Subgroups — #62686

Order 256.

3613 conjugacy classes of subgroups.

The subgroup lattice has 183 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3461	C₂	1	12	yes	maximal, complemented
2	#3460	C₂	1	12	yes	maximal, complemented
3	#3459	C₂	1	8	yes	maximal, complemented
4	#3458	C₂	1	1	yes	maximal, characteristic, complemented
5	#3455	C₂	1	12	yes	maximal, complemented
6	#3449	C₂	1	3	yes	maximal, complemented
7	#3444	C₂	1	12	yes	maximal, complemented
8	C₂×C₄×#41	C₂	1	3	yes	maximal, complemented
9	C₂²×#41	C₂²	1	3	yes	—
10	#544	C₂²	1	12	yes	complemented
11	#543	C₂²	1	4	yes	—
12	#542	C₂²	1	48	yes	complemented
13	#541	C₂²	1	24	yes	complemented
14	#540	C₂²	1	24	yes	complemented
15	#539	C₂²	1	48	yes	complemented
16	#538	C₂²	1	48	yes	complemented
17	#537	C₂²	1	12	yes	—
18	#536	C₂²	1	12	yes	complemented
19	#535	C₂²	1	12	yes	complemented
20	#534	C₂²	1	12	yes	—
21	#533	C₂²	1	12	yes	complemented
22	#533	C₂²	1	24	yes	complemented
23	#532	C₂²	1	12	yes	—
24	#532	C₂²	1	24	yes	complemented
25	#529	C₂²	1	48	yes	complemented
26	#529	C₂²	1	48	yes	complemented
27	#529	C₂²	1	48	yes	complemented
28	#525	C₂²	1	24	yes	complemented
29	#525	C₂²	1	48	yes	complemented
30	C₄×#41	C₂²	1	12	yes	complemented
31	C₄×#41	C₂²	1	12	yes	complemented
32	C₄×#41	C₂²	1	24	yes	complemented
33	C₄×#41	C₂²	1	16	yes	complemented
34	C₂×C₄×Q₈	C₂²	1	6	yes	complemented
35	C₂×C₄×D₄	C₂²	1	12	yes	—
36	C₂×C₄×D₄	C₂²	1	6	yes	complemented
37	C₂×#117	C₂²	1	3	yes	—
38	C₂×#117	C₂²	1	6	yes	complemented
39	C₂²×C₄²	C₂²	1	1	yes	characteristic
40	C₂²×C₄²	C₂²	1	6	yes	complemented
41	C₂×#41	—	2	24	no	—
42	C₂×#41	C₂³	1	12	yes	—
43	C₂×#41	C₂³	1	12	yes	—
44	C₂×#41	C₂³	1	16	yes	—
45	C₂²×Q₈	C₂³	1	3	yes	—
46	C₂²×D₄	C₂³	1	6	yes	—
47	C₂²×D₄	C₂³	1	3	yes	—
48	C₂³×C₄	C₂³	1	6	yes	—
49	C₂³×C₄	C₂³	1	2	yes	—
50	C₂³×C₄	C₂³	1	12	yes	—
51	C₂³×C₄	C₂³	1	1	yes	characteristic
52	C₄⋊Q₈	C₂³	1	12	yes	—
53	C₄⋊Q₈	C₂³	1	24	yes	—
54	#127	C₂³	1	12	yes	—
55	#126	C₂³	1	96	yes	—
56	#125	C₂³	1	48	yes	—
57	#125	C₂³	1	24	yes	—
58	#124	C₂³	1	24	yes	—
59	#124	C₂³	1	48	yes	—
