Subgroups — C₂²×#16839516 #29677363

Order 15616.

3086 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#92418	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄×Dic₆₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#16839516	C₂	1	12	yes	maximal, direct product factor
4	C₂⁴×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×Dic₆₁	C₄	1	2	yes	—
6	C₂³×#92418	—	2	4	no	—
7	C₂³×#92418	—	2	4	no	—
8	C₂³×#92418	C₄	1	8	yes	complemented
9	C₂³×#92418	C₂²	1	6	yes	—
10	C₂²×C₄×Dic₆₁	—	2	4	no	—
11	C₂²×C₄×Dic₆₁	—	2	4	no	—
12	C₂²×C₄×Dic₆₁	C₄	1	6	yes	—
13	C₂²×C₄×Dic₆₁	C₂²	1	6	yes	—
14	#16839516	C₂²	1	16	yes	direct product factor
15	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₆₁	—	2	4	no	—
17	C₂³×C₈×C₆₁	—	2	4	no	—
18	C₂³×C₈×C₆₁	C₄	1	8	yes	complemented
19	C₂³×C₈×C₆₁	C₂²	1	6	yes	—
20	C₂³×Dic₆₁	—	2	8	no	—
21	C₂³×Dic₆₁	—	2	8	no	—
22	C₂³×Dic₆₁	C₈	1	16	yes	complemented
23	C₂³×Dic₆₁	C₂×C₄	1	12	yes	—
24	C₂²×#92418	—	2	32	no	—
25	C₂²×#92418	—	2	12	no	—
26	C₂²×#92418	—	2	12	no	—
27	C₂²×#92418	C₂×C₄	1	24	yes	complemented
28	C₂²×#92418	C₂³	1	4	yes	—
29	C₂×C₄×Dic₆₁	—	2	24	no	—
30	C₂×C₄×Dic₆₁	—	2	12	no	—
31	C₂×C₄×Dic₆₁	—	2	12	no	—
32	C₂×C₄×Dic₆₁	C₂×C₄	1	12	yes	—
33	C₂×C₄×Dic₆₁	C₂³	1	4	yes	—
34	C₂⁵×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₆₁	C₂×C₄	1	3	yes	—
36	C₂³×C₄×C₆₁	C₂×C₄	1	4	yes	—
37	C₂³×C₄×C₆₁	C₂×C₄	1	4	yes	—
38	C₂³×C₄×C₆₁	D₄	1	4	yes	—
39	C₂³×C₄×C₆₁	D₄	1	4	yes	—
40	C₂³×C₄×C₆₁	D₄	1	4	yes	—
41	C₂³×C₄×C₆₁	Q₈	1	4	yes	—
42	C₂³×C₄×C₆₁	C₂³	1	3	yes	—
43	C₂²×C₈×C₆₁	—	2	32	no	—
44	C₂²×C₈×C₆₁	—	2	12	no	—
45	C₂²×C₈×C₆₁	—	2	12	no	—
46	C₂²×C₈×C₆₁	C₂×C₄	1	24	yes	complemented
47	C₂²×C₈×C₆₁	C₂³	1	4	yes	—
48	C₂⁴×C₆₁	C₄²	1	4	yes	—
49	C₂⁴×C₆₁	C₂²⋊C₄	1	4	yes	—
50	C₂⁴×C₆₁	C₄⋊C₄	1	4	yes	—
51	C₂⁴×C₆₁	C₂×C₈	1	8	yes	—
52	C₂⁴×C₆₁	M₁₆	1	8	yes	—
53	C₂⁴×C₆₁	C₂²×C₄	1	3	yes	—
54	C₂²×C₄×C₆₁	C₄²	1	12	yes	—
55	C₂²×C₄×C₆₁	C₂²⋊C₄	1	12	yes	—
56	C₂²×C₄×C₆₁	C₂²⋊C₄	1	16	yes	—
57	C₂²×C₄×C₆₁	C₂²⋊C₄	1	16	yes	—
58	C₂²×C₄×C₆₁	C₄⋊C₄	1	12	yes	—
59	C₂²×C₄×C₆₁	C₄⋊C₄	1	16	yes	—
60	C₂²×C₄×C₆₁	C₄⋊C₄	1	16	yes	—
61	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
62	C₂²×C₄×C₆₁	C₂²×C₄	1	6	yes	—
63	C₂²×C₄×C₆₁	C₂²×C₄	1	6	yes	—
64	C₂²×C₄×C₆₁	C₂×D₄	1	6	yes	—
65	C₂²×C₄×C₆₁	C₂×D₄	1	6	yes	—
66	C₂²×C₄×C₆₁	C₂×D₄	1	6	yes	—
67	C₂²×C₄×C₆₁	C₂×Q₈	1	6	yes	—
68	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
69	C₂²×Dic₆₁	—	2	64	no	—
70	C₂²×Dic₆₁	—	2	24	no	—
71	C₂²×Dic₆₁	—	2	24	no	—
72	C₂²×Dic₆₁	C₂×C₈	1	48	yes	complemented
73	C₂²×Dic₆₁	C₂²×C₄	1	8	yes	—
74	C₂×C₈×C₆₁	—	2	48	no	—
75	C₂×C₈×C₆₁	—	2	4	no	—
76	C₂×C₈×C₆₁	—	2	4	no	—
