Subgroups — #3505

Order 128.

576 conjugacy classes of subgroups.

The subgroup lattice has 229 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#553	C₂	1	2	yes	maximal, complemented
2	#553	C₂	1	4	yes	maximal, complemented
3	#552	C₂	1	2	yes	maximal, complemented
4	#551	C₂	1	4	yes	maximal, complemented
5	#548	C₂	1	2	yes	maximal, complemented
6	#546	C₂	1	2	yes	maximal, complemented
7	#539	C₂	1	2	yes	maximal, complemented
8	#538	C₂	1	2	yes	maximal, complemented
9	#534	C₂	1	2	yes	maximal, complemented
10	#529	C₂	1	2	yes	maximal, complemented
11	#525	C₂	1	2	yes	maximal, complemented
12	C₂×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#121	C₂	1	1	yes	maximal, characteristic, complemented
14	#518	C₂	1	2	yes	maximal, complemented
15	C₂×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#41	C₂²	1	2	yes	complemented
17	C₂²×Q₈	C₂²	1	1	yes	characteristic, complemented
18	C₂²×D₄	C₂²	1	1	yes	characteristic
19	C₂²×D₄	C₂²	1	2	yes	—
20	C₂²×D₄	C₂²	1	1	yes	characteristic, complemented
21	C₂³×C₄	C₂²	1	1	yes	characteristic
22	C₂³×C₄	C₂²	1	1	yes	characteristic, complemented
23	#126	C₂²	1	4	yes	complemented
24	#126	C₂²	1	4	yes	complemented
25	#126	C₂²	1	4	yes	complemented
26	#125	C₂²	1	2	yes	complemented
27	#125	C₂²	1	4	yes	complemented
28	#124	C₂²	1	2	yes	complemented
29	#124	C₂²	1	4	yes	complemented
30	#123	C₂²	1	4	yes	—
31	#123	C₂²	1	4	yes	complemented
32	#123	C₂²	1	4	yes	complemented
33	#123	C₂²	1	4	yes	complemented
34	#123	C₂²	1	4	yes	complemented
35	C₂²⋊Q₈	C₂²	1	2	yes	complemented
36	C₂²⋊Q₈	C₂²	1	2	yes	complemented
37	C₂²⋊Q₈	C₂²	1	2	yes	complemented
38	C₂²⋊Q₈	C₂²	1	2	yes	complemented
39	C₂²⋊Q₈	C₂²	1	2	yes	complemented
40	C₂²⋊Q₈	C₂²	1	4	yes	complemented
41	#121	C₂²	1	2	yes	—
42	#121	C₂²	1	2	yes	—
43	#121	C₂²	1	2	yes	complemented
44	#121	C₂²	1	2	yes	complemented
45	#121	C₂²	1	2	yes	complemented
46	#121	C₂²	1	4	yes	complemented
47	#121	C₂²	1	4	yes	complemented
48	#121	C₂²	1	4	yes	complemented
49	#120	C₂²	1	4	yes	—
50	#120	C₂²	1	4	yes	complemented
51	C₄×Q₈	C₂²	1	2	yes	complemented
52	C₄×D₄	C₂²	1	2	yes	—
53	C₄×D₄	C₂²	1	2	yes	—
54	C₄×D₄	C₂²	1	2	yes	complemented
55	C₄×D₄	C₂²	1	2	yes	complemented
56	C₄×D₄	C₂²	1	4	yes	complemented
57	C₄×D₄	C₂²	1	4	yes	complemented
58	C₄×D₄	C₂²	1	4	yes	complemented
59	C₄×D₄	C₂²	1	4	yes	complemented
60	C₄×D₄	C₂²	1	4	yes	complemented
61	C₄×D₄	C₂²	1	2	yes	complemented
62	#117	C₂²	1	4	yes	complemented
63	#117	C₂²	1	2	yes	complemented
64	#117	C₂²	1	4	yes	complemented
65	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
66	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
67	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
68	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
