Subgroups — #3540

Order 128.

529 conjugacy classes of subgroups.

The subgroup lattice has 215 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(3,𝔽₄)	C₂	1	2	yes	maximal, complemented
2	#560	C₂	1	2	yes	maximal, complemented
3	#559	C₂	1	2	yes	maximal, complemented
4	#556	C₂	1	2	yes	maximal, complemented
5	#555	C₂	1	2	yes	maximal, complemented
6	#551	C₂	1	2	yes	maximal, complemented
7	#548	C₂	1	2	yes	maximal, complemented
8	#546	C₂	1	2	yes	maximal, complemented
9	#542	C₂	1	2	yes	maximal, complemented
10	#542	C₂	1	4	yes	maximal, complemented
11	#538	C₂	1	4	yes	maximal, complemented
12	#536	C₂	1	1	yes	maximal, characteristic, complemented
13	#535	C₂	1	1	yes	maximal, characteristic, complemented
14	#525	C₂	1	1	yes	maximal, characteristic, complemented
15	#525	C₂	1	2	yes	maximal, complemented
16	C₂×#41	C₂²	1	1	yes	characteristic
17	C₂×#41	C₂²	1	2	yes	—
18	C₂²×Q₈	C₂²	1	1	yes	characteristic, complemented
19	C₂²×D₄	C₂²	1	1	yes	characteristic, complemented
20	C₂³×C₄	C₂²	1	1	yes	characteristic, complemented
21	C₄⋊Q₈	C₂²	1	2	yes	complemented
22	C₄⋊Q₈	C₂²	1	2	yes	complemented
23	#127	C₂²	1	2	yes	—
24	#126	C₂²	1	4	yes	complemented
25	#126	C₂²	1	4	yes	complemented
26	#126	C₂²	1	4	yes	complemented
27	#124	C₂²	1	2	yes	—
28	#124	C₂²	1	4	yes	—
29	#124	C₂²	1	2	yes	complemented
30	#124	C₂²	1	4	yes	complemented
31	#124	C₂²	1	2	yes	complemented
32	#124	C₂²	1	4	yes	complemented
33	#124	C₂²	1	4	yes	complemented
34	#124	C₂²	1	4	yes	complemented
35	#123	C₂²	1	1	yes	characteristic
36	#123	C₂²	1	4	yes	complemented
37	#123	C₂²	1	1	yes	characteristic, complemented
38	#123	C₂²	1	2	yes	complemented
39	#123	C₂²	1	2	yes	complemented
40	#123	C₂²	1	4	yes	complemented
41	#123	C₂²	1	4	yes	complemented
42	C₂²⋊Q₈	C₂²	1	2	yes	complemented
43	C₂²⋊Q₈	C₂²	1	2	yes	complemented
44	C₂²⋊Q₈	C₂²	1	2	yes	complemented
45	C₂²⋊Q₈	C₂²	1	2	yes	complemented
46	C₂²⋊Q₈	C₂²	1	4	yes	complemented
47	C₂²⋊Q₈	C₂²	1	4	yes	complemented
48	#121	C₂²	1	2	yes	—
49	#121	C₂²	1	2	yes	—
50	#121	C₂²	1	2	yes	—
51	#121	C₂²	1	4	yes	—
52	#121	C₂²	1	2	yes	complemented
53	#121	C₂²	1	4	yes	complemented
54	#121	C₂²	1	4	yes	complemented
55	#120	C₂²	1	1	yes	characteristic
56	#120	C₂²	1	2	yes	—
57	#120	C₂²	1	4	yes	—
58	#120	C₂²	1	1	yes	characteristic, complemented
59	#120	C₂²	1	4	yes	complemented
60	#120	C₂²	1	2	yes	complemented
61	C₄×Q₈	C₂²	1	2	yes	complemented
62	C₄×Q₈	C₂²	1	4	yes	complemented
63	C₄×D₄	C₂²	1	2	yes	—
64	C₄×D₄	C₂²	1	4	yes	complemented
65	C₄×D₄	C₂²	1	2	yes	complemented
66	C₄×D₄	C₂²	1	4	yes	complemented
67	C₄×D₄	C₂²	1	4	yes	complemented
68	C₄×D₄	C₂²	1	2	yes	complemented
69	#117	C₂²	1	1	yes	characteristic, complemented
70	#117	C₂²	1	2	yes	complemented
71	#117	C₂²	1	1	yes	characteristic, complemented
72	#117	C₂²	1	1	yes	characteristic, complemented
73	#117	C₂²	1	4	yes	complemented
74	C₂×C₂²⋊C₄	C₂²	1	4	yes	complemented
75	C₂⁴	C₂³	1	2	yes	—
76	C₂⁴	C₂³	1	1	yes	characteristic
77	#41	—	4	2	no	—
78	#41	—	4	4	no	—
79	C₂×Q₈	—	4	2	no	—
80	C₂×Q₈	C₂³	1	2	yes	—
81	C₂×Q₈	C₂³	1	4	yes	—