77	C₂×C₈×C₆₁	C₂²×C₄	1	8	yes	complemented
78	C₄×Dic₆₁	—	2	24	no	—
79	C₄×Dic₆₁	—	2	4	no	—
80	C₄×Dic₆₁	—	2	4	no	—
81	C₂×#92418	—	2	48	no	—
82	C₂×#92418	—	2	4	no	—
83	C₂×#92418	—	2	4	no	—
84	C₂×#92418	C₂²×C₄	1	8	yes	complemented
85	C₂³×C₆₁	#95	1	16	yes	—
86	C₂³×C₆₁	C₄×C₈	1	16	yes	—
87	C₂³×C₆₁	#97	1	16	yes	—
88	C₂³×C₆₁	C₂²⋊C₈	1	32	yes	—
89	C₂³×C₆₁	C₄⋊C₈	1	32	yes	—
90	C₂³×C₆₁	C₂×C₄²	1	6	yes	—
91	C₂³×C₆₁	C₂×C₂²⋊C₄	1	6	yes	—
92	C₂³×C₆₁	C₂×C₄⋊C₄	1	6	yes	—
93	C₂³×C₆₁	C₂²×C₈	1	12	yes	—
94	C₂³×C₆₁	C₂×M₁₆	1	12	yes	—
95	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄×C₆₁	#95	1	48	yes	—
97	C₂×C₄×C₆₁	C₂×C₄²	1	6	yes	—
98	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	12	yes	—
99	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	12	yes	—
100	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	6	yes	—
101	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	12	yes	—
102	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	12	yes	—
103	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	6	yes	—
104	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
110	C₂×Dic₆₁	—	2	8	no	—
111	C₂×Dic₆₁	—	2	8	no	—
112	C₂×Dic₆₁	—	2	96	no	—
113	C₂×Dic₆₁	C₂²×C₈	1	16	yes	complemented
114	C₈×C₆₁	—	2	8	no	—
115	#92418	—	2	8	no	—
116	C₂²×#336	—	61	1	no	maximal, Sylow
117	C₂²×C₆₁	#336	1	64	yes	complemented
118	C₂²×C₆₁	C₂×#95	1	12	yes	—
119	C₂²×C₆₁	C₂×C₄×C₈	1	12	yes	—
120	C₂²×C₆₁	C₂×#97	1	12	yes	—
121	C₂²×C₆₁	C₂×C₂²⋊C₈	1	24	yes	—
122	C₂²×C₆₁	C₂×C₄⋊C₈	1	24	yes	—
123	C₂²×C₆₁	C₂²×C₄²	1	1	yes	characteristic
124	C₂²×C₆₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
125	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×C₆₁	C₂³×C₈	1	2	yes	—
127	C₂²×C₆₁	C₂²×M₁₆	1	2	yes	—
128	C₄×C₆₁	C₂×#95	1	12	yes	—
129	C₄×C₆₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
130	C₄×C₆₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
131	C₄×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
132	C₄×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
133	Dic₆₁	—	2	16	no	—
134	C₂⁴×C₈	—	61	1	no	—
135	C₂⁴×C₈	D₆₁	1	1	yes	characteristic
136	C₂³×C₄²	—	61	1	no	—
137	C₂×#336	—	61	12	no	—
138	C₂×C₆₁	C₂×#336	1	24	yes	complemented
139	C₂×C₆₁	C₂²×#95	1	1	yes	characteristic
140	C₂×C₆₁	C₂²×C₄×C₈	1	1	yes	characteristic
141	C₂×C₆₁	C₂²×#97	1	1	yes	characteristic
142	C₂×C₆₁	C₂²×C₂²⋊C₈	1	2	yes	—
143	C₂×C₆₁	C₂²×C₄⋊C₈	1	2	yes	—
144	C₂⁴×C₄	—	61	2	no	—
145	C₂⁴×C₄	C₂×D₆₁	1	1	yes	characteristic
146	C₂³×C₈	—	2	4	no	—
147	C₂³×C₈	—	2	4	no	—
148	C₂³×C₈	—	61	8	no	—
149	C₂³×C₈	—	61	6	no	—
150	C₂³×C₈	—	122	4	no	—
151	C₂³×C₈	—	122	4	no	—
152	C₂³×C₈	Dic₆₁	1	8	yes	complemented
