Subgroups — #3520

Order 128.

562 conjugacy classes of subgroups.

The subgroup lattice has 231 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#556	C₂	1	1	yes	maximal, characteristic, complemented
2	#554	C₂	1	2	yes	maximal, complemented
3	#552	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	#547	C₂	1	1	yes	maximal, characteristic, complemented
8	#547	C₂	1	1	yes	maximal, characteristic, complemented
9	#546	C₂	1	2	yes	maximal, complemented
10	#542	C₂	1	2	yes	maximal, complemented
11	#539	C₂	1	4	yes	maximal, complemented
12	#533	C₂	1	1	yes	maximal, characteristic, complemented
13	#529	C₂	1	1	yes	maximal, characteristic, complemented
14	#525	C₂	1	2	yes	maximal, complemented
15	C₂×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
16	#520	C₂	1	2	yes	maximal, complemented
17	C₂×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#41	C₂²	1	2	yes	—
19	C₂×#41	C₂²	1	2	yes	complemented
20	C₂²×Q₈	C₂²	1	2	yes	complemented
21	C₂²×D₄	C₂²	1	1	yes	characteristic
22	C₂³×C₄	C₂²	1	2	yes	—
23	C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
24	C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
25	C₄⋊Q₈	C₂²	1	2	yes	complemented
26	#126	C₂²	1	2	yes	complemented
27	#126	C₂²	1	4	yes	complemented
28	#126	C₂²	1	4	yes	complemented
29	#125	C₂²	1	1	yes	characteristic, complemented
30	#125	C₂²	1	2	yes	complemented
31	#125	C₂²	1	4	yes	complemented
32	#124	C₂²	1	2	yes	—
33	#124	C₂²	1	1	yes	characteristic, complemented
34	#124	C₂²	1	4	yes	complemented
35	#123	C₂²	1	2	yes	—
36	#123	C₂²	1	4	yes	—
37	#123	C₂²	1	4	yes	complemented
38	#123	C₂²	1	4	yes	complemented
39	#123	C₂²	1	4	yes	complemented
40	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
41	C₂²⋊Q₈	C₂²	1	2	yes	—
42	C₂²⋊Q₈	C₂²	1	4	yes	—
43	C₂²⋊Q₈	C₂²	1	2	yes	—
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	C₂²⋊Q₈	C₂²	1	2	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	#121	C₂²	1	1	yes	characteristic
52	#121	C₂²	1	4	yes	—
53	#121	C₂²	1	2	yes	—
54	#121	C₂²	1	2	yes	complemented
55	#120	C₂²	1	4	yes	—
56	C₄×Q₈	C₂²	1	2	yes	complemented
57	C₄×Q₈	C₂²	1	2	yes	complemented
58	C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
59	C₄×Q₈	C₂²	1	2	yes	complemented
60	C₄×D₄	C₂²	1	1	yes	characteristic
61	C₄×D₄	C₂²	1	2	yes	—
62	C₄×D₄	C₂²	1	1	yes	characteristic
63	C₄×D₄	C₂²	1	1	yes	characteristic
64	C₄×D₄	C₂²	1	1	yes	characteristic
65	C₄×D₄	C₂²	1	2	yes	—
66	C₄×D₄	C₂²	1	2	yes	—
67	C₄×D₄	C₂²	1	4	yes	—
68	C₄×D₄	C₂²	1	2	yes	—
69	C₄×D₄	C₂²	1	4	yes	complemented
70	C₄×D₄	C₂²	1	4	yes	complemented
71	C₄×D₄	C₂²	1	1	yes	characteristic, complemented
72	#117	C₂²	1	2	yes	—
73	#117	C₂²	1	2	yes	complemented