82	C₂×Q₈	C₂³	1	2	yes	—
83	C₂×Q₈	C₂³	1	1	yes	characteristic
84	C₂×D₄	—	4	2	no	—
85	C₂×D₄	C₂³	1	2	yes	—
86	C₂×D₄	C₂³	1	2	yes	—
87	C₂×D₄	C₂³	1	4	yes	—
88	C₂×D₄	C₂³	1	4	yes	—
89	C₂×D₄	C₂³	1	2	yes	—
90	C₂×D₄	C₂³	1	4	yes	—
91	C₂×D₄	C₂³	1	1	yes	characteristic
92	C₂×D₄	C₂³	1	4	yes	—
93	C₂×D₄	C₂³	1	1	yes	characteristic
94	C₂×D₄	C₂³	1	1	yes	characteristic
95	C₂×D₄	C₂³	1	2	yes	—
96	C₂²×C₄	—	4	2	no	—
97	C₂²×C₄	C₂³	1	1	yes	characteristic
98	C₂²×C₄	C₂³	1	2	yes	—
99	C₂²×C₄	C₂³	1	1	yes	characteristic
100	C₂²×C₄	C₂³	1	1	yes	characteristic
101	C₂²×C₄	C₂³	1	4	yes	—
102	C₂²×C₄	C₂³	1	4	yes	—
103	C₂²×C₄	C₂³	1	1	yes	characteristic
104	C₂²×C₄	C₂³	1	2	yes	—
105	C₂²×C₄	C₂³	1	2	yes	—
106	C₄⋊C₄	C₂³	1	2	yes	—
107	C₄⋊C₄	C₂³	1	2	yes	—
108	C₄⋊C₄	C₂³	1	2	yes	—
109	C₄⋊C₄	C₂³	1	2	yes	—
110	C₄⋊C₄	C₂³	1	4	yes	—
111	C₄⋊C₄	C₂³	1	2	yes	—
112	C₄⋊C₄	C₂³	1	4	yes	—
113	C₄⋊C₄	C₂³	1	4	yes	—
114	C₄⋊C₄	C₂³	1	4	yes	—
115	C₄⋊C₄	C₂³	1	4	yes	—
116	C₂²⋊C₄	C₂³	1	2	yes	—
117	C₂²⋊C₄	C₂³	1	4	yes	—
118	C₂²⋊C₄	C₂³	1	4	yes	—
119	C₂²⋊C₄	C₂³	1	4	yes	—
120	C₂²⋊C₄	C₂³	1	2	yes	—
121	C₂²⋊C₄	C₂³	1	4	yes	—
122	C₂²⋊C₄	C₂³	1	1	yes	characteristic
123	C₂²⋊C₄	C₂³	1	4	yes	—
124	C₂²⋊C₄	C₂³	1	1	yes	characteristic
125	C₂²⋊C₄	C₂³	1	2	yes	—
126	C₂²⋊C₄	C₂³	1	2	yes	—
127	C₂²⋊C₄	C₂³	1	2	yes	—
128	C₂²⋊C₄	C₂³	1	4	yes	—
129	C₂²⋊C₄	C₂³	1	2	yes	—
130	C₂²⋊C₄	C₂³	1	4	yes	—
131	C₂²⋊C₄	C₂³	1	4	yes	—
132	C₂²⋊C₄	C₂³	1	4	yes	—
133	C₂²⋊C₄	C₂³	1	4	yes	—
134	C₄²	C₂³	1	2	yes	—
135	C₄²	C₂³	1	4	yes	—
136	C₄²	C₂³	1	2	yes	—
137	C₄²	C₂³	1	2	yes	—
138	C₄²	C₂³	1	4	yes	—
139	C₂³	—	4	1	no	—
140	C₂³	—	4	2	no	—
141	C₂³	—	4	2	no	—
142	C₂³	—	4	4	no	—
143	C₂³	C₂⁴	1	1	yes	characteristic
144	C₂³	C₂⁴	1	1	yes	characteristic
145	C₂³	C₂⁴	1	2	yes	—
146	C₂³	C₂⁴	1	4	yes	—
147	C₂³	C₂⁴	1	1	yes	characteristic
148	Q₈	—	2	4	no	—
149	Q₈	—	4	2	no	—
150	Q₈	—	4	4	no	—
151	Q₈	—	4	1	no	—
152	D₄	—	2	4	no	—
153	D₄	—	4	2	no	—
154	D₄	—	4	2	no	—
155	D₄	—	4	4	no	—
156	D₄	—	4	4	no	—
157	D₄	—	4	4	no	—
158	D₄	—	4	2	no	—
159	D₄	—	4	1	no	—
160	D₄	—	4	1	no	—
161	D₄	—	4	1	no	—
162	D₄	—	4	4	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	C₂⁴	1	2	yes	—
173	C₂×C₄	C₂⁴	1	4	yes	—
174	C₂×C₄	C₂⁴	1	4	yes	—
175	C₂×C₄	C₂⁴	1	1	yes	characteristic
176	C₂×C₄	C₂⁴	1	2	yes	—
177	C₂×C₄	C₂⁴	1	1	yes	characteristic
178	C₂×C₄	C₂⁴	1	2	yes	—
179	C₂×C₄	C₂⁴	1	4	yes	—
180	C₂×C₄	C₂⁴	1	1	yes	characteristic
181	C₂×C₄	C₂⁴	1	1	yes	characteristic
182	C₂²	—	2	1	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	2	1	no	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	4	no	—
187	C₂²	—	2	4	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	2	no	—
192	C₂²	—	2	4	no	—
193	C₂²	—	2	2	no	—
194	C₂²	—	4	4	no	—
195	C₂²	—	8	4	no	—
196	C₂²	C₂⁵	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	—	2	1	no	—
201	C₄	—	2	4	no	—
202	C₄	—	2	2	no	—
203	C₄	—	2	4	no	—
204	C₄	—	2	4	no	—
205	C₄	—	2	2	no	—
206	C₄	H(5,𝔽₂)	1	2	yes	—
207	C₂	—	4	1	no	—
208	C₂	—	4	2	no	—
209	C₂	—	4	1	no	—
210	C₂	—	4	4	no	—
211	C₂	—	4	1	no	—
212	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic
213	C₂	#585	1	2	yes	—

