Subgroups — #3546

Order 128.

575 conjugacy classes of subgroups.

The subgroup lattice has 241 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#560	C₂	1	4	yes	maximal, complemented
2	#556	C₂	1	1	yes	maximal, characteristic, complemented
3	#553	C₂	1	2	yes	maximal, complemented
4	#550	C₂	1	2	yes	maximal, complemented
5	#550	C₂	1	1	yes	maximal, characteristic, complemented
6	#547	C₂	1	1	yes	maximal, characteristic, complemented
7	#546	C₂	1	2	yes	maximal, complemented
8	#546	C₂	1	2	yes	maximal, complemented
9	D₄²	C₂	1	1	yes	maximal, characteristic, complemented
10	#542	C₂	1	2	yes	maximal, complemented
11	#538	C₂	1	4	yes	maximal, complemented
12	#537	C₂	1	2	yes	maximal, complemented
13	#535	C₂	1	2	yes	maximal, complemented
14	#535	C₂	1	2	yes	maximal, complemented
15	#532	C₂	1	1	yes	maximal, characteristic, complemented
16	#532	C₂	1	1	yes	maximal, characteristic, complemented
17	#529	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#41	C₂²	1	2	yes	complemented
19	C₂×#41	C₂²	1	2	yes	complemented
20	C₂×#41	C₂²	1	2	yes	complemented
21	C₂×#41	C₂²	1	2	yes	complemented
22	C₂²×D₄	C₂²	1	2	yes	—
23	C₂²×D₄	C₂²	1	2	yes	—
24	C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
25	C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
26	#127	C₂²	1	1	yes	characteristic
27	#127	C₂²	1	1	yes	characteristic
28	#127	C₂²	1	2	yes	complemented
29	#127	C₂²	1	4	yes	complemented
30	#126	C₂²	1	2	yes	complemented
31	#126	C₂²	1	4	yes	complemented
32	#126	C₂²	1	4	yes	complemented
33	#125	C₂²	1	1	yes	characteristic, complemented
34	#124	C₂²	1	1	yes	characteristic
35	#124	C₂²	1	2	yes	—
36	#124	C₂²	1	2	yes	complemented
37	#124	C₂²	1	2	yes	complemented
38	#124	C₂²	1	4	yes	complemented
39	#123	C₂²	1	2	yes	complemented
40	#123	C₂²	1	4	yes	complemented
41	#123	C₂²	1	4	yes	complemented
42	#123	C₂²	1	4	yes	complemented
43	C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
44	C₂²⋊Q₈	C₂²	1	4	yes	complemented
45	C₂²⋊Q₈	C₂²	1	2	yes	complemented
46	C₂²⋊Q₈	C₂²	1	2	yes	complemented
47	#121	C₂²	1	1	yes	characteristic
48	#121	C₂²	1	1	yes	characteristic
49	#121	C₂²	1	2	yes	—
50	#121	C₂²	1	2	yes	—
51	#121	C₂²	1	2	yes	—
52	#121	C₂²	1	2	yes	complemented
53	#121	C₂²	1	1	yes	characteristic, complemented
54	#121	C₂²	1	2	yes	complemented
55	#121	C₂²	1	2	yes	complemented
56	#121	C₂²	1	4	yes	complemented
57	#121	C₂²	1	2	yes	complemented
58	#121	C₂²	1	4	yes	complemented
59	#121	C₂²	1	4	yes	complemented
60	#121	C₂²	1	4	yes	complemented
61	#121	C₂²	1	4	yes	complemented
62	#121	C₂²	1	4	yes	complemented
63	#120	C₂²	1	4	yes	—
64	#120	C₂²	1	4	yes	complemented
65	#120	C₂²	1	4	yes	complemented
66	#120	C₂²	1	4	yes	complemented
67	C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
68	C₄×Q₈	C₂²	1	2	yes	complemented
69	C₄×D₄	C₂²	1	1	yes	characteristic
70	C₄×D₄	C₂²	1	1	yes	characteristic
71	C₄×D₄	C₂²	1	2	yes	complemented
72	C₄×D₄	C₂²	1	2	yes	complemented
