Subgroups — #3561

Order 128.

516 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#563	C₂	1	2	yes	maximal, complemented
2	#562	C₂	1	2	yes	maximal, complemented
3	#556	C₂	1	2	yes	maximal, complemented
4	#551	C₂	1	4	yes	maximal, complemented
5	D₄×Q₈	C₂	1	2	yes	maximal, complemented
6	#548	C₂	1	2	yes	maximal, complemented
7	#546	C₂	1	2	yes	maximal, complemented
8	#543	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	2	yes	maximal, complemented
12	#536	C₂	1	2	yes	maximal, complemented
13	#533	C₂	1	1	yes	maximal, characteristic, complemented
14	#529	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#124	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#41	C₂²	1	2	yes	—
17	C₂²×Q₈	C₂²	1	1	yes	characteristic
18	C₂²×Q₈	C₂²	1	2	yes	complemented
19	C₂²×D₄	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
21	C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
22	C₄⋊Q₈	C₂²	1	2	yes	complemented
23	C₄⋊Q₈	C₂²	1	4	yes	complemented
24	#126	C₂²	1	2	yes	complemented
25	#126	C₂²	1	4	yes	complemented
26	#126	C₂²	1	4	yes	complemented
27	#126	C₂²	1	4	yes	complemented
28	#125	C₂²	1	1	yes	characteristic, complemented
29	#125	C₂²	1	2	yes	complemented
30	#124	C₂²	1	2	yes	—
31	#124	C₂²	1	2	yes	—
32	#124	C₂²	1	2	yes	—
33	#124	C₂²	1	2	yes	—
34	#124	C₂²	1	2	yes	—
35	#124	C₂²	1	1	yes	characteristic, complemented
36	#124	C₂²	1	4	yes	complemented
37	#124	C₂²	1	4	yes	complemented
38	#123	C₂²	1	2	yes	—
39	#123	C₂²	1	4	yes	—
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
44	C₂²⋊Q₈	C₂²	1	2	yes	—
45	C₂²⋊Q₈	C₂²	1	2	yes	—
46	C₂²⋊Q₈	C₂²	1	4	yes	—
47	C₂²⋊Q₈	C₂²	1	4	yes	complemented
48	C₂²⋊Q₈	C₂²	1	4	yes	complemented
49	C₂²⋊Q₈	C₂²	1	4	yes	complemented
50	C₂²⋊Q₈	C₂²	1	2	yes	complemented
51	C₂²⋊Q₈	C₂²	1	4	yes	complemented
52	C₂²⋊Q₈	C₂²	1	4	yes	complemented
53	C₂²⋊Q₈	C₂²	1	4	yes	complemented
54	#121	C₂²	1	1	yes	characteristic
55	#121	C₂²	1	2	yes	—
56	#121	C₂²	1	4	yes	—
57	#121	C₂²	1	4	yes	complemented
58	#120	C₂²	1	4	yes	—
59	C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
60	C₄×Q₈	C₂²	1	2	yes	complemented
61	C₄×Q₈	C₂²	1	2	yes	complemented
62	C₄×Q₈	C₂²	1	2	yes	complemented
63	C₄×D₄	C₂²	1	2	yes	—
64	C₄×D₄	C₂²	1	4	yes	—
65	C₄×D₄	C₂²	1	2	yes	—
66	C₄×D₄	C₂²	1	1	yes	characteristic, complemented
67	C₄×D₄	C₂²	1	4	yes	complemented
68	#117	C₂²	1	2	yes	—
