Subgroups — #3514

Order 128.

571 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#559	C₂	1	1	yes	maximal, characteristic, complemented
2	#556	C₂	1	1	yes	maximal, characteristic, complemented
3	#555	C₂	1	2	yes	maximal, complemented
4	#551	C₂	1	4	yes	maximal, complemented
5	D₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#548	C₂	1	2	yes	maximal, 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	#532	C₂	1	1	yes	maximal, characteristic, complemented
13	#529	C₂	1	1	yes	maximal, characteristic, complemented
14	#529	C₂	1	2	yes	maximal, complemented
15	C₂×#124	C₂	1	2	yes	maximal, complemented
16	#518	C₂	1	2	yes	maximal, complemented
17	C₄×#41	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#41	C₂²	1	1	yes	characteristic
19	C₂×#41	C₂²	1	2	yes	complemented
20	C₂²×Q₈	C₂²	1	2	yes	complemented
21	C₂²×D₄	C₂²	1	2	yes	—
22	C₂²×D₄	C₂²	1	2	yes	—
23	C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
24	C₄⋊Q₈	C₂²	1	2	yes	complemented
25	#127	C₂²	1	1	yes	characteristic
26	#126	C₂²	1	4	yes	complemented
27	#126	C₂²	1	2	yes	complemented
28	#126	C₂²	1	4	yes	complemented
29	#125	C₂²	1	1	yes	characteristic, complemented
30	#124	C₂²	1	1	yes	characteristic
31	#124	C₂²	1	2	yes	—
32	#124	C₂²	1	2	yes	complemented
33	#124	C₂²	1	4	yes	complemented
34	#124	C₂²	1	2	yes	complemented
35	#124	C₂²	1	4	yes	complemented
36	#124	C₂²	1	4	yes	complemented
37	#124	C₂²	1	2	yes	complemented
38	#123	C₂²	1	2	yes	complemented
39	#123	C₂²	1	4	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	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	4	yes	complemented
47	#121	C₂²	1	2	yes	—
48	#121	C₂²	1	1	yes	characteristic
49	#121	C₂²	1	2	yes	—
50	#121	C₂²	1	2	yes	complemented
51	#121	C₂²	1	2	yes	complemented
52	#121	C₂²	1	4	yes	complemented
53	#121	C₂²	1	4	yes	complemented
54	#120	C₂²	1	4	yes	—
55	#120	C₂²	1	4	yes	complemented
56	C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
57	C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
58	C₄×Q₈	C₂²	1	2	yes	complemented
59	C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
60	C₄×D₄	C₂²	1	1	yes	characteristic
61	C₄×D₄	C₂²	1	1	yes	characteristic
62	C₄×D₄	C₂²	1	2	yes	—
63	C₄×D₄	C₂²	1	1	yes	characteristic
64	C₄×D₄	C₂²	1	2	yes	complemented
65	C₄×D₄	C₂²	1	2	yes	complemented
66	C₄×D₄	C₂²	1	4	yes	complemented
67	C₄×D₄	C₂²	1	2	yes	complemented
68	C₄×D₄	C₂²	1	4	yes	complemented
69	C₄×D₄	C₂²	1	2	yes	complemented
70	C₄×D₄	C₂²	1	2	yes	complemented
71	C₄×D₄	C₂²	1	2	yes	complemented
72	C₄×D₄	C₂²	1	2	yes	complemented
73	#117	C₂²	1	2	yes	complemented
74	#117	C₂²	1	1	yes	characteristic, complemented
75	#117	C₂²	1	2	yes	complemented