73	C₄×D₄	C₂²	1	2	yes	complemented
74	C₄×D₄	C₂²	1	2	yes	complemented
75	C₄×D₄	C₂²	1	1	yes	characteristic, complemented
76	C₄×D₄	C₂²	1	4	yes	complemented
77	C₄×D₄	C₂²	1	2	yes	complemented
78	#117	C₂²	1	2	yes	complemented
79	#117	C₂²	1	2	yes	complemented
80	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
81	C₂×C₂²⋊C₄	C₂²	1	4	yes	complemented
82	C₂×C₄²	C₂²	1	1	yes	characteristic, complemented
83	C₂⁴	C₂³	1	4	yes	—
84	#41	—	4	4	no	—
85	#41	—	4	4	no	—
86	#41	—	4	4	no	—
87	#41	—	4	4	no	—
88	C₂×Q₈	C₂³	1	1	yes	characteristic
89	C₂×Q₈	C₂³	1	2	yes	—
90	C₂×Q₈	C₂³	1	2	yes	—
91	C₂×D₄	—	4	4	no	—
92	C₂×D₄	—	4	4	no	—
93	C₂×D₄	C₂³	1	1	yes	characteristic
94	C₂×D₄	C₂³	1	1	yes	characteristic
95	C₂×D₄	C₂³	1	2	yes	—
96	C₂×D₄	C₂³	1	2	yes	—
97	C₂×D₄	C₂³	1	1	yes	characteristic
98	C₂×D₄	C₂³	1	2	yes	—
99	C₂×D₄	C₂³	1	2	yes	—
100	C₂×D₄	C₂³	1	2	yes	—
101	C₂×D₄	C₂³	1	4	yes	—
102	C₂×D₄	C₂³	1	4	yes	—
103	C₂×D₄	C₂³	1	4	yes	—
104	C₂×D₄	C₂³	1	2	yes	—
105	C₂×D₄	C₂³	1	4	yes	—
106	C₂×D₄	C₂³	1	2	yes	—
107	C₂×D₄	C₂³	1	4	yes	—
108	C₂×D₄	C₂³	1	4	yes	—
109	C₂×D₄	C₂³	1	2	yes	—
110	C₂²×C₄	C₂³	1	1	yes	characteristic
111	C₂²×C₄	C₂³	1	1	yes	characteristic
112	C₂²×C₄	C₂³	1	2	yes	—
113	C₂²×C₄	C₂³	1	2	yes	—
114	C₂²×C₄	C₂³	1	2	yes	—
115	C₂²×C₄	C₂³	1	1	yes	characteristic
116	C₂²×C₄	C₂³	1	4	yes	—
117	C₂²×C₄	C₂³	1	4	yes	—
118	C₂²×C₄	C₂³	1	4	yes	—
119	C₄⋊C₄	C₂³	1	1	yes	characteristic
120	C₄⋊C₄	C₂³	1	2	yes	—
121	C₄⋊C₄	C₂³	1	4	yes	—
122	C₄⋊C₄	C₂³	1	1	yes	characteristic
123	C₄⋊C₄	C₂³	1	2	yes	—
124	C₄⋊C₄	C₂³	1	2	yes	—
125	C₄⋊C₄	C₂³	1	1	yes	characteristic
126	C₄⋊C₄	C₂³	1	1	yes	characteristic
127	C₄⋊C₄	C₂³	1	2	yes	—
128	C₄⋊C₄	C₂³	1	2	yes	—
129	C₄⋊C₄	C₂³	1	2	yes	—
130	C₄⋊C₄	C₂³	1	4	yes	—
131	C₄⋊C₄	C₂³	1	2	yes	—
132	C₂²⋊C₄	C₂³	1	2	yes	—
133	C₂²⋊C₄	C₂³	1	2	yes	—
134	C₂²⋊C₄	C₂³	1	4	yes	—
135	C₂²⋊C₄	C₂³	1	4	yes	—
136	C₂²⋊C₄	C₂³	1	2	yes	—
137	C₂²⋊C₄	C₂³	1	4	yes	—
138	C₂²⋊C₄	C₂³	1	2	yes	—
139	C₂²⋊C₄	C₂³	1	2	yes	—
140	C₂²⋊C₄	C₂³	1	2	yes	—
141	C₂²⋊C₄	C₂³	1	2	yes	—
142	C₂²⋊C₄	C₂³	1	4	yes	—
143	C₂²⋊C₄	C₂³	1	4	yes	—
144	C₂²⋊C₄	C₂³	1	4	yes	—
145	C₂²⋊C₄	C₂³	1	4	yes	—
146	C₂²⋊C₄	C₂³	1	2	yes	—
147	C₂²⋊C₄	C₂³	1	2	yes	—
148	C₄²	C₂³	1	1	yes	characteristic
149	C₄²	C₂³	1	1	yes	characteristic
150	C₄²	C₂³	1	1	yes	characteristic
151	C₄²	C₂³	1	1	yes	characteristic
152	C₄²	C₂³	1	2	yes	—
153	C₄²	C₂³	1	4	yes	—
154	C₂³	—	4	4	no	—
155	C₂³	—	4	4	no	—
156	C₂³	—	4	4	no	—
157	C₂³	C₂⁴	1	2	yes	—
158	C₂³	C₂⁴	1	2	yes	—
159	C₂³	C₂⁴	1	4	yes	—
160	C₂³	C₂⁴	1	1	yes	characteristic
161	C₂³	C₂⁴	1	2	yes	—
162	Q₈	—	2	2	no	—
163	Q₈	—	2	4	no	—
164	Q₈	—	4	2	no	—
165	D₄	—	2	2	no	—
166	D₄	—	2	2	no	—
167	D₄	—	2	4	no	—
168	D₄	—	2	4	no	—
169	D₄	—	2	2	no	—
170	D₄	—	4	4	no	—
171	D₄	—	4	4	no	—
172	D₄	—	4	2	no	—
173	D₄	—	4	2	no	—
174	D₄	—	4	2	no	—
175	D₄	—	4	4	no	—
176	D₄	—	4	2	no	—
177	D₄	—	4	2	no	—
178	D₄	—	4	2	no	—
179	D₄	—	4	4	no	—
180	D₄	—	4	4	no	—
181	D₄	—	4	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	4	1	no	—
