Subgroups — #3536

Order 128.

567 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#562	C₂	1	4	yes	maximal, complemented
2	#554	C₂	1	2	yes	maximal, complemented
3	#548	C₂	1	2	yes	maximal, complemented
4	#547	C₂	1	2	yes	maximal, complemented
5	#546	C₂	1	4	yes	maximal, complemented
6	#546	C₂	1	2	yes	maximal, complemented
7	#543	C₂	1	2	yes	maximal, complemented
8	#542	C₂	1	2	yes	maximal, complemented
9	#539	C₂	1	2	yes	maximal, complemented
10	#538	C₂	1	2	yes	maximal, complemented
11	#537	C₂	1	2	yes	maximal, complemented
12	#537	C₂	1	2	yes	maximal, complemented
13	#525	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#121	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#41	C₂²	1	1	yes	characteristic, complemented
17	C₂×#41	C₂²	1	4	yes	complemented
18	C₂×#41	C₂²	1	2	yes	complemented
19	C₂²×Q₈	C₂²	1	1	yes	characteristic, complemented
20	C₂²×D₄	C₂²	1	1	yes	characteristic
21	C₂²×D₄	C₂²	1	2	yes	—
22	C₂³×C₄	C₂²	1	1	yes	characteristic
23	C₄⋊Q₈	C₂²	1	2	yes	complemented
24	#126	C₂²	1	4	yes	complemented
25	#125	C₂²	1	4	yes	complemented
26	#124	C₂²	1	4	yes	complemented
27	#124	C₂²	1	2	yes	complemented
28	#124	C₂²	1	4	yes	complemented
29	#123	C₂²	1	2	yes	complemented
30	#123	C₂²	1	4	yes	complemented
31	#123	C₂²	1	4	yes	complemented
32	#123	C₂²	1	4	yes	complemented
33	#123	C₂²	1	4	yes	complemented
34	C₂²⋊Q₈	C₂²	1	2	yes	—
35	C₂²⋊Q₈	C₂²	1	2	yes	—
36	C₂²⋊Q₈	C₂²	1	1	yes	characteristic
37	C₂²⋊Q₈	C₂²	1	4	yes	—
38	C₂²⋊Q₈	C₂²	1	2	yes	complemented
39	C₂²⋊Q₈	C₂²	1	2	yes	complemented
40	C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
41	C₂²⋊Q₈	C₂²	1	4	yes	complemented
42	C₂²⋊Q₈	C₂²	1	4	yes	complemented
43	C₂²⋊Q₈	C₂²	1	2	yes	complemented
44	C₂²⋊Q₈	C₂²	1	4	yes	complemented
45	C₂²⋊Q₈	C₂²	1	4	yes	complemented
46	C₂²⋊Q₈	C₂²	1	4	yes	complemented
47	#121	C₂²	1	2	yes	—
48	#121	C₂²	1	2	yes	—
49	#121	C₂²	1	2	yes	—
50	#121	C₂²	1	2	yes	—
51	#121	C₂²	1	1	yes	characteristic
52	#121	C₂²	1	1	yes	characteristic, complemented
53	#121	C₂²	1	4	yes	complemented
54	#121	C₂²	1	2	yes	complemented
55	#121	C₂²	1	4	yes	complemented
56	#121	C₂²	1	4	yes	complemented
57	#121	C₂²	1	4	yes	complemented
58	#121	C₂²	1	4	yes	complemented
59	#120	C₂²	1	2	yes	—
60	#120	C₂²	1	4	yes	—
61	C₄×Q₈	C₂²	1	2	yes	complemented
62	C₄×D₄	C₂²	1	2	yes	—
63	C₄×D₄	C₂²	1	4	yes	—
64	C₄×D₄	C₂²	1	2	yes	complemented
65	C₄×D₄	C₂²	1	4	yes	complemented
66	C₄×D₄	C₂²	1	2	yes	complemented
67	#117	C₂²	1	1	yes	characteristic, complemented
