Subgroups — #72339816

Order 8192.

At least 26775 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

The subgroup lattice has 215 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72339812	C₂	1	1	yes	maximal, characteristic, complemented
2	#72339801	C₂	1	1	yes	maximal, characteristic, complemented
3	#72339792	C₂	1	1	yes	maximal, characteristic, complemented
4	#72339790	C₂	1	1	yes	maximal, characteristic, complemented
5	#72339787	C₂	1	1	yes	maximal, characteristic, complemented
6	#72339785	C₂	1	1	yes	maximal, characteristic, complemented
7	#72339783	C₂	1	1	yes	maximal, characteristic, complemented
8	#72339781	C₂²	1	1	yes	characteristic, complemented
9	#72339780	C₂²	1	1	yes	characteristic, complemented
10	#72339777	C₂	1	1	yes	maximal, characteristic, complemented
11	#72339775	C₂²	1	1	yes	characteristic, complemented
12	#72339772	C₂²	1	1	yes	characteristic, complemented
13	#72339752	C₂	1	1	yes	maximal, characteristic, complemented
14	#72339750	C₂²	1	1	yes	characteristic, complemented
15	#72339748	C₂²	1	1	yes	characteristic, complemented
16	#72339746	C₂	1	1	yes	maximal, characteristic, complemented
17	#72339744	C₂²	1	1	yes	characteristic, complemented
18	#72339738	C₂²	1	1	yes	characteristic, complemented
19	#72339720	C₂	1	1	yes	maximal, characteristic, complemented
20	#72339716	C₂²	1	1	yes	characteristic
21	#72339716	C₂²	1	1	yes	characteristic
22	#72339714	C₂²	1	1	yes	characteristic, complemented
23	#72339711	C₂²	1	1	yes	characteristic, complemented
24	#72339706	C₂	1	1	yes	maximal, characteristic, complemented
25	#72339703	C₂²	1	1	yes	characteristic
26	#72339701	C₂²	1	1	yes	characteristic
27	#72339699	C₂²	1	1	yes	characteristic, complemented
28	#72339695	C₂²	1	1	yes	characteristic, complemented
29	#72339694	C₂³	1	1	yes	characteristic
30	#72339689	C₂²	1	1	yes	characteristic, complemented
31	#72339687	C₂³	1	1	yes	characteristic
32	#72339674	D₄	1	2	yes	complemented
33	#72339670	D₄	1	2	yes	complemented
34	#72339664	D₄	1	2	yes	complemented
35	#72339658	D₄	1	2	yes	complemented
36	#72338406	C₂	1	1	yes	maximal, characteristic, complemented
37	#72338405	C₂	1	1	yes	maximal, characteristic, complemented
38	#72338399	C₂²	1	1	yes	characteristic, complemented
39	#72338388	C₂²	1	1	yes	characteristic, complemented
40	#72338367	C₂²	1	1	yes	characteristic, complemented
41	#72338363	C₂²	1	1	yes	characteristic, complemented
42	#72338346	C₂²	1	1	yes	characteristic, complemented
43	#72338343	C₂²	1	1	yes	characteristic, complemented
44	#72338338	C₂²	1	1	yes	characteristic, complemented
45	#72338337	C₂²	1	1	yes	characteristic, complemented
46	#72338326	C₂²	1	1	yes	characteristic, complemented
47	#72338326	C₂²	1	1	yes	characteristic, complemented
48	#72338323	C₂³	1	1	yes	characteristic
49	#72338323	C₂³	1	1	yes	characteristic
50	#72338318	C₂³	1	1	yes	characteristic
51	#72338318	C₂³	1	1	yes	characteristic
52	#72338302	C₂²	1	1	yes	characteristic, complemented
