#107	1	1	yes	characteristic
115	C₂×C₃×C₄	#107	1	1	yes	characteristic
116	C₂×C₃×C₄	#107	1	1	yes	characteristic
117	C₂×C₃×C₄	#107	1	1	yes	characteristic
118	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₃×C₄	#121	1	1	yes	characteristic
120	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
122	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₃×C₄	#137	1	1	yes	characteristic
124	C₃×C₈	—	2	1	no	—
125	C₃×C₈	—	2	1	no	—
126	C₃×C₈	C₄×D₄	1	1	yes	characteristic
127	C₃×C₈	C₄×Q₈	1	1	yes	characteristic
128	C₃×C₈	#125	1	1	yes	characteristic
129	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
130	C₃⋊C₈	—	4	2	no	—
131	C₃⋊C₈	—	4	2	no	—
132	C₃⋊C₈	—	8	2	no	—
133	C₂²×C₄	Dic₁₂	1	1	yes	characteristic
134	C₂²×C₄	Dic₁₂	1	1	yes	characteristic
135	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
136	M₁₆	—	6	2	no	—
137	M₁₆	—	6	2	no	—
138	M₁₆	—	12	1	no	—
139	M₁₆	—	12	1	no	—
140	M₁₆	—	12	2	no	—
141	M₁₆	—	12	2	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	12	1	no	—
145	C₂×C₈	—	12	1	no	—
146	C₂×C₈	—	12	2	no	—
147	C₂×C₈	C₄⋊Dic₃	1	1	yes	characteristic
148	C₂×C₈	C₄⋊Dic₃	1	1	yes	characteristic
149	C₂×C₈	C₂×Dic₆	1	1	yes	characteristic
150	C₂×C₈	D₃×D₄	1	1	yes	characteristic
151	C₂×C₈	#237	1	1	yes	characteristic
152	C₂×C₈	D₃×Q₈	1	1	yes	characteristic
153	C₂×C₈	#239	1	1	yes	characteristic
154	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
155	C₄²	D₂₄	1	1	yes	characteristic
156	C₄²	D₂₄	1	1	yes	characteristic
157	C₄²	C₄⋊Dic₃	1	1	yes	characteristic
158	C₄²	C₄⋊Dic₃	1	1	yes	characteristic
159	C₁₆	—	2	4	no	—
160	C₂²×C₃	—	2	1	no	—
161	C₂²×C₃	—	2	1	no	—
162	C₂²×C₃	C₂×#107	1	1	yes	characteristic
163	C₂²×C₃	#462	1	1	yes	characteristic
164	C₂²×C₃	#477	1	1	yes	characteristic
165	C₃×C₄	—	2	1	no	—
166	C₃×C₄	—	2	1	no	—
167	C₃×C₄	—	2	1	no	—
168	C₃×C₄	#368	1	2	yes	complemented
169	C₃×C₄	#389	1	1	yes	characteristic
170	C₃×C₄	#428	1	1	yes	characteristic
171	C₃×C₄	#459	1	1	yes	characteristic
172	C₃×C₄	#474	1	1	yes	characteristic
173	C₂³	C₂×Dic₁₂	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	#818	1	1	yes	characteristic
177	C₂×C₄	#818	1	1	yes	characteristic
178	C₂×C₄	#818	1	1	yes	characteristic
179	C₂×C₄	#818	1	1	yes	characteristic
180	C₂×C₄	#895	1	1	yes	characteristic
181	C₂×C₄	#897	1	1	yes	characteristic
182	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
183	C₂×C₄	#908	1	1	yes	characteristic
184	C₂×C₄	#909	1	1	yes	characteristic
185	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
186	C₈	—	2	1	no	—
187	C₈	—	2	1	no	—
188	C₈	—	12	2	no	—
189	C₈	—	12	2	no	—
190	C₈	—	24	2	no	—
191	C₈	#888	1	1	yes	characteristic
192	C₈	#890	1	1	yes	characteristic
193	C₈	D₄×Dic₃	1	1	yes	characteristic
194	C₈	Q₈×Dic₃	1	1	yes	characteristic
195	C₂×C₃	—	2	1	no	—
196	C₂×C₃	—	2	1	no	—
197	C₂×C₃	#1947	1	1	yes	characteristic
198	C₂×C₃	C₂×#368	1	1	yes	characteristic
199	C₂×C₃	#2161	1	1	yes	characteristic
200	C₂²	—	2	1	no	—
201	C₂²	—	2	1	no	—
202	C₂²	#4849	1	1	yes	characteristic
203	C₂²	#4850	1	1	yes	characteristic
204	C₂²	C₂×#818	1	1	yes	characteristic
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	#4506	1	2	yes	complemented
209	C₄	#4843	1	1	yes	characteristic
210	C₄	#4846	1	1	yes	characteristic
211	C₄	#4984	1	1	yes	characteristic
212	C₄	#5121	1	1	yes	characteristic
213	C₃	#13179	1	1	yes	characteristic, complemented, Sylow
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	#70453	1	1	yes	characteristic
217	C₂	C₂×#4506	1	1	yes	characteristic
218	C₂	#71429	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₄
commutator	C₂×C₃×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₂×C₃×C₄⋊C₁₆	C₂
socle	C₂²×C₃	C₂×#107
prime radical	C₂³×C₃	C₂×Q₁₆
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₃
hypocenter	C₃	#13179
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
168. C₃×C₄	#368	semidirect
208. C₄	#4506	semidirect
213. C₃	#13179	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×#4506	2
#69792	2
C₄×#819	2
#67808	2
#13179	3