Subgroups — C₂×#72216336 #72337602

Order 2048.

At least 6635 conjugacy classes of subgroups; only the 664 normal subgroups are shown.

The subgroup lattice has 290 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#135516	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#144639	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#347047	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#139759	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#493480	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#511811	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#144633	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#139718	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#144386	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#7266798	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#139750	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#139743	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#139743	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#144378	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#139731	C₂	1	1	yes	maximal, characteristic, complemented
16	#72216336	C₂	1	16	yes	maximal, direct product factor
17	C₂×#62832	C₂²	1	1	yes	characteristic
18	#7266798	C₂²	1	8	yes	complemented
19	C₂×#32873	C₂²	1	1	yes	characteristic, complemented
20	C₂×#23895	C₂²	1	1	yes	characteristic
21	C₂×#23895	C₂²	1	1	yes	characteristic
22	C₂×#23900	C₂²	1	1	yes	characteristic
23	C₂×#23900	C₂²	1	1	yes	characteristic
24	C₂×#23900	C₂²	1	1	yes	characteristic, complemented
25	C₂×#23906	C₂²	1	1	yes	characteristic
26	C₂×#21776	C₂²	1	1	yes	characteristic
27	C₂×#21776	C₂²	1	1	yes	characteristic
28	C₂×#21776	C₂²	1	1	yes	characteristic, complemented
29	C₂×#21746	C₂²	1	1	yes	characteristic, complemented
30	C₂×#15258	C₂²	1	1	yes	characteristic
31	#511811	C₂²	1	8	yes	complemented
32	#493480	C₂²	1	8	yes	complemented
33	C₂×#12707	C₂²	1	1	yes	characteristic
34	C₂×#12707	C₂²	1	1	yes	characteristic
35	C₂×#12707	C₂²	1	1	yes	characteristic, complemented
36	C₂×#12705	C₂²	1	1	yes	characteristic
37	C₂×#12705	C₂²	1	1	yes	characteristic
38	C₂×#12693	C₂²	1	1	yes	characteristic
39	C₂×#12693	C₂²	1	1	yes	characteristic
40	C₂×#12693	C₂²	1	1	yes	characteristic
41	C₂×#12687	C₂²	1	1	yes	characteristic, complemented
42	#347047	C₂²	1	8	yes	complemented
43	C₂×#8546	C₂²	1	1	yes	characteristic
44	C₂×#8541	C₂²	1	1	yes	characteristic, complemented
45	C₂×#8534	C₂²	1	1	yes	characteristic
46	C₂×#8534	C₂²	1	1	yes	characteristic
47	C₂×#8534	C₂²	1	1	yes	characteristic, complemented
48	#144639	C₂²	1	8	yes	complemented
49	#144633	C₂²	1	8	yes	complemented
50	#144386	C₂²	1	8	yes	complemented
51	#144378	C₂²	1	8	yes	complemented
52	#139759	C₂²	1	8	yes	complemented
53	#139750	C₂²	1	8	yes	complemented
54	#139743	C₂²	1	8	yes	complemented
55	#139743	C₂²	1	8	yes	complemented
56	#139731	C₂²	1	8	yes	complemented
57	#139718	C₂²	1	8	yes	complemented
