Subgroups — #72259010

Order 2048.

At least 6587 conjugacy classes of subgroups; only the 400 normal subgroups are shown.

The subgroup lattice has 269 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72259008	C₂	1	1	yes	maximal, characteristic, complemented
2	#72259005	C₂	1	2	yes	maximal, complemented
3	#72235718	C₂	1	1	yes	maximal, characteristic, complemented
4	#72234515	C₂	1	2	yes	maximal, complemented
5	#72232723	C₂	1	2	yes	maximal, complemented
6	#72226097	C₂	1	2	yes	maximal, complemented
7	#72226091	C₂	1	2	yes	maximal, complemented
8	C₂×#139716	C₂	1	2	yes	maximal, complemented
9	#72219264	C₂	1	1	yes	maximal, characteristic
10	C₂²×#3532	C₄	1	1	yes	characteristic, complemented
11	C₂×#34980	C₂²	1	1	yes	characteristic
12	C₂×#21746	C₂²	1	2	yes	complemented
13	#5025789	C₂²	1	1	yes	characteristic
14	#5025782	C₂²	1	1	yes	characteristic
15	#5021140	C₄	1	1	yes	characteristic, complemented
16	#5021138	C₄	1	1	yes	characteristic, complemented
17	#4422949	C₂²	1	1	yes	characteristic
18	#4420586	C₂²	1	1	yes	characteristic
19	#763333	C₄	1	1	yes	characteristic, complemented
20	#686187	C₄	1	1	yes	characteristic, complemented
21	#647656	C₄	1	1	yes	characteristic, complemented
22	#616961	C₂²	1	1	yes	characteristic
23	#616960	C₄	1	1	yes	characteristic, complemented
24	C₂×#15947	C₂²	1	1	yes	characteristic
25	#552670	C₄	1	1	yes	characteristic, complemented
26	C₂×#8547	C₂²	1	2	yes	complemented
27	#139716	C₂²	1	4	yes	complemented
28	#139716	C₂²	1	4	yes	complemented
29	#139504	C₂²	1	4	yes	complemented
30	#139486	C₂²	1	4	yes	complemented
31	#134036	C₂²	1	2	yes	complemented
32	#134036	C₂²	1	2	yes	complemented
33	#115631	C₂²	1	2	yes	complemented
34	#115631	C₂²	1	2	yes	complemented
35	C₂⁵×D₄	D₄	1	1	yes	characteristic
36	C₂²×#551	C₂×C₄	1	1	yes	characteristic, complemented
37	C₂²×D₄²	C₂×C₄	1	1	yes	characteristic, complemented
38	C₂²×H(3,𝔽₄)	C₂×C₄	1	1	yes	characteristic, complemented
39	C₂²×#560	C₂×C₄	1	1	yes	characteristic, complemented
40	C₂²×#535	C₂×C₄	1	1	yes	characteristic
41	C₂²×#534	C₂×C₄	1	1	yes	characteristic
42	C₂²×#534	C₂×C₄	1	1	yes	characteristic
43	C₂³×#120	C₂×C₄	1	1	yes	characteristic
44	C₂³×#120	D₄	1	2	yes	—
45	C₂³×#120	C₂³	1	1	yes	characteristic
46	C₂×#2849	C₂×C₄	1	1	yes	characteristic
47	C₂×#2849	D₄	1	1	yes	characteristic
48	C₂×#2846	C₂×C₄	1	1	yes	characteristic
49	C₂×#2846	D₄	1	1	yes	characteristic
50	C₂×#2404	D₄	1	2	yes	—
51	C₂×#2613	D₄	1	1	yes	characteristic
52	C₂×#2613	D₄	1	1	yes	characteristic
53	C₂×#2613	C₂³	1	1	yes	characteristic
54	C₂×#2613	C₂³	1	1	yes	characteristic
55	C₂×#2308	D₄	1	1	yes	characteristic
56	C₂×#2308	D₄	1	1	yes	characteristic
57	C₂²×#379	D₄	1	1	yes	characteristic
58	#21746	C₂³	1	4	yes	—
59	#15948	C₂×C₄	1	1	yes	characteristic, complemented
60	#15948	C₂×C₄	1	1	yes	characteristic, complemented
61	#15947	C₂×C₄	1	1	yes	characteristic
62	#15947	C₂×C₄	1	1	yes	characteristic
63	#15947	C₂×C₄	1	1	yes	characteristic, complemented
64	#15947	C₂×C₄	1	1	yes	characteristic, complemented
65	#15947	D₄	1	2	yes	—
66	#15947	D₄	1	2	yes	—
67	#15947	C₂³	1	1	yes	characteristic
68	#15947	C₂³	1	1	yes	characteristic
69	#15262	C₂×C₄	1	1	yes	characteristic, complemented
70	#15261	C₂×C₄	1	1	yes	characteristic, complemented
