Subgroups — C₂×#21461412 #30612289

Order 15616.

718 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#92420	C₂	1	2	yes	maximal
2	#21461412	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16840623	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊#92420	C₂	1	2	yes	maximal
5	C₂×#16840603	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₁₆×Dic₆₁	C₂	1	1	yes	maximal, characteristic
7	Q₈×#92420	—	2	4	no	—
8	Q₈×#92420	C₂²	1	8	yes	—
9	C₂×C₄⋊#92420	—	2	2	no	—
10	C₂×C₄⋊#92420	C₂²	1	2	yes	—
11	C₂×C₄×#92420	—	2	2	no	—
12	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
13	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
14	#16840623	C₂²	1	4	yes	—
15	Q₈⋊#92420	C₂²	1	8	yes	—
16	#16840603	C₂²	1	4	yes	—
17	C₂×Q₈×Dic₆₁	C₂²	1	2	yes	—
18	Q₁₆×Dic₆₁	C₄	1	4	yes	—
19	Q₁₆×Dic₆₁	C₂²	1	4	yes	—
20	C₂×#14519711	C₄	1	1	yes	characteristic
21	C₂×#14519701	C₄	1	2	yes	—
22	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
23	C₂²×Q₁₆×C₆₁	C₄	1	1	yes	characteristic
24	C₂²×#92420	—	2	2	no	—
25	C₂²×#92420	—	2	1	no	—
26	C₂²×#92420	D₄	1	2	yes	—
27	C₄⋊#92420	—	2	8	no	—
28	C₄⋊#92420	—	2	2	no	—
29	C₄⋊#92420	C₂³	1	4	yes	—
30	C₄×#92420	—	2	8	no	—
31	C₄×#92420	—	2	1	no	—
32	C₄×#92420	D₄	1	2	yes	—
33	C₄×#92420	C₂³	1	2	yes	—
34	C₄×#92420	C₂³	1	2	yes	—
35	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
36	Q₈×Dic₆₁	—	2	2	no	—
37	Q₈×Dic₆₁	—	2	2	no	—
38	Q₈×Dic₆₁	C₂×C₄	1	4	yes	—
39	Q₈×Dic₆₁	C₂³	1	4	yes	—
40	C₂×C₄⋊Dic₆₁	—	2	2	no	—
41	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
42	C₂×C₄×Dic₆₁	—	2	2	no	—
43	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
44	#14519711	C₂×C₄	1	4	yes	—
45	#14519701	C₂×C₄	1	8	yes	—
46	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
47	C₈×Dic₆₁	C₂³	1	2	yes	—
48	C₂²×Q₈×C₆₁	C₂×C₄	1	2	yes	—
49	C₂×Q₁₆×C₆₁	C₈	1	8	yes	complemented
50	C₂×Q₁₆×C₆₁	C₂×C₄	1	4	yes	—
51	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
52	C₂×Q₈×C₆₁	—	2	2	no	—
53	C₂×Q₈×C₆₁	—	2	4	no	—
54	C₂×Q₈×C₆₁	C₂×C₈	1	8	yes	—
55	C₂×Q₈×C₆₁	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₆₁	—	2	2	no	—
57	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
58	C₂²×Dic₆₁	—	2	2	no	—
59	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₆₁	#41	1	1	yes	characteristic
61	C₂×#92420	—	2	4	no	—
62	C₂×#92420	—	2	4	no	—
63	C₂×#92420	—	2	2	no	—
64	C₂×#92420	—	4	2	no	—
65	C₂×#92420	—	4	4	no	—
66	C₂×#92420	C₂×D₄	1	4	yes	—
67	Q₁₆×C₆₁	C₂×C₈	1	16	yes	complemented
68	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
69	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
70	C₄⋊Dic₆₁	—	2	2	no	—
71	C₄⋊Dic₆₁	—	2	4	no	—
72	C₄⋊Dic₆₁	—	2	4	no	—
73	C₄⋊Dic₆₁	C₂²×C₄	1	4	yes	—
74	C₄×Dic₆₁	—	2	4	no	—
75	C₄×Dic₆₁	—	2	4	no	—
76	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
77	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