58	#135516	C₂²	1	8	yes	complemented
59	C₂×#7304	C₂²	1	1	yes	characteristic, complemented
60	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
61	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
62	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
63	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
64	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
65	C₂×#7307	C₂²	1	1	yes	characteristic, complemented
66	C₂×#7307	C₂²	1	1	yes	characteristic, complemented
67	#62832	C₂³	1	4	yes	—
68	C₂×#3484	C₂³	1	1	yes	characteristic
69	C₂×#3484	C₂³	1	1	yes	characteristic
70	C₂×#3484	C₂³	1	1	yes	characteristic
71	C₂³×#127	C₂³	1	1	yes	characteristic
72	C₂³×#120	C₂³	1	1	yes	characteristic
73	C₂×#3026	D₄	1	2	yes	complemented
74	C₂×#2883	D₄	1	2	yes	complemented
75	C₂×#2846	D₄	1	2	yes	complemented
76	#32873	C₂³	1	4	yes	complemented
77	#23906	C₂³	1	4	yes	—
78	#23900	D₄	1	4	yes	complemented
79	#23900	C₂³	1	4	yes	—
80	#23900	C₂³	1	4	yes	—
81	#23900	C₂³	1	4	yes	complemented
82	#23895	D₄	1	4	yes	complemented
83	#23895	C₂³	1	4	yes	—
84	#23895	C₂³	1	4	yes	—
85	#21776	D₄	1	4	yes	complemented
86	#21776	C₂³	1	4	yes	—
87	#21776	C₂³	1	4	yes	—
88	#21776	C₂³	1	4	yes	complemented
89	#21746	C₂³	1	4	yes	complemented
90	C₂×#1913	D₄	1	2	yes	complemented
91	C₂×#1898	D₄	1	2	yes	complemented
92	C₂×#1828	D₄	1	2	yes	complemented
93	C₂×#1783	C₂³	1	1	yes	characteristic
94	C₂×#1781	C₂³	1	1	yes	characteristic
95	C₂×#1781	C₂³	1	1	yes	characteristic
96	C₂×#1781	C₂³	1	1	yes	characteristic
97	C₂×#1781	C₂³	1	1	yes	characteristic
98	C₂×#1781	C₂³	1	1	yes	characteristic
99	C₂×#1781	C₂³	1	1	yes	characteristic
100	C₂×#1781	C₂³	1	1	yes	characteristic
101	C₂×#1781	C₂³	1	1	yes	characteristic
102	#15258	C₂³	1	4	yes	—
103	C₂×#1438	C₂³	1	1	yes	characteristic
104	#12707	C₂³	1	4	yes	—
105	#12707	C₂³	1	4	yes	—
106	#12707	C₂³	1	4	yes	complemented
107	#12705	C₂³	1	4	yes	—
108	#12705	C₂³	1	4	yes	—
109	#12693	C₂³	1	4	yes	—
110	#12693	C₂³	1	4	yes	—
111	#12693	C₂³	1	4	yes	—
112	#12687	C₂³	1	4	yes	complemented
113	#8546	C₂³	1	4	yes	—
114	#8541	C₂³	1	4	yes	complemented
115	#8534	C₂³	1	4	yes	—
116	#8534	C₂³	1	4	yes	—
117	#8534	C₂³	1	4	yes	complemented
118	#7316	C₂³	1	4	yes	complemented
119	#7314	C₂³	1	4	yes	complemented
120	#7310	C₂³	1	4	yes	complemented
121	#7310	C₂³	1	4	yes	complemented
122	#7310	C₂³	1	4	yes	complemented
123	#7307	C₂³	1	4	yes	complemented
124	#7307	C₂³	1	4	yes	complemented
125	#7304	C₂³	1	4	yes	complemented
126	C₂⁷	C₂×D₄	1	1	yes	characteristic
127	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
128	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
129	C₂²×H(5,𝔽₂)	#41	1	1	yes	characteristic, complemented
130	C₂⁴×D₄	#41	1	1	yes	characteristic, complemented
131	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
132	#3484	C₂×D₄	1	2	yes	—
133	#3484	C₂×D₄	1	2	yes	—
134	#3484	C₂×D₄	1	2	yes	—
135	#3484	C₂×D₄	1	2	yes	—