71	#15259	C₂×C₄	1	1	yes	characteristic
72	#15259	C₂×C₄	1	1	yes	characteristic, complemented
73	#15259	D₄	1	2	yes	—
74	#15259	C₂³	1	1	yes	characteristic
75	#15258	C₂×C₄	1	1	yes	characteristic
76	#15258	C₂×C₄	1	1	yes	characteristic, complemented
77	#15258	D₄	1	2	yes	—
78	#15258	C₂³	1	1	yes	characteristic
79	#14658	C₂×C₄	1	1	yes	characteristic, complemented
80	#14657	C₂×C₄	1	1	yes	characteristic, complemented
81	#14335	C₂×C₄	1	1	yes	characteristic
82	#14167	C₂×C₄	1	1	yes	characteristic, complemented
83	#14134	C₂×C₄	1	1	yes	characteristic, complemented
84	#13868	D₄	1	2	yes	—
85	C₂×#1438	C₂×C₄	1	1	yes	characteristic
86	C₂×#1438	D₄	1	2	yes	—
87	#8547	C₂³	1	4	yes	—
88	C₂⁷	C₂²⋊C₄	1	1	yes	characteristic
89	C₂⁷	C₂²⋊C₄	1	1	yes	characteristic
90	C₂⁷	C₂×D₄	1	1	yes	characteristic
91	C₂³×#41	C₂²⋊C₄	1	1	yes	characteristic
92	C₂⁴×D₄	C₂²⋊C₄	1	1	yes	characteristic
93	C₂⁴×D₄	C₂²⋊C₄	1	1	yes	characteristic
94	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
95	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
96	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
97	C₂⁴×D₄	C₂×D₄	1	2	yes	—
98	C₂⁵×C₄	C₂²⋊C₄	1	1	yes	characteristic
99	C₂²×#127	C₂²⋊C₄	1	1	yes	characteristic
100	C₂²×#127	C₂²×C₄	1	1	yes	characteristic
101	C₂²×#124	C₂²⋊C₄	1	1	yes	characteristic
102	C₂²×#124	C₂²×C₄	1	1	yes	characteristic
103	C₂²×#123	C₂²⋊C₄	1	1	yes	characteristic
104	C₂²×#123	C₂²⋊C₄	1	1	yes	characteristic
105	C₂²×#123	C₂²×C₄	1	1	yes	characteristic
106	C₂²×#123	C₂²×C₄	1	1	yes	characteristic
107	C₂²×#120	C₂²⋊C₄	1	1	yes	characteristic
108	C₂²×#120	C₂²⋊C₄	1	2	yes	—
109	C₂²×#120	C₂²⋊C₄	1	1	yes	characteristic
110	C₂²×#120	C₂²⋊C₄	1	1	yes	characteristic
111	C₂²×#120	C₂²⋊C₄	1	1	yes	characteristic
112	C₂²×#120	C₂²⋊C₄	1	2	yes	—
113	C₂²×#120	C₂²×C₄	1	1	yes	characteristic
114	C₂²×#120	C₂²×C₄	1	1	yes	characteristic
115	C₂²×#120	C₂²×C₄	1	1	yes	characteristic
116	C₂²×#120	C₂²×C₄	1	1	yes	characteristic
117	C₂²×#120	C₂²×C₄	1	1	yes	characteristic
118	C₂²×#120	C₂×D₄	1	2	yes	—
119	C₂²×#120	C₂×D₄	1	2	yes	—
120	C₂²×#120	C₂×D₄	1	1	yes	characteristic
121	C₂²×#120	C₂×D₄	1	2	yes	—
122	C₂²×#120	C₂×D₄	1	2	yes	—
123	C₂²×#120	C₂×D₄	1	1	yes	characteristic
124	C₂²×#120	C₂⁴	1	1	yes	characteristic
125	C₂²×#117	C₂²⋊C₄	1	1	yes	characteristic
126	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
127	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
128	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
129	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
130	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
131	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
132	C₂×#393	C₂²⋊C₄	1	2	yes	—
133	C₂×#393	C₂×D₄	1	2	yes	—
134	C₂×#392	C₂²⋊C₄	1	2	yes	—
135	C₂×#392	C₂×D₄	1	2	yes	—
136	C₂×#381	C₂²⋊C₄	1	2	yes	—
137	C₂×#381	C₂²⋊C₄	1	1	yes	characteristic
138	C₂×#381	C₂²⋊C₄	1	1	yes	characteristic
139	C₂×#381	C₂×D₄	1	1	yes	characteristic
140	C₂×#381	C₂×D₄	1	1	yes	characteristic
141	C₂×#379	C₂²⋊C₄	1	1	yes	characteristic
142	C₂×#379	C₂²⋊C₄	1	1	yes	characteristic
143	C₂×#379	C₂²⋊C₄	1	1	yes	characteristic
144	C₂×#379	C₂²⋊C₄	1	1	yes	characteristic
145	C₂×#379	C₂×D₄	1	1	yes	characteristic
146	C₂×#379	C₂×D₄	1	1	yes	characteristic
147	C₂×#379	C₂×D₄	1	1	yes	characteristic
148	C₂×#379	C₂×D₄	1	1	yes	characteristic