78	C₄×Dic₆₁	#41	1	1	yes	characteristic
79	C₄×Dic₆₁	C₂⁴	1	1	yes	characteristic
80	C₂×#92418	—	2	2	no	—
81	C₂×#92418	C₂²×C₄	1	2	yes	—
82	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
83	Q₈×C₆₁	—	2	4	no	—
84	Q₈×C₆₁	—	2	8	no	—
85	Q₈×C₆₁	C₂²×C₈	1	8	yes	—
86	C₂×C₄×C₆₁	—	2	4	no	—
87	C₂×C₄×C₆₁	—	2	8	no	—
88	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
90	C₂×C₄×C₆₁	#131	1	2	yes	—
91	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
92	C₂×Dic₆₁	—	2	2	no	—
93	C₂×Dic₆₁	—	2	4	no	—
94	C₂×Dic₆₁	—	4	4	no	—
95	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
96	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
97	C₂×Dic₆₁	#137	1	4	yes	—
98	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
99	C₂×Dic₆₁	C₂×#41	1	1	yes	characteristic
100	#92420	—	2	4	no	—
101	#92420	—	4	2	no	—
102	#92420	—	4	4	no	—
103	C₈×C₆₁	C₂²×C₈	1	4	yes	—
104	#92418	—	2	2	no	—
105	C₂×#1579	—	61	1	no	maximal, Sylow
106	C₂²×C₆₁	C₈×D₄	1	2	yes	—
107	C₂²×C₆₁	#441	1	2	yes	—
108	C₂²×C₆₁	#444	1	2	yes	—
109	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
110	C₄×C₆₁	—	2	8	no	—
111	C₄×C₆₁	C₈×D₄	1	2	yes	—
112	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
113	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
114	Dic₆₁	—	2	1	no	—
115	Dic₆₁	—	2	1	no	—
116	Dic₆₁	—	4	4	no	—
117	Dic₆₁	#441	1	2	yes	—
118	Dic₆₁	C₂×#137	1	2	yes	—
119	C₂×C₈×Q₈	—	61	2	no	—
120	C₂×C₄×Q₁₆	—	61	1	no	—
121	#1579	—	61	8	no	—
122	C₂×#334	—	61	1	no	—
123	C₂×Q₈⋊C₈	—	61	2	no	—
124	C₂×#322	—	61	1	no	—
125	C₂×C₆₁	#1579	1	4	yes	complemented
126	C₂×C₆₁	C₂×C₈×D₄	1	1	yes	characteristic
127	C₂×C₆₁	C₂×#441	1	1	yes	characteristic
128	C₂×C₆₁	C₂×#444	1	1	yes	characteristic
129	C₂²×Q₁₆	#6923	1	1	yes	characteristic
130	C₂×C₄×Q₈	—	61	2	no	—
131	C₈×Q₈	—	61	8	no	—
132	C₈×Q₈	—	122	4	no	—
133	C₄×Q₁₆	—	61	4	no	—
134	C₄×Q₁₆	—	61	4	no	—
135	C₂×#107	—	61	1	no	—
136	C₂×C₄⋊C₈	—	61	2	no	—
137	C₂×C₄⋊C₈	—	122	2	no	—
138	C₂×#103	—	61	2	no	—
139	C₂×C₄×C₈	—	61	1	no	—
140	C₂×C₄×C₈	—	61	1	no	—
141	C₂×C₄×C₈	—	61	1	no	—
142	C₂×C₄×C₈	—	122	2	no	—
143	#334	—	61	4	no	—
144	Q₈⋊C₈	—	61	8	no	—
145	#322	—	61	4	no	—
146	C₆₁	C₂×#1579	1	1	yes	characteristic, complemented, Sylow
147	C₂²×Q₈	C₂×#6923	1	2	yes	—
148	C₂×Q₁₆	#92420	1	8	yes	complemented
149	C₂×Q₁₆	C₂×#6923	1	4	yes	—
150	C₂²×C₈	—	61	2	no	—
151	C₂²×C₈	—	61	1	no	—
152	C₂²×C₈	—	122	1	no	—
153	C₂²×C₈	—	122	2	no	—
154	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
155	C₄×Q₈	—	61	4	no	—
156	C₄×Q₈	—	61	4	no	—
157	C₄×Q₈	—	122	2	no	—
158	C₄×Q₈	—	122	2	no	—
159	C₂×C₄⋊C₄	—	61	2	no	—
160	C₂×C₄⋊C₄	—	122	2	no	—
161	C₂×C₄²	—	61	1	no	—
162	C₂×C₄²	—	122	2	no	—
163	#107	—	61	4	no	—
164	C₄⋊C₈	—	61	4	no	—
165	C₄⋊C₈	—	122	2	no	—
166	C₄⋊C₈	—	122	8	no	—
167	#103	—	61	8	no	—
168	C₄×C₈	—	61	2	no	—
169	C₄×C₈	—	61	2	no	—
170	C₄×C₈	—	61	2	no	—