136	#3484	#41	1	2	yes	complemented
137	#3484	C₂⁴	1	2	yes	—
138	#3484	C₂⁴	1	2	yes	—
139	#3484	C₂⁴	1	2	yes	—
140	C₂×#534	C₂×D₄	1	2	yes	—
141	C₂²×#127	#41	1	2	yes	complemented
142	C₂²×#127	C₂⁴	1	2	yes	—
143	C₂²×#120	C₂×D₄	1	2	yes	—
144	C₂²×#120	C₂×D₄	1	2	yes	—
145	C₂²×#120	C₂⁴	1	2	yes	—
146	C₂×#518	C₂×D₄	1	1	yes	characteristic
147	C₂×#518	C₂×D₄	1	1	yes	characteristic
148	C₂×#518	#41	1	1	yes	characteristic, complemented
149	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
150	C₂³×C₄²	#41	1	1	yes	characteristic, complemented
151	#3026	C₂×D₄	1	8	yes	complemented
152	#2883	C₂×D₄	1	8	yes	complemented
153	C₂²×#99	C₂×D₄	1	2	yes	—
154	C₂²×#99	C₂×D₄	1	2	yes	—
155	C₂²×#99	C₂×D₄	1	2	yes	—
156	#2846	C₂×D₄	1	8	yes	complemented
157	C₂×#379	C₂×D₄	1	2	yes	—
158	#1913	C₂×D₄	1	8	yes	complemented
159	#1898	C₂×D₄	1	8	yes	complemented
160	#1828	C₂×D₄	1	8	yes	complemented
161	#1783	C₂⁴	1	2	yes	—
162	#1781	C₂×D₄	1	2	yes	—
163	#1781	C₂×D₄	1	2	yes	—
164	#1781	C₂×D₄	1	2	yes	—
165	#1781	C₂×D₄	1	2	yes	—
166	#1781	C₂⁴	1	2	yes	—
167	#1781	C₂⁴	1	2	yes	—
168	#1781	C₂⁴	1	2	yes	—
169	#1781	C₂⁴	1	2	yes	—
170	#1781	C₂⁴	1	2	yes	—
171	#1781	C₂⁴	1	2	yes	—
172	#1781	C₂⁴	1	2	yes	—
173	#1781	C₂⁴	1	2	yes	—
174	#1438	C₂⁴	1	2	yes	—
175	C₂⁶	C₂²×D₄	1	1	yes	characteristic
176	C₂⁶	C₂²×D₄	1	2	yes	—
177	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
178	C₂×H(5,𝔽₂)	#120	1	2	yes	—
179	C₂×H(5,𝔽₂)	C₂²×D₄	1	2	yes	—
180	C₂×H(5,𝔽₂)	C₂²×D₄	1	2	yes	—
181	C₂×H(5,𝔽₂)	C₂×#41	1	2	yes	complemented
182	C₂³×D₄	#120	1	2	yes	—
183	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
184	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
185	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
186	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
187	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
188	C₂³×D₄	C₂×#41	1	1	yes	characteristic
189	C₂³×D₄	C₂×#41	1	1	yes	characteristic
190	C₂³×D₄	C₂×#41	1	1	yes	characteristic
191	C₂³×D₄	C₂×#41	1	2	yes	complemented
192	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
193	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
194	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
195	C₂³×D₄	C₂⁵	1	1	yes	characteristic
196	C₂⁴×C₄	C₂×#41	1	1	yes	characteristic
197	#534	#120	1	4	yes	—
198	#534	C₂²×D₄	1	4	yes	—
199	C₂×#120	#120	1	4	yes	—
200	C₂×#120	C₂²×D₄	1	4	yes	—
201	#518	#120	1	2	yes	—
202	#518	C₂²×D₄	1	2	yes	—
203	#518	C₂²×D₄	1	2	yes	—
204	#518	C₂×#41	1	2	yes	complemented
205	C₂²×C₂²⋊C₄	#120	1	2	yes	—
206	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
207	C₂²×C₄²	C₂×#41	1	2	yes	complemented
208	C₂×#99	C₂²×D₄	1	4	yes	—
209	C₂×#99	C₂²×D₄	1	4	yes	—
210	C₂×#99	C₂²×D₄	1	4	yes	—
211	#379	C₂²×D₄	1	4	yes	—
212	C₂⁵	B(4,𝔽₂)	1	2	yes	—
213	C₂⁵	C₂×#120	1	1	yes	characteristic
214	C₂⁵	C₂×#120	1	1	yes	characteristic
215	C₂⁵	#535	1	1	yes	characteristic
216	C₂⁵	#535	1	1	yes	characteristic
217	C₂⁵	#546	1	1	yes	characteristic