69	#117	C₂²	1	2	yes	complemented
70	#117	C₂²	1	4	yes	complemented
71	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
72	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
73	C₂×C₂²⋊C₄	C₂²	1	4	yes	—
74	C₂×C₄²	C₂²	1	1	yes	characteristic
75	C₂⁴	C₂³	1	2	yes	—
76	#41	—	4	4	no	—
77	C₂×Q₈	—	4	2	no	—
78	C₂×Q₈	—	4	4	no	—
79	C₂×Q₈	C₂³	1	2	yes	—
80	C₂×Q₈	C₂³	1	2	yes	—
81	C₂×Q₈	C₂³	1	1	yes	characteristic
82	C₂×Q₈	C₂³	1	2	yes	—
83	C₂×Q₈	C₂³	1	4	yes	—
84	C₂×Q₈	C₂³	1	2	yes	—
85	C₂×D₄	—	4	2	no	—
86	C₂×D₄	C₂³	1	2	yes	—
87	C₂×D₄	C₂³	1	2	yes	—
88	C₂×D₄	C₂³	1	4	yes	—
89	C₂×D₄	C₂³	1	1	yes	characteristic
90	C₂×D₄	C₂³	1	4	yes	—
91	C₂×D₄	C₂³	1	2	yes	—
92	C₂²×C₄	C₂³	1	1	yes	characteristic
93	C₂²×C₄	C₂³	1	1	yes	characteristic
94	C₂²×C₄	C₂³	1	2	yes	—
95	C₂²×C₄	C₂³	1	1	yes	characteristic
96	C₂²×C₄	C₂³	1	2	yes	—
97	C₂²×C₄	C₂³	1	4	yes	—
98	C₂²×C₄	C₂³	1	4	yes	—
99	C₂²×C₄	C₂³	1	4	yes	—
100	C₄⋊C₄	C₂³	1	2	yes	—
101	C₄⋊C₄	C₂³	1	2	yes	—
102	C₄⋊C₄	C₂³	1	2	yes	—
103	C₄⋊C₄	C₂³	1	2	yes	—
104	C₄⋊C₄	C₂³	1	4	yes	—
105	C₄⋊C₄	C₂³	1	2	yes	—
106	C₄⋊C₄	C₂³	1	2	yes	—
107	C₄⋊C₄	C₂³	1	4	yes	—
108	C₄⋊C₄	C₂³	1	2	yes	—
109	C₄⋊C₄	C₂³	1	1	yes	characteristic
110	C₄⋊C₄	C₂³	1	4	yes	—
111	C₄⋊C₄	C₂³	1	4	yes	—
112	C₄⋊C₄	C₂³	1	1	yes	characteristic
113	C₄⋊C₄	C₂³	1	4	yes	—
114	C₄⋊C₄	C₂³	1	4	yes	—
115	C₄⋊C₄	C₂³	1	2	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	4	yes	—
121	C₂²⋊C₄	C₂³	1	4	yes	—
122	C₂²⋊C₄	C₂³	1	4	yes	—
123	C₂²⋊C₄	C₂³	1	4	yes	—
124	C₂²⋊C₄	C₂³	1	4	yes	—
125	C₂²⋊C₄	C₂³	1	2	yes	—
126	C₂²⋊C₄	C₂³	1	2	yes	—
127	C₂²⋊C₄	C₂³	1	4	yes	—
128	C₂²⋊C₄	C₂³	1	2	yes	—
129	C₂²⋊C₄	C₂³	1	4	yes	—
130	C₂²⋊C₄	C₂³	1	2	yes	—
131	C₄²	C₂³	1	1	yes	characteristic
132	C₄²	C₂³	1	1	yes	characteristic
133	C₄²	C₂³	1	1	yes	characteristic
134	C₄²	C₂³	1	1	yes	characteristic
135	C₄²	C₂³	1	4	yes	—
136	C₄²	C₂³	1	2	yes	—
137	C₄²	C₂³	1	4	yes	—
138	C₂³	—	2	4	no	—
139	C₂³	—	4	2	no	—
140	C₂³	—	4	2	no	—
141	C₂³	C₂⁴	1	4	yes	—
142	C₂³	C₂⁴	1	2	yes	—
143	C₂³	C₂⁴	1	1	yes	characteristic
144	Q₈	—	2	4	no	—
145	Q₈	—	2	4	no	—
146	Q₈	—	4	4	no	—
147	Q₈	—	4	2	no	—
148	Q₈	—	4	1	no	—
149	Q₈	—	4	2	no	—
150	D₄	—	2	4	no	—
151	D₄	—	2	4	no	—
152	D₄	—	4	2	no	—
153	D₄	—	4	4	no	—
154	D₄	—	4	1	no	—
155	D₄	—	4	4	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	C₂⁴	1	1	yes	characteristic
165	C₂×C₄	C₂⁴	1	2	yes	—
166	C₂×C₄	C₂⁴	1	1	yes	characteristic