68	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
69	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
70	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
71	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
72	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
73	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
74	C₂×C₂²⋊C₄	C₂²	1	4	yes	—
75	C₂⁴	C₂³	1	1	yes	characteristic
76	C₂⁴	C₂³	1	2	yes	—
77	#41	—	4	2	no	—
78	#41	—	4	8	no	—
79	#41	—	4	4	no	—
80	C₂×Q₈	—	4	2	no	—
81	C₂×Q₈	C₂³	1	2	yes	—
82	C₂×Q₈	C₂³	1	1	yes	characteristic
83	C₂×Q₈	C₂³	1	2	yes	—
84	C₂×Q₈	C₂³	1	4	yes	—
85	C₂×D₄	—	4	2	no	—
86	C₂×D₄	—	4	4	no	—
87	C₂×D₄	C₂³	1	2	yes	—
88	C₂×D₄	C₂³	1	2	yes	—
89	C₂×D₄	C₂³	1	4	yes	—
90	C₂×D₄	C₂³	1	2	yes	—
91	C₂×D₄	C₂³	1	2	yes	—
92	C₂×D₄	C₂³	1	4	yes	—
93	C₂×D₄	C₂³	1	4	yes	—
94	C₂×D₄	C₂³	1	2	yes	—
95	C₂×D₄	C₂³	1	4	yes	—
96	C₂×D₄	C₂³	1	1	yes	characteristic
97	C₂²×C₄	—	4	2	no	—
98	C₂²×C₄	C₂³	1	2	yes	—
99	C₂²×C₄	C₂³	1	1	yes	characteristic
100	C₂²×C₄	C₂³	1	2	yes	—
101	C₂²×C₄	C₂³	1	1	yes	characteristic
102	C₂²×C₄	C₂³	1	1	yes	characteristic
103	C₂²×C₄	C₂³	1	1	yes	characteristic
104	C₂²×C₄	C₂³	1	2	yes	—
105	C₂²×C₄	C₂³	1	2	yes	—
106	C₂²×C₄	C₂³	1	4	yes	—
107	C₂²×C₄	C₂³	1	4	yes	—
108	C₂²×C₄	C₂³	1	1	yes	characteristic
109	C₂²×C₄	C₂³	1	4	yes	—
110	C₂²×C₄	C₂³	1	1	yes	characteristic
111	C₄⋊C₄	C₂³	1	2	yes	—
112	C₄⋊C₄	C₂³	1	2	yes	—
113	C₄⋊C₄	C₂³	1	2	yes	—
114	C₄⋊C₄	C₂³	1	4	yes	—
115	C₄⋊C₄	C₂³	1	4	yes	—
116	C₄⋊C₄	C₂³	1	4	yes	—
117	C₄⋊C₄	C₂³	1	2	yes	—
118	C₄⋊C₄	C₂³	1	1	yes	characteristic
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	4	yes	—
123	C₄⋊C₄	C₂³	1	2	yes	—
124	C₄⋊C₄	C₂³	1	4	yes	—
125	C₂²⋊C₄	C₂³	1	4	yes	—
126	C₂²⋊C₄	C₂³	1	2	yes	—
127	C₂²⋊C₄	C₂³	1	4	yes	—
128	C₂²⋊C₄	C₂³	1	4	yes	—
129	C₂²⋊C₄	C₂³	1	4	yes	—
130	C₂²⋊C₄	C₂³	1	4	yes	—
131	C₂²⋊C₄	C₂³	1	4	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	2	yes	—
136	C₂²⋊C₄	C₂³	1	4	yes	—
137	C₂²⋊C₄	C₂³	1	4	yes	—
138	C₂²⋊C₄	C₂³	1	2	yes	—
139	C₄²	C₂³	1	2	yes	—
140	C₄²	C₂³	1	4	yes	—
141	C₂³	—	2	2	no	—
142	C₂³	—	2	2	no	—
143	C₂³	—	2	4	no	—
144	C₂³	—	4	1	no	—
145	C₂³	—	4	2	no	—
146	C₂³	—	4	2	no	—
147	C₂³	C₂⁴	1	1	yes	characteristic
148	C₂³	C₂⁴	1	4	yes	—
149	C₂³	C₂⁴	1	2	yes	—
150	C₂³	C₂⁴	1	1	yes	characteristic
151	C₂³	C₂⁴	1	1	yes	characteristic
152	Q₈	—	2	4	no	—
153	Q₈	—	4	2	no	—
154	Q₈	—	4	1	no	—
155	Q₈	—	4	4	no	—
156	D₄	—	2	8	no	—
157	D₄	—	2	4	no	—
158	D₄	—	2	4	no	—
159	D₄	—	2	4	no	—
160	D₄	—	4	2	no	—
161	D₄	—	4	4	no	—
162	D₄	—	4	2	no	—