74	#117	C₂²	1	4	yes	complemented
75	#117	C₂²	1	2	yes	complemented
76	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
77	C₂×C₄⋊C₄	C₂²	1	4	yes	complemented
78	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
79	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
80	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
81	C₂×C₂²⋊C₄	C₂²	1	4	yes	—
82	C₂×C₄²	C₂²	1	1	yes	characteristic
83	C₂⁴	C₂³	1	2	yes	—
84	#41	—	2	8	no	—
85	#41	—	4	4	no	—
86	C₂×Q₈	—	4	4	no	—
87	C₂×Q₈	C₂³	1	2	yes	—
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	4	no	—
92	C₂×D₄	C₂³	1	2	yes	—
93	C₂×D₄	C₂³	1	2	yes	—
94	C₂×D₄	C₂³	1	1	yes	characteristic
95	C₂×D₄	C₂³	1	4	yes	—
96	C₂×D₄	C₂³	1	1	yes	characteristic
97	C₂×D₄	C₂³	1	4	yes	—
98	C₂×D₄	C₂³	1	1	yes	characteristic
99	C₂²×C₄	—	4	4	no	—
100	C₂²×C₄	C₂³	1	1	yes	characteristic
101	C₂²×C₄	C₂³	1	1	yes	characteristic
102	C₂²×C₄	C₂³	1	1	yes	characteristic
103	C₂²×C₄	C₂³	1	4	yes	—
104	C₂²×C₄	C₂³	1	2	yes	—
105	C₂²×C₄	C₂³	1	2	yes	—
106	C₂²×C₄	C₂³	1	2	yes	—
107	C₂²×C₄	C₂³	1	2	yes	—
108	C₂²×C₄	C₂³	1	4	yes	—
109	C₂²×C₄	C₂³	1	2	yes	—
110	C₂²×C₄	C₂³	1	4	yes	—
111	C₂²×C₄	C₂³	1	2	yes	—
112	C₄⋊C₄	C₂³	1	1	yes	characteristic
113	C₄⋊C₄	C₂³	1	2	yes	—
114	C₄⋊C₄	C₂³	1	1	yes	characteristic
115	C₄⋊C₄	C₂³	1	2	yes	—
116	C₄⋊C₄	C₂³	1	4	yes	—
117	C₄⋊C₄	C₂³	1	1	yes	characteristic
118	C₄⋊C₄	C₂³	1	2	yes	—
119	C₄⋊C₄	C₂³	1	2	yes	—
120	C₄⋊C₄	C₂³	1	4	yes	—
121	C₄⋊C₄	C₂³	1	4	yes	—
122	C₄⋊C₄	C₂³	1	4	yes	—
123	C₄⋊C₄	C₂³	1	2	yes	—
124	C₄⋊C₄	C₂³	1	2	yes	—
125	C₄⋊C₄	C₂³	1	2	yes	—
126	C₄⋊C₄	C₂³	1	1	yes	characteristic
127	C₄⋊C₄	C₂³	1	4	yes	—
128	C₄⋊C₄	C₂³	1	2	yes	—
129	C₄⋊C₄	C₂³	1	4	yes	—
130	C₄⋊C₄	C₂³	1	2	yes	—
131	C₂²⋊C₄	C₂³	1	2	yes	—
132	C₂²⋊C₄	C₂³	1	4	yes	—
133	C₂²⋊C₄	C₂³	1	2	yes	—
134	C₂²⋊C₄	C₂³	1	2	yes	—
135	C₂²⋊C₄	C₂³	1	2	yes	—
136	C₂²⋊C₄	C₂³	1	2	yes	—
137	C₂²⋊C₄	C₂³	1	4	yes	—
138	C₂²⋊C₄	C₂³	1	4	yes	—
139	C₂²⋊C₄	C₂³	1	4	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	2	yes	—
144	C₂²⋊C₄	C₂³	1	4	yes	—
145	C₂²⋊C₄	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	1	yes	characteristic
150	C₄²	C₂³	1	4	yes	—
151	C₄²	C₂³	1	4	yes	—
152	C₄²	C₂³	1	2	yes	—
153	C₂³	—	2	4	no	—
154	C₂³	—	2	4	no	—
155	C₂³	—	4	2	no	—
156	C₂³	C₂⁴	1	2	yes	—
157	C₂³	C₂⁴	1	2	yes	—
158	C₂³	C₂⁴	1	2	yes	—
159	C₂³	C₂⁴	1	1	yes	characteristic
160	Q₈	—	2	4	no	—
161	Q₈	—	2	2	no	—
162	Q₈	—	2	4	no	—
163	Q₈	—	4	4	no	—
164	D₄	—	2	4	no	—
165	D₄	—	2	8	no	—
166	D₄	—	2	4	no	—
167	D₄	—	2	2	no	—
168	D₄	—	2	2	no	—
169	D₄	—	4	4	no	—
170	D₄	#41	1	4	yes	complemented
171	C₂×C₄	—	2	8	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	C₂⁴	1	1	yes	characteristic
184	C₂×C₄	C₂⁴	1	2	yes	—
185	C₂×C₄	C₂⁴	1	1	yes	characteristic