218	C₂⁵	#546	1	1	yes	characteristic
219	C₂⁵	#546	1	1	yes	characteristic
220	C₂⁵	#546	1	1	yes	characteristic
221	C₂⁵	#546	1	1	yes	characteristic
222	C₂⁵	#546	1	1	yes	characteristic
223	C₂⁵	#546	1	1	yes	characteristic
224	C₂⁵	C₂³×D₄	1	1	yes	characteristic
225	C₂⁵	C₂×H(5,𝔽₂)	1	1	yes	characteristic
226	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
227	C₂²×D₄	C₂×#120	1	1	yes	characteristic
228	C₂²×D₄	C₂×#120	1	1	yes	characteristic
229	C₂²×D₄	C₂×#120	1	4	yes	—
230	C₂²×D₄	#525	1	1	yes	characteristic
231	C₂²×D₄	#525	1	1	yes	characteristic
232	C₂²×D₄	#535	1	1	yes	characteristic
233	C₂²×D₄	#535	1	1	yes	characteristic
234	C₂²×D₄	#546	1	1	yes	characteristic
235	C₂²×D₄	#546	1	1	yes	characteristic
236	C₂²×D₄	#546	1	1	yes	characteristic
237	C₂²×D₄	#546	1	1	yes	characteristic
238	C₂²×D₄	#546	1	1	yes	characteristic
239	C₂²×D₄	#546	1	1	yes	characteristic
240	C₂²×D₄	C₂³×D₄	1	1	yes	characteristic
241	C₂²×D₄	C₂³×D₄	1	1	yes	characteristic
242	C₂²×D₄	C₂²×#41	1	1	yes	characteristic
243	C₂²×D₄	C₂×H(5,𝔽₂)	1	1	yes	characteristic
244	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
245	C₂³×C₄	#525	1	1	yes	characteristic
246	C₂³×C₄	#525	1	1	yes	characteristic
247	C₂³×C₄	#546	1	1	yes	characteristic
248	C₂³×C₄	#546	1	1	yes	characteristic
249	C₂³×C₄	#546	1	1	yes	characteristic
250	C₂³×C₄	C₂²×#41	1	1	yes	characteristic
251	C₂×C₂²⋊C₄	C₂×#120	1	4	yes	—
252	C₂⁴	#2404	1	1	yes	characteristic
253	C₂⁴	#2404	1	1	yes	characteristic
254	C₂⁴	#2404	1	1	yes	characteristic
255	C₂⁴	#2404	1	1	yes	characteristic
256	C₂⁴	#2404	1	1	yes	characteristic
257	C₂⁴	#2404	1	1	yes	characteristic
258	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
259	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
260	C₂⁴	C₂×B(4,𝔽₂)	1	4	yes	—
261	C₂⁴	C₂²×#120	1	1	yes	characteristic
262	C₂⁴	C₂×#535	1	1	yes	characteristic
263	C₂⁴	C₂×#546	1	1	yes	characteristic
264	C₂⁴	C₂×#546	1	1	yes	characteristic
265	C₂⁴	C₂×#546	1	1	yes	characteristic
266	C₂²×C₄	#2404	1	1	yes	characteristic
267	C₂²×C₄	#2404	1	1	yes	characteristic
268	C₂²×C₄	C₂×B(4,𝔽₂)	1	4	yes	—
269	C₂²×C₄	C₂×#525	1	1	yes	characteristic
270	C₂²×C₄	C₂×#546	1	1	yes	characteristic
271	C₂³	#23984	1	1	yes	characteristic
272	C₂³	#23984	1	2	yes	—
273	C₂³	#23984	1	1	yes	characteristic
274	C₂³	#23996	1	1	yes	characteristic
275	C₂³	#23996	1	1	yes	characteristic
276	C₂³	#23996	1	1	yes	characteristic
277	C₂³	#23996	1	1	yes	characteristic
278	C₂³	C₂×#2404	1	1	yes	characteristic
279	C₂³	C₂²×B(4,𝔽₂)	1	2	yes	—
280	C₂²	#251827	1	1	yes	characteristic
281	C₂²	#251827	1	1	yes	characteristic
282	C₂²	#251827	1	1	yes	characteristic
283	C₂²	#251827	1	1	yes	characteristic
284	C₂²	C₂×#23996	1	1	yes	characteristic
285	C₂²	C₂×#23996	1	1	yes	characteristic
286	C₂²	C₂×#23984	1	1	yes	characteristic
287	C₂	#72216336	1	2	yes	direct product factor
288	C₂	C₂×#251827	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#251827
commutator, Frattini, cosocle	C₂³×D₄	C₂⁵