153	C₂³×C₈	C₂×D₆₁	1	6	yes	—
154	C₂²×C₄²	—	61	6	no	—
155	C₂²×C₄²	—	61	6	no	—
156	C₂²×C₄²	—	122	4	no	—
157	C₂²×C₄²	—	122	4	no	—
158	#336	—	61	16	no	—
159	C₆₁	C₂²×#336	1	1	yes	characteristic, complemented, Sylow
160	C₂⁵	C₄×D₆₁	1	1	yes	characteristic
161	C₂³×C₄	—	61	16	no	—
162	C₂³×C₄	—	61	12	no	—
163	C₂³×C₄	—	122	8	no	—
164	C₂³×C₄	—	122	8	no	—
165	C₂³×C₄	Dic₁₂₂	1	4	yes	—
166	C₂³×C₄	C₄×D₆₁	1	3	yes	—
167	C₂³×C₄	C₄×D₆₁	1	4	yes	—
168	C₂³×C₄	D₂₄₄	1	4	yes	—
169	C₂³×C₄	C₂×Dic₆₁	1	4	yes	—
170	C₂³×C₄	C₂²⋊D₆₁	1	4	yes	—
171	C₂³×C₄	C₂²⋊D₆₁	1	4	yes	—
172	C₂³×C₄	C₂²×D₆₁	1	3	yes	—
173	C₂²×C₈	—	2	32	no	—
174	C₂²×C₈	—	2	12	no	—
175	C₂²×C₈	—	2	12	no	—
176	C₂²×C₈	—	61	24	no	—
177	C₂²×C₈	—	61	4	no	—
178	C₂²×C₈	—	122	32	no	—
179	C₂²×C₈	—	122	12	no	—
180	C₂²×C₈	—	122	12	no	—
181	C₂²×C₈	C₂×Dic₆₁	1	24	yes	complemented
182	C₂²×C₈	C₂²×D₆₁	1	4	yes	—
183	C₂×C₄²	—	61	12	no	—
184	C₂×C₄²	—	61	4	no	—
185	C₂×C₄²	—	122	24	no	—
186	C₂×C₄²	—	122	12	no	—
187	C₂×C₄²	—	122	12	no	—
188	C₂⁴	C₈×D₆₁	1	8	yes	—
189	C₂⁴	#11758189	1	8	yes	—
190	C₂⁴	C₄×Dic₆₁	1	4	yes	—
191	C₂⁴	#11758196	1	4	yes	—
192	C₂⁴	#11758198	1	4	yes	—
193	C₂⁴	C₂×C₄×D₆₁	1	3	yes	—
194	C₂²×C₄	—	61	8	no	—
195	C₂²×C₄	—	61	48	no	—
196	C₂²×C₄	—	122	64	no	—
197	C₂²×C₄	—	122	24	no	—
198	C₂²×C₄	—	122	24	no	—
199	C₂²×C₄	C₄×Dic₆₁	1	12	yes	—
200	C₂²×C₄	#11758196	1	12	yes	—
201	C₂²×C₄	#11758196	1	16	yes	—
202	C₂²×C₄	C₄⋊Dic₆₁	1	16	yes	—
203	C₂²×C₄	#11758198	1	12	yes	—
204	C₂²×C₄	#11758198	1	16	yes	—
205	C₂²×C₄	C₂²⋊Dic₆₁	1	16	yes	—
206	C₂²×C₄	C₂×Dic₁₂₂	1	6	yes	—
207	C₂²×C₄	C₂×C₄×D₆₁	1	6	yes	—
208	C₂²×C₄	C₂×C₄×D₆₁	1	3	yes	—
209	C₂²×C₄	C₂×D₂₄₄	1	6	yes	—
210	C₂²×C₄	C₂²×Dic₆₁	1	6	yes	—
211	C₂²×C₄	C₂×C₂²⋊D₆₁	1	6	yes	—
212	C₂²×C₄	C₂×C₂²⋊D₆₁	1	6	yes	—
213	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	48	no	—
217	C₂×C₈	—	61	8	no	—
218	C₂×C₈	—	122	4	no	—
219	C₂×C₈	—	122	4	no	—
220	C₂×C₈	—	122	48	no	—
221	C₂×C₈	C₂²×Dic₆₁	1	8	yes	complemented
222	C₄²	—	122	4	no	—
223	C₄²	—	122	4	no	—
224	C₄²	—	122	24	no	—
225	C₂³	#14519684	1	16	yes	—
226	C₂³	C₈×Dic₆₁	1	16	yes	—
227	C₂³	#14519687	1	16	yes	—
228	C₂³	#14519689	1	32	yes	—
229	C₂³	#14519706	1	32	yes	—
230	C₂³	C₂×C₄×Dic₆₁	1	6	yes	—
231	C₂³	C₂×#11758198	1	6	yes	—
232	C₂³	C₂×#11758196	1	6	yes	—
233	C₂³	C₂×C₈×D₆₁	1	12	yes	—
234	C₂³	C₂×#11758189	1	12	yes	—
235	C₂³	C₂²×C₄×D₆₁	1	1	yes	characteristic
236	C₂×C₄	—	61	16	no	—
237	C₂×C₄	—	122	8	no	—
238	C₂×C₄	—	122	96	no	—
239	C₂×C₄	—	122	8	no	—
240	C₂×C₄	#14519684	1	48	yes	—
241	C₂×C₄	C₂×C₄×Dic₆₁	1	6	yes	—
242	C₂×C₄	C₂×#11758198	1	12	yes	—
243	C₂×C₄	C₂×#11758198	1	6	yes	—
