Subgroups — #3488

Order 128.

829 conjugacy classes of subgroups.

The subgroup lattice has 161 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#41	C₂	1	1	yes	maximal, characteristic, complemented
3	#550	C₂	1	2	yes	maximal, complemented
4	#548	C₂	1	2	yes	maximal, complemented
5	#546	C₂	1	6	yes	maximal, complemented
6	D₄²	C₂	1	6	yes	maximal, complemented
7	#537	C₂	1	3	yes	maximal, complemented
8	#535	C₂	1	3	yes	maximal, complemented
9	#525	C₂	1	3	yes	maximal, complemented
10	C₂×#121	C₂	1	3	yes	maximal, complemented
11	#520	C₂	1	1	yes	maximal, characteristic, complemented
12	H(5,𝔽₂)	—	2	8	no	—
13	C₂×#41	—	2	8	no	—
14	C₂×#41	C₂²	1	1	yes	characteristic
15	C₂×#41	C₂²	1	3	yes	complemented
16	C₂×#41	C₂²	1	3	yes	complemented
17	C₂×#41	C₂²	1	3	yes	complemented
18	C₂×#41	C₂²	1	2	yes	complemented
19	C₂×#41	C₂²	1	1	yes	characteristic, complemented
20	C₂²×Q₈	C₂²	1	1	yes	characteristic, complemented
21	C₂²×D₄	C₂²	1	3	yes	complemented
22	C₂²×D₄	C₂²	1	3	yes	complemented
23	C₂²×D₄	C₂²	1	6	yes	complemented
24	C₂²×D₄	C₂²	1	6	yes	complemented
25	C₂³×C₄	C₂²	1	3	yes	complemented
26	#127	C₂²	1	6	yes	complemented
27	#124	C₂²	1	6	yes	complemented
28	#123	C₂²	1	6	yes	complemented
29	C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
30	C₂²⋊Q₈	C₂²	1	3	yes	complemented
31	C₂²⋊Q₈	C₂²	1	6	yes	complemented
32	#121	C₂²	1	3	yes	complemented
33	#121	C₂²	1	6	yes	complemented
34	#121	C₂²	1	6	yes	complemented
35	#121	C₂²	1	6	yes	complemented
36	#121	C₂²	1	3	yes	complemented
37	#121	C₂²	1	12	yes	complemented
38	#121	C₂²	1	6	yes	complemented
39	#120	C₂²	1	6	yes	complemented
40	#120	C₂²	1	12	yes	complemented
41	C₄×Q₈	C₂²	1	2	yes	complemented
42	C₄×D₄	C₂²	1	6	yes	complemented
43	C₄×D₄	C₂²	1	6	yes	complemented
44	C₄×D₄	C₂²	1	6	yes	complemented
45	#117	C₂²	1	3	yes	complemented
46	C₂×C₄⋊C₄	C₂²	1	3	yes	complemented
47	C₂×C₂²⋊C₄	C₂²	1	6	yes	complemented
48	C₂⁴	C₂³	1	3	yes	—
49	C₂⁴	C₂³	1	6	yes	—
50	#41	—	2	4	no	—
51	#41	—	2	8	no	—
52	#41	—	2	12	no	—
53	#41	—	2	12	no	—
54	#41	—	2	12	no	—
55	#41	D₄	1	8	yes	complemented
56	C₂×Q₈	—	2	4	no	—
57	C₂×Q₈	C₂³	1	1	yes	characteristic
58	C₂×Q₈	C₂³	1	3	yes	—
59	C₂×Q₈	C₂³	1	1	yes	characteristic
60	C₂×D₄	—	2	12	no	—
61	C₂×D₄	—	2	24	no	—
62	C₂×D₄	—	2	12	no	—
63	C₂×D₄	—	4	12	no	—
64	C₂×D₄	C₂³	1	6	yes	—
65	C₂×D₄	C₂³	1	6	yes	—
66	C₂×D₄	C₂³	1	6	yes	—
67	C₂×D₄	C₂³	1	3	yes	—
68	C₂×D₄	C₂³	1	6	yes	—
69	C₂×D₄	C₂³	1	3	yes	—
70	C₂×D₄	C₂³	1	6	yes	—
71	C₂×D₄	C₂³	1	12	yes	—
72	C₂×D₄	C₂³	1	3	yes	—
73	C₂²×C₄	—	2	12	no	—
74	C₂²×C₄	C₂³	1	3	yes	—
75	C₂²×C₄	C₂³	1	3	yes	—
76	C₂²×C₄	C₂³	1	3	yes	—
77	C₂²×C₄	C₂³	1	6	yes	—
78	C₂²×C₄	C₂³	1	1	yes	characteristic
79	C₂²×C₄	C₂³	1	3	yes	—
80	C₂²×C₄	C₂³	1	6	yes	—
81	C₂²×C₄	C₂³	1	3	yes	—
82	C₄⋊C₄	C₂³	1	3	yes	—
83	C₄⋊C₄	C₂³	1	6	yes	—
84	C₄⋊C₄	C₂³	1	1	yes	characteristic
85	C₄⋊C₄	C₂³	1	2	yes	—
86	C₄⋊C₄	C₂³	1	6	yes	—
87	C₂²⋊C₄	C₂³	1	6	yes	—
88	C₂²⋊C₄	C₂³	1	12	yes	—
89	C₂²⋊C₄	C₂³	1	6	yes	—
90	C₂²⋊C₄	C₂³	1	2	yes	—
91	C₂²⋊C₄	C₂³	1	6	yes	—
92	C₂²⋊C₄	C₂³	1	6	yes	—