163	D₄	—	4	1	no	—
164	D₄	—	4	4	no	—
165	D₄	—	4	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	C₂⁴	1	2	yes	—
180	C₂×C₄	C₂⁴	1	2	yes	—
181	C₂×C₄	C₂⁴	1	4	yes	—
182	C₂×C₄	C₂⁴	1	4	yes	—
183	C₂×C₄	C₂⁴	1	1	yes	characteristic
184	C₂×C₄	C₂⁴	1	2	yes	—
185	C₂×C₄	C₂⁴	1	4	yes	—
186	C₂×C₄	C₂⁴	1	1	yes	characteristic
187	C₂×C₄	C₂⁴	1	1	yes	characteristic
188	C₂×C₄	C₂⁴	1	1	yes	characteristic
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	4	no	—
197	C₂²	—	2	4	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	2	1	no	—
201	C₂²	—	2	4	no	—
202	C₂²	—	4	4	no	—
203	C₂²	—	8	4	no	—
204	C₂²	H(5,𝔽₂)	1	2	yes	—
205	C₂²	2₋¹⁺⁴	1	2	yes	—
206	C₂²	C₂⁵	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	4	no	—
209	C₄	—	2	1	no	—
210	C₄	—	2	2	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	4	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	4	no	—
215	C₄	—	2	2	no	—
216	C₄	H(5,𝔽₂)	1	2	yes	—
217	C₂	—	2	2	no	—
218	C₂	—	2	2	no	—
219	C₂	—	4	2	no	—
220	C₂	—	4	1	no	—
221	C₂	—	4	4	no	—
222	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic
223	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic
224	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. #562	C₂	semidirect
2. #554	C₂	semidirect
3. #548	C₂	semidirect
4. #547	C₂	semidirect
5. #546	C₂	semidirect
6. #546	C₂	semidirect
7. #543	C₂	semidirect
8. #542	C₂	semidirect
9. #539	C₂	semidirect
10. #538	C₂	semidirect
11. #537	C₂	semidirect
12. #537	C₂	semidirect
13. #525	C₂	semidirect
14. C₂×C₂²⋊Q₈	C₂	semidirect
15. C₂×#121	C₂	semidirect
16. C₂×#41	C₂²	semidirect
17. C₂×#41	C₂²	semidirect
18. C₂×#41	C₂²	semidirect
19. C₂²×Q₈	C₂²	semidirect
23. C₄⋊Q₈	C₂²	semidirect
24. #126	C₂²	semidirect
25. #125	C₂²	semidirect
26. #124	C₂²	semidirect
27. #124	C₂²	semidirect
28. #124	C₂²	semidirect
29. #123	C₂²	semidirect
30. #123	C₂²	semidirect
31. #123	C₂²	semidirect
32. #123	C₂²	semidirect
33. #123	C₂²	semidirect
38. C₂²⋊Q₈	C₂²	semidirect
39. C₂²⋊Q₈	C₂²	semidirect
40. C₂²⋊Q₈	C₂²	semidirect
41. C₂²⋊Q₈	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
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
61. C₄×Q₈	C₂²	semidirect
64. C₄×D₄	C₂²	semidirect
65. C₄×D₄	C₂²	semidirect
66. C₄×D₄	C₂²	semidirect
67. #117	C₂²	semidirect
69. C₂×C₄⋊C₄	C₂²	semidirect
70. C₂×C₄⋊C₄	C₂²	semidirect
71. C₂×C₄⋊C₄	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#562	2
#554	2
#548	2
#547	2
#546	2
#543	2
#542	2
#539	2
#538	2
#537	2
#525	2
C₂×C₂²⋊Q₈	2
C₂×#121	2