Subgroups — #29578906

Order 15616.

482 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21446756	C₂	1	2	yes	maximal
2	C₂×#16839515	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#92418	C₂	1	2	yes	maximal
4	C₂×#16839484	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Q₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#92418	C₂²	1	2	yes	—
7	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
8	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
9	#16839517	—	2	4	no	—
10	#16839515	—	2	2	no	—
11	Q₈⋊#92418	—	2	4	no	—
12	#16839484	—	2	2	no	—
13	C₂²×Q₁₆×C₆₁	C₄	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₆₁	C₂²	1	2	yes	—
15	C₄×Q₁₆×C₆₁	—	2	2	no	—
16	C₄×Q₁₆×C₆₁	—	2	2	no	—
17	C₂×#107×C₆₁	C₄	1	1	yes	characteristic
18	C₂×#103×C₆₁	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂²×#92418	—	2	1	no	—
21	C₂²×#92418	—	2	2	no	—
22	C₂²×#92418	D₄	1	2	yes	—
23	C₄⋊#92418	—	2	2	no	—
24	C₄⋊#92418	—	2	2	no	—
25	C₄×#92418	—	2	1	no	—
26	C₄×#92418	—	2	1	no	—
27	C₄×#92418	—	2	1	no	—
28	C₄×#92418	D₄	1	2	yes	—
29	C₂²×Q₈×C₆₁	C₂×C₄	1	2	yes	—
30	C₂×Q₁₆×C₆₁	—	2	4	no	—
31	C₂×Q₁₆×C₆₁	—	2	2	no	—
32	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₄×Q₈×C₆₁	—	2	2	no	—
35	C₄×Q₈×C₆₁	—	2	2	no	—
36	C₄×Q₈×C₆₁	—	2	2	no	—
37	C₄×Q₈×C₆₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₆₁	—	2	2	no	—
41	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
42	#107×C₆₁	—	2	2	no	—
43	#103×C₆₁	—	2	4	no	—
44	C₄×C₈×C₆₁	D₄	1	2	yes	—
45	C₄×C₈×C₆₁	D₄	1	2	yes	—
46	C₂×Q₈×C₆₁	—	2	2	no	—
47	C₂×Q₈×C₆₁	—	2	2	no	—
48	C₂×Q₈×C₆₁	—	2	2	no	—
49	C₂×Q₈×C₆₁	—	2	2	no	—
50	C₂×Q₈×C₆₁	—	2	2	no	—
51	C₂×Q₈×C₆₁	M₁₆	1	4	yes	—
52	C₂²×C₄×C₆₁	—	2	2	no	—
53	C₂²×C₄×C₆₁	—	2	2	no	—
54	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
57	Q₁₆×C₆₁	—	2	8	no	—
58	C₂×C₈×C₆₁	—	2	2	no	—
59	C₂×C₈×C₆₁	—	2	2	no	—
60	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
62	C₄⋊C₄×C₆₁	—	2	2	no	—
63	C₄⋊C₄×C₆₁	—	2	2	no	—
64	C₄⋊C₄×C₆₁	—	4	2	no	—
65	C₄⋊C₄×C₆₁	—	4	2	no	—
66	C₄²×C₆₁	—	4	2	no	—
67	C₄²×C₆₁	—	4	2	no	—
68	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₆₁	#41	1	1	yes	characteristic
72	C₂×#92418	—	2	2	no	—
73	C₂×#92418	—	2	2	no	—
74	C₂×#92418	—	2	2	no	—
75	C₂×#92418	—	4	2	no	—
76	C₂×#92418	—	4	2	no	—
77	C₂×#92418	—	4	4	no	—
78	C₂×#92418	Q₁₆	1	4	yes	complemented
79	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
80	Q₈×C₆₁	—	2	4	no	—
81	Q₈×C₆₁	—	2	4	no	—
82	Q₈×C₆₁	—	4	2	no	—
83	Q₈×C₆₁	—	4	2	no	—
84	C₂×C₄×C₆₁	—	2	2	no	—
85	C₂×C₄×C₆₁	—	2	4	no	—
86	C₂×C₄×C₆₁	—	4	2	no	—
87	C₂×C₄×C₆₁	—	4	2	no	—
88	C₂×C₄×C₆₁	—	4	2	no	—
89	C₂×C₄×C₆₁	—	4	4	no	—
90	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
93	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
94	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
95	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
97	C₂×C₄×C₆₁	#131	1	2	yes	—
98	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
99	C₂×C₄×C₆₁	#135	1	2	yes	—
100	C₈×C₆₁	—	2	2	no	—
101	C₈×C₆₁	—	2	2	no	—
102	#92418	—	2	4	no	—
103	#92418	—	4	2	no	—
104	#92418	—	8	2	no	—
105	#9662	—	61	1	no	maximal, Sylow
106	C₂²×C₆₁	#390	1	1	yes	characteristic
107	C₂²×C₆₁	#435	1	2	yes	—
108	C₂²×C₆₁	C₄×Q₁₆	1	2	yes	—
109	C₂²×C₆₁	#444	1	2	yes	—
110	C₄×C₆₁	—	2	1	no	—
111	C₄×C₆₁	—	2	1	no	—
112	C₄×C₆₁	—	4	4	no	—
113	C₄×C₆₁	—	8	2	no	—
114	C₄×C₆₁	#407	1	1	yes	characteristic
115	C₄×C₆₁	#408	1	1	yes	characteristic
116	C₄×C₆₁	#435	1	2	yes	—
117	C₄×C₆₁	#467	1	2	yes	—
118	C₄×C₆₁	#494	1	1	yes	characteristic
119	C₄×C₆₁	#495	1	1	yes	characteristic
120	C₂×C₄×Q₁₆	D₆₁	1	1	yes	characteristic
121	#1984	—	61	2	no	—
122	C₂×#335	—	61	1	no	—
123	C₂×Q₈⋊C₈	—	61	2	no	—
124	C₂×#322	—	61	1	no	—
125	C₂×C₆₁	#1584	1	4	yes	—
126	C₂×C₆₁	#1955	1	1	yes	characteristic
127	C₂×C₆₁	#1969	1	1	yes	characteristic
128	C₂×C₆₁	#1975	1	1	yes	characteristic