93	C₄²	C₂³	1	6	yes	—
94	C₂³	—	2	6	no	—
95	C₂³	—	2	6	no	—
96	C₂³	—	2	12	no	—
97	C₂³	—	4	3	no	—
98	C₂³	—	4	6	no	—
99	C₂³	—	4	6	no	—
100	C₂³	C₂⁴	1	6	yes	—
101	C₂³	C₂⁴	1	3	yes	—
102	C₂³	C₂⁴	1	3	yes	—
103	C₂³	C₂⁴	1	1	yes	characteristic
104	Q₈	—	2	6	no	—
105	Q₈	—	2	2	no	—
106	Q₈	C₂×D₄	1	4	yes	complemented
107	D₄	—	2	12	no	—
108	D₄	—	2	12	no	—
109	D₄	—	2	12	no	—
110	D₄	—	2	6	no	—
111	D₄	—	2	12	no	—
112	D₄	—	2	6	no	—
113	D₄	—	4	12	no	—
114	D₄	—	4	6	no	—
115	D₄	C₂×D₄	1	12	yes	complemented
116	C₂×C₄	—	2	6	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	6	no	—
119	C₂×C₄	—	2	6	no	—
120	C₂×C₄	—	2	6	no	—
121	C₂×C₄	—	2	12	no	—
122	C₂×C₄	—	4	6	no	—
123	C₂×C₄	C₂×D₄	1	12	yes	complemented
124	C₂×C₄	C₂⁴	1	2	yes	—
125	C₂×C₄	C₂⁴	1	6	yes	—
126	C₂×C₄	C₂⁴	1	1	yes	characteristic
127	C₂×C₄	C₂⁴	1	3	yes	—
128	C₂×C₄	C₂⁴	1	2	yes	—
129	C₂×C₄	C₂⁴	1	3	yes	—
130	C₂×C₄	C₂⁴	1	1	yes	characteristic
131	C₂²	—	2	1	no	—
132	C₂²	—	2	3	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	2	3	no	—
135	C₂²	—	2	3	no	—
136	C₂²	—	2	6	no	—
137	C₂²	—	2	6	no	—
138	C₂²	—	2	3	no	—
139	C₂²	—	2	3	no	—
140	C₂²	—	2	6	no	—
141	C₂²	—	4	12	no	—
142	C₂²	—	8	12	no	—
143	C₂²	C₂²×D₄	1	6	yes	—
144	C₂²	H(5,𝔽₂)	1	2	yes	complemented
145	C₂²	C₂⁵	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	2	2	no	—
148	C₄	—	2	3	no	—
149	C₄	—	2	6	no	—
150	C₄	—	2	2	no	—
151	C₄	C₂²×D₄	1	2	yes	—
152	C₄	C₂²×D₄	1	6	yes	—
153	C₂	—	2	2	no	—
154	C₂	—	2	6	no	—
155	C₂	—	4	3	no	—
156	C₂	—	4	6	no	—
157	C₂	C₂³×D₄	1	1	yes	characteristic
158	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic
159	C₂	#585	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×H(5,𝔽₂)	C₂	semidirect
2. C₂²×#41	C₂	semidirect
3. #550	C₂	semidirect
4. #548	C₂	semidirect
5. #546	C₂	semidirect
6. D₄²	C₂	semidirect
7. #537	C₂	semidirect
8. #535	C₂	semidirect
9. #525	C₂	semidirect
10. C₂×#121	C₂	semidirect
11. #520	C₂	semidirect
15. C₂×#41	C₂²	semidirect
16. C₂×#41	C₂²	semidirect
17. C₂×#41	C₂²	semidirect
18. C₂×#41	C₂²	semidirect
19. C₂×#41	C₂²	semidirect
20. C₂²×Q₈	C₂²	semidirect
21. C₂²×D₄	C₂²	semidirect
22. C₂²×D₄	C₂²	semidirect
23. C₂²×D₄	C₂²	semidirect
24. C₂²×D₄	C₂²	semidirect
25. C₂³×C₄	C₂²	semidirect
26. #127	C₂²	semidirect
27. #124	C₂²	semidirect
28. #123	C₂²	semidirect
29. C₂²⋊Q₈	C₂²	semidirect
30. C₂²⋊Q₈	C₂²	semidirect
31. C₂²⋊Q₈	C₂²	semidirect
32. #121	C₂²	semidirect
33. #121	C₂²	semidirect
34. #121	C₂²	semidirect
35. #121	C₂²	semidirect
36. #121	C₂²	semidirect
37. #121	C₂²	semidirect
38. #121	C₂²	semidirect
39. #120	C₂²	semidirect
40. #120	C₂²	semidirect
41. C₄×Q₈	C₂²	semidirect
42. C₄×D₄	C₂²	semidirect
43. C₄×D₄	C₂²	semidirect
44. C₄×D₄	C₂²	semidirect
45. #117	C₂²	semidirect
46. C₂×C₄⋊C₄	C₂²	semidirect
47. C₂×C₂²⋊C₄	C₂²	semidirect
55. #41	D₄	semidirect
106. Q₈	C₂×D₄	semidirect
115. D₄	C₂×D₄	semidirect
123. C₂×C₄	C₂×D₄	semidirect
144. C₂²	H(5,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
C₂×H(5,𝔽₂)	2
C₂²×#41	2
#550	2
#548	2
#546	2
D₄²	2
#537	2
#535	2
#525	2
C₂×#121	2
#520	2