Subgroups — #72323874

Order 4096.

At least 8601 conjugacy classes of subgroups; only the 231 normal subgroups are shown.

The subgroup lattice has 205 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72323217	C₂	1	1	yes	maximal, characteristic, complemented
2	#72314618	C₂	1	1	yes	maximal, characteristic, complemented
3	#72302137	C₂	1	1	yes	maximal, characteristic
4	#72266446	C₂	1	1	yes	maximal, characteristic
5	#72265766	C₂²	1	1	yes	characteristic
6	#72240132	C₄	1	1	yes	characteristic, complemented
7	#72238601	C₂	1	1	yes	maximal, characteristic, complemented
8	#72237588	C₂	1	1	yes	maximal, characteristic, complemented
9	#72235186	C₄	1	1	yes	characteristic, complemented
10	#72234668	C₂²	1	1	yes	characteristic
11	#72226043	C₂	1	1	yes	maximal, characteristic
12	#72226031	C₄	1	2	yes	complemented
13	#72226027	C₄	1	2	yes	complemented
14	#72225870	C₂²	1	1	yes	characteristic
15	#72219311	C₂²	1	1	yes	characteristic
16	#72219311	C₂²	1	1	yes	characteristic
17	#72219310	C₂²	1	1	yes	characteristic
18	C₂×#139716	C₄	1	1	yes	characteristic, complemented
19	#72219264	C₂²	1	1	yes	characteristic
20	#72219241	C₄	1	1	yes	characteristic, complemented
21	#72216481	C₄	1	2	yes	complemented
22	#72211163	C₄	1	2	yes	complemented
23	C₂²×#3532	D₄	1	1	yes	characteristic
24	C₂×#34980	C₂³	1	1	yes	characteristic
25	#7626202	C₂×C₄	1	1	yes	characteristic, complemented
26	C₂×#21746	C₂×C₄	1	1	yes	characteristic, complemented
27	#5025789	Q₈	1	1	yes	characteristic
28	#5025782	D₄	1	1	yes	characteristic
29	#5021140	C₂×C₄	1	1	yes	characteristic
30	#5021138	C₂×C₄	1	1	yes	characteristic
31	#4422949	Q₈	1	1	yes	characteristic
32	#4420586	D₄	1	1	yes	characteristic
33	#763333	D₄	1	1	yes	characteristic
34	#686187	C₂×C₄	1	1	yes	characteristic
35	#647656	C₂×C₄	1	1	yes	characteristic
36	#616961	C₂×C₄	1	1	yes	characteristic
37	#616960	D₄	1	1	yes	characteristic
38	C₂×#15947	C₂×C₄	1	1	yes	characteristic
39	#552670	D₄	1	1	yes	characteristic
40	#392109	D₄	1	2	yes	—
41	#190116	D₄	1	2	yes	—
42	C₂×#8547	C₂×C₄	1	1	yes	characteristic, complemented
43	C₂×#8541	C₂×C₄	1	1	yes	characteristic, complemented
44	#134035	C₂×C₄	1	2	yes	complemented
45	#115697	C₂×C₄	1	2	yes	complemented
46	#115690	C₂×C₄	1	2	yes	—
47	#115690	C₂×C₄	1	2	yes	—
48	#115690	D₄	1	2	yes	—
49	#115631	D₄	1	2	yes	—
50	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
51	C₂²×#551	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×D₄²	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×H(3,𝔽₄)	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×#560	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×#535	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×#534	C₂×D₄	1	1	yes	characteristic
57	C₂²×#534	C₂×D₄	1	1	yes	characteristic
58	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
59	C₂³×#120	C₂²×C₄	1	1	yes	characteristic
60	C₂×#2849	C₂²×C₄	1	1	yes	characteristic
61	C₂×#2849	#41	1	1	yes	characteristic
62	C₂×#2846	C₂²×C₄	1	1	yes	characteristic
63	C₂×#2846	C₂×D₄	1	1	yes	characteristic
64	C₂×#2613	C₂×D₄	1	1	yes	characteristic
65	C₂×#2613	C₂×D₄	1	1	yes	characteristic
66	C₂×#2613	C₂×Q₈	1	1	yes	characteristic
67	C₂×#2613	#41	1	1	yes	characteristic
68	C₂×#2308	C₂×D₄	1	1	yes	characteristic
69	C₂×#2308	#41	1	1	yes	characteristic
