Subgroups — #3519

Order 128.

626 conjugacy classes of subgroups.

The subgroup lattice has 245 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#555	C₂	1	1	yes	maximal, characteristic, complemented
2	#554	C₂	1	1	yes	maximal, characteristic, complemented
3	#553	C₂	1	2	yes	maximal, complemented
4	#553	C₂	1	2	yes	maximal, complemented
5	#550	C₂	1	1	yes	maximal, characteristic, complemented
6	#546	C₂	1	4	yes	maximal, complemented
7	#546	C₂	1	2	yes	maximal, complemented
8	D₄²	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄²	C₂	1	2	yes	maximal, complemented
10	#540	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	#525	C₂	1	2	yes	maximal, complemented
15	C₂×#121	C₂	1	2	yes	maximal, complemented
16	#518	C₂	1	2	yes	maximal, complemented
17	C₂×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#41	C₂²	1	2	yes	complemented
19	C₂×#41	C₂²	1	2	yes	complemented
20	C₂²×D₄	C₂²	1	1	yes	characteristic
21	C₂²×D₄	C₂²	1	2	yes	—
22	C₂²×D₄	C₂²	1	2	yes	complemented
23	C₂²×D₄	C₂²	1	4	yes	complemented
24	C₂³×C₄	C₂²	1	2	yes	complemented
25	C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
26	#127	C₂²	1	1	yes	characteristic, complemented
27	#127	C₂²	1	2	yes	complemented
28	#126	C₂²	1	2	yes	complemented
29	#126	C₂²	1	4	yes	complemented
30	#125	C₂²	1	1	yes	characteristic, complemented
31	#125	C₂²	1	2	yes	complemented
32	#124	C₂²	1	1	yes	characteristic, complemented
33	#124	C₂²	1	2	yes	complemented
34	#124	C₂²	1	4	yes	complemented
35	#123	C₂²	1	4	yes	complemented
36	#123	C₂²	1	2	yes	complemented
37	#123	C₂²	1	4	yes	complemented
38	#123	C₂²	1	4	yes	complemented
39	C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
40	C₂²⋊Q₈	C₂²	1	2	yes	complemented
41	C₂²⋊Q₈	C₂²	1	4	yes	complemented
42	#121	C₂²	1	1	yes	characteristic
43	#121	C₂²	1	2	yes	complemented
44	#121	C₂²	1	4	yes	complemented
45	#121	C₂²	1	4	yes	complemented
46	#121	C₂²	1	2	yes	complemented
47	#121	C₂²	1	2	yes	complemented
48	#121	C₂²	1	2	yes	complemented
49	#121	C₂²	1	2	yes	complemented
50	#121	C₂²	1	4	yes	complemented
51	#121	C₂²	1	4	yes	complemented
52	#121	C₂²	1	2	yes	complemented
53	#121	C₂²	1	2	yes	complemented
54	#121	C₂²	1	4	yes	complemented
55	#120	C₂²	1	4	yes	complemented
56	#120	C₂²	1	4	yes	complemented
57	#120	C₂²	1	4	yes	complemented
58	C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
59	C₄×D₄	C₂²	1	1	yes	characteristic
60	C₄×D₄	C₂²	1	1	yes	characteristic
61	C₄×D₄	C₂²	1	2	yes	—
62	C₄×D₄	C₂²	1	2	yes	complemented
63	C₄×D₄	C₂²	1	1	yes	characteristic, complemented
64	C₄×D₄	C₂²	1	2	yes	complemented
65	C₄×D₄	C₂²	1	1	yes	characteristic, 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	4	yes	complemented
71	C₄×D₄	C₂²	1	2	yes	complemented
72	C₄×D₄	C₂²	1	1	yes	characteristic, complemented