149	#1438	C₂²⋊C₄	1	2	yes	—
150	#1438	C₂²⋊C₄	1	2	yes	—
151	#1438	C₂²⋊C₄	1	2	yes	—
152	#1438	C₂²⋊C₄	1	2	yes	—
153	#1438	C₂²×C₄	1	1	yes	characteristic
154	#1438	C₂²×C₄	1	1	yes	characteristic
155	#1438	C₂²×C₄	1	1	yes	characteristic
156	#1438	C₂²×C₄	1	1	yes	characteristic
157	#1438	C₂×D₄	1	2	yes	—
158	#1438	C₂×D₄	1	2	yes	—
159	#1437	C₂²⋊C₄	1	2	yes	—
160	#1437	C₂²⋊C₄	1	2	yes	—
161	#1437	C₂×D₄	1	2	yes	—
162	#1437	C₂×D₄	1	2	yes	—
163	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
164	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
165	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
166	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
167	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
168	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
169	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
170	C₂⁶	#120	1	2	yes	—
171	C₂⁶	#120	1	2	yes	—
172	C₂⁶	C₂²×D₄	1	1	yes	characteristic
173	C₂³×Q₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
174	C₂³×D₄	#99	1	2	yes	—
175	C₂³×D₄	#99	1	2	yes	—
176	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
177	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
178	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
179	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
180	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
181	C₂³×D₄	#120	1	2	yes	—
182	C₂³×D₄	#120	1	2	yes	—
183	C₂³×D₄	C₂³×C₄	1	1	yes	characteristic
184	C₂³×D₄	C₂²×D₄	1	2	yes	—
185	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
186	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
187	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
188	C₂⁴×C₄	#120	1	2	yes	—
189	C₂²×C₂²⋊C₄	#99	1	2	yes	—
190	C₂²×C₂²⋊C₄	#99	1	2	yes	—
191	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
192	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
193	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
194	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
195	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
196	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
197	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
198	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
199	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
200	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
201	C₂²×C₂²⋊C₄	#120	1	2	yes	—
202	C₂²×C₂²⋊C₄	#120	1	2	yes	—
203	C₂²×C₂²⋊C₄	#120	1	2	yes	—
204	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
205	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
206	C₂²×C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
207	C₂²×C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
208	C₂×#95	C₂×C₂²⋊C₄	1	2	yes	—
209	C₂×#95	C₂×C₂²⋊C₄	1	2	yes	—
210	C₂⁵	#379	1	1	yes	characteristic
211	C₂⁵	#379	1	2	yes	—
212	C₂⁵	#379	1	1	yes	characteristic
213	C₂⁵	#379	1	1	yes	characteristic
214	C₂⁵	C₂×#99	1	1	yes	characteristic
215	C₂⁵	C₂×#99	1	1	yes	characteristic
216	C₂⁵	C₂×#99	1	1	yes	characteristic
217	C₂⁵	C₂×#99	1	1	yes	characteristic
218	C₂⁵	C₂×#99	1	2	yes	—
219	C₂⁵	B(4,𝔽₂)	1	2	yes	—
220	C₂⁵	B(4,𝔽₂)	1	2	yes	—
221	C₂⁵	#458	1	2	yes	—
222	C₂⁵	#458	1	2	yes	—
223	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
224	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