76	#117	C₂²	1	2	yes	complemented
77	#117	C₂²	1	4	yes	complemented
78	C₂×C₂²⋊C₄	C₂²	1	4	yes	complemented
79	C₂×C₂²⋊C₄	C₂²	1	4	yes	complemented
80	C₂×C₂²⋊C₄	C₂²	1	4	yes	complemented
81	C₂×C₄²	C₂²	1	2	yes	complemented
82	C₂×C₄²	C₂²	1	1	yes	characteristic, complemented
83	C₂⁴	C₂³	1	4	yes	—
84	#41	—	2	4	no	—
85	#41	—	4	4	no	—
86	C₂×Q₈	—	4	4	no	—
87	C₂×Q₈	C₂³	1	1	yes	characteristic
88	C₂×Q₈	C₂³	1	1	yes	characteristic
89	C₂×Q₈	C₂³	1	2	yes	—
90	C₂×Q₈	C₂³	1	4	yes	—
91	C₂×D₄	—	2	8	no	—
92	C₂×D₄	—	4	4	no	—
93	C₂×D₄	C₂³	1	1	yes	characteristic
94	C₂×D₄	C₂³	1	4	yes	—
95	C₂×D₄	C₂³	1	2	yes	—
96	C₂×D₄	C₂³	1	2	yes	—
97	C₂×D₄	C₂³	1	2	yes	—
98	C₂×D₄	C₂³	1	2	yes	—
99	C₂×D₄	C₂³	1	2	yes	—
100	C₂×D₄	C₂³	1	4	yes	—
101	C₂×D₄	C₂³	1	4	yes	—
102	C₂×D₄	C₂³	1	1	yes	characteristic
103	C₂²×C₄	C₂³	1	2	yes	—
104	C₂²×C₄	C₂³	1	1	yes	characteristic
105	C₂²×C₄	C₂³	1	2	yes	—
106	C₂²×C₄	C₂³	1	1	yes	characteristic
107	C₂²×C₄	C₂³	1	2	yes	—
108	C₂²×C₄	C₂³	1	2	yes	—
109	C₂²×C₄	C₂³	1	1	yes	characteristic
110	C₂²×C₄	C₂³	1	4	yes	—
111	C₂²×C₄	C₂³	1	4	yes	—
112	C₂²×C₄	C₂³	1	4	yes	—
113	C₄⋊C₄	C₂³	1	1	yes	characteristic
114	C₄⋊C₄	C₂³	1	2	yes	—
115	C₄⋊C₄	C₂³	1	2	yes	—
116	C₄⋊C₄	C₂³	1	1	yes	characteristic
117	C₄⋊C₄	C₂³	1	2	yes	—
118	C₄⋊C₄	C₂³	1	2	yes	—
119	C₄⋊C₄	C₂³	1	2	yes	—
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	1	yes	characteristic
124	C₄⋊C₄	C₂³	1	2	yes	—
125	C₄⋊C₄	C₂³	1	4	yes	—
126	C₄⋊C₄	C₂³	1	2	yes	—
127	C₂²⋊C₄	C₂³	1	2	yes	—
128	C₂²⋊C₄	C₂³	1	2	yes	—
129	C₂²⋊C₄	C₂³	1	4	yes	—
130	C₂²⋊C₄	C₂³	1	2	yes	—
131	C₂²⋊C₄	C₂³	1	2	yes	—
132	C₂²⋊C₄	C₂³	1	4	yes	—
133	C₂²⋊C₄	C₂³	1	4	yes	—
134	C₂²⋊C₄	C₂³	1	4	yes	—
135	C₂²⋊C₄	C₂³	1	4	yes	—
136	C₂²⋊C₄	C₂³	1	4	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	4	yes	—
141	C₂²⋊C₄	C₂³	1	2	yes	—
142	C₂²⋊C₄	C₂³	1	4	yes	—
143	C₂²⋊C₄	C₂³	1	2	yes	—
144	C₄²	C₂³	1	1	yes	characteristic
145	C₄²	C₂³	1	2	yes	—
146	C₄²	C₂³	1	1	yes	characteristic
147	C₄²	C₂³	1	1	yes	characteristic
148	C₄²	C₂³	1	1	yes	characteristic
149	C₄²	C₂³	1	2	yes	—
150	C₄²	C₂³	1	2	yes	—
151	C₄²	C₂³	1	4	yes	—
152	C₄²	C₂³	1	2	yes	—
153	C₂³	—	2	8	no	—
154	C₂³	—	4	4	no	—
155	C₂³	—	4	4	no	—
156	C₂³	C₂⁴	1	2	yes	—
157	C₂³	C₂⁴	1	4	yes	—
158	C₂³	C₂⁴	1	1	yes	characteristic
159	C₂³	C₂⁴	1	2	yes	—
160	Q₈	—	2	2	no	—
161	Q₈	—	2	2	no	—
162	Q₈	—	2	4	no	—
163	Q₈	—	4	4	no	—
164	D₄	—	2	8	no	—
165	D₄	—	2	4	no	—
166	D₄	—	2	4	no	—
167	D₄	—	2	2	no	—
168	D₄	—	2	4	no	—
169	D₄	—	4	4	no	—
170	D₄	—	4	2	no	—
171	D₄	—	4	2	no	—
172	D₄	—	4	4	no	—
173	D₄	#41	1	4	yes	complemented
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	C₂⁴	1	1	yes	characteristic
185	C₂×C₄	C₂⁴	1	1	yes	characteristic
186	C₂×C₄	C₂⁴	1	2	yes	—
187	C₂×C₄	C₂⁴	1	2	yes	—