70	C₂²×#379	#41	1	1	yes	characteristic
71	#15948	C₂²⋊C₄	1	1	yes	characteristic
72	#15948	C₂²⋊C₄	1	1	yes	characteristic
73	#15947	C₄²	1	1	yes	characteristic
74	#15947	C₄²	1	1	yes	characteristic
75	#15947	C₄⋊C₄	1	1	yes	characteristic
76	#15947	C₄⋊C₄	1	1	yes	characteristic
77	#15947	C₄⋊C₄	1	1	yes	characteristic
78	#15947	C₄⋊C₄	1	1	yes	characteristic
79	#15262	C₂²⋊C₄	1	1	yes	characteristic
80	#15261	C₂²⋊C₄	1	1	yes	characteristic
81	#15259	C₄²	1	1	yes	characteristic
82	#15259	C₄⋊C₄	1	1	yes	characteristic
83	#15259	C₄⋊C₄	1	1	yes	characteristic
84	#15258	C₄²	1	1	yes	characteristic
85	#15258	C₄⋊C₄	1	1	yes	characteristic
86	#15258	C₄⋊C₄	1	1	yes	characteristic
87	#14679	C₂²⋊C₄	1	4	yes	complemented
88	#14658	C₄⋊C₄	1	1	yes	characteristic
89	#14657	C₂²⋊C₄	1	1	yes	characteristic
90	#14335	C₂²⋊C₄	1	1	yes	characteristic
91	#14167	C₄⋊C₄	1	1	yes	characteristic
92	#14134	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×#1438	C₂²⋊C₄	1	1	yes	characteristic
94	#8101	C₂²⋊C₄	1	4	yes	complemented
95	C₂⁷	#99	1	1	yes	characteristic
96	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂⁷	C₄×D₄	1	1	yes	characteristic
98	C₂³×#41	#99	1	1	yes	characteristic
99	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂⁴×D₄	#120	1	1	yes	characteristic
101	C₂⁴×D₄	#120	1	1	yes	characteristic
102	C₂⁴×D₄	#121	1	1	yes	characteristic
103	C₂⁴×D₄	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂⁵×C₄	C₄×D₄	1	1	yes	characteristic
105	C₂²×#127	#95	1	1	yes	characteristic
106	C₂²×#127	#99	1	1	yes	characteristic
107	C₂²×#124	#95	1	1	yes	characteristic
108	C₂²×#124	#99	1	1	yes	characteristic
109	C₂²×#123	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂²×#123	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂²×#123	#121	1	1	yes	characteristic
112	C₂²×#123	#123	1	1	yes	characteristic
113	C₂²×#120	#95	1	1	yes	characteristic
114	C₂²×#120	#95	1	1	yes	characteristic
115	C₂²×#120	#99	1	1	yes	characteristic
116	C₂²×#120	#99	1	1	yes	characteristic
117	C₂²×#120	C₂×C₄²	1	1	yes	characteristic
118	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂²×#120	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×#120	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×#120	C₄×D₄	1	1	yes	characteristic
122	C₂²×#120	C₄×D₄	1	1	yes	characteristic
123	C₂²×#120	#120	1	1	yes	characteristic
124	C₂²×#120	#121	1	1	yes	characteristic
125	C₂²×#117	#99	1	1	yes	characteristic
126	C₂³×C₂²⋊C₄	#99	1	1	yes	characteristic
127	C₂³×C₂²⋊C₄	C₄×D₄	1	1	yes	characteristic
128	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
129	C₂³×C₂²⋊C₄	#123	1	1	yes	characteristic
130	C₂×#381	C₄×D₄	1	1	yes	characteristic
131	C₂×#381	C₄×D₄	1	1	yes	characteristic
132	C₂×#381	C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×#381	#123	1	1	yes	characteristic
134	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×#379	#117	1	1	yes	characteristic
136	C₂×#379	#117	1	1	yes	characteristic
137	C₂×#379	C₄×D₄	1	1	yes	characteristic
138	C₂×#379	#120	1	1	yes	characteristic
139	C₂×#379	C₂²⋊Q₈	1	1	yes	characteristic
140	C₂×#379	C₂²⋊Q₈	1	1	yes	characteristic
141	C₂×#379	#123	1	1	yes	characteristic
142	#1438	#95	1	1	yes	characteristic
143	#1438	#95	1	1	yes	characteristic
144	#1438	#95	1	1	yes	characteristic
145	#1438	#95	1	1	yes	characteristic