69	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic
70	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
71	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
72	C₂×C₂²⋊C₄	C₂²	1	4	yes	complemented
73	C₂×C₂²⋊C₄	C₂²	1	2	yes	complemented
74	C₂×C₄²	C₂²	1	1	yes	characteristic, complemented
75	C₂⁴	C₂³	1	2	yes	—
76	C₂⁴	C₂³	1	1	yes	characteristic
77	C₂⁴	C₂³	1	1	yes	characteristic
78	#41	—	4	4	no	—
79	C₂×Q₈	—	4	2	no	—
80	C₂×Q₈	C₂³	1	2	yes	—
81	C₂×Q₈	C₂³	1	2	yes	—
82	C₂×D₄	—	2	4	no	—
83	C₂×D₄	—	2	8	no	—
84	C₂×D₄	—	4	2	no	—
85	C₂×D₄	C₂³	1	2	yes	—
86	C₂×D₄	C₂³	1	4	yes	—
87	C₂×D₄	C₂³	1	2	yes	—
88	C₂×D₄	C₂³	1	4	yes	—
89	C₂×D₄	C₂³	1	2	yes	—
90	C₂×D₄	C₂³	1	4	yes	—
91	C₂×D₄	C₂³	1	4	yes	—
92	C₂×D₄	C₂³	1	2	yes	—
93	C₂²×C₄	—	4	2	no	—
94	C₂²×C₄	—	4	2	no	—
95	C₂²×C₄	C₂³	1	1	yes	characteristic
96	C₂²×C₄	C₂³	1	1	yes	characteristic
97	C₂²×C₄	C₂³	1	2	yes	—
98	C₂²×C₄	C₂³	1	1	yes	characteristic
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	2	yes	—
103	C₂²×C₄	C₂³	1	1	yes	characteristic
104	C₂²×C₄	C₂³	1	2	yes	—
105	C₂²×C₄	C₂³	1	1	yes	characteristic
106	C₂²×C₄	C₂³	1	1	yes	characteristic
107	C₂²×C₄	C₂³	1	4	yes	—
108	C₂²×C₄	C₂³	1	4	yes	—
109	C₂²×C₄	C₂³	1	1	yes	characteristic
110	C₂²×C₄	C₂³	1	2	yes	—
111	C₄⋊C₄	C₂³	1	2	yes	—
112	C₄⋊C₄	C₂³	1	4	yes	—
113	C₄⋊C₄	C₂³	1	2	yes	—
114	C₄⋊C₄	C₂³	1	2	yes	—
115	C₄⋊C₄	C₂³	1	2	yes	—
116	C₄⋊C₄	C₂³	1	2	yes	—
117	C₄⋊C₄	C₂³	1	4	yes	—
118	C₄⋊C₄	C₂³	1	2	yes	—
119	C₄⋊C₄	C₂³	1	4	yes	—
120	C₄⋊C₄	C₂³	1	4	yes	—
121	C₄⋊C₄	C₂³	1	4	yes	—
122	C₄⋊C₄	C₂³	1	4	yes	—
123	C₂²⋊C₄	C₂³	1	2	yes	—
124	C₂²⋊C₄	C₂³	1	4	yes	—
125	C₂²⋊C₄	C₂³	1	2	yes	—
126	C₂²⋊C₄	C₂³	1	4	yes	—
127	C₂²⋊C₄	C₂³	1	4	yes	—
128	C₂²⋊C₄	C₂³	1	2	yes	—
129	C₂²⋊C₄	C₂³	1	2	yes	—
130	C₂²⋊C₄	C₂³	1	4	yes	—
131	C₂²⋊C₄	C₂³	1	4	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	C₄²	C₂³	1	2	yes	—
138	C₄²	C₂³	1	2	yes	—
139	C₄²	C₂³	1	4	yes	—
140	C₄²	C₂³	1	4	yes	—
141	C₂³	—	2	2	no	—
142	C₂³	—	2	4	no	—
143	C₂³	—	4	1	no	—
144	C₂³	—	4	1	no	—
145	C₂³	—	4	1	no	—
146	C₂³	—	4	1	no	—
147	C₂³	—	4	2	no	—
148	C₂³	—	4	2	no	—
149	C₂³	#41	1	4	yes	complemented
150	C₂³	C₂⁴	1	2	yes	—
151	C₂³	C₂⁴	1	4	yes	—
152	C₂³	C₂⁴	1	1	yes	characteristic
153	C₂³	C₂⁴	1	1	yes	characteristic
154	C₂³	C₂⁴	1	1	yes	characteristic
155	Q₈	—	2	4	no	—
156	Q₈	—	4	2	no	—
157	D₄	—	2	4	no	—
158	D₄	—	2	8	no	—
159	D₄	—	2	4	no	—
160	D₄	—	2	8	no	—
161	D₄	—	2	4	no	—
162	D₄	—	4	2	no	—
163	D₄	—	4	4	no	—
164	D₄	—	4	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	2	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	2	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	4	1	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	C₂⁴	1	2	yes	—
182	C₂×C₄	C₂⁴	1	2	yes	—
183	C₂×C₄	C₂⁴	1	4	yes	—