53	#72338299	C₂³	1	1	yes	characteristic
54	#72338294	C₂³	1	1	yes	characteristic
55	#72338260	C₂	1	1	yes	maximal, characteristic, complemented
56	#72338259	C₂²	1	1	yes	characteristic
57	#72338250	C₂²	1	1	yes	characteristic
58	#72338247	C₂²	1	1	yes	characteristic
59	#72338243	C₂²	1	1	yes	characteristic
60	#72338242	D₄	1	1	yes	characteristic, complemented
61	#72338241	C₂³	1	1	yes	characteristic
62	#72338233	C₂²	1	1	yes	characteristic
63	#72338232	D₄	1	1	yes	characteristic, complemented
64	#72338231	C₂³	1	1	yes	characteristic
65	#72338211	D₄	1	1	yes	characteristic, complemented
66	#72338209	C₂²	1	1	yes	characteristic
67	#72338208	D₄	1	1	yes	characteristic
68	#72338208	C₂³	1	1	yes	characteristic
69	#72338207	D₄	1	1	yes	characteristic
70	#72317033	C₂²	1	1	yes	characteristic
71	#72316812	C₂³	1	1	yes	characteristic
72	#72316805	D₄	1	1	yes	characteristic, complemented
73	#72310296	D₄	1	1	yes	characteristic, complemented
74	#72302242	C₂³	1	1	yes	characteristic
75	#72280381	D₄	1	1	yes	characteristic, complemented
76	#72279737	D₄	1	1	yes	characteristic
77	#72279737	C₂³	1	1	yes	characteristic
78	#72257769	D₄	1	1	yes	characteristic, complemented
79	#72257649	D₄	1	1	yes	characteristic, complemented
80	#72239916	C₂³	1	1	yes	characteristic
81	C₂²×#63011	D₄	1	1	yes	characteristic
82	C₂²×#3549	C₂×D₄	1	1	yes	characteristic
83	C₂²×#3498	C₂×D₄	1	1	yes	characteristic
84	C₂²×#3481	C₂×D₄	1	1	yes	characteristic
85	C₂×#60968	C₂×D₄	1	1	yes	characteristic
86	#7628157	C₂×D₄	1	1	yes	characteristic
87	#7513958	C₂×D₄	1	1	yes	characteristic, complemented
88	#7503371	C₂×D₄	1	1	yes	characteristic, complemented
89	#6374317	C₂×D₄	1	1	yes	characteristic
90	#6374317	C₂⁴	1	1	yes	characteristic
91	#6374304	C₂×D₄	1	1	yes	characteristic
92	#6374304	C₂×D₄	1	1	yes	characteristic
93	#6370950	C₂×D₄	1	1	yes	characteristic
94	#6370937	C₂×D₄	1	1	yes	characteristic
95	#6343804	C₂×D₄	1	1	yes	characteristic
96	#6343804	C₂×D₄	1	1	yes	characteristic
97	#387626	C₂×D₄	1	1	yes	characteristic, complemented
98	#387625	C₂×D₄	1	1	yes	characteristic, complemented
99	#187400	C₂×D₄	1	2	yes	—
100	#187400	C₂×D₄	1	2	yes	—
101	#181777	C₂×D₄	1	1	yes	characteristic
102	#181771	C₂×D₄	1	1	yes	characteristic
103	#181392	C₂×D₄	1	1	yes	characteristic
104	#181339	C₂×D₄	1	1	yes	characteristic
105	#180871	C₂×D₄	1	1	yes	characteristic
106	#180855	C₂×D₄	1	1	yes	characteristic
107	#151856	C₂×D₄	1	2	yes	complemented
108	#151854	C₂×D₄	1	2	yes	complemented
109	C₂×#3549	#120	1	2	yes	—
110	C₂×#3549	#135	1	2	yes	—
111	C₂²×#559	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₂²×#555	AGL(1,ℤ/8)	1	2	yes	—
113	C₂²×D₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂²×#532	C₂²×D₄	1	1	yes	characteristic
115	C₂³×#127	#120	1	1	yes	characteristic
116	C₂²×C₄×#41	#120	1	1	yes	characteristic
117	C₂×#2920	#120	1	1	yes	characteristic
118	C₂×#2920	C₂²×D₄	1	1	yes	characteristic
119	#34012	#120	1	1	yes	characteristic
120	#34012	C₂²×D₄	1	1	yes	characteristic