60	#123	C₂³	1	48	yes	—
61	#123	C₂³	1	96	yes	—
62	C₂²⋊Q₈	C₂³	1	48	yes	—
63	C₂²⋊Q₈	C₂³	1	48	yes	—
64	C₂²⋊Q₈	C₂³	1	48	yes	—
65	#121	C₂³	1	48	yes	—
66	#121	C₂³	1	48	yes	—
67	#121	C₂³	1	48	yes	—
68	#120	C₂³	1	48	yes	—
69	C₄×Q₈	C₂³	1	24	yes	—
70	C₄×Q₈	C₂³	1	24	yes	—
71	C₄×Q₈	C₂³	1	16	yes	—
72	C₄×Q₈	C₂³	1	16	yes	—
73	C₄×D₄	C₂³	1	48	yes	—
74	C₄×D₄	C₂³	1	48	yes	—
75	C₄×D₄	C₂³	1	24	yes	—
76	C₄×D₄	C₂³	1	24	yes	—
77	C₄×D₄	C₂³	1	48	yes	—
78	C₄×D₄	C₂³	1	48	yes	—
79	#117	C₂³	1	12	yes	—
80	#117	C₂³	1	12	yes	—
81	#117	C₂³	1	24	yes	—
82	#117	C₂³	1	24	yes	—
83	#117	C₂³	1	48	yes	—
84	C₂×C₄⋊C₄	C₂³	1	6	yes	—
85	C₂×C₄⋊C₄	C₂³	1	12	yes	—
86	C₂×C₂²⋊C₄	C₂³	1	6	yes	—
87	C₂×C₂²⋊C₄	C₂³	1	12	yes	—
88	C₂×C₄²	C₂³	1	6	yes	—
89	C₂×C₄²	C₂³	1	3	yes	—
90	C₂×C₄²	C₂³	1	12	yes	—
91	C₂×C₄²	C₂³	1	6	yes	—
92	C₂×C₄²	C₂³	1	12	yes	—
93	C₂×C₄²	C₂³	1	24	yes	—
94	C₂×C₄²	C₂³	1	12	yes	—
95	C₂×C₄²	C₂³	1	3	yes	—
96	C₂⁴	C₂⁴	1	6	yes	—
97	C₂⁴	C₂⁴	1	1	yes	characteristic
98	#41	—	2	48	no	—
99	#41	—	2	48	no	—
100	#41	—	4	32	no	—
101	C₂×Q₈	—	2	12	no	—
102	C₂×Q₈	C₂⁴	1	12	yes	—
103	C₂×Q₈	C₂⁴	1	16	yes	—
104	C₂×D₄	—	2	24	no	—
105	C₂×D₄	—	2	12	no	—
106	C₂×D₄	C₂⁴	1	24	yes	—
107	C₂×D₄	C₂⁴	1	12	yes	—
108	C₂×D₄	C₂⁴	1	48	yes	—
109	C₂²×C₄	—	2	24	no	—
110	C₂²×C₄	—	4	4	no	—
111	C₂²×C₄	—	4	24	no	—
112	C₂²×C₄	#41	1	8	yes	complemented
113	C₂²×C₄	C₂⁴	1	6	yes	—
114	C₂²×C₄	C₂⁴	1	6	yes	—
115	C₂²×C₄	C₂⁴	1	12	yes	—
116	C₂²×C₄	C₂⁴	1	24	yes	—
117	C₂²×C₄	C₂⁴	1	12	yes	—
118	C₂²×C₄	C₂⁴	1	6	yes	—
119	C₂²×C₄	C₂⁴	1	12	yes	—
120	C₂²×C₄	C₂⁴	1	24	yes	—
121	C₂²×C₄	C₂⁴	1	3	yes	—
122	C₂²×C₄	C₂⁴	1	3	yes	—
123	C₄⋊C₄	C₂⁴	1	24	yes	—
124	C₄⋊C₄	C₂⁴	1	48	yes	—
125	C₄⋊C₄	C₂⁴	1	48	yes	—
126	C₄⋊C₄	C₂⁴	1	48	yes	—
127	C₂²⋊C₄	C₂⁴	1	24	yes	—
128	C₂²⋊C₄	C₂⁴	1	48	yes	—
129	C₂²⋊C₄	C₂⁴	1	96	yes	—
130	C₄²	C₂⁴	1	3	yes	—
131	C₄²	C₂⁴	1	6	yes	—
132	C₄²	C₂⁴	1	6	yes	—
133	C₄²	C₂⁴	1	12	yes	—
134	C₄²	C₂⁴	1	24	yes	—
135	C₄²	C₂⁴	1	12	yes	—
136	C₄²	C₂⁴	1	24	yes	—
137	C₄²	C₂⁴	1	1	yes	characteristic
138	C₂³	—	2	12	no	—
139	C₂³	—	4	2	no	—
140	C₂³	—	4	12	no	—
141	C₂³	C₂×#41	1	4	yes	—
142	C₂³	C₂⁵	1	12	yes	—