184	C₂×C₄	C₂⁴	1	1	yes	characteristic
185	C₂×C₄	C₂⁴	1	2	yes	—
186	C₂×C₄	C₂⁴	1	4	yes	—
187	C₂×C₄	C₂⁴	1	4	yes	—
188	C₂×C₄	C₂⁴	1	1	yes	characteristic
189	C₂×C₄	C₂⁴	1	1	yes	characteristic
190	C₂×C₄	C₂⁴	1	1	yes	characteristic
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	4	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	2	4	no	—
201	C₂²	—	2	4	no	—
202	C₂²	—	2	2	no	—
203	C₂²	—	4	4	no	—
204	C₂²	—	4	4	no	—
205	C₂²	—	8	4	no	—
206	C₂²	C₂×#41	1	2	yes	—
207	C₂²	C₂×#41	1	2	yes	—
208	C₂²	C₂×#41	1	2	yes	—
209	C₂²	C₂⁵	1	1	yes	characteristic
210	C₄	—	2	4	no	—
211	C₄	—	2	2	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	4	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	4	no	—
219	C₄	H(5,𝔽₂)	1	2	yes	—
220	C₂	—	2	2	no	—
221	C₂	—	2	2	no	—
222	C₂	—	2	2	no	—
223	C₂	—	4	2	no	—
224	C₂	—	4	4	no	—
225	C₂	C₂²×#41	1	1	yes	characteristic
226	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic
227	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. #553	C₂	semidirect
2. #553	C₂	semidirect
3. #552	C₂	semidirect
4. #551	C₂	semidirect
5. #548	C₂	semidirect
6. #546	C₂	semidirect
7. #539	C₂	semidirect
8. #538	C₂	semidirect
9. #534	C₂	semidirect
10. #529	C₂	semidirect
11. #525	C₂	semidirect
12. C₂×C₂²⋊Q₈	C₂	semidirect
13. C₂×#121	C₂	semidirect
14. #518	C₂	semidirect
15. C₂×C₄×D₄	C₂	semidirect
16. C₂×#41	C₂²	semidirect
17. C₂²×Q₈	C₂²	semidirect
20. C₂²×D₄	C₂²	semidirect
22. C₂³×C₄	C₂²	semidirect
23. #126	C₂²	semidirect
24. #126	C₂²	semidirect
25. #126	C₂²	semidirect
26. #125	C₂²	semidirect
27. #125	C₂²	semidirect
28. #124	C₂²	semidirect
29. #124	C₂²	semidirect
31. #123	C₂²	semidirect
32. #123	C₂²	semidirect
33. #123	C₂²	semidirect
34. #123	C₂²	semidirect
35. C₂²⋊Q₈	C₂²	semidirect
36. C₂²⋊Q₈	C₂²	semidirect
37. C₂²⋊Q₈	C₂²	semidirect
38. C₂²⋊Q₈	C₂²	semidirect
39. C₂²⋊Q₈	C₂²	semidirect
40. 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
50. #120	C₂²	semidirect
51. C₄×Q₈	C₂²	semidirect
54. C₄×D₄	C₂²	semidirect
55. C₄×D₄	C₂²	semidirect
56. C₄×D₄	C₂²	semidirect
57. C₄×D₄	C₂²	semidirect
58. C₄×D₄	C₂²	semidirect
59. C₄×D₄	C₂²	semidirect
60. C₄×D₄	C₂²	semidirect
61. C₄×D₄	C₂²	semidirect
62. #117	C₂²	semidirect
63. #117	C₂²	semidirect
64. #117	C₂²	semidirect
65. C₂×C₄⋊C₄	C₂²	semidirect
66. C₂×C₄⋊C₄	C₂²	semidirect
67. C₂×C₄⋊C₄	C₂²	semidirect
68. C₂×C₄⋊C₄	C₂²	semidirect
71. C₂×C₂²⋊C₄	C₂²	semidirect
72. C₂×C₂²⋊C₄	C₂²	semidirect
73. C₂×C₂²⋊C₄	C₂²	semidirect
74. C₂×C₄²	C₂²	semidirect
149. C₂³	#41	semidirect

Maximal subgroups

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

subgroup	index
#553	2
#552	2
#551	2
#548	2
#546	2
#539	2
#538	2
#534	2
#529	2
#525	2
C₂×C₂²⋊Q₈	2
C₂×#121	2
#518	2
C₂×C₄×D₄	2