	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. H(3,𝔽₄)	C₂	semidirect
2. #560	C₂	semidirect
3. #559	C₂	semidirect
4. #556	C₂	semidirect
5. #555	C₂	semidirect
6. #551	C₂	semidirect
7. #548	C₂	semidirect
8. #546	C₂	semidirect
9. #542	C₂	semidirect
10. #542	C₂	semidirect
11. #538	C₂	semidirect
12. #536	C₂	semidirect
13. #535	C₂	semidirect
14. #525	C₂	semidirect
15. #525	C₂	semidirect
18. C₂²×Q₈	C₂²	semidirect
19. C₂²×D₄	C₂²	semidirect
20. C₂³×C₄	C₂²	semidirect
21. C₄⋊Q₈	C₂²	semidirect
22. C₄⋊Q₈	C₂²	semidirect
24. #126	C₂²	semidirect
25. #126	C₂²	semidirect
26. #126	C₂²	semidirect
29. #124	C₂²	semidirect
30. #124	C₂²	semidirect
31. #124	C₂²	semidirect
32. #124	C₂²	semidirect
33. #124	C₂²	semidirect
34. #124	C₂²	semidirect
36. #123	C₂²	semidirect
37. #123	C₂²	semidirect
38. #123	C₂²	semidirect
39. #123	C₂²	semidirect
40. #123	C₂²	semidirect
41. #123	C₂²	semidirect
42. C₂²⋊Q₈	C₂²	semidirect
43. C₂²⋊Q₈	C₂²	semidirect
44. C₂²⋊Q₈	C₂²	semidirect
45. C₂²⋊Q₈	C₂²	semidirect
46. C₂²⋊Q₈	C₂²	semidirect
47. C₂²⋊Q₈	C₂²	semidirect
52. #121	C₂²	semidirect
53. #121	C₂²	semidirect
54. #121	C₂²	semidirect
58. #120	C₂²	semidirect
59. #120	C₂²	semidirect
60. #120	C₂²	semidirect
61. C₄×Q₈	C₂²	semidirect
62. C₄×Q₈	C₂²	semidirect
64. C₄×D₄	C₂²	semidirect
65. C₄×D₄	C₂²	semidirect
66. C₄×D₄	C₂²	semidirect
67. C₄×D₄	C₂²	semidirect
68. C₄×D₄	C₂²	semidirect
69. #117	C₂²	semidirect
70. #117	C₂²	semidirect
71. #117	C₂²	semidirect
72. #117	C₂²	semidirect
73. #117	C₂²	semidirect
74. C₂×C₂²⋊C₄	C₂²	semidirect

Maximal subgroups

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

subgroup	index
H(3,𝔽₄)	2
#560	2
#559	2
#556	2
#555	2
#551	2
#548	2
#546	2
#542	2
#538	2
#536	2
#535	2
#525	2