171	C₄×C₈	—	61	2	no	—
172	C₄×C₈	—	61	2	no	—
173	C₄×C₈	—	122	1	no	—
174	C₄×C₈	—	122	8	no	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	C₂×#92420	1	8	yes	—
178	C₂×Q₈	C₂²×#6923	1	4	yes	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	61	1	no	—
181	C₂²×C₄	—	61	1	no	—
182	C₂²×C₄	—	122	2	no	—
183	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
184	Q₁₆	C₂×#92420	1	16	yes	complemented
185	C₂×C₈	—	61	4	no	—
186	C₂×C₈	—	61	2	no	—
187	C₂×C₈	—	122	4	no	—
188	C₂×C₈	—	122	2	no	—
189	C₂×C₈	—	122	2	no	—
190	C₂×C₈	—	122	4	no	—
191	C₂×C₈	—	244	4	no	—
192	C₂×C₈	—	244	2	no	—
193	C₂×C₈	C₂×#92420	1	4	yes	—
194	C₂×C₈	C₂²×#6923	1	2	yes	—
195	C₄⋊C₄	—	61	4	no	—
196	C₄⋊C₄	—	122	4	no	—
197	C₄⋊C₄	—	122	4	no	—
198	C₄⋊C₄	—	122	2	no	—
199	C₄²	—	61	1	no	—
200	C₄²	—	61	1	no	—
201	C₄²	—	61	1	no	—
202	C₄²	—	61	1	no	—
203	C₄²	—	122	4	no	—
204	C₄²	—	122	4	no	—
205	C₂³	D₄×#6923	1	1	yes	characteristic
206	Q₈	—	2	8	no	—
207	Q₈	—	2	4	no	—
208	Q₈	C₂²×#92420	1	8	yes	—
209	C₂×C₄	—	2	8	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	61	4	no	—
212	C₂×C₄	—	61	1	no	—
213	C₂×C₄	—	61	1	no	—
214	C₂×C₄	—	61	1	no	—
215	C₂×C₄	—	61	1	no	—
216	C₂×C₄	—	122	2	no	—
217	C₂×C₄	—	122	4	no	—
218	C₂×C₄	—	244	4	no	—
219	C₂×C₄	D₄×#6923	1	1	yes	characteristic
220	C₂×C₄	C₂²×#92420	1	2	yes	—
221	C₂×C₄	#14519864	1	2	yes	—
222	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
223	C₈	—	122	4	no	—
224	C₈	—	122	2	no	—
225	C₈	—	244	4	no	—
226	C₈	—	244	2	no	—
227	C₈	C₂²×#92420	1	4	yes	—
228	C₂²	D₄×#92420	1	2	yes	—
229	C₂²	#16840802	1	2	yes	—
230	C₂²	#16840808	1	2	yes	—
231	C₂²	C₂×D₄×#6923	1	1	yes	characteristic
232	C₄	—	2	8	no	—
233	C₄	—	61	2	no	—
234	C₄	—	61	2	no	—
235	C₄	—	122	1	no	—
236	C₄	—	122	1	no	—
237	C₄	—	244	4	no	—
238	C₄	D₄×#92420	1	2	yes	—
239	C₄	C₂³×#92420	1	1	yes	characteristic
240	C₄	C₂×#14519864	1	1	yes	characteristic
241	C₂	#21461412	1	4	yes	direct product factor
242	C₂	C₂×D₄×#92420	1	1	yes	characteristic
243	C₂	C₂×#16840802	1	1	yes	characteristic
244	C₂	C₂×#16840808	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#6923
commutator	C₄×C₆₁	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₂²×Q₁₆×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₄×D₄
cosocle	C₄×Dic₆₁	C₂⁴
hypercenter	C₂²×Q₁₆	#6923
hypocenter	C₆₁	C₂×#1579
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #21461412	C₂	direct (N × H)
49. C₂×Q₁₆×C₆₁	C₈	semidirect
67. Q₁₆×C₆₁	C₂×C₈	semidirect
125. C₂×C₆₁	#1579	semidirect
146. C₆₁	C₂×#1579	semidirect
148. C₂×Q₁₆	#92420	semidirect
184. Q₁₆	C₂×#92420	semidirect
241. C₂	#21461412	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₈×#92420	2
#21461412	2
C₂×#16840623	2
C₂×Q₈⋊#92420	2
C₂×#16840603	2
C₂×Q₁₆×Dic₆₁	2
C₂×#1579	61

Subgroups — C2×#21461412 #30612289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21461412 #30612289