188	C₂×C₄	—	4	4	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	C₂⁴	1	2	yes	—
192	C₂×C₄	C₂⁴	1	1	yes	characteristic
193	C₂×C₄	C₂⁴	1	1	yes	characteristic
194	C₂×C₄	C₂⁴	1	1	yes	characteristic
195	C₂×C₄	C₂⁴	1	1	yes	characteristic
196	C₂×C₄	C₂⁴	1	2	yes	—
197	C₂×C₄	C₂⁴	1	2	yes	—
198	C₂×C₄	C₂⁴	1	4	yes	—
199	C₂×C₄	C₂⁴	1	2	yes	—
200	C₂×C₄	C₂⁴	1	2	yes	—
201	C₂×C₄	C₂⁴	1	1	yes	characteristic
202	C₂×C₄	C₂⁴	1	1	yes	characteristic
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	8	8	no	—
219	C₂²	C₂⁵	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	2	no	—
222	C₄	—	2	2	no	—
223	C₄	—	2	2	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	4	no	—
226	C₄	—	2	2	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	2	no	—
230	C₄	H(5,𝔽₂)	1	2	yes	—
231	C₄	H(5,𝔽₂)	1	2	yes	—
232	C₂	—	4	1	no	—
233	C₂	—	4	2	no	—
234	C₂	—	4	4	no	—
235	C₂	—	4	2	no	—
236	C₂	—	4	2	no	—
237	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic
238	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic
239	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. #560	C₂	semidirect
2. #556	C₂	semidirect
3. #553	C₂	semidirect
4. #550	C₂	semidirect
5. #550	C₂	semidirect
6. #547	C₂	semidirect
7. #546	C₂	semidirect
8. #546	C₂	semidirect
9. D₄²	C₂	semidirect
10. #542	C₂	semidirect
11. #538	C₂	semidirect
12. #537	C₂	semidirect
13. #535	C₂	semidirect
14. #535	C₂	semidirect
15. #532	C₂	semidirect
16. #532	C₂	semidirect
17. #529	C₂	semidirect
18. C₂×#41	C₂²	semidirect
19. C₂×#41	C₂²	semidirect
20. C₂×#41	C₂²	semidirect
21. C₂×#41	C₂²	semidirect
24. C₄⋊Q₈	C₂²	semidirect
25. C₄⋊Q₈	C₂²	semidirect
28. #127	C₂²	semidirect
29. #127	C₂²	semidirect
30. #126	C₂²	semidirect
31. #126	C₂²	semidirect
32. #126	C₂²	semidirect
33. #125	C₂²	semidirect
36. #124	C₂²	semidirect
37. #124	C₂²	semidirect
38. #124	C₂²	semidirect
39. #123	C₂²	semidirect
40. #123	C₂²	semidirect
41. #123	C₂²	semidirect
42. #123	C₂²	semidirect
43. C₂²⋊Q₈	C₂²	semidirect
44. C₂²⋊Q₈	C₂²	semidirect
45. C₂²⋊Q₈	C₂²	semidirect
46. C₂²⋊Q₈	C₂²	semidirect
52. #121	C₂²	semidirect
53. #121	C₂²	semidirect
54. #121	C₂²	semidirect
55. #121	C₂²	semidirect
56. #121	C₂²	semidirect
57. #121	C₂²	semidirect
58. #121	C₂²	semidirect
59. #121	C₂²	semidirect
60. #121	C₂²	semidirect
61. #121	C₂²	semidirect
62. #121	C₂²	semidirect
64. #120	C₂²	semidirect
65. #120	C₂²	semidirect
66. #120	C₂²	semidirect
67. C₄×Q₈	C₂²	semidirect
68. C₄×Q₈	C₂²	semidirect
71. C₄×D₄	C₂²	semidirect
72. C₄×D₄	C₂²	semidirect
73. C₄×D₄	C₂²	semidirect
74. C₄×D₄	C₂²	semidirect
75. C₄×D₄	C₂²	semidirect
76. C₄×D₄	C₂²	semidirect
77. C₄×D₄	C₂²	semidirect
78. #117	C₂²	semidirect
79. #117	C₂²	semidirect
80. C₂×C₄⋊C₄	C₂²	semidirect
81. C₂×C₂²⋊C₄	C₂²	semidirect
82. C₂×C₄²	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#560	2
#556	2
#553	2
#550	2
#547	2
#546	2
D₄²	2
#542	2
#538	2
#537	2
#535	2
#532	2
#529	2