186	C₂×C₄	C₂⁴	1	4	yes	—
187	C₂×C₄	C₂⁴	1	1	yes	characteristic
188	C₂×C₄	C₂⁴	1	1	yes	characteristic
189	C₂×C₄	C₂⁴	1	2	yes	—
190	C₂×C₄	C₂⁴	1	2	yes	—
191	C₂×C₄	C₂⁴	1	4	yes	—
192	C₂×C₄	C₂⁴	1	2	yes	—
193	C₂×C₄	C₂⁴	1	2	yes	—
194	C₂×C₄	C₂⁴	1	1	yes	characteristic
195	C₂×C₄	C₂⁴	1	1	yes	characteristic
196	C₂²	—	2	1	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	2	2	no	—
201	C₂²	—	2	2	no	—
202	C₂²	—	2	2	no	—
203	C₂²	—	2	2	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	4	8	no	—
207	C₂²	C₂×#41	1	4	yes	—
208	C₂²	2₋¹⁺⁴	1	2	yes	—
209	C₂²	C₂⁵	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	4	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	2	no	—
215	C₄	—	2	2	no	—
216	C₄	—	2	4	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	C₂×#41	1	2	yes	—
222	C₄	H(5,𝔽₂)	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	2	4	no	—
225	C₂	—	4	2	no	—
226	C₂	—	4	2	no	—
227	C₂	C₂²×#41	1	1	yes	characteristic
228	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic
229	C₂	C₂×2₋¹⁺⁴	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. #556	C₂	semidirect
2. #554	C₂	semidirect
3. #552	C₂	semidirect
4. #551	C₂	semidirect
5. D₄×Q₈	C₂	semidirect
6. #548	C₂	semidirect
7. #547	C₂	semidirect
8. #547	C₂	semidirect
9. #546	C₂	semidirect
10. #542	C₂	semidirect
11. #539	C₂	semidirect
12. #533	C₂	semidirect
13. #529	C₂	semidirect
14. #525	C₂	semidirect
15. C₂×C₂²⋊Q₈	C₂	semidirect
16. #520	C₂	semidirect
17. C₂×C₄×D₄	C₂	semidirect
19. C₂×#41	C₂²	semidirect
20. C₂²×Q₈	C₂²	semidirect
23. C₄⋊Q₈	C₂²	semidirect
24. C₄⋊Q₈	C₂²	semidirect
25. C₄⋊Q₈	C₂²	semidirect
26. #126	C₂²	semidirect
27. #126	C₂²	semidirect
28. #126	C₂²	semidirect
29. #125	C₂²	semidirect
30. #125	C₂²	semidirect
31. #125	C₂²	semidirect
33. #124	C₂²	semidirect
34. #124	C₂²	semidirect
37. #123	C₂²	semidirect
38. #123	C₂²	semidirect
39. #123	C₂²	semidirect
44. C₂²⋊Q₈	C₂²	semidirect
45. C₂²⋊Q₈	C₂²	semidirect
46. C₂²⋊Q₈	C₂²	semidirect
47. C₂²⋊Q₈	C₂²	semidirect
48. C₂²⋊Q₈	C₂²	semidirect
49. C₂²⋊Q₈	C₂²	semidirect
50. C₂²⋊Q₈	C₂²	semidirect
54. #121	C₂²	semidirect
56. C₄×Q₈	C₂²	semidirect
57. C₄×Q₈	C₂²	semidirect
58. C₄×Q₈	C₂²	semidirect
59. C₄×Q₈	C₂²	semidirect
69. C₄×D₄	C₂²	semidirect
70. C₄×D₄	C₂²	semidirect
71. C₄×D₄	C₂²	semidirect
73. #117	C₂²	semidirect
74. #117	C₂²	semidirect
75. #117	C₂²	semidirect
77. C₂×C₄⋊C₄	C₂²	semidirect
78. C₂×C₄⋊C₄	C₂²	semidirect
79. C₂×C₄⋊C₄	C₂²	semidirect
170. D₄	#41	semidirect

Maximal subgroups

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

subgroup	index
#556	2
#554	2
#552	2
#551	2
D₄×Q₈	2
#548	2
#547	2
#546	2
#542	2
#539	2
#533	2
#529	2
#525	2
C₂×C₂²⋊Q₈	2
#520	2
C₂×C₄×D₄	2