121	#21209	C₂²×D₄	1	1	yes	characteristic
122	#21209	C₂²×D₄	1	1	yes	characteristic
123	#21209	H(5,𝔽₂)	1	1	yes	characteristic
124	#21209	H(5,𝔽₂)	1	1	yes	characteristic
125	#17663	#120	1	2	yes	—
126	C₂³×#41	B(4,𝔽₂)	1	1	yes	characteristic
127	#3549	#453	1	4	yes	—
128	#3549	B(4,𝔽₂)	1	4	yes	—
129	C₂×#555	#449	1	4	yes	—
130	C₂×#532	C₂×#120	1	1	yes	characteristic
131	C₂×#532	#534	1	1	yes	characteristic
132	C₂×#532	C₂×#135	1	1	yes	characteristic
133	C₂×#532	#575	1	1	yes	characteristic
134	C₂²×C₄⋊Q₈	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂²×#127	B(4,𝔽₂)	1	1	yes	characteristic
136	C₂²×#127	B(4,𝔽₂)	1	1	yes	characteristic
137	C₂²×#127	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂²×#117	B(4,𝔽₂)	1	1	yes	characteristic
139	C₂³×C₄²	C₂×#120	1	1	yes	characteristic
140	C₂×#423	D₄²	1	1	yes	characteristic
141	C₂×#423	D₄²	1	1	yes	characteristic
142	C₄×#117	D₄²	1	1	yes	characteristic
143	C₄×#117	D₄²	1	1	yes	characteristic
144	C₂³×Q₈	#2196	1	1	yes	characteristic
145	C₂³×D₄	#2196	1	1	yes	characteristic
146	C₂⁴×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
147	#555	#1600	1	8	yes	—
148	#532	C₂×#453	1	2	yes	—
149	#532	C₂×B(4,𝔽₂)	1	2	yes	—
150	C₂×C₄⋊Q₈	C₂×#449	1	1	yes	characteristic
151	C₂×C₄⋊Q₈	#3194	1	1	yes	characteristic
152	C₂×#127	#2196	1	2	yes	—
153	C₂×#127	#2196	1	2	yes	—
154	C₂×#127	#3006	1	1	yes	characteristic
155	C₂×#127	#3188	1	1	yes	characteristic
156	C₂²×C₂²⋊C₄	#2196	1	2	yes	—
157	C₂²×C₄²	#2403	1	1	yes	characteristic
158	C₂²×C₄²	#3002	1	1	yes	characteristic
159	C₂²×C₄²	#3003	1	1	yes	characteristic
160	C₂²×C₄²	#3019	1	1	yes	characteristic
161	C₂²×C₄²	#3019	1	1	yes	characteristic
162	C₂²×C₄²	C₂×B(4,𝔽₂)	1	1	yes	characteristic
163	C₂⁵	C₂×#2196	1	1	yes	characteristic
164	C₂³×C₄	#18132	1	1	yes	characteristic
165	C₂³×C₄	#18192	1	1	yes	characteristic
166	C₂³×C₄	#23900	1	1	yes	characteristic
167	C₂³×C₄	C₂×#2196	1	1	yes	characteristic
168	C₂³×C₄	#33551	1	1	yes	characteristic
169	C₂³×C₄	#33551	1	1	yes	characteristic
170	C₄⋊Q₈	C₂×#1600	1	2	yes	—
171	#127	#18162	1	2	yes	—
172	C₂×C₄²	#18132	1	1	yes	characteristic
173	C₂×C₄²	#18192	1	1	yes	characteristic
174	C₂×C₄²	#23895	1	1	yes	characteristic
175	C₂×C₄²	#23897	1	1	yes	characteristic
176	C₂×C₄²	#23900	1	1	yes	characteristic
177	C₂×C₄²	#23906	1	1	yes	characteristic
178	C₂×C₄²	#23912	1	1	yes	characteristic
179	C₂×C₄²	#23920	1	1	yes	characteristic
180	C₂×C₄²	C₂×#2196	1	1	yes	characteristic
181	C₂×C₄²	C₂×#2196	1	1	yes	characteristic
182	C₂×C₄²	#33551	1	1	yes	characteristic
183	C₂×C₄²	#33551	1	1	yes	characteristic
184	C₂⁴	#485039	1	1	yes	characteristic
185	C₂⁴	#486139	1	1	yes	characteristic
186	C₂⁴	#499787	1	1	yes	characteristic
187	C₂²×C₄	#245225	1	1	yes	characteristic
188	C₂²×C₄	#248913	1	1	yes	characteristic
189	C₂²×C₄	#249319	1	1	yes	characteristic
190	C₂²×C₄	#485047	1	1	yes	characteristic
191	C₂²×C₄	#486139	1	1	yes	characteristic
192	C₂²×C₄	#499787	1	1	yes	characteristic
193	C₂²×C₄	#499787	1	1	yes	characteristic