73	C₄×D₄	C₂²	1	2	yes	complemented
74	#117	C₂²	1	2	yes	complemented
75	#117	C₂²	1	2	yes	complemented
76	#117	C₂²	1	2	yes	complemented
77	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
78	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
79	C₂×C₂²⋊C₄	C₂²	1	2	yes	complemented
80	C₂×C₂²⋊C₄	C₂²	1	4	yes	complemented
81	C₂×C₂²⋊C₄	C₂²	1	4	yes	complemented
82	C₂×C₄²	C₂²	1	1	yes	characteristic, complemented
83	C₂⁴	C₂³	1	2	yes	—
84	C₂⁴	C₂³	1	4	yes	—
85	#41	—	4	4	no	—
86	#41	—	4	4	no	—
87	C₂×Q₈	C₂³	1	1	yes	characteristic
88	C₂×Q₈	C₂³	1	2	yes	—
89	C₂×D₄	—	2	4	no	—
90	C₂×D₄	—	2	8	no	—
91	C₂×D₄	—	4	4	no	—
92	C₂×D₄	—	4	8	no	—
93	C₂×D₄	C₂³	1	2	yes	—
94	C₂×D₄	C₂³	1	4	yes	—
95	C₂×D₄	C₂³	1	1	yes	characteristic
96	C₂×D₄	C₂³	1	2	yes	—
97	C₂×D₄	C₂³	1	4	yes	—
98	C₂×D₄	C₂³	1	2	yes	—
99	C₂×D₄	C₂³	1	1	yes	characteristic
100	C₂×D₄	C₂³	1	2	yes	—
101	C₂×D₄	C₂³	1	4	yes	—
102	C₂×D₄	C₂³	1	2	yes	—
103	C₂×D₄	C₂³	1	4	yes	—
104	C₂×D₄	C₂³	1	4	yes	—
105	C₂×D₄	C₂³	1	4	yes	—
106	C₂×D₄	C₂³	1	1	yes	characteristic
107	C₂²×C₄	—	4	4	no	—
108	C₂²×C₄	C₂³	1	1	yes	characteristic
109	C₂²×C₄	C₂³	1	1	yes	characteristic
110	C₂²×C₄	C₂³	1	1	yes	characteristic
111	C₂²×C₄	C₂³	1	2	yes	—
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	4	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	2	yes	—
120	C₄⋊C₄	C₂³	1	1	yes	characteristic
121	C₄⋊C₄	C₂³	1	1	yes	characteristic
122	C₄⋊C₄	C₂³	1	2	yes	—
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	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	2	yes	—
131	C₄⋊C₄	C₂³	1	4	yes	—
132	C₄⋊C₄	C₂³	1	2	yes	—
133	C₂²⋊C₄	C₂³	1	2	yes	—
134	C₂²⋊C₄	C₂³	1	2	yes	—
135	C₂²⋊C₄	C₂³	1	4	yes	—
136	C₂²⋊C₄	C₂³	1	2	yes	—
137	C₂²⋊C₄	C₂³	1	2	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	4	yes	—
142	C₂²⋊C₄	C₂³	1	2	yes	—
143	C₂²⋊C₄	C₂³	1	4	yes	—
144	C₂²⋊C₄	C₂³	1	2	yes	—
145	C₂²⋊C₄	C₂³	1	2	yes	—
146	C₂²⋊C₄	C₂³	1	2	yes	—
147	C₂²⋊C₄	C₂³	1	4	yes	—
148	C₂²⋊C₄	C₂³	1	4	yes	—
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	1	yes	characteristic
153	C₄²	C₂³	1	4	yes	—
154	C₄²	C₂³	1	2	yes	—
155	C₂³	—	2	4	no	—
156	C₂³	—	2	4	no	—
157	C₂³	—	2	8	no	—
158	C₂³	—	4	2	no	—
159	C₂³	—	4	4	no	—
160	C₂³	—	4	4	no	—
161	C₂³	C₂⁴	1	2	yes	—
162	C₂³	C₂⁴	1	2	yes	—
163	C₂³	C₂⁴	1	4	yes	—
164	C₂³	C₂⁴	1	2	yes	—
165	C₂³	C₂⁴	1	1	yes	characteristic
166	Q₈	—	2	2	no	—
167	Q₈	—	4	2	no	—
168	D₄	—	2	4	no	—
169	D₄	—	2	8	no	—
170	D₄	—	2	4	no	—
171	D₄	—	2	8	no	—
172	D₄	—	2	2	no	—
173	D₄	—	2	4	no	—
174	D₄	—	2	2	no	—
175	D₄	—	2	4	no	—
176	D₄	—	4	4	no	—
177	D₄	—	4	4	no	—
178	D₄	—	4	4	no	—
179	D₄	—	4	2	no	—
180	D₄	—	4	4	no	—
181	D₄	#41	1	4	yes	complemented
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	C₂⁴	1	1	yes	characteristic
195	C₂×C₄	C₂⁴	1	2	yes	—
196	C₂×C₄	C₂⁴	1	1	yes	characteristic
197	C₂×C₄	C₂⁴	1	1	yes	characteristic