146	C₂⁶	#342	1	1	yes	characteristic
147	C₂⁶	#342	1	1	yes	characteristic
148	C₂⁶	C₄×C₂²⋊C₄	1	1	yes	characteristic
149	C₂⁶	#379	1	1	yes	characteristic
150	C₂⁶	#380	1	1	yes	characteristic
151	C₂⁶	#385	1	1	yes	characteristic
152	C₂⁶	#386	1	1	yes	characteristic
153	C₂⁶	C₂×#99	1	1	yes	characteristic
154	C₂³×Q₈	#342	1	1	yes	characteristic
155	C₂³×D₄	#342	1	1	yes	characteristic
156	C₂³×D₄	C₂×#95	1	1	yes	characteristic
157	C₂³×D₄	#386	1	1	yes	characteristic
158	C₂³×D₄	C₂×#99	1	1	yes	characteristic
159	C₂³×D₄	C₂×#99	1	1	yes	characteristic
160	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
161	C₂³×D₄	#458	1	2	yes	—
162	C₂⁴×C₄	#385	1	1	yes	characteristic
163	C₂⁴×C₄	#386	1	1	yes	characteristic
164	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
165	C₂²×C₂²⋊C₄	#342	1	1	yes	characteristic
166	C₂²×C₂²⋊C₄	#342	1	1	yes	characteristic
167	C₂²×C₂²⋊C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
168	C₂²×C₂²⋊C₄	#380	1	1	yes	characteristic
169	C₂²×C₂²⋊C₄	#381	1	1	yes	characteristic
170	C₂²×C₂²⋊C₄	#385	1	1	yes	characteristic
171	C₂²×C₂²⋊C₄	#385	1	1	yes	characteristic
172	C₂²×C₂²⋊C₄	#386	1	1	yes	characteristic
173	C₂²×C₄²	#342	1	1	yes	characteristic
174	C₂²×C₄²	#342	1	1	yes	characteristic
175	C₂⁵	C₂²⋊#95	1	1	yes	characteristic
176	C₂⁵	C₂×#342	1	1	yes	characteristic
177	C₂⁵	#1790	1	2	yes	—
178	C₂⁵	#1889	1	1	yes	characteristic
179	C₂⁵	#1890	1	1	yes	characteristic
180	C₂⁵	#1898	1	1	yes	characteristic
181	C₂⁵	#1898	1	1	yes	characteristic
182	C₂⁵	#1898	1	1	yes	characteristic
183	C₂⁵	#1913	1	1	yes	characteristic
184	C₂⁵	#1913	1	1	yes	characteristic
185	C₂³×C₄	C₂×#342	1	1	yes	characteristic
186	C₂³×C₄	#1867	1	2	yes	—
187	C₂³×C₄	#1898	1	1	yes	characteristic
188	C₂⁴	#9495	1	1	yes	characteristic
189	C₂⁴	#9495	1	1	yes	characteristic
190	C₂⁴	#9499	1	1	yes	characteristic
191	C₂⁴	#9499	1	1	yes	characteristic
192	C₂⁴	#10735	1	1	yes	characteristic
193	C₂⁴	#10745	1	1	yes	characteristic
194	C₂⁴	#10754	1	1	yes	characteristic
195	C₂⁴	#12753	1	4	yes	—
196	C₂³	#106542	1	1	yes	characteristic
197	C₂³	#139749	1	1	yes	characteristic
198	C₂³	#139750	1	1	yes	characteristic
199	C₂³	#146593	1	2	yes	—
200	C₂²	#72241271	1	1	yes	characteristic
201	C₂²	#72246453	1	1	yes	characteristic
202	C₂²	#72255793	1	1	yes	characteristic
203	C₂	#72266320	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266320
commutator	C₂²×#120	C₂×C₄²
Frattini, cosocle	C₂×#34980	C₂³
prime radical	#72219264	C₂²
LCS	G ▷ C₂²×#120 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#34980 ◁ G
LECS	G ▷ C₂×#34980 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72323217	C₂	semidirect
2. #72314618	C₂	semidirect
6. #72240132	C₄	semidirect
7. #72238601	C₂	semidirect
8. #72237588	C₂	semidirect
9. #72235186	C₄	semidirect
12. #72226031	C₄	semidirect
13. #72226027	C₄	semidirect
18. C₂×#139716	C₄	semidirect
20. #72219241	C₄	semidirect
21. #72216481	C₄	semidirect
22. #72211163	C₄	semidirect
25. #7626202	C₂×C₄	semidirect
26. C₂×#21746	C₂×C₄	semidirect
42. C₂×#8547	C₂×C₄	semidirect
43. C₂×#8541	C₂×C₄	semidirect
44. #134035	C₂×C₄	semidirect
45. #115697	C₂×C₄	semidirect
87. #14679	C₂²⋊C₄	semidirect
94. #8101	C₂²⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
#72323217	2
#72314618	2
#72302137	2
#72266446	2
#72238601	2
#72237588	2
#72226043	2