143	C₂³	C₂⁵	1	3	yes	—
144	Q₈	—	2	24	no	—
145	Q₈	—	4	16	no	—
146	D₄	—	2	48	no	—
147	D₄	—	2	24	no	—
148	D₄	—	4	48	no	—
149	C₂×C₄	—	2	12	no	—
150	C₂×C₄	—	2	24	no	—
151	C₂×C₄	—	2	24	no	—
152	C₂×C₄	—	2	12	no	—
153	C₂×C₄	—	2	48	no	—
154	C₂×C₄	—	4	6	no	—
155	C₂×C₄	—	4	24	no	—
156	C₂×C₄	—	4	12	no	—
157	C₂×C₄	C₂×#41	1	12	yes	—
158	C₂×C₄	C₂×#41	1	12	yes	—
159	C₂×C₄	C₂⁵	1	6	yes	—
160	C₂×C₄	C₂⁵	1	12	yes	—
161	C₂×C₄	C₂⁵	1	24	yes	—
162	C₂×C₄	C₂⁵	1	1	yes	characteristic
163	C₂×C₄	C₂⁵	1	3	yes	—
164	C₂×C₄	C₂⁵	1	2	yes	—
165	C₂²	—	2	6	no	—
166	C₂²	—	2	12	no	—
167	C₂²	—	2	24	no	—
168	C₂²	—	4	4	no	—
169	C₂²	—	4	24	no	—
170	C₂²	C₂²×#41	1	6	yes	—
171	C₂²	C₂⁶	1	1	yes	characteristic
172	C₄	—	2	12	no	—
173	C₄	—	2	6	no	—
174	C₄	—	2	24	no	—
175	C₄	C₂²×#41	1	2	yes	—
176	C₄	C₂²×#41	1	6	yes	—
177	C₄	#585	1	4	yes	—
178	C₂	—	2	6	no	—
179	C₂	—	4	12	no	—
180	C₂	C₂³×#41	1	1	yes	characteristic
181	C₂	C₂×#585	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂⁴
commutator, Frattini, socle, cosocle	C₂²	C₂⁶
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3461	C₂	semidirect
2. #3460	C₂	semidirect
3. #3459	C₂	semidirect
4. #3458	C₂	semidirect
5. #3455	C₂	semidirect
6. #3449	C₂	semidirect
7. #3444	C₂	semidirect
8. C₂×C₄×#41	C₂	semidirect
10. #544	C₂²	semidirect
12. #542	C₂²	semidirect
13. #541	C₂²	semidirect
14. #540	C₂²	semidirect
15. #539	C₂²	semidirect
16. #538	C₂²	semidirect
18. #536	C₂²	semidirect
19. #535	C₂²	semidirect
21. #533	C₂²	semidirect
22. #533	C₂²	semidirect
24. #532	C₂²	semidirect
25. #529	C₂²	semidirect
26. #529	C₂²	semidirect
27. #529	C₂²	semidirect
28. #525	C₂²	semidirect
29. #525	C₂²	semidirect
30. C₄×#41	C₂²	semidirect
31. C₄×#41	C₂²	semidirect
32. C₄×#41	C₂²	semidirect
33. C₄×#41	C₂²	semidirect
34. C₂×C₄×Q₈	C₂²	semidirect
36. C₂×C₄×D₄	C₂²	semidirect
38. C₂×#117	C₂²	semidirect
40. C₂²×C₄²	C₂²	semidirect
112. C₂²×C₄	#41	semidirect

Maximal subgroups

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

subgroup	index
#3461	2
#3460	2
#3459	2
#3458	2
#3455	2
#3449	2
#3444	2
C₂×C₄×#41	2