244	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	12	yes	—
245	C₂×C₄	C₂×#11758196	1	12	yes	—
246	C₂×C₄	C₂×#11758196	1	6	yes	—
247	C₂×C₄	C₂×C₄⋊Dic₆₁	1	12	yes	—
248	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
249	C₂×C₄	C₂³×Dic₆₁	1	1	yes	characteristic
250	C₂×C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
251	C₂×C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
252	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
253	C₂×C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
254	C₈	—	2	8	no	—
255	C₈	—	122	8	no	—
256	C₂²	#16839516	1	64	yes	direct product factor
257	C₂²	C₂×#14519684	1	12	yes	—
258	C₂²	C₂×C₈×Dic₆₁	1	12	yes	—
259	C₂²	C₂×#14519687	1	12	yes	—
260	C₂²	C₂×#14519689	1	24	yes	—
261	C₂²	C₂×#14519706	1	24	yes	—
262	C₂²	C₂²×C₄×Dic₆₁	1	1	yes	characteristic
263	C₂²	C₂²×#11758198	1	1	yes	characteristic
264	C₂²	C₂²×#11758196	1	1	yes	characteristic
265	C₂²	C₂²×C₈×D₆₁	1	2	yes	—
266	C₂²	C₂²×#11758189	1	2	yes	—
267	C₄	—	122	16	no	—
268	C₄	C₂×#14519684	1	12	yes	—
269	C₄	C₂²×#11758198	1	1	yes	characteristic
270	C₄	C₂²×C₂²⋊Dic₆₁	1	1	yes	characteristic
271	C₄	C₂²×#11758196	1	1	yes	characteristic
272	C₄	C₂²×C₄⋊Dic₆₁	1	1	yes	characteristic
273	C₂	C₂×#16839516	1	24	yes	direct product factor
274	C₂	C₂²×#14519684	1	1	yes	characteristic
275	C₂	C₂²×C₈×Dic₆₁	1	1	yes	characteristic
276	C₂	C₂²×#14519687	1	1	yes	characteristic
277	C₂	C₂²×#14519689	1	2	yes	—
278	C₂	C₂²×#14519706	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₆₁
commutator	C₂×C₆₁	C₂²×C₄×C₈
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₂⁴×C₈×C₆₁	C₂
socle, prime radical	C₂⁵×C₆₁	C₂×C₄
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	C₂⁴×C₈	D₆₁
hypocenter	C₆₁	C₂²×#336
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#16839516	C₂	direct (N × H)
8. C₂³×#92418	C₄	semidirect
14. #16839516	C₂²	direct (N × H)
18. C₂³×C₈×C₆₁	C₄	semidirect
22. C₂³×Dic₆₁	C₈	semidirect
27. C₂²×#92418	C₂×C₄	semidirect
46. C₂²×C₈×C₆₁	C₂×C₄	semidirect
72. C₂²×Dic₆₁	C₂×C₈	semidirect
77. C₂×C₈×C₆₁	C₂²×C₄	semidirect
84. C₂×#92418	C₂²×C₄	semidirect
113. C₂×Dic₆₁	C₂²×C₈	semidirect
117. C₂²×C₆₁	#336	semidirect
138. C₂×C₆₁	C₂×#336	semidirect
152. C₂³×C₈	Dic₆₁	semidirect
159. C₆₁	C₂²×#336	semidirect
181. C₂²×C₈	C₂×Dic₆₁	semidirect
221. C₂×C₈	C₂²×Dic₆₁	semidirect
256. C₂²	#16839516	direct (N × H)
273. C₂	C₂×#16839516	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×#92418	2
C₂³×C₄×Dic₆₁	2
C₂×#16839516	2
C₂⁴×C₈×C₆₁	2
C₂²×#336	61

Subgroups — C22×#16839516 #29677363

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16839516 #29677363