225	C₂⁵	C₂×#120	1	2	yes	—
226	C₂²×D₄	AΣL(1,𝔽₁₆)	1	8	yes	complemented
227	C₂³×C₄	#379	1	2	yes	—
228	C₂³×C₄	#379	1	1	yes	characteristic
229	C₂³×C₄	C₂×#99	1	2	yes	—
230	C₂³×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
231	C₂³×C₄	C₂×#120	1	2	yes	—
232	C₂×C₂²⋊C₄	C₂×#99	1	2	yes	—
233	C₂×C₂²⋊C₄	C₂×#99	1	2	yes	—
234	C₂⁴	#1781	1	1	yes	characteristic
235	C₂⁴	#1781	1	1	yes	characteristic
236	C₂⁴	#1781	1	1	yes	characteristic
237	C₂⁴	#1781	1	1	yes	characteristic
238	C₂⁴	#1790	1	2	yes	—
239	C₂⁴	#1790	1	2	yes	—
240	C₂⁴	#1790	1	2	yes	—
241	C₂⁴	#1790	1	2	yes	—
242	C₂⁴	C₂×AΣL(1,𝔽₁₆)	1	4	yes	—
243	C₂⁴	C₂×AΣL(1,𝔽₁₆)	1	4	yes	complemented
244	C₂⁴	C₂×#379	1	1	yes	characteristic
245	C₂⁴	C₂²×#99	1	1	yes	characteristic
246	C₂⁴	C₂²×#99	1	1	yes	characteristic
247	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
248	C₂⁴	C₂×#458	1	2	yes	—
249	C₂²×C₄	C₂×AΣL(1,𝔽₁₆)	1	4	yes	complemented
250	C₂³	#8534	1	1	yes	characteristic
251	C₂³	#8534	1	1	yes	characteristic
252	C₂³	#8534	1	2	yes	—
253	C₂³	#12687	1	2	yes	—
254	C₂³	#12687	1	2	yes	—
255	C₂³	#12705	1	2	yes	—
256	C₂³	#12753	1	4	yes	—
257	C₂³	C₂×#1781	1	1	yes	characteristic
258	C₂³	C₂×#1790	1	2	yes	—
259	C₂³	C₂²×AΣL(1,𝔽₁₆)	1	2	yes	—
260	C₂²	#133278	1	4	yes	—
261	C₂²	C₂×#8534	1	1	yes	characteristic
262	C₂²	C₂×#12687	1	1	yes	characteristic
263	C₂²	C₂×#12705	1	1	yes	characteristic
264	C₂²	C₂×#12753	1	2	yes	—
265	C₂²	#493594	1	2	yes	—
266	C₂	C₂×#133278	1	1	yes	characteristic
267	C₂	#72258234	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×#8534
commutator	C₂³×D₄	C₂³×C₄
Frattini, cosocle	C₂²×#120	C₂⁴
prime radical	#72219264	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×#120 ◁ G
LECS	G ▷ C₂²×#120 ▷ C₂⁴ ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72259008	C₂	semidirect
2. #72259005	C₂	semidirect
3. #72235718	C₂	semidirect
4. #72234515	C₂	semidirect
5. #72232723	C₂	semidirect
6. #72226097	C₂	semidirect
7. #72226091	C₂	semidirect
8. C₂×#139716	C₂	semidirect
10. C₂²×#3532	C₄	semidirect
12. C₂×#21746	C₂²	semidirect
15. #5021140	C₄	semidirect
16. #5021138	C₄	semidirect
19. #763333	C₄	semidirect
20. #686187	C₄	semidirect
21. #647656	C₄	semidirect
23. #616960	C₄	semidirect
25. #552670	C₄	semidirect
26. C₂×#8547	C₂²	semidirect
27. #139716	C₂²	semidirect
28. #139716	C₂²	semidirect
29. #139504	C₂²	semidirect
30. #139486	C₂²	semidirect
31. #134036	C₂²	semidirect
32. #134036	C₂²	semidirect
33. #115631	C₂²	semidirect
34. #115631	C₂²	semidirect
36. C₂²×#551	C₂×C₄	semidirect
37. C₂²×D₄²	C₂×C₄	semidirect
38. C₂²×H(3,𝔽₄)	C₂×C₄	semidirect
39. C₂²×#560	C₂×C₄	semidirect
59. #15948	C₂×C₄	semidirect
60. #15948	C₂×C₄	semidirect
63. #15947	C₂×C₄	semidirect
64. #15947	C₂×C₄	semidirect
69. #15262	C₂×C₄	semidirect
70. #15261	C₂×C₄	semidirect
72. #15259	C₂×C₄	semidirect
76. #15258	C₂×C₄	semidirect
79. #14658	C₂×C₄	semidirect
80. #14657	C₂×C₄	semidirect
82. #14167	C₂×C₄	semidirect
83. #14134	C₂×C₄	semidirect
226. C₂²×D₄	AΣL(1,𝔽₁₆)	semidirect
243. C₂⁴	C₂×AΣL(1,𝔽₁₆)	semidirect
249. C₂²×C₄	C₂×AΣL(1,𝔽₁₆)	semidirect

Maximal subgroups

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

subgroup	index
#72259008	2
#72259005	2
#72235718	2
#72234515	2
#72232723	2
#72226097	2
#72226091	2
C₂×#139716	2
#72219264	2