129	C₂²×Q₁₆	Dic₆₁	1	1	yes	characteristic
130	C₂×C₄×Q₈	C₂×D₆₁	1	2	yes	—
131	C₄×Q₁₆	—	2	2	no	—
132	C₄×Q₁₆	—	2	2	no	—
133	C₂×#107	Dic₆₁	1	1	yes	characteristic
134	C₂×C₄⋊C₈	—	61	2	no	—
135	C₂×#103	Dic₆₁	1	2	yes	—
136	C₂×C₄×C₈	—	61	1	no	—
137	C₂×C₄×C₈	—	61	1	no	—
138	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
139	#336	—	122	4	no	—
140	#335	—	122	2	no	—
141	Q₈⋊C₈	—	122	4	no	—
142	#322	—	122	2	no	—
143	C₆₁	#9662	1	1	yes	characteristic, complemented, Sylow
144	C₂²×Q₈	C₂×Dic₆₁	1	2	yes	—
145	C₂×Q₁₆	—	2	4	no	—
146	C₂×Q₁₆	—	2	2	no	—
147	C₂²×C₈	—	61	2	no	—
148	C₂²×C₈	—	122	1	no	—
149	C₂²×C₈	—	122	2	no	—
150	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
151	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
152	C₄×Q₈	—	2	2	no	—
153	C₄×Q₈	—	2	2	no	—
154	C₄×Q₈	—	2	2	no	—
155	C₄×Q₈	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
158	C₂×C₄²	—	2	2	no	—
159	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
160	#107	—	2	2	no	—
161	C₄⋊C₈	—	122	2	no	—
162	C₄⋊C₈	—	122	2	no	—
163	#103	—	2	4	no	—
164	C₄×C₈	—	61	2	no	—
165	C₄×C₈	—	122	1	no	—
166	C₄×C₈	—	122	1	no	—
167	C₄×C₈	—	122	1	no	—
168	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
169	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	2	2	no	—
172	C₂×Q₈	—	2	2	no	—
173	C₂×Q₈	—	2	2	no	—
174	C₂×Q₈	—	2	2	no	—
175	C₂×Q₈	#11758194	1	4	yes	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
179	C₂²×C₄	#11758224	1	1	yes	characteristic
180	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
181	Q₁₆	—	2	8	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	61	4	no	—
185	C₂×C₈	—	122	2	no	—
186	C₂×C₈	—	122	2	no	—
187	C₂×C₈	—	122	2	no	—
188	C₂×C₈	—	244	2	no	—
189	C₂×C₈	—	244	4	no	—
190	C₂×C₈	—	244	2	no	—
191	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
192	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄²	—	4	2	no	—
198	C₄²	—	4	2	no	—
199	C₄²	D₄×D₆₁	1	1	yes	characteristic
200	C₄²	#11758224	1	1	yes	characteristic
201	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
202	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
203	C₂³	D₄×Dic₆₁	1	1	yes	characteristic
204	Q₈	—	2	4	no	—
205	Q₈	—	2	4	no	—
206	Q₈	—	4	2	no	—
207	Q₈	—	4	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
215	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
216	C₂×C₄	#14519748	1	1	yes	characteristic
217	C₂×C₄	#14519748	1	1	yes	characteristic
218	C₂×C₄	#14519764	1	1	yes	characteristic
219	C₂×C₄	#14519773	1	1	yes	characteristic
220	C₂×C₄	C₂×#11758194	1	2	yes	—
221	C₂×C₄	#14519784	1	2	yes	—
222	C₂×C₄	Q₁₆×D₆₁	1	2	yes	—
223	C₂×C₄	#14519800	1	2	yes	—
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	122	4	no	—
227	C₈	—	488	2	no	—
228	C₈	—	244	2	no	—
229	C₂²	#16839665	1	1	yes	characteristic
230	C₂²	#16839834	1	2	yes	—
231	C₂²	Q₁₆×Dic₆₁	1	2	yes	—
232	C₂²	#16839884	1	2	yes	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	4	no	—
236	C₄	—	8	2	no	—
237	C₄	#16839726	1	1	yes	characteristic
238	C₄	#16839729	1	1	yes	characteristic
239	C₄	#16839834	1	2	yes	—
240	C₄	#16840000	1	2	yes	—
241	C₄	#16840140	1	1	yes	characteristic
242	C₄	#16840147	1	1	yes	characteristic
243	C₂	#21444670	1	4	yes	—
244	C₂	#21446618	1	1	yes	characteristic
245	C₂	#21446681	1	1	yes	characteristic
246	C₂	#21446713	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₁
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₄×Q₁₆×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₄×D₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×C₄×Q₁₆	D₆₁
hypocenter	C₆₁	#9662
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
78. C₂×#92418	Q₁₆	semidirect
143. C₆₁	#9662	semidirect

Maximal subgroups

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

subgroup	index
#21446756	2
C₂×#16839515	2
C₂×Q₈⋊#92418	2
C₂×#16839484	2
C₂×C₄×Q₁₆×C₆₁	2
#9662	61