194	C₂²×C₄	#499787	1	1	yes	characteristic
195	C₄²	#245091	1	1	yes	characteristic
196	C₄²	#245168	1	1	yes	characteristic
197	C₄²	#248891	1	1	yes	characteristic
198	C₄²	#249355	1	1	yes	characteristic
199	C₄²	#485039	1	1	yes	characteristic
200	C₄²	#485047	1	1	yes	characteristic
201	C₄²	#486139	1	1	yes	characteristic
202	C₄²	#486139	1	1	yes	characteristic
203	C₂³	#72210587	1	1	yes	characteristic
204	C₂³	#72213827	1	1	yes	characteristic
205	C₂³	#72339605	1	1	yes	characteristic
206	C₂×C₄	#72210587	1	1	yes	characteristic
207	C₂×C₄	#72213827	1	1	yes	characteristic
208	C₂×C₄	#72339605	1	1	yes	characteristic
209	C₂×C₄	#72339606	1	1	yes	characteristic
210	C₂²	#72339610	1	1	yes	characteristic
211	C₄	#72251819	1	1	yes	characteristic
212	C₄	#72339616	1	1	yes	characteristic
213	C₂	#72339650	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72339650
commutator, Frattini, cosocle	#6374317	C₂⁴
LCS	G ▷ #6374317 ▷ C₂²×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₄² ◁ #72302242 ◁ G
LECS	G ▷ #6374317 ▷ C₂²×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72339812	C₂	semidirect
2. #72339801	C₂	semidirect
3. #72339792	C₂	semidirect
4. #72339790	C₂	semidirect
5. #72339787	C₂	semidirect
6. #72339785	C₂	semidirect
7. #72339783	C₂	semidirect
8. #72339781	C₂²	semidirect
9. #72339780	C₂²	semidirect
10. #72339777	C₂	semidirect
11. #72339775	C₂²	semidirect
12. #72339772	C₂²	semidirect
13. #72339752	C₂	semidirect
14. #72339750	C₂²	semidirect
15. #72339748	C₂²	semidirect
16. #72339746	C₂	semidirect
17. #72339744	C₂²	semidirect
18. #72339738	C₂²	semidirect
19. #72339720	C₂	semidirect
22. #72339714	C₂²	semidirect
23. #72339711	C₂²	semidirect
24. #72339706	C₂	semidirect
27. #72339699	C₂²	semidirect
28. #72339695	C₂²	semidirect
30. #72339689	C₂²	semidirect
32. #72339674	D₄	semidirect
33. #72339670	D₄	semidirect
34. #72339664	D₄	semidirect
35. #72339658	D₄	semidirect
36. #72338406	C₂	semidirect
37. #72338405	C₂	semidirect
38. #72338399	C₂²	semidirect
39. #72338388	C₂²	semidirect
40. #72338367	C₂²	semidirect
41. #72338363	C₂²	semidirect
42. #72338346	C₂²	semidirect
43. #72338343	C₂²	semidirect
44. #72338338	C₂²	semidirect
45. #72338337	C₂²	semidirect
46. #72338326	C₂²	semidirect
47. #72338326	C₂²	semidirect
52. #72338302	C₂²	semidirect
55. #72338260	C₂	semidirect
60. #72338242	D₄	semidirect
63. #72338232	D₄	semidirect
65. #72338211	D₄	semidirect
72. #72316805	D₄	semidirect
73. #72310296	D₄	semidirect
75. #72280381	D₄	semidirect
78. #72257769	D₄	semidirect
79. #72257649	D₄	semidirect
87. #7513958	C₂×D₄	semidirect
88. #7503371	C₂×D₄	semidirect
97. #387626	C₂×D₄	semidirect
98. #387625	C₂×D₄	semidirect
107. #151856	C₂×D₄	semidirect
108. #151854	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72339812	2
#72339801	2
#72339792	2
#72339790	2
#72339787	2
#72339785	2
#72339783	2
#72339777	2
#72339752	2
#72339746	2
#72339720	2
#72339706	2
#72338406	2
#72338405	2
#72338260	2