167	C₂×C₄	C₂⁴	1	1	yes	characteristic
168	C₂×C₄	C₂⁴	1	1	yes	characteristic
169	C₂×C₄	C₂⁴	1	4	yes	—
170	C₂×C₄	C₂⁴	1	1	yes	characteristic
171	C₂×C₄	C₂⁴	1	2	yes	—
172	C₂×C₄	C₂⁴	1	4	yes	—
173	C₂×C₄	C₂⁴	1	4	yes	—
174	C₂×C₄	C₂⁴	1	2	yes	—
175	C₂×C₄	C₂⁴	1	1	yes	characteristic
176	C₂²	—	2	1	no	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	2	4	no	—
180	C₂²	—	2	2	no	—
181	C₂²	—	2	4	no	—
182	C₂²	—	2	4	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	8	4	no	—
185	C₂²	2₋¹⁺⁴	1	2	yes	—
186	C₂²	C₂⁵	1	1	yes	characteristic
187	C₄	—	2	2	no	—
188	C₄	—	2	1	no	—
189	C₄	—	2	4	no	—
190	C₄	—	2	4	no	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	2	no	—
195	C₄	—	2	4	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	H(5,𝔽₂)	1	2	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	4	2	no	—
201	C₂	—	4	4	no	—
202	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic
203	C₂	C₂×2₋¹⁺⁴	1	1	yes	characteristic
204	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. #563	C₂	semidirect
2. #562	C₂	semidirect
3. #556	C₂	semidirect
4. #551	C₂	semidirect
5. D₄×Q₈	C₂	semidirect
6. #548	C₂	semidirect
7. #546	C₂	semidirect
8. #543	C₂	semidirect
9. #542	C₂	semidirect
10. #542	C₂	semidirect
11. #538	C₂	semidirect
12. #536	C₂	semidirect
13. #533	C₂	semidirect
14. #529	C₂	semidirect
15. C₂×#124	C₂	semidirect
18. C₂²×Q₈	C₂²	semidirect
20. C₄⋊Q₈	C₂²	semidirect
21. C₄⋊Q₈	C₂²	semidirect
22. C₄⋊Q₈	C₂²	semidirect
23. C₄⋊Q₈	C₂²	semidirect
24. #126	C₂²	semidirect
25. #126	C₂²	semidirect
26. #126	C₂²	semidirect
27. #126	C₂²	semidirect
28. #125	C₂²	semidirect
29. #125	C₂²	semidirect
35. #124	C₂²	semidirect
36. #124	C₂²	semidirect
37. #124	C₂²	semidirect
40. #123	C₂²	semidirect
41. #123	C₂²	semidirect
42. #123	C₂²	semidirect
47. C₂²⋊Q₈	C₂²	semidirect
48. C₂²⋊Q₈	C₂²	semidirect
49. C₂²⋊Q₈	C₂²	semidirect
50. C₂²⋊Q₈	C₂²	semidirect
51. C₂²⋊Q₈	C₂²	semidirect
52. C₂²⋊Q₈	C₂²	semidirect
53. C₂²⋊Q₈	C₂²	semidirect
57. #121	C₂²	semidirect
59. C₄×Q₈	C₂²	semidirect
60. C₄×Q₈	C₂²	semidirect
61. C₄×Q₈	C₂²	semidirect
62. C₄×Q₈	C₂²	semidirect
66. C₄×D₄	C₂²	semidirect
67. C₄×D₄	C₂²	semidirect
69. #117	C₂²	semidirect
70. #117	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#563	2
#562	2
#556	2
#551	2
D₄×Q₈	2
#548	2
#546	2
#543	2
#542	2
#538	2
#536	2
#533	2
#529	2
C₂×#124	2