198	C₂×C₄	C₂⁴	1	2	yes	—
199	C₂×C₄	C₂⁴	1	1	yes	characteristic
200	C₂×C₄	C₂⁴	1	4	yes	—
201	C₂×C₄	C₂⁴	1	2	yes	—
202	C₂×C₄	C₂⁴	1	2	yes	—
203	C₂×C₄	C₂⁴	1	2	yes	—
204	C₂×C₄	C₂⁴	1	1	yes	characteristic
205	C₂×C₄	C₂⁴	1	1	yes	characteristic
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	4	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	4	8	no	—
220	C₂²	—	8	8	no	—
221	C₂²	C₂×#41	1	4	yes	—
222	C₂²	H(5,𝔽₂)	1	2	yes	—
223	C₂²	C₂⁵	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	4	no	—
233	C₄	—	2	2	no	—
234	C₄	C₂×#41	1	2	yes	—
235	C₄	H(5,𝔽₂)	1	2	yes	—
236	C₂	—	2	2	no	—
237	C₂	—	2	4	no	—
238	C₂	—	4	2	no	—
239	C₂	—	4	4	no	—
240	C₂	—	4	2	no	—
241	C₂	C₂²×#41	1	1	yes	characteristic
242	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic
243	C₂	C₂×H(5,𝔽₂)	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. #555	C₂	semidirect
2. #554	C₂	semidirect
3. #553	C₂	semidirect
4. #553	C₂	semidirect
5. #550	C₂	semidirect
6. #546	C₂	semidirect
7. #546	C₂	semidirect
8. D₄²	C₂	semidirect
9. D₄²	C₂	semidirect
10. #540	C₂	semidirect
11. #538	C₂	semidirect
12. #532	C₂	semidirect
13. #529	C₂	semidirect
14. #525	C₂	semidirect
15. C₂×#121	C₂	semidirect
16. #518	C₂	semidirect
17. C₂×C₄×D₄	C₂	semidirect
18. C₂×#41	C₂²	semidirect
19. C₂×#41	C₂²	semidirect
22. C₂²×D₄	C₂²	semidirect
23. C₂²×D₄	C₂²	semidirect
24. C₂³×C₄	C₂²	semidirect
25. C₄⋊Q₈	C₂²	semidirect
26. #127	C₂²	semidirect
27. #127	C₂²	semidirect
28. #126	C₂²	semidirect
29. #126	C₂²	semidirect
30. #125	C₂²	semidirect
31. #125	C₂²	semidirect
32. #124	C₂²	semidirect
33. #124	C₂²	semidirect
34. #124	C₂²	semidirect
35. #123	C₂²	semidirect
36. #123	C₂²	semidirect
37. #123	C₂²	semidirect
38. #123	C₂²	semidirect
39. C₂²⋊Q₈	C₂²	semidirect
40. C₂²⋊Q₈	C₂²	semidirect
41. C₂²⋊Q₈	C₂²	semidirect
43. #121	C₂²	semidirect
44. #121	C₂²	semidirect
45. #121	C₂²	semidirect
46. #121	C₂²	semidirect
47. #121	C₂²	semidirect
48. #121	C₂²	semidirect
49. #121	C₂²	semidirect
50. #121	C₂²	semidirect
51. #121	C₂²	semidirect
52. #121	C₂²	semidirect
53. #121	C₂²	semidirect
54. #121	C₂²	semidirect
55. #120	C₂²	semidirect
56. #120	C₂²	semidirect
57. #120	C₂²	semidirect
58. C₄×Q₈	C₂²	semidirect
62. C₄×D₄	C₂²	semidirect
63. C₄×D₄	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. C₄×D₄	C₂²	semidirect
74. #117	C₂²	semidirect
75. #117	C₂²	semidirect
76. #117	C₂²	semidirect
77. C₂×C₄⋊C₄	C₂²	semidirect
78. C₂×C₄⋊C₄	C₂²	semidirect
79. C₂×C₂²⋊C₄	C₂²	semidirect
80. C₂×C₂²⋊C₄	C₂²	semidirect
81. C₂×C₂²⋊C₄	C₂²	semidirect
82. C₂×C₄²	C₂²	semidirect
181. D₄	#41	semidirect

Maximal subgroups

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

subgroup	index
#555	2
#554	2
#553	2
#550	2
#546	2
D₄²	2
#540	2
#538	2
#532	2
#529	2
#525	2
C₂×#121	2
#518	2
C₂×C₄×D₄	2