188	C₂×C₄	C₂⁴	1	2	yes	—
189	C₂×C₄	C₂⁴	1	1	yes	characteristic
190	C₂×C₄	C₂⁴	1	1	yes	characteristic
191	C₂×C₄	C₂⁴	1	2	yes	—
192	C₂×C₄	C₂⁴	1	2	yes	—
193	C₂×C₄	C₂⁴	1	2	yes	—
194	C₂×C₄	C₂⁴	1	4	yes	—
195	C₂×C₄	C₂⁴	1	1	yes	characteristic
196	C₂×C₄	C₂⁴	1	1	yes	characteristic
197	C₂²	—	2	2	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	2	1	no	—
200	C₂²	—	2	2	no	—
201	C₂²	—	2	4	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	2	4	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	8	8	no	—
209	C₂²	C₂×#41	1	4	yes	—
210	C₂²	C₂⁵	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	2	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	2	no	—
216	C₄	—	2	4	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	2	1	no	—
222	C₄	C₂×#41	1	2	yes	—
223	C₄	H(5,𝔽₂)	1	2	yes	—
224	C₂	—	2	4	no	—
225	C₂	—	4	1	no	—
226	C₂	—	4	4	no	—
227	C₂	—	4	2	no	—
228	C₂	C₂²×#41	1	1	yes	characteristic
229	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic
230	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. #559	C₂	semidirect
2. #556	C₂	semidirect
3. #555	C₂	semidirect
4. #551	C₂	semidirect
5. D₄×Q₈	C₂	semidirect
6. #548	C₂	semidirect
7. #546	C₂	semidirect
8. #546	C₂	semidirect
9. D₄²	C₂	semidirect
10. #542	C₂	semidirect
11. #538	C₂	semidirect
12. #532	C₂	semidirect
13. #529	C₂	semidirect
14. #529	C₂	semidirect
15. C₂×#124	C₂	semidirect
16. #518	C₂	semidirect
17. C₄×#41	C₂	semidirect
19. C₂×#41	C₂²	semidirect
20. C₂²×Q₈	C₂²	semidirect
23. C₄⋊Q₈	C₂²	semidirect
24. C₄⋊Q₈	C₂²	semidirect
26. #126	C₂²	semidirect
27. #126	C₂²	semidirect
28. #126	C₂²	semidirect
29. #125	C₂²	semidirect
32. #124	C₂²	semidirect
33. #124	C₂²	semidirect
34. #124	C₂²	semidirect
35. #124	C₂²	semidirect
36. #124	C₂²	semidirect
37. #124	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
50. #121	C₂²	semidirect
51. #121	C₂²	semidirect
52. #121	C₂²	semidirect
53. #121	C₂²	semidirect
55. #120	C₂²	semidirect
56. C₄×Q₈	C₂²	semidirect
57. C₄×Q₈	C₂²	semidirect
58. C₄×Q₈	C₂²	semidirect
59. 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. C₄×D₄	C₂²	semidirect
70. C₄×D₄	C₂²	semidirect
71. C₄×D₄	C₂²	semidirect
72. C₄×D₄	C₂²	semidirect
73. #117	C₂²	semidirect
74. #117	C₂²	semidirect
75. #117	C₂²	semidirect
76. #117	C₂²	semidirect
77. #117	C₂²	semidirect
78. C₂×C₂²⋊C₄	C₂²	semidirect
79. C₂×C₂²⋊C₄	C₂²	semidirect
80. C₂×C₂²⋊C₄	C₂²	semidirect
81. C₂×C₄²	C₂²	semidirect
82. C₂×C₄²	C₂²	semidirect
173. D₄	#41	semidirect

Maximal subgroups

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

subgroup	index
#559	2
#556	2
#555	2
#551	2
D₄×Q₈	2
#548	2
#546	2
D₄²	2
#542	2
#538	2
#532	2
#529	2
C₂×#124	2
#518	2
C₄×#41	2