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#135516	C₂	semidirect
2. C₂×#144639	C₂	semidirect
3. C₂×#347047	C₂	semidirect
4. C₂×#139759	C₂	semidirect
5. C₂×#493480	C₂	semidirect
6. C₂×#511811	C₂	semidirect
7. C₂×#144633	C₂	semidirect
8. C₂×#139718	C₂	semidirect
9. C₂×#144386	C₂	semidirect
10. C₂×#7266798	C₂	semidirect
11. C₂×#139750	C₂	semidirect
12. C₂×#139743	C₂	semidirect
13. C₂×#139743	C₂	semidirect
14. C₂×#144378	C₂	semidirect
15. C₂×#139731	C₂	semidirect
16. #72216336	C₂	direct (N × H)
18. #7266798	C₂²	semidirect
19. C₂×#32873	C₂²	semidirect
24. C₂×#23900	C₂²	semidirect
28. C₂×#21776	C₂²	semidirect
29. C₂×#21746	C₂²	semidirect
31. #511811	C₂²	semidirect
32. #493480	C₂²	semidirect
35. C₂×#12707	C₂²	semidirect
41. C₂×#12687	C₂²	semidirect
42. #347047	C₂²	semidirect
44. C₂×#8541	C₂²	semidirect
47. C₂×#8534	C₂²	semidirect
48. #144639	C₂²	semidirect
49. #144633	C₂²	semidirect
50. #144386	C₂²	semidirect
51. #144378	C₂²	semidirect
52. #139759	C₂²	semidirect
53. #139750	C₂²	semidirect
54. #139743	C₂²	semidirect
55. #139743	C₂²	semidirect
56. #139731	C₂²	semidirect
57. #139718	C₂²	semidirect
58. #135516	C₂²	semidirect
59. C₂×#7304	C₂²	semidirect
60. C₂×#7316	C₂²	semidirect
61. C₂×#7314	C₂²	semidirect
62. C₂×#7310	C₂²	semidirect
63. C₂×#7310	C₂²	semidirect
64. C₂×#7310	C₂²	semidirect
65. C₂×#7307	C₂²	semidirect
66. C₂×#7307	C₂²	semidirect
73. C₂×#3026	D₄	semidirect
74. C₂×#2883	D₄	semidirect
75. C₂×#2846	D₄	semidirect
76. #32873	C₂³	semidirect
78. #23900	D₄	semidirect
81. #23900	C₂³	semidirect
82. #23895	D₄	semidirect
85. #21776	D₄	semidirect
88. #21776	C₂³	semidirect
89. #21746	C₂³	semidirect
90. C₂×#1913	D₄	semidirect
91. C₂×#1898	D₄	semidirect
92. C₂×#1828	D₄	semidirect
106. #12707	C₂³	semidirect
112. #12687	C₂³	semidirect
114. #8541	C₂³	semidirect
117. #8534	C₂³	semidirect
118. #7316	C₂³	semidirect
119. #7314	C₂³	semidirect
120. #7310	C₂³	semidirect
121. #7310	C₂³	semidirect
122. #7310	C₂³	semidirect
123. #7307	C₂³	semidirect
124. #7307	C₂³	semidirect
125. #7304	C₂³	semidirect
129. C₂²×H(5,𝔽₂)	#41	semidirect
130. C₂⁴×D₄	#41	semidirect
136. #3484	#41	semidirect
141. C₂²×#127	#41	semidirect
148. C₂×#518	#41	semidirect
150. C₂³×C₄²	#41	semidirect
151. #3026	C₂×D₄	semidirect
152. #2883	C₂×D₄	semidirect
156. #2846	C₂×D₄	semidirect
158. #1913	C₂×D₄	semidirect
159. #1898	C₂×D₄	semidirect
160. #1828	C₂×D₄	semidirect
181. C₂×H(5,𝔽₂)	C₂×#41	semidirect
191. C₂³×D₄	C₂×#41	semidirect
204. #518	C₂×#41	semidirect
207. C₂²×C₄²	C₂×#41	semidirect
287. C₂	#72216336	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×#135516	2
C₂×#144639	2
C₂×#347047	2
C₂×#139759	2
C₂×#493480	2
C₂×#511811	2
C₂×#144633	2
C₂×#139718	2
C₂×#144386	2
C₂×#7266798	2
C₂×#139750	2
C₂×#139743	2
C₂×#144378	2
C₂×#139731	2
#72216336	2

Subgroups — C2×#72216336 #